Le Superficie Algebriche

Federigo Enriques

With Forewords by Guido Castelnuovo (1949) and Mickey Mouse (1999)

English translation by Fabrizio Catanese,

Ciro Ciliberto and Rita Pardini Edited by Miles Reid to appear CUP 2000

Contents

0.1 A reminder of elementary notions . . . . . . . . . . . . . . . . 1 0.2 Rational transformations . . . . . . . . . . . . . . . . . . . . . 2 0.3 Surfaces in hyperspaces S_{r} and resolution of singularities … 5 0.4 Surfaces with normal singularities . . . . . . . . . . . . . . . . 6 0.5 Note . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

1 Linear systems of curves 13 1.1 Linear pencils . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 1.2 Irrational pencils . . . . . . . . . . . . . . . . . . . . . . . . . 15 1.3 Linear systems . . . . . . . . . . . . . . . . . . . . . . . . . . 16 1.4 The characteristic property of linear systems . . . . . . . . . . 18 1.5 Extension of the theorems of Bertini . . . . . . . . . . . . . . 19 1.6 The image surfaces of linear systems . . . . . . . . . . . . . . 20 1.7 Equivalent curves and complete linear systems . . . . . . . . . 22 1.8 Sum and difference of linear systems: the residue theorem . . 25 1.9 The image surface of the sum system . . . . . . . . . . . . . . 28 1.10 Virtual characters . . . . . . . . . . . . . . . . . . . . . . . . . 29 1.11 Exceptional curves . . . . . . . . . . . . . . . . . . . . . . . . 31 1.12 A note on the reducible exceptional curves . . . . . . . . . . . 35 1.13 The virtual characteristic series . . . . . . . . . . . . . . . . . 37

2 Covariant and invariant systems 39 2.1 Jacobian curves . . . . . . . . . . . . . . . . . . . . . . . . . . 39 2.2 The Jacobian system . . . . . . . . . . . . . . . . . . . . . . . 40 2.3 The fundamental theorem . . . . . . . . . . . . . . . . . . . . 45 2.4 Canonical curves . . . . . . . . . . . . . . . . . . . . . . . . . 47 2.5 Properties of canonical curves . . . . . . . . . . . . . . . . . . 49 2.6 Geometric genus and linear genus . . . . . . . . . . . . . . . . 50 2.7 The exceptional curves as fixed parts of the canonical system . 54 2.8 Bicanonical and pluricanonical curves . . . . . . . . . . . . . . 57 2.9 A historical note . . . . . . . . . . . . . . . . . . . . . . . . . 59 3 Adjoint surfaces 63

3.1 Surfaces of order n − 3 adjoint to a surface of order n without

proper multiple points . . . . . . . . . . . . . . . . . . . . . . 63 3.2 Adjoint surfaces of arbitrary order . . . . . . . . . . . . . . . . 65 3.3 Ruled surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . 66 3.4 Characteristic properties of canonical curves and of the curves

adjoint to a linear system . . . . . . . . . . . . . . . . . . . . 67 3.5 Isolated multiple points . . . . . . . . . . . . . . . . . . . . . . 68 3.6 Adjoints and subadjoints . . . . . . . . . . . . . . . . . . . . . 69 3.7 Multiple points of the first kind . . . . . . . . . . . . . . . . . 71 3.8 Double planes . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 3.9 The genus of the fundamental curves and their influence on the

adjoints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75 3.10 Point singularities of a higher kind . . . . . . . . . . . . . . . 77 3.11 Bicanonical curves and biadjoint surfaces . . . . . . . . . . . . 84 3.12 Surfaces of genus p = 0 and bigenus P > 0 . . . . . . . . . . . 87 3.13 Equivalence criteria . . . . . . . . . . . . . . . . . . . . . . . . 91

4 The numerical genus and the Riemann–Roch theorem for surfaces 96 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96 4.2 Conditions imposed on a surface that contains a curve: postu-

lation formulas . . . . . . . . . . . . . . . . . . . . . . . . . . 97 4.3 Dimension of the systems of adjoint surfaces to a given one . . 102 4.4 The numerical genus . . . . . . . . . . . . . . . . . . . . . . . 104 4.5 Complements . . . . . . . . . . . . . . . . . . . . . . . . . . . 108 4.6 Remarks on regularity conditions for a surface . . . . . . . . . 112 4.7 Examples of irregular surfaces . . . . . . . . . . . . . . . . . . 115 4.8 The theorem of Riemann–Roch for surfaces: systems larger

than the canonical . . . . . . . . . . . . . . . . . . . . . . . . 118 4.9 Elimination of the exceptional curves . . . . . . . . . . . . . . 121 4.10 Completeness of the series cut out by a small linear system on

the curves of a large system . . . . . . . . . . . . . . . . . . . 123 4.11 The theorem of Riemann–Roch for arbitrary linear systems . . 126 4.12 The deficiency of the characteristic series . . . . . . . . . . . . 128 4.13 Historical note . . . . . . . . . . . . . . . . . . . . . . . . . . . 132 4.14 Virtual curves . . . . . . . . . . . . . . . . . . . . . . . . . . . 136 4.15 Regular and superabundant systems . . . . . . . . . . . . . . . 141 4.16 More on the regularity of the adjoint system of a curve on a

regular surface . . . . . . . . . . . . . . . . . . . . . . . . . . . 144 4.17 The adjoint system of a connected curve consisting of multiple

components on a regular surface (continued) . . . . . . . . . . 146 4.18 On the reducibility of the canonical system . . . . . . . . . . . 150 4.19 On the regularity of the bicanonical system . . . . . . . . . . . 153 4.20 Remark on the irreducibility of the bicanonical system …. 157 4.21 Genera of a reducible surface . . . . . . . . . . . . . . . . . . . 158

5 Numerical invariants and multiple planes 161 5.1 The Zeuthen–Segre invariant . . . . . . . . . . . . . . . . . . . 161 5.2 Irrational pencil . . . . . . . . . . . . . . . . . . . . . . . . . . 166 5.3 Expressing the Zeuthen–Segre invariant in terms of the genera 167 5.4 Cuspidal curves of a net . . . . . . . . . . . . . . . . . . . . . 172 5.5 Historical note . . . . . . . . . . . . . . . . . . . . . . . . . . . 174 5.6 Characters of a net and multiple planes . . . . . . . . . . . . . 175 5.7 (1, n) correspondence between two surfaces: double planes . . 177 5.8 Correspondence formulas . . . . . . . . . . . . . . . . . . . . . 179 5.9 The existence theorem . . . . . . . . . . . . . . . . . . . . . . 186 5.10 Complements: limit forms of the branch curve according to

Chisini . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192 5.11 The moduli of a class of algebraic surfaces . . . . . . . . . . . 197 5.12 Digression on the completeness of

the characteristic series of a complete system of plane curves with nodes and cusps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 200

5.13 The moduli of the regular surfaces of genus p > 3 . . . . . . . 202 5.14 Historical note and complements . . . . . . . . . . . . . . . . 206

6 Regular surfaces: the minimum for the genera and rationality conditions 209 6.1 The lower bound for the linear genus . . . . . . . . . . . . . . 209 6.2 The bigenus and the plurigenera . . . . . . . . . . . . . . . . . 212 6.3 Examples: double planes with linear genus p^{(1)} = 1 . . . . . . 216 6.4 Rationality conditions for a surface . . . . . . . . . . . . . . . 222

7 Classification of surfaces of linear genus p^{(1)} = 1 228

7.1 Surfaces of genus zero and bigenus one, with bicanonical curve

of order zero . . . . . . . . . . . . . . . . . . . . . . . . . . . . 228 7.2 Surfaces with all the genera equal to one . . . . . . . . . . . . 238 7.3 A historical note . . . . . . . . . . . . . . . . . . . . . . . . . 245 7.4 Surfaces of linear genus p^{(1)} = 1 with P > 1 . . . . . . . . . . 248

8 Regular canonical and pluricanonical surfaces 258 8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 258 8.2 Canonical surfaces of genus p = 4 and linear genus p^{(1)} = 6, 5 . 259 8.3 Surfaces with p = 4 and p^{(1)} = 7 . . . . . . . . . . . . . . . . . 262 8.4 Surfaces with p = 4 and p^{(1)} = 8 . . . . . . . . . . . . . . . . . 264 8.5 Surfaces with p = 4 and p^{(1)} = 9 . . . . . . . . . . . . . . . . . 271 8.6 Surfaces with p = 4 and p^{(1)} > 9 . . . . . . . . . . . . . . . . . 274 8.7 Canonical surfaces with p = 5 and p^{(1)} = 9 . . . . . . . . . . . 275 8.8 Surfaces with p = 5 and p^{(1)} = 10 . . . . . . . . . . . . . . . . 276 8.9 Surfaces on a variety of dimension 3 or more . . . . . . . . . . 279 8.10 Examples of canonical surfaces in hyperspace . . . . . . . . . . 282 8.11 Minimal value of the linear genus with respect to the superficial

genus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 285 8.12 Upper bound for the linear genus p^{(1)} in terms of the superficial

genus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 289 8.13 Canonical systems belonging to an involution . . . . . . . . . 290 8.14 Surfaces with linear genus p^{(1)} = 2: first case p = 2 . . . . . . 294 8.15 Surfaces with p^{(1)} = 2 and p = 1 (P = 3, P_{3} = 5) . . . . . . . . 295 8.16 Surfaces with p^{(1)} = 2 and p = 0 (P = 2, P_{3} = 4) . . . . . . . . 298 8.17 Surfaces with linear genus p^{(1)} = 3: first case p = 3 . . . . . . 302 8.18 Surfaces with p^{(1)} = 3 and p = 2 (P = 5) . . . . . . . . . . . . 303 8.19 Surfaces with p^{(1)} = 3 and p = 1 (P = 4) . . . . . . . . . . . . 307 8.20 Surfaces with p = 0 and p^{(1)} = 3 . . . . . . . . . . . . . . . . . 311 8.21 Simple and multiple pluricanonical surfaces . . . . . . . . . . . 312

9 Irregular surfaces and continuous systems of inequivalent curves 315 9.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 315 9.2 The arithmetic condition for a curve on a surface with geo-

metric irregularity q to belong to an ∞^{q} continuous series of inequivalent curves . . . . . . . . . . . . . . . . . . . . . . . . 317

9.3 The Picard variety corresponding to an irregular surface … 318 9.4 The characteristic property of irregular algebraic surfaces: the

fundamental theorem for p_{g} = 0 . . . . . . . . . . . . . . . . . 319 9.5 The fundamental theorem for p_{g} > 0 . . . . . . . . . . . . . . 320 9.6 The history of the theory of continuous systems . . . . . . . . 329 9.7 On various attempts to prove and extend the fundamental the-

orem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 336 9.8 The paracanonical system . . . . . . . . . . . . . . . . . . . . 343 9.9 A digression on Abelian varieties . . . . . . . . . . . . . . . . 346 9.10 Continuation: the genus of the Abelian varieties . . . . . . . . 350 9.11 Irrational pencil on the surfaces with geometric genus zero . . 353 9.12 A note on the surfaces of irregularity one . . . . . . . . . . . . 357 10 Surfaces with geometric genus zero 359

10.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 359 10.2 Surfaces containing a system of curves of genus π and degree

n > 2π − 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 360 10.3 A lemma on the reducible curves in a pencil . . . . . . . . . . 364 10.4 Surfaces with genus p_{g} = 0 and p_{a} < −1 . . . . . . . . . . . . 366 10.5 Surfaces F with p_{g} = 0 and p_{a} = −1, possessing an elliptic

pencil of curves K of genus π > 1: Lemma I . . . . . . . . . . 367 10.6 Lemma II: a sketch of the proof . . . . . . . . . . . . . . . . . 369 10.7 The first case: the genus π of the curves K is even . . . . . . . 372 10.8 The second case: the genus π of the curves K is odd . . . . . 378 10.9 Lemma III . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 380 10.10Conclusion: the surfaces F also possess a linear pencil of ellip-

tic curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 381 10.11Surfaces with genera p_{g} = 0 and p_{a} = −1 with an elliptic pencil

of curves of genus π = 1 . . . . . . . . . . . . . . . . . . . . . 382 10.12Elliptic surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . 386 10.13Construction of the elliptic surfaces of genus p_{g} = 0 . . . . . . 389 10.14Surfaces with all plurigenera equal to zero: characterization of

the ruled surfaces . . . . . . . . . . . . . . . . . . . . . . . . . 393 10.15A historical note . . . . . . . . . . . . . . . . . . . . . . . . . 400 10.16Surfaces with pluricanonical curves of order zero: the types

with elliptic curves K with general modulus . . . . . . . . . . 402 10.17Continuation: The harmonic case . . . . . . . . . . . . . . . . 405 10.18The equianharmonic case . . . . . . . . . . . . . . . . . . . . . 409 10.19Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 411

11 The general classification of surfaces 414 11.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 414 11.2 Surfaces admitting a continuous series of birational transfor-

mations into themselves: the cases leading to ruled surfaces . . 416 11.3 Elliptic and hyperelliptic surfaces . . . . . . . . . . . . . . . . 420 11.4 Characterization of elliptic and hyperelliptic surfaces by means

of the values of the genera . . . . . . . . . . . . . . . . . . . . 425 11.5 Surfaces with genera p_{g} = 1 and p^{(1)} = 1 . . . . . . . . . . . . 433 11.6 A note on the geometric theory of hyperelliptic surfaces …. 438 11.7 The general classification of algebraic surfaces . . . . . . . . . 442 11.8 Surfaces with negative arithmetic genus . . . . . . . . . . . . . 446 11.9 Summary of the above classification . . . . . . . . . . . . . . . 450

Index 453

Introduction

0.1 A reminder of elementary notions

In this book we will usually consider an algebraic surface in ordinary 3-space and assume it is irreducible. The surface f shall be defined, in a system of Cartesian coordinates, by an equation of the form

f(x, y, z) = 0;

or else, in homogeneous coordinates

f(x_{1}, x_{2}, x_{3}, x_{4}) = 0,

where in the former case f denotes a polynomial in the variables x, y, z, and in the latter a homogeneous polynomial, that is, a form.

We assume the fundamentals of the projective theory of surfaces, including the notions of order, class, polar surfaces, and so on, to be well known^{(1)}. We recall now some properties that we will use later.

Rational transformations 0.2 2

A point O of the surface f of order n is said to be simple if the lines passing through O cut the surface in n−1 points outside of O. The neighbourhood of a simple point of the surface can be represented in an elementary way upon a plane with an arbitrarily high order of approximation, that is, keeping track of infinitesimals of orders 1, 2, … and so on. To start with, it is clear that, at the first order of approximation, the surface can be replaced by the tangent plane at the given point. But if one wants to consider the second order of approximation, one may assume the Cartesian frame of reference to be chosen in a suitable fashion and instead replace the surface by the osculating paraboloid of the same order z = φ_{2}(x, y), which can be projected one-to-one onto the plane (x, y) from the point at infinity on the z-axis. One can proceed similarly for the third order neighbourhood, and so on. Therefore, given a curve on our surface with an arbitrary singularity at O, one can view this singularity as being composed of multiple points and infinitely near multiple points, exactly as on the plane^{(2)}.

(1)Cf. Enriques–Chisini, Teoria geometrica delle equazioni e delle funzioni algebriche, Libro III, Vol. II. Bologna, Zanichelli, 1918.

Singular (or multiple) points for the surface f are defined as follows: a multiple point O of order r (for r > 1) is such that the lines through O cut the surface at n − r residual points. It is convenient to distinguish between multiple curves and isolated multiple points (without excluding consideration of remarkable multiple points belonging to a multiple curve, which occur as a superposition of both cases); we shall have more to say about these distinctions later. A multiple curve of order r can be regarded in general as the intersection of r surfaces or branches, which are differentiably distinct at a generic point of the curve and correspond to the r tangent planes at the point. On the other hand, an isolated multiple point usually has a conical aspect, in the sense that the lines having r + 1 intersections with the surface form a cone of order r which is generally irreducible.

But particular cases can occur. For instance, two or more branches of the surface along a multiple curve can touch each other or even come together in a higher order branch: the simplest example is given by the double cuspidal curve. On the other hand, the tangent cone at an isolated r-tuple point can split or even reduce to a plane counted r times: the simplest example is provided by the uniplanar double point.

In order to master all the intricacies which can occur when dealing with singularities of surfaces, it is convenient to define infinitely near multiple points and the (proper or infinitesimal) curves to which they give rise, as one does for the branches of curves passing through a point^{(3)}. In this order of ideas, a uniplanar point appears as a double point having three infinitely near double points in different directions. The simplest example of an infinitesimal double curve (line), infinitely near to an isolated multiple point, is the tacnode: that is, a double point O such that the plane sections through O have another double point infinitely near to O. But we do not want to spend more time on this topic, for which we refer to the volume of Enriques–Chisini quoted above.

0.2 Rational transformations

A rational transformation of the space to which the surface f belongs is defined by letting the new coordinates X, Y, Z be rational functions of the coordinates x, y, z:

X = \frac{φ_{1}(x, y, z)}{φ_{0}(x, y, z)}, Y = \frac{φ_{2}(x, y, z)}{φ_{0}(x, y, z)}, Z = \frac{φ_{3}(x, y, z)}{φ_{0}(x, y, z)},  (1)

(2)Cf. Enriques–Chisini, Ibidem, Libro IV, Vol. II.

(3)Cf. Enriques–Chisini, Ibidem, Libro IV, Vol. IV.

where the φ are polynomials, which we can assume to be of the same order. Through this transformation, which we assume to be nondegenerate, to every generic point of the space (xyz) there corresponds a point of the space (XY Z). Conversely, to a point (XY Z) corresponds in general a certain finite number n (≥ 1) of points (xyz), which can be obtained by solving the equations (??), where the XY Z are regarded as given, and the xyz as unknown. For n > 1 the groups of points thus obtained form, in the space (xyz), an involution I_{n} of order n, that is, a series of groups of n points such that every point belongs in general to one and only one group.

Now, in general, the transformation given by (??) takes the ∞^{2} points of the surface f(x, y, z) = 0 into ∞^{2} distinct points, describing a surface F (X, Y, Z) = 0, the transform of f. To a generic point of f shall therefore correspond a single point of F, and conversely, to a generic point of F shall usually correspond a single point of f, the transformation turning out to be one-to-one invertible for the points of the surface F.^{(4)} This invertibility, and the particular circumstances under which it fails to hold, require some explanation. We saw indeed that the transformation (??), regarded as a map from the space (xyz) to the space (XY Z), is not in general one-to-one invertible, but, in fact, carries the points (XY Z) to the groups of n points of an involution I_{n} in the space (xyz). But if f is chosen in a general way inside this space, it will not belong to the involution I_{n}, and thus a point of f shall have n − 1 conjugate points in I_{n}, lying outside the surface. Therefore to a generic point of F shall correspond, via (??), one point of f and n − 1 points outside of f, which shall describe the conjugate surface of f with respect to I_{n}.

Only in the case when f belongs to the involution I_{n}, or to an involution partially contained in it, will the surface in which F is transformed reduce to a multiple surface, in [1, n] or [1, m] correspondence with f (with m < n).

Now, considering the general case, in which f and F are in a one-to-one correspondence, we should note that the transformation given by (??) is not in general everywhere defined without exceptions^{(5)}. Indeed, the formulas expressing the transformation become indeterminate at all those points (xyz) for which all the φ vanish simultaneously (that is, for the base points of the transforming system, which is determined by the surfaces φ_{0} = 0, φ_{1} = 0, φ_{2} = 0, φ_{3} = 0). An exception of this sort can take place first of all at a simple point of f; then such a point is in general transformed into a line or into a curve of F, and this curve, which is the transform of a simple point, is called an exceptional curve of the surface F.

(4)Hence the distinction between a birational transformation relating two surfaces and a Cremona transformation, that extend to the space containing them.

(5)One should bear in mind analogous considerations concerning transformations of curves: cf. Enriques–Chisini, Ibidem, Libro V, Vol. III.

Alternatively, it may happen that a base point of the transforming system φ occurs at a singular point of f. And if there is a multiple curve of f on which the φ all vanish simultaneously, then in general this curve will be transformed into a simple curve: to every r-tuple point of the curve will correspond r distinct points, related to the r branches of f through the point. If instead a base point of the φ occurs at an isolated multiple point O of f, this is in general transformed into a curve whose points correspond to the points infinitely near to O.

These statements fail when multiple points and curves infinitely near to the singular points under consideration intervene. In such cases the overall effect of the transformation is to simplify the singularity, transforming infinitely near multiple points of the first order into proper multiple points, and more generally, reducing from r to r − 1 the order of infinitesimal neighbourhood of the points under consideration – at least, under the assumption that the surfaces φ passing through a point O do not have any fixed tangent in common.

Nevertheless, the transformation which achieves a simplification of the singularities of a surface does in general introduce new singularities. In fact, there will in general be ∞^{1} pairs of points of f at which the X, Y, Z take the same values, so that these points will correspond to the points of a double curve of F. This follows essentially from an easy count of constants. For X, Y, Z to take the same values at points (x, y, z) and (x′, y′, z′), the following five equations in six unknowns have to be satisfied:

φ_{1}(x, y, z)φ_{0}(x′, y′, z′) − φ_{1}(x′, y′, z′)φ_{0}(x, y, z) = 0,
φ_{2}(x, y, z)φ_{0}(x′, y′, z′) − φ_{2}(x′, y′, z′)φ_{0}(x, y, z) = 0,
φ_{3}(x, y, z)φ_{0}(x′, y′, z′) − φ_{3}(x′, y′, z′)φ_{0}(x, y, z) = 0,
f(x, y, z) = 0 f(x′, y′, z′) = 0.

In general, these admit ∞^{1} solutions (x, y, z, x′, y′, z′), in addition to the ∞^{2} trivial solutions^{(6)}

x = x′, y = y′, z = z′.

In particular cases, one may have ∞^{1} groups of more than two points of f corresponding to a multiple curve of F. But there may also be curves of f on which the XY Z are constant, corresponding to isolated multiple points of F. And as the singularities of f get more complicated, and as other particular circumstances occur, we cannot exclude the genesis of higher singularities for F.

(6)The first three equations define three hypersurfaces of the space S_{6}, which have in common the S_{3} given by (x = x′, y = y′, z = z′) and, outside this, a certain 3-dimensional variety V; this V is intersected by the cylindrical varieties f(x, y, z) = 0 and f(x′, y′, z′) = 0 along a curve.

Surfaces in S_{r} and resolution of singularities 0.3 5

0.3 Surfaces in hyperspaces S_{r} and resolution of singularities

The transformation of ordinary space considered above can be generalized, thus defining a transformation carrying the surface f into a projective space S_{r} of higher dimension:

X_{1} = \frac{φ_{1}(x, y)}{φ_{0}(x, y)}, X_{2} = \frac{φ_{2}(x, y)}{φ_{0}(x, y)}, …, X_{r} = \frac{φ_{r}(x, y)}{φ_{0}(x, y)}.

The surface f(x, y, z) = 0 will be transformed into a surface F of S_{r} defined by the above formulas, but may also be defined in other ways as a partial intersection of varieties of the hyperspace S_{r}. ^{(7)}

Most of what we said for a transformation of S_{3} can be repeated for this transformation; it usually turns out to be one-to-one for the points of the two surfaces (but not for the points of the spaces containing them). An ordinary multiple curve of f that is a base curve for the transforming system may, in general, be resolved, and becomes the locus of the r simple points corresponding to the r-tuple points of f that make up the curve. The same can be said for isolated multiple points that map to curves, and so on.

But here, at least when r ≥ 5, no new singular points for F are produced in general. Indeed when r ≥ 5, there do not exist, in general, pairs of points of f where the X_{1}, X_{2}, …, X_{r} take the same values, whereas for r = 4 one has, in general, a finite number of such pairs, giving rise to the same number of double points for F.

Surfaces with normal singularities 0.4 6

One can guess in this way that it will always be possible to resolve the singularities of a surface f and to transform it into another surface of S_{r} completely free of singularities, that is, having only simple points. But this reduction, which turns out to be easy for surfaces only having ordinary multiple curves and points (multiple curves with distinct branches and isolated multiple points with completely general osculating cone), requires a more detailed examination when f possesses arbitrarily complicated infinitely near singularities. However the analysis required for this purpose can be decomposed into two parts: a first analysis in the differential geometric sense, where the main issue is to resolve the singularities defined in a neighbourhood of a point O belonging to a curve in ordinary space, and where it is allowed to use birational or Cremona transformations of the entire space S_{3}; and secondly a critical analysis of the new singularities which can be produced by the transformation itself. But this last step, which involves delicate difficulties when dealing with the reduction of singularities through birational transformations of S_{3}^{(8)}, can be avoided via a simple remark of B. Levi. In fact, if the surface f(x, y, z) = 0 is transformed into another F (X, Y, Z) = 0, the surface of 6dimensional space, which is the locus of the points (x, y, z, X, Y, Z), will only have singularities corresponding to the common singularities of f and F.

(7)Cf. Enriques–Chisini, Ibidem, Libro I, §23.

(8)A problem solved by O. Chisini, in Mem. Acc. Bologna, 1921.

We do not propose to dwell here on an explanation of the reduction of surface singularities, as first rigorously obtained by B. Levi. We shall limit ourselves to stating the main result that can be obtained, referring for a proof to the most important Memoirs which contain and develop it in different manners^{(9)}.

A surface f admitting arbitrary singularities can always be birationally transformed into another one, free of singularities, and belonging to a 5-dimensional space.

0.4 Surfaces with normal singularities

Given a surface with arbitrary singularities, we begin by transforming it into a surface F of S_{5} completely free of multiple points. If we project F successively from two points to an ordinary space, we shall obtain in the latter space an image Φ of F. We deal with the question of what the singularities of Φ will be when the projection of F is made from two generic points. We want to show that Φ shall have only a nodal double curve (with generally distinct tangent planes), endowed with triple points which are also triple for the surface.

It is clear first of all that this is the type of normal singularities, which will in general appear on a surface of ordinary space, obtained as a projection of a surface free from singular points in a higher dimensional space; this comes out of a simple computation of constants. Observe in fact that the chords of a surface of the space S_{r} are always ∞^{4}, hence for r = 5 there is a finite number of chords passing through a point, whereas for r = 4 there are ∞^{1} such chords. Hence, by projecting the surface F from a generic point of S_{5}, the projected surface F′ shall acquire a finite number of double points, corresponding to the chords of F passing through the centre. By then projecting F′ from a generic point of S_{4}, the projected surface acquires an (apparent) double curve to which the double points of F′ will map. In other words, through two points of S_{5}, or through the line joining them, there passes an ∞^{1} series of planes meeting F at two points and a finite number of planes meeting it at three points. Hence the projection of F performed from those centres shall indeed possess a nodal double curve endowed with triple points, triple both for the double curve and for the surface.

(9)B. Levi, Risoluzione delle singolarità delle superficie algebriche, Atti Acc. Torino, 1897 (cf. Ann. di Mat., 1897).

O. Chisini, La risoluzione delle singolaritá delle superficie mediante trasformazioni birazionali dello spazio. Mem. Acc. Bologna, 1921.

G. Albanese, Trasformazione birazionale di una superficie algebrica qualunque in un’altra priva di punti multipli, Rendic. Circ. Mat. Palermo, 1924.

R. J. Walker, Reduction of singularities of an algebraic surface, Ann. of Math., 1935.

O. Zariski, The reduction of the singularities of an algebraic surface, Ann. of Math., 1939. cf. Ann. of Math., 1941.

One might nevertheless wonder whether the projection from two generic points of S_{5} of a particular surface of S_{5}, even if free from multiple points, might acquire higher singularities, that is, a multiple curve instead of a double curve, or multiple points of multiplicity greater than 3 for the double curve. In order to exclude the first doubt, it is enough to remark that the ∞^{4} chords of a surface of any space of dimension greater than 3 cannot be trisecants, since otherwise the chords of a twisted curve, hyperplane section of the surface, would also be trisecants, which is impossible^{(10)}. It follows from this that no trisecants of F can pass through a generic point of S_{5}, and that, when F is projected from a generic point to a surface F′ of S_{4}, only finitely many trisecants can pass through a generic point of S_{4}.

It remains to prove that, if the centres of projection in S_{5} are chosen in a generic way, it is impossible for the trisecant planes through the centres to meet the surface F in more than three points and therefore that the double curve of the projected surface Φ possesses points of higher multiplicity rather than triple points^{(11)}. For this, it is enough to exclude the possibility that all planes subject to the condition of meeting the surface at three points would, as a consequence, meet it at four or more points. In fact, if every trisecant plane of F were quadrisecant, then the same would happen for every twisted curve, hyperplane section of F (which cannot belong to some S_{3}). Therefore the projection of this curve from one of its points to S_{3} should also give a curve every chord of which is a trisecant, which, as we already recalled, is impossible.

In such a way one can rigorously prove a fundamental theorem already discovered by Noether (1888):

A surface with arbitrary singularities can always be transformed into another surface of ordinary space having only normal singularities, that is, a nodal double curve possessing in general a finite number of triple points that are also triple points for the surface.

The double curve can also possess a finite number of double points (crossings), arising from the chords of the surface of S_{4} being projected to S_{3} that are double for the cone of projecting chords.

Note 0.5 8

It is also useful to recall that, on the above nodal curve, there will in general be a finite number of pinch points (punti cuspidali) at which the two linear branches of the surface come together in a second order branch, and which turn out to be simple base points for the curves obtained as variable intersections with the polar surfaces.

(10)Cf. Enriques–Chisini, Ibidem, Libro III, Chapter IV, §43, Vol. II, page 289.

(11)Cf. Enriques, Sulle singolarità che nascono per proiezione di una superficie o varietà algebrica, in Scritti Matematici offerti a L. Berzolari, Pavia, 1936, page 351.

In what follows, as we develop a theory of surfaces concerned with the properties that are invariant under birational transformations, we may always consider either a model surface free from singularities of 5-dimensional space, or a surface of ordinary space endowed with the normal singularities, as in the preceding statement. In this last case a biplanar point of the double curve should be viewed as the superposition of two simple points whose neighbourhoods correspond to the two branches, as in the invariant theory of curves a node may be considered as the superposition of two points belonging to the two branches.

0.5 Note

What we have said about the transformation from a surface to another surface with normal singularities in ordinary space can be made more precise in the following sense: we may suppose that the transform of any surface is endowed with an irreducible double curve, which, as we said above, possesses a certain number of triple points, triple both for the surface and for its nodal curve.

We present the proof of this property,^{(12)} assuming known the principles of the theory of linear systems, which we treat in the first sections of this book.

For this purpose, it is convenient to start from a surface free from singularities in the space S_{5}, or, even better, from a surface F which is a generic projection of the former in S_{4}, and is therefore endowed only with ordinary improper double points, each of which may be viewed as an apparent singularity, representing a pair of distinct superimposed points. It is convenient to analyze the singularities of a surface Φ of ordinary space, which is a projection of F from a generic point O of S_{4}: in which case it could happen that the double curve of Φ always turns out to be reducible as the centre of projection O varies.

This examination leads to establishing that: if a surface F, endowed with improper ordinary double points in S_{4}, is projected from a generic point to a surface Φ of S_{3}, whose double curve is reducible, then Φ is a projection of the Veronese surface (the surface of the fourth order in S_{5}, represented on the plane by the ∞^{5} system of all conics).

To obtain this result it is convenient first to state the following:

(12)Cf. A. Franchetta, Sulla curva doppia della proiezione di una superficie dello S_{4} da un punto generico su uno S_{3}. Rendic. R. Acc. d’Italia, 1941, see also Rendic. Acc. Lincei, s. VIII, Vol. II, (3), 1947.

Lemma 1 There do not exist surfaces of S_{4} possessing ∞^{3} pairs of tangent planes incident along a line, in such a way that the lines joining their points of contact fill up the whole space.

Let P be a generic point of F and π the tangent plane at P. By assumption there are ∞^{1} tangent planes to F that meet π along a line; their points of contact fill up a curve, possibly reducible, of which we consider a component C. Let Q be one of its points and Q′ the point infinitely near to Q on C. The tangent planes to F at Q and Q′ each cut π in a line; hence each of them lies together with π in some S_{3}. But the tangent plane at Q′ contains Q, hence the two above planes determine, together with π, a unique S_{3}, which is tangent to F at all points of C. As P varies, the curve C describes an ∞^{1} system (at least); and if, as we are assuming, the surface F does not belong to some S_{3}, each of these curves C, which must lie in both the S_{3} determined by it and in the S_{3} determined by the infinitely near curve, is contained in a plane (that could, in particular, reduce to a line); the planes of the two infinitely near curves C lie in an S_{3}. We conclude that a surface for which our assumption holds is either ruled or contains ∞^{1} plane curves whose planes form a developable variety. In the former case, the lines joining the points of contact of two planes incident along lines are the generators of the ruling; in the latter, they are the lines of the planes of the developable. In neither case do they fill up S_{4}.

With this setting we assume we have a surface F in S_{4}, endowed only with finitely many improper ordinary double points, whose projection Φ from a generic point O possesses a reducible double curve. We claim first that this curve must be connected, that is, formed of two or more parts having some common point. Indeed, the apparent double curves

C = ∑ K

corresponding to the sets of those parts K form a rational (or at least unirational) system whose elements correspond to the points O of the space S_{4}, hence the C belong to a linear system of curves on F.^{(13)} If this system is irreducible, then the reducible curves C which belong to it must be connected^{(14)}; if instead all the C in the above linear system are reducible, then they must be composed with the curves K of a pencil, which do not have variable intersections.

But this case cannot occur, because two curves K containing two pairs of points such as AA_{1} and AA_{2} certainly have (in A) a common point. Note that they cannot coincide with one and the same curve since, varying A_{2}, this curve would contain all the points of the surface.

(13)This theorem has been established for rational series of groups of points over a curve and extended to varieties by Enriques. cf. Enriques–Chisini, Ibidem, Libro V, Chapter I, §9, Vol. III, page 78 (cf. page 485).

(14)The principle of degeneration, cf. Enriques–Chisini, Ibidem, Libro V, Chapter III, §36, Vol. III, page 405.

We are led therefore to examine whether and how the surface Φ, a projection of F from a generic point O, may have a double curve composed of two or more parts connected between them.

A connection point of such parts can be either:

(1) a simple crossing, that is, a double point that is common to two simple branches, which shall therefore be a tacnode for the surface, or

(2) a triple point, through which pass three branches of the double curve, two of which may a priori also belong to the same irreducible component.

(1) A common point \overline{P} of two simple branches of the double curve \overline{C} of F will in general be a projection of two points P and P′ which are collinear with the centre O of S_{4}. One sees immediately that the tangent planes to F at P and P′ belong to the same hyperplane whose trace is the tangent plane to Φ at \overline{P}, and thus they are incident along a line. But since the double point \overline{P} is assumed to exist for every position of the centre O, the surface F should possess ∞^{3} pairs of incident tangent planes, and the lines joining the contact points of these pairs of planes should fill up the whole space; by the above lemma this is absurd.

A special examination is required in the case where the two points P and P′ are infinitely near, thus specializing to points which a tangent line through O has in common with F. In this case the point \overline{P}, a crossing point for the two branches of the double curve of Φ, shall again be a tacnode for Φ; hence every plane through \overline{P} shall cut Φ along a curve still having a double point infinitely near to \overline{P}. Hence it follows that the hyperplane joining O with the tacnodal plane of Φ must cut F along a curve which has a fourfold intersection with every plane through the tangent line P P′, and thus it has P and P′ as double points. Note that a separate examination is needed in the particular cases where the effective multiplicity of P becomes bigger than the virtual multiplicity 2, which however decreases the multiplicity of P′. The assumption under consideration implies that, as O moves in the space S_{4}, there is always a tangent through O, call it p, meeting F at two infinitely near points, that would be double points for the section with the tangent hyperplane at P. Therefore, each of the ∞^{3} tangent hyperplanes of F should touch the surface at two infinitely near points P and P′. But it is easy to see that this property cannot hold for a surface F of S_{4}. In fact, a generic projection of such a surface in S_{3} would be such that each of its tangent planes would also be tangent to it at an infinitely near point, hence it would turn out to be a developable surface. In which case the surface in S_{4} of which it is the projection would also be a developable surface. However a developable surface in S_{4} admits only ∞^{1} and not ∞^{3} hyperplanes touching it at two infinitely near points.

The conclusion we draw from the above discussion is the impossibility of the above assumption, namely that the projection of our surface F from a generic point O of S_{4} is a surface Φ whose double curve is composed of two components meeting in a point \overline{P} which is the image of two infinitely near points P and P′ of F. But one should really examine the particular case which we previously mentioned: that instead of a tangent hyperplane at P whose intersection with F has two double points P and P′, we encounter a tangent hyperplane whose section has a triple point at P and a simple point at P′, or just a quadruple point at P. Of the two cases we are confronted with, the first does not give rise to any exception to the previous argument, since there will still be ∞^{3} tangent hyperplanes to F that would also (virtually) touch it at an infinitely near point. It is instead necessary to dwell for a moment on the second exception, according to which we would a priori have only ∞^{2}, rather than ∞^{3}, hyperplanes cutting F along curves endowed with a quadruple point. As a matter of fact, this exception cannot occur because, if we have a surface of S_{4} such that for each of its points one has a tangent hyperplane cutting it along a curve endowed with a quadruple point, then that surface cannot belong to S_{4}, but is entirely contained in a space S_{3}. In fact, its hyperplane sections are curves such that the osculating plane at any point has a fourfold contact with the curve itself, hence they are plane curves.

(2) A point \overline{P} which is a triple point for the double curve C of Φ and which is a point connecting two irreducible components of it may be either:

a) a simple point common to three irreducible components of \overline{C}; or

b) a point which is simple for an irreducible component of \overline{C} and is double for another component.

a) The point \overline{P} is the projection of three points of F, say P_{1}, P_{2}, P_{3}; and the components \overline{K}_{1}, \overline{K}_{2}, \overline{K}_{3} of C for which \overline{P} is a triple point are the projection of three curves, respectively K_{1}, K_{2}, K_{3}, for which we may assume that the first passes through the points P_{2} and P_{3}; the second through P_{1} and P_{3}, the third through P_{1} and P_{2}.

We recall that the surface Φ obtained as the projection of F must have a singularity similar to that at \overline{P}, no matter how the centre O of the projection varies in S_{4}; in particular, we let it move along the line P_{1}P_{2}P_{3}. Now, corresponding to such a variation in O, it can either happen that the apparent double curves K_{1}, K_{2}, K_{3} on F remain fixed, or that they vary with O. If one of them, say K_{1}, remains fixed, we deduce that it is a conic. In fact, for \overline{K}_{1} to project always to the same curve, we must have that every line joining a point of \overline{K}_{1} with a point of the line P_{2}P_{3} meets K_{1} at another point, and thus K_{1} is contained in a plane through the line P_{2}P_{3}. A plane curve that projects, as this one does, to a double curve (and not to a curve of higher multiplicity) must be a conic. Thus the surface F, possessing ∞^{2} conics, must be a projection of the Veronese surface.

We should now examine the hypothesis that the curve K_{1} varies as O varies on the line P_{2}P_{3}. In such a case, for a suitable position of O, K_{1} will also pass through the point P_{1} of this line, and correspondingly we shall have a surface Φ endowed with a particular triple point \overline{P}, where two branches, say those corresponding to the neighbourhoods of P_{1}, P_{2}, are tangent to each other. But this implies that the tangent planes to F at P_{1} and P_{2} meet along a line. Then under our assumptions, F will have ∞^{3} pairs of incident tangent planes in such a way that the lines joining the contact points of these pair of planes fill up the space S_{4}; this is excluded by the above lemma.

b) We examine now the hypothesis that the triple point \overline{P} of Φ is simple for one component \overline{K}_{1} and double for a component \overline{K}_{2} of the double curve. \overline{K}_{1} and \overline{K}_{2} are projections of two apparent double curves K_{1} and K_{2} of F, and we may assume that the former passes simply through P_{2} and P_{3}, and the latter passes simply through P_{2} and P_{3} and doubly through P_{1}. We now let the centre of projection O vary on the line P_{1}P_{2}P_{3}; it may happen either that the curve K_{1} stays fixed or that it varies with O. If it stays fixed, we conclude as before that it is a plane curve, and thus a conic, whence it follows that Φ is a projection of the Veronese surface. If instead K_{1} varies with O, for a suitable position of O, it will also pass through P_{1}. The singularity that Φ presents at \overline{P} is now a triple point in which at least two branches touch each other, and therefore the tangent planes to F at two such points as P_{1} and P_{2} will turn out to meet along a line. The surface F possesses ∞^{3} pairs of incident tangent planes, and we fall back to the case we already excluded.

From all the preceding discussion we conclude that

the projection to the space S_{3} of a surface of S_{5} free from singularities has in general an irreducible double curve; the only exception being the case of the Veronese surface which in fact is projected onto the Steiner surface, having three double lines passing through a triple point.

Chapter 1

Linear systems of curves

1.1 Linear pencils

Let F be a surface, for the sake of simplicity assumed to belong to ordinary space and endowed with normal singularities, as defined in the preceding introduction.

Consider a rational function t defined on F, which is not reduced to a constant:

t = \frac{φ(x, y, z)}{ψ(x, y, z)}.  (1.1)

For any given value t_{0} of t there will be points on F at which t takes this value: the locus C of these points is said to be a level curve of the rational function t on F. As t varies, we shall have on F a simply infinite system of level curves which is said to form a linear pencil. Among the curves C of the pencil there is the curve of zeros, namely the curve φ = 0 which corresponds to t = 0, and the curve of poles, namely that defined by ψ = 0, for which t = ∞.

In the case that these two curves have a common part K, on which t has indeterminate form, this curve K can either be kept as a fixed component of the C or can be omitted from all them.

The definitions given above are perfectly analogous to those concerning curves and linear series g_{n}^{1} on them, and give rise to similar remarks. The pencil of curves C defined by (??) is equally well given by the rational function

τ = \frac{αt + β}{γt + δ} ;

the only effect of the rational substitution is to change the value of t corresponding to each individual curve C. Thus all the curves C still appear in the pencil, so that the curve of zeros and the curve of poles is of no particular

Linear pencils 1.1 14

significance. The rational function corresponding to a given pencil is determined by the arbitrary choice of the curves τ = 0 and τ = ∞, together with that of the unit level curve τ = 1, since these choices determine the preceding substitution. If two functions t and τ have the same curve of zeros and poles, they differ by a multiplicative constant.

We remark moreover that the rational function (??) is defined only modulo F. Thus two different functions φ/ψ and φ_{1}/ψ_{1} are to be identified on F if we have

\frac{φ}{ψ} = \frac{φ_{1}}{ψ_{1}}

for all points of F. That is,

φψ_{1} − ψφ_{1} ≡ 0 mod F,  (1.2)

or

φψ_{1} − ψφ_{1} = AF,

where A represents a polynomial.

All the curves C of a linear pencil have the same order; if the order of one of them apparently drops, it means that the curve has become reducible and contains one part at infinity. To eliminate this apparent reduction of the order, if we do not want to use a projective transformation of the surface, we have only to adopt homogeneous coordinates x_{1}, x_{2}, x_{3}, x_{4}. Therefore, if for instance the degree of the polynomial φ in (??) were less than that of ψ, then, on passing to homogeneous coordinates, φ gets multiplied by a suitable power of x_{4} and then clearly, the curve of zeros contains also the section of F with the plane at infinity (x_{4} = 0) counted a certain number of times.

The definition of a linear pencil of curves C has a simple projective meaning: indeed, the equation

φ(x, y, z) − tψ(x, y, z) = 0  (1.3)

represents a pencil of surfaces which cut out the curves C on F; and (??) says that

the same pencil of curves can be cut out on F by different pencils of surfaces.

The pencil of surfaces (??) may have a base curve Γ which in general does not belong to F and meets F in a finite number of points. These base points of the linear pencil of curves C are points of essential indeterminacy of the rational function (??). In fact, a function of the form 0/0 at a base point, has a nonremovable singularity there: indeed, if one tries to define the value of the function at such a point by a limiting argument, then one finds that

Irrational pencils 1.2 15

the values so obtained by continuity will be different when one approaches the base point along different curves C.

A base point for the curves C of a linear pencil may have a certain multiplicity i > 1 for the C, in which case it will be said to be an i-tuple base point of the pencil.

Finally, it may happen that the base curve Γ of the pencil (??), or some part K of Γ if it is reducible, lies on F; then, as we have already said, K may either be considered as a fixed component of the curves C of the pencil (which is thus reducible), or one can forget about K and keep only the pencil of curves C consisting only of the variable intersection of F with the surfaces (??).

1.2 Irrational pencils

A linear pencil of curves C on F is characterized geometrically by the following properties:

1) it is an ∞^{1} (algebraic) series of index 1, that is, only one C passes through any generic point of F;

2) this series is rational, that is, it can be put into one-to-one algebraic correspondence with the points of a line or with the set of values of a parameter t.

In fact the function t defined in this was is single valued, and is therefore a rational function of the points of F, and the curves C are its level curves. Thus if one calls a pencil a series of curves of index 1, namely a series enjoying property (1), we can say that

any rational pencil is linear.

We may also have irrational pencils on a surface, whose curves are elements of an algebraic but not rational system, having a certain genus p > 0. Thus, for example, the generators of a scroll of genus p (that is, with plane sections of genus p) form an irrational pencil of lines.

We can construct an example of a surface with an irrational pencil of genus p of curves by intersecting any hypersurface of S_{4} with a double cone, formed by the planes of the same S_{4} projecting the points of a curve of genus p from a line vertex.

We should point out that

a pencil of curves belonging to the surface F is certainly rational if it has a base point at a simple point of the surface.

Linear systems 1.3 16

In fact, if the base point O is simple for the curves C of a pencil, then the series of these curves is in one-to-one correspondence with the pencil of tangent lines at O. On the other hand, if the point has multiplicity i > 1 for the C, then they correspond in general to the groups of an involution of order i in the pencil of tangent lines at O, and this involution is rational (Lüroth’s theorem^{(1)}).

However, this argument may fail if the curves C have fixed tangent lines at O, that is, if they have as base points not only O, but also other points infinitely near to O. In order to remove this exception, we need only remark that, proceeding on a branch of a curve C up to a suitably high order r, one finds a first point O_{r} that belongs to the rth neighbourhood of O, and moves freely through the neighbourhood of the base point O_{r−1}; since this is a neighbourhood of O, it is a rational system. If O_{r−1} is a simple base point, then the curves C are in one-to-one correspondence with the points O_{r} of this neighbourhood; if O_{r−1} is an i-tuple point and is succeeded by several free points O_{r}, then the C correspond to the groups of an involution described by these groups of points.

1.3 Linear systems

The definition of a linear pencil of curves on F can be generalized by considering linear systems of dimension r > 1. An ∞^{r} linear system |C| of curves C is cut out on F by a linear system of surfaces (or hypersurfaces)

λ_{0}φ_{0} + λ_{1}φ_{1} + ⋯ + λ_{R}φ_{R} = 0;  (1.4)

the dimension r turns out to be smaller than R when the system (??) contains ∞^{h} surfaces that contain F as a part, and then r = R − h − 1. One defines the curves which are fixed components of the C and the base points of |C| for linear systems |C| with several degrees of freedom in the same way as for a linear pencil; a base point is a common point of all the C, where all the φ vanish simultaneously.

A linear system |C| can be defined in an intrinsic way, independently of the projective realization of the surface, in the same way as we did for the linear pencil. Indeed, consider r rational functions of the variable point of F, all having the same curve of poles, and represent them as

t_{1} = \frac{φ_{1}}{φ_{0}} , t_{2} = \frac{φ_{2}}{φ_{0}} , …, t_{r} = \frac{φ_{r}}{φ_{0}} ,

where we assume, for the sake of simplicity, that they are linearly independent on F; in other words, it is not possible to find r + 1 constants a_{0}, a_{1}, …, a_{r}, not all zero, such that

(1)See Enriques–Chisini, Ibidem, Libro II, Chapter I, §3, Vol. I, page 168.

a_{0}φ_{0} + a_{1}φ_{1} + ⋯ + a_{r}φ_{r} ≡ 0 mod F

holds identically. Then the level curves of the function

t_{0} = λ_{1}t_{1} + λ_{2}t_{2} + ⋯ + λ_{r}t_{r}

describe the linear system |C| of dimension r.

It might seem that the curve of poles φ_{0} assumes a special position in this system; but we may choose any curve of the system to be the curve of poles, since one and the same system |C| may correspond to infinitely many linear systems of functions, derived from one other by linear substitutions.

The dimension r of the linear system |C| turns out to be defined geometrically by the following property:

there is one and only one curve of the system passing through r generic points of the surface.

A linear system is said to be irreducible if its generic curve C is irreducible. For an irreducible linear system |C|, the effective genus is the genus of its generic curve, and the effective degree is the number of variable intersections of two generic curves C. For r = 1 the degree is 0, since any common point of two curves of the pencil is a base point belonging to all the C.

The dimension r and the degree n of a linear system |C| are related by the inequality r ≤ n + 1; indeed, the series cut out on any curve C of |C| by the remaining curves of the system, away from the base points, is a g_{n}^{r−1}, from which we get r − 1 ≤ n. The linear series g_{n}^{r−1} defined on a generic curve C is of fundamental importance in the study of the linear system and is called the characteristic series^{(2)} of |C|.

The effective degree of a linear system of curves is also defined for a system of reducible curves that has no fixed part. Now it can be proved that the degree is n > 0, unless either the system itself is a linear pencil, or its curves are composed with those of a pencil, rational or otherwise. In fact, if n = 0 then either all the curves C of the system that pass through a generic point must coincide with a single curve (dimension r = 1), or they must have a common component K, and the set of the K will form a pencil, rational or irrational, and all the C will be composed of this.

(2)The characteristic series of a linear system was introduced by C. Segre and G. Castelnuovo around 1890. The term characteristic series was adopted for the first time by G. Castelnuovo, Ricerche generali sopra i sistemi lineari di curve piane, Mem. Acc. di Torino, s. II, t. XLII, 1891, reproduced on page 137 ffoll. of Castelnuovo’s Memorie scelte, Zanichelli, Bologna, 1937.

The characteristic property of linear systems 1.4 18

1.4 The characteristic property of linear systems

We observed in Section ?? that if |C| is a linear system of dimension r on a surface F, there is one and only one curve C passing through r generic points of F. For r > 1, this is a characteristic property^{(3)}, whereas we have seen that for r = 1 one may also have nonlinear pencils.

The proof of this theorem is given as follows.

Assume for simplicity that the curves of the system |C| are irreducible, and consider first the case r = 2. The ∞^{2} curves intersect pairwise in the groups of an involution I_{n} of order n equal to the degree of |C|. Let us take these groups as elements (points) of a new surface F′; then on F′ we have an ∞^{2} system of curves C′, the transforms of the curves of |C|, which intersect pairwise at one point. The curves C′ passing through a point A′ form a pencil with a base point, and is thus a rational pencil (see Section ??). In addition to this pencil, let us consider another with a base point B′. The two pencils of C′ with centres at A′ and B′ can be set in one-to-one correspondence respectively with two pencils of lines A and B in the plane, by means of projective transformations that make the curve C′ joining A′ and B′ correspond to the line AB. This assignment gives a one-to-one correspondence between the points of F′ and the points of the plane, in which the curves C′ correspond to the lines of the plane and the pencils of C′ correspond to the pencils of lines; this implies that the C′ on F′, and so also the C on F, form a linear system (a net).

Once the theorem is established for nets, it extends to the case of linear systems that are ∞^{3}, and so in general, passing from dimension r to dimension r + 1. If |C| is an ∞^{3} system of curves such that only one curve C passes through three generic points, then the C passing through a fixed point A will form a net. Now nets defined by two base points A and B may be set in a projective one-to-one correspondence with two nets of planes A′ and B′ of ordinary space in such a way that the curves of the pencil common to the two nets correspond, in the same way for the two correspondences, to the planes common to the two nets of planes. From this one gets a bijective (homographic) correspondence between the system of the C and the planes of the space S_{3}, where the planes of a pencil correspond to a pencil of the C, and the planes of a net correspond to the curves C of a net; this yields that the system |C| is a linear system. Q.E.D.

(3)F. Enriques, Una questione sulla linearità ecc., Atti Acc. Lincei, 1893. Cf. C. Segre, Introduzione alla geometria sopra un ente algebrico semplicemente infinito, Ann. di Mat., s. II, t. 22 (1894).

Extension of the theorems of Bertini 1.5 19

1.5 Extension of the theorems of Bertini

Bertini has given two fundamental theorems^{(4)} for linear systems of plane curves: the first says that the curves of a linear system cannot have variable double or multiple points not belonging to a fixed component; the second one, which is a corollary of the first, states that if the variable parts of a linear system are reducible, then they are composed of the curves of a pencil. These theorems extend easily to linear systems of curves belonging to any surface^{(5)}. The extension turns out to be immediate, following the proof indicated in the lectures of Enriques–Chisini^{(6)}.

If the generic curve C of the linear system |C| possesses a variable double or multiple point O, then the curve C has at least two intersections at O with any infinitely near curve, and thus with the curves of all the pencils to which it belongs. This means that O is either a base point or a point belonging to a curve K which is a fixed component of the C. We therefore state:

the curves of a linear system on a surface cannot have double or multiple variable points (away from the base points) not belonging to a fixed component.

The second theorem of Bertini follows as a corollary.

Suppose that the curves C of the system |C| without fixed components are reducible, for instance, in two variable parts K and K′. If K and K′ do not belong to a common pencil, of which the curves C are composed, then there will be at least one curve K, and another K′ distinct from K passing through the generic point O. As a consequence the K and K′ will have to meet in at least one variable point, and the C must possess at least one double point freely varying on the surface, which is absurd. Indeed, if the common points of the K and K′ that form a C always occurred at base points of the system |C|, then all the K and all the K′ would pass through these base points and would not have any variable intersections. It is also impossible for the complementary curves K and K′ to meet at the points of a fixed curve, since this would have to be a fixed component of the C, which we suppose does not exist.

We therefore state the following result:

if the generic curve of a linear system |C| on F is reducible then:

Image surfaces of linear systems 1.6 20

either 1) |C| contains a fixed part, and, after we remove this, the re-

(4)E. Bertini, Rendic. Ist. Lombardo, 1880.

(5)See F. Enriques, Ricerche di geometria ecc., Mem. Acc. di Torino, 1893 and Introduzione alla geometria sopra una superficie algebrica, Mem. della Società Italiana delle Scienze (detta dei XL), 1896. The reducibility theorem was already observed in Noether, Math. Ann., VII, 1874.

(6)Ibidem, Libro II, Chapter I, §5, Vol. I, page 180.

maining system is irreducible, or

2) the curves C are composed with r > 1 curves variable in a pencil, rational or otherwise, (and in this pencil, regarded as an ∞^{1} system, they form a g_{r}), or finally

3) the curves C are composed of the curves of a pencil as in case 2) together with some fixed curves added to these.

Cases 2) and 3) include all the linear systems of dimension r > 1, and of degree n = 0 (see Section ??).

1.6 The image surfaces of linear systems

Consider a linear system |C| that is irreducible of dimension r ≥ 3, and therefore of degree n > 0. Referring |C| projectively to the system of the hyperplanes of a space S_{r} gives rise to a transformation of our surface F into a surface F′ of order n, on which the transforms of the C are cut out by the hyperplanes. In fact the C may be viewed abstractly as the elements (hyperplanes) of a space S_{r}; thus the projective transformation we have indicated sends the ∞^{r−1} linear system of the C passing through a point P to the star of hyperplanes passing through a point P′ of S_{r} and, as P varies on F, the point P′ describes a surface F′.

The single-valued correspondence between F and F′ is, in general, oneto-one invertible, because the star with centre P′ will correspond to a linear system of curves C with a (variable) base point P and in general, this system will not have any other base point as a consequence of the existence of P.

Actually, one cannot exclude the possibility that the condition that the C pass through a generic point of F implies that they pass through other points P_{1}P_{2} … P_{m−1}; in this case the groups analogous to P P_{1}P_{2} … P_{m−1} will form an involution on F of order m, and that the C belong to this involution. The resulting surface F′ of S_{r} determined above will then be a surface of order n/m, to be considered as a multiple surface of order m.

It is in fact a particular circumstance for a linear system |C| of dimension r ≥ 3 to belong to an involution, as follows from an easy count of constants: there are at least three curves passing through P that are linearly independent, and in general there is no reason for one of these to pass through the points of intersection of the other two. In contrast, an ∞^{2} linear system (that is, a net) always belongs to an involution of order n, in which case it is of degree n > 1; in fact, only two linearly independent curves C pass through a point P, and all the other C through P form a pencil, having their intersection as base points. Thus a net |C| of degree n leads to a multiple plane of order n, with the C having multiple lines of order n as images; here the branch curve is of special importance, that is, the locus of points in the plane corresponding to groups of the involution endowed with a double point.

We may give an example of a linear system |C| of dimension r ≥ 3 belonging to an involution, by considering the system of the curves C which are intersections of the surface F with the cones of a certain order n having vertex at a fixed point of the space; the linear system |C| will now belong to an involution of order m and will be of degree mn, where we denote the order of the surface F by m.

When we want to exclude that a linear system |C| belongs to an involution, we say for short that it is a simple system.

Going back in general to the transformation of the surface F determined by a linear system |C| of dimension r ≥ 3, we note that the projective correspondence discussed above can be translated into formulas as follows: if the system |C| is represented by

λ_{0}φ_{0} + λ_{1}φ_{1} + ⋯ + λ_{r}φ_{r} = 0,

we should take the coordinates of the points of S_{r} as

X_{0} ≡ φ_{0}, X_{1} ≡ φ_{1}, …, X_{r} ≡ φ_{r}.

We conclude by stating a theorem which extends to surfaces a well-known statement about curves and linear series on them:

when we study the invariant relations of surfaces using a simple linear system (at least ∞^{3}), the geometry of birational transformations on the surfaces is reflected in the projective geometry of the image surface of the system.

In this vein, the relation between two given linear systems |C| and |K| (both simple and at least ∞^{3}) on F that the first should contain the second is reflected in the fact that the image surface of |K| can be obtained as the projection of the image surface of |C| from a certain linear space. In general, this space will intersect the image surface at a group of points if |K| is deduced from |C| by imposing them as base points. On the other hand, it will intersect the surface along a curve L if |K| is partially contained in |C| in such a way that one obtains curves C by adding the fixed curve L to the K.

Equivalent curves and complete linear systems 1.7 22

Concerning base points, it will be useful to take into account the following remarks. If O is an i-tuple base point on the surface F of a system |C| (that is simple for F), then it will be a fundamental point for the transformation which sends F into the image surface Φ whose hyperplane sections are the C; and if the C have no fixed tangent line at O, then Φ will have an exceptional curve of order i corresponding to O. However, if O, as an i-tuple base point of |C| with i ≥ 1, is such that the C have a fixed tangent line (that is, a simple base point O_{1} infinitely near to O) without a higher contact at O, then we can still say that an exceptional curve of order i on Φ corresponds to O. But this curve will, in general, consist of a curve of order i − 1 (whose points correspond to the variable points of the first order neighbourhood of O) and of a line which is the image of the neighbourhood of O_{1} (whose points belong to the second order neighbourhood of O). In particular if i = 1, then the whole neighbourhood of the simple point O is mapped to the neighbourhood of a conic double point on Φ, to which the exceptional line image of O_{1} has to be added (consider, for instance, the plane representation of the quadric cone).

Note In general, if the simple point O of the surface F is an arbitrarily complicated singularity for the curves C of |C|, one may resolve this singularity, transforming F into a surface Φ on which the neighbourhood of the point O (taken in its entirety) will be represented by an exceptional curve consisting of as many parts as the branches (linear or otherwise) that form the singularity of the curves C at O. The connections between these parts will reflect the way in which the stated singularity is composed of multiple or simple infinitely near points, and their proximity relations^{(7)}. This has been an object of analysis of E. Barber and O. Zariski^{(8)}. We will have occasion to return to this argument later.

1.7 Equivalent curves and complete linear systems

The following considerations are a step-by-step extension of the well-known results for linearly equivalent groups of points and complete linear series on a curve^{(9)} to linear systems of curves on a surface.

Two curves (of the same order) on a surface F are said to be equivalent if they both belong to a linear system of dimension r ≥ 0: if they are distinct, this means that they belong to a linear pencil; allowing systems of dimension 0 to be considered means that a curve is to be taken as equivalent to itself.

The equivalence of two curves C and K is expressed by writing

C ≡ K, or sometimes C = K.

(7)Enriques–Chisini, Ibidem, Libro IV, Vol. II, page 327 and following.

(8)Cf. E. Barber and O. Zariski, Reducible exceptional curves of the first kind, Amer. J.

Math., 1935 (page 119). P. Du Val, Amer. J. of Math., 1936.

(9)Enriques–Chisini, Ibidem, Libro V, Vol. III (1924).

Two equivalent curves C and K may always be considered as the curve of zeros and the curve of poles of a rational function t on F; the two curves are interchanged if one substitutes 1/t for t. The relation of equivalence between curves on a surface enjoys the three fundamental properties of equality:

1) the reflexive property, expressed by C ≡ C;

2) the symmetric property: if C ≡ K then also K ≡ C;

3) the transitive property: if C ≡ K and K ≡ L then also C ≡ L.

To prove this last property, one constructs a rational function t having K as the curve of zeros and C as the curve of poles, then a rational function τ having K as the curve of zeros and L as the curve of poles; the function τ/t then has C as the curve of zeros and L as the curve of poles. This proves that C and K are indeed equivalent.

As a consequence of the above properties, one has the fundamental theorem:

On a surface F the set of all curves equivalent to a given curve C forms a linear system, which is said to be complete.

This system is obtained as the set of level curves of the rational functions that arise as linear combinations of the independent rational functions having C as the curve of poles.

The complete system C thus defined will in general be free from base points, and in any case, unless the contrary is explicitly stated, we will mean that it is virtually base point free, in a sense that we will make precise later; but one may also define the complete linear system relative to a group of assigned base points on F. Notice that, if one transforms the surface in such a way that the base points of C become curves (exceptional curves), then one has to bear in mind that these curves are subtracted from the curve corresponding to C, that would otherwise appear as fixed components of the transformed system. Thus the notion of a complete linear system relative to a group of base points can be reduced to the general notion of a complete base point free linear system on a transformed surface.

Be that as it may, the irreducible complete system, with or without base points, can be defined starting from a linear system that is at least ∞^{2}, as the largest linear system of the same degree containing the given one. From the transitive property of equivalence, it follows that the complete linear system |C|, with or without assigned base points, is uniquely determined by any one of its curves.

It is obvious that the complete system |C| can be constructed as follows. One starts by constructing an arbitrary linear system that contains C (totally) and then one enlarges it successively as far as possible; since the dimension cannot exceed the degree plus 1, in this way one will eventually obtain a system |C| not contained in a larger linear system of the same order, and this will be the complete linear system defined by C. Therefore, however one proceeds in constructing larger and larger linear systems as just described, in the end we obtain the same complete linear system.

The concept of complete linear system corresponds to that of normal surface, that is, a surface of a space S_{r} that cannot be obtained as a projection of another surface of the same order belonging to a higher dimensional space: for a surface to be normal means that the (irreducible) system of its hyperplane sections is complete, and vice-versa.

This is therefore the projective meaning of the theorem on complete linear systems. Consider, for example, a surface F of a certain order n in ordinary space; it may happen that F can be obtained as a projection of two surfaces F_{1} and F′_{1} of the same order in S_{4}, the projections being performed from two points O and O′ respectively. Now if F_{1} and F′_{1} are normal surfaces, then it follows that they will be projective to one another, that is, projectively equivalent. If not, it means that F_{1} can be thought of as a projection of a surface F_{2} of S_{5} and F′_{1} as a projection of a surface F′_{2} of S_{5}, and so on. Proceeding in any possible way from a surface to another of a higher dimensional space, we eventually arrive at a surface F_{r} of the space S_{r+3} which will be normal. And this F_{r} will always be projectively well defined, whatever sequences of surfaces (F_{1}, F_{2}, …) or (F′_{1}, F′_{2}, …), derived from one another by projection, we used to arrive at it.

Examples of normal surfaces have already appeared in the theory of rational surfaces^{(10)}. Thus a surface F_{5} of order 5 in ordinary space, with elliptic plane sections and a double curve of order 5 belonging to a quadric cone and having a triple point at the vertex of the cone that is also a triple point for the surface can be considered as a projection of several surfaces F_{5} of the same order of the space S_{4}. These surfaces are not projective to one another, but they are all projections of a normal surface of S_{5} that is uniquely defined up to projective equivalence. These assertions can be justified by looking at the plane representation of the F_{5} by means of an ∞^{3} linear system of cubics C_{3} with four simple base points A_{1}A_{2}A_{3}A_{4}.

Sum and difference of linear systems: the residue theorem 1.8 25

Indeed such an ∞^{3} system is contained in ∞^{1} 4-dimensional linear systems of curves C_{3} with the same base points A_{1}A_{2}A_{3}A_{4}, which are homographically (and also birationally) distinct in the plane, and therefore represent surfaces F_{5} of S_{4} that are not projective to one another. By contrast the normal surface of S_{5} of which these F_{5} are projections (from an external point) is defined up to projective transformations by the complete system of the C_{3} through A_{1}A_{2}A_{3}A_{4}.

(10)See Enriques–Conforto, Le superficie razionali, Bologna, 1939 (1945).

1.8 Sum and difference of linear systems: the residue theorem

Curves composed of equivalent curves, namely sums of equivalent curves, are equivalent. Indeed, let C_{1} and C_{2}, K_{1} and K_{2} be two pairs of equivalent curves. Take t a rational function having C_{1} as curve of poles, C_{2} as the curve of zeros, and τ a rational function having K_{1} as curve of poles and K_{2} as curve of zeros. Then the function τt has C_{1} + K_{1} as curve of poles and C_{2} + K_{2} as curve of zeros. Thus from

C_{1} ≡ C_{2} and K_{1} ≡ K_{2}

one deduces

C_{1} + K_{1} ≡ C_{2} + K_{2}.

This yields that, given any two linear systems |C| and |K|, there exists a well determined complete system |C + K|, which contains all the reducible curves C + K; this will be called the complete system sum of |C| and |K|.

It is convenient to notice that if |C| and |K| are irreducible linear systems of dimension bigger than zero, which do not coincide with one and the same pencil, then the complete system |C + K| is irreducible. Moreover if the irreducible systems |C| and |K| have assigned base points, then these are always base points for |C+K|. This is an obvious consequence of the extension of Bertini theorem. In particular the double, and in general the system |sC|, the s-multiple of the system |C|, is irreducible, except in the case in which |C| is a pencil.

Let |C| and |K| be again two systems which do not coincide with one and the same pencil. Let us consider all reducible curves C + K; as we said, they are contained in the complete system |C + K|; but in general they can also be contained in systems of lower dimension, and in particular there will be a minimal sum of the two given systems.

For instance consider a surface F of order n of ordinary space, without singularities; there are ∞^{2} planes passing through one of its points O, which cut out a complete net; the minimal double of this is the ∞^{35} system cut out by the quadric cones with the vertex at O; but this system, which has a double base point at O, is not complete, since it is contained in the ∞^{6} system cut out on F by all quadrics touching F at O.

The subtraction of two linear systems, one of which partially contains the other, is defined as the inverse operation of the sum and precisely as follows.

Let |C| be a linear system to which the reducible curve C ≡ L+K belongs; then one calls residual system of K with respect to |C|, the system of all the curves L which added to K give a curve of |C|. By the properties of the equivalence of curves one may claim that:

if a particular curve C=L+K of the complete system |C| partially contains a K, then any other curve equivalent to K is a part of a C. Thus one has the Residue theorem: the linear system |L| residual of a K with respect to a system |C| is also residual of any other curve equivalent to K.

In conclusion the complete linear system difference of |C| and |K|:

|L| = |C − K|

is defined by the relation

|C| = |L + K|.

Note The theorem just stated has a projective complement which we will prove later on. We shall in fact see that on a surface F of ordinary space with normal singularities, the (adjoint) surfaces of a given order which pass through the double curve, cut out, away from this, a complete linear system.

In particular, on a surface of ordinary space without singularities, the surfaces φ of a given order cut out a complete system.

Let us refer to the general case. If we consider the adjoint surfaces φ of a certain order passing through a curve C of F, and their residual intersections K with F, then any curve C of the complete system |C| will be in the same way residual of any K with respect to the system cut out by the φ (away from the double curve), and conversely any K is residual of any C.

This is M. Noether’s Restsatz^{(11)}, which, as in the case of the geometry of curves (as remarked by G. Castelnuovo, 1890), contains two distinct statements: the former concerns the subtraction of complete systems, the latter adds a projective complement to the first and concerns the completeness of the system cut out on a surface F of the space S_{3} by its adjoints φ of a given order.

It is now worth remarking that, while the operation of sum leads in general, as we said, from irreducible linear systems to irreducible systems, this no longer holds for subtraction: if one subtracts from an irreducible system |C| a curve K (which may be irreducible or not), then the complete residual system |C − K| may turn out to be reducible. A simple example is obtained by considering a linear pencil |L| that, for simplicity, we assume to be base point free, and a linear system |K|: if we add |K| to |2L| we get a system |C| = |K + 2L|. And subtracting from this |K|, one gets the double of the system |L|, which is reducible, composed with the curves of the pencil |L|.

(11)Math. Ann., Band VIII. For the case of curves see: Brill and Noether, Math. Ann., Band VII; Enriques–Chisini, Ibidem, Libro V.

It may also happen that, by subtracting a K from the irreducible system |C|, one gets a linear system containing a fixed component K′, which turns out to be a part of the C as a consequence of the fact that they contain K; and similarly it may happen that, by subtracting a curve K from |C|, the residual system |L| consequently has some new base point O, besides the base points of |C| which do not belong to K. This case can be reduced to the previous one by a transformation of the surface changing O into an exceptional curve, since on the transformed surface such a curve will appear as a fixed part of |L|.

Remark The definition of the system |L| = |C − K| requires an existence assumption, since the difference has an effective meaning only if the system |C| partially contains |K|. But it will also be convenient in what follows to talk about virtual curves, defined by subtraction, even if this is not actually possible. Thus we shall look at the symbolic relation

|C − K| = |L − M|

as equivalent to

|C + M| = |K + L|.

The virtual curves are defined as the negative numbers in Peano’s theory of operators, namely as possible summands. To add the virtual curve C − K to a system |D|, means adding C and then subtracting K from it, and this will turn out to be possible for sufficiently ample systems. In any case for the virtual curves C − K and L − M to be equivalent means that the linear systems

|D + C − K| = |D + L − M|

are equivalent, where

|D + (C − K)| = |D + C − K|.

One can also give, as we shall see later on, a more restricted definition of virtual curves as curves which do not effectively exist, but such that the double or a multiple does exist. And this concept, rather than to the notion of negative numbers, is related to the notion of ideals, which are met in the theory of algebraic numbers fields.

NOTE The idea, if not the name, of “virtual curves”, was considered by Enriques in connection with the notion of canonical curves. Enriques realized that the curve C − K, defined as the difference of two linear systems, should correspond to a real mathematical object, even in the case the subtraction does not lead to an effective curve. This is indeed the main theme of the

The image surface of the sum system 1.9 28

new development of the theory in passing from the Ricerche of 1893 to the Introduzione of 1896, where one basically looks for the invariant properties of the aforesaid canonical curves in the most general setting, even when the genus p is equal to 0. The said author defines the virtual characters, degree and genus (see Section ??), of the difference curve C − K, and notices that, even if this curve does not exist, it may have an effective double. The word “virtual”, referred to the curve as well as to its characters (which were so named by Enriques), can be found in Severi’s note, Sulle curve algebriche virtuali appartenenti ad una superficie algebrica, Rendic. Ist. Lombardo, 1905. The topological interpretation of the virtual curves as two dimensional cycles has been given by S. Lefschetz.

1.9 The image surface of the sum system

Let us assume we have on the surface F two irreducible linear systems |C| and |K| of dimension r and s respectively, where r ≥ 1 and s ≥ 1. One may construct in general a transformed surface Φ, the image of the sum system |C + K|, in a space of dimension R, where R > r + s; the sum system which we are considering may be either the minimal sum system, or, if one prefers, the complete system.

In any case, there will be two linear series of spaces in S_{r}, respectively of dimensions R − r − 1 and R − s − 1, which cut out the K and the C, and such that the ∞^{r} hyperplanes passing through one S_{R−r−1} of the first series cuts out the C and contain the S_{R−s−1} to which they belong, and conversely the same holds for the hyperplanes passing through a S_{R−s−1} of the second series.

Let us fix a S_{R−r−1} of the first series cutting out a curve C of |C| and a S_{R−s−1} of the second series, cutting out a curve K of |K|; these are contained in a hyperplane of S_{R}, and therefore have a S_{R−r−s−1} in common; projecting from this space down to a S_{r+s} one obtains there an image F of the given surface, containing a particular K, section with a S_{s−1}, and a particular C, section with a S_{r−1}; the hyperplanes through the S_{s−1} cut out the ∞^{r} curves C, and the hyperplanes through the S_{r−1} cut out the ∞^{s} curves K.

Let us take in particular r = 2, s = 1: we will be led to a surface of ordinary space on which the net of curves C is cut out by the planes through a point O, and the pencil of the K is cut out by the planes through a line a (not passing through O); if |C| and |K| do not belong to one and the same involution, then the surface thus obtained will certainly be in one-to-one correspondence with the surface we started from.

Virtual characters 1.10 29

The construction of the surface F can be also obtained directly, by giving a projective correspondence between the net |C| and the star of planes through O, and between the pencil |K| and the pencil of planes through a. Then the curve \overline{K} which is added to the C of the net turns out to be in correspondence with the neighbourhood of the multiple point O, which will be precisely of multiplicity i, where one denotes by i the intersection number of a curve C and a K; indeed the neighbourhood of O must have (as well as \overline{K}) i common points with a plane section C. By contrast, to the curve \overline{C}, which is added to the K of the pencil, will correspond the line a, which is multiple for F, and precisely of multiplicity n, where we denote by n the degree of |C|. The plane Oa will cut out i exceptional lines passing through O on F, away from a. The surface F will be therefore of order n + i and one sees that it is the image of a particular ∞^{3} system contained in |C + K|, which has i base points at the common points of C and K: these are just the points which give rise to the above mentioned exceptional lines through O.

If now one wants to construct on F the sum system of the net |C| and of the pencil |K|, it is clear that it will be convenient to consider the system of surfaces obtained by summing a plane through O and a plane through a, namely the ∞^{5} linear system of all quadrics through O and a: one obtains in this way the minimal sum system |C + K|.

1.10 Virtual characters

Let |C_{1}| and |C_{2}| be two irreducible linear systems on the surface F, and let us denote by n_{1} and n_{2} their effective degrees, by π_{1} and π_{2} their genera, and by i the number of variable intersections of a curve C_{1} with a C_{2}. The degree of the system |C_{1} + C_{2}| will be the number of intersections

(C_{1} + C_{2})^{2} = C_{1}^{2} + C_{2}^{2} + 2C_{1}C_{2},

and therefore its value will be

n = n_{1} + n_{2} + 2i.

The genus of the sum system will be

π = π_{1} + π_{2} + i − 1,

since this is the expression of the genus of a curve which becomes reducible in two parts; this expression has been determined by Noether by an argument of continuity^{(12)}.

These formulas lead to the definition of virtual degree and virtual genus of a curve C on F, independently of the possibility of extending the ∞^{0} linear system which they determine. In order to do this, let us associate to C an irreducible system |K| such that the system |C + K| turns out to be irreducible; this can be easily achieved by taking surfaces of sufficiently high order through C and then the residual intersections as the curves K. Now, denoting by n the degree of C which we want to define, by m the degree of K and by i the intersection number of a C with a K, the degree of |C + K| will be given by

(12)See Enriques–Chisini, Ibidem, Libro V, Vol. III, page 399.

N = n + m + 2i,

whence one deduces

n = N − m − 2i.

The virtual degree of C thus defined is independent of the choice of the auxiliary system |K| which was associated to C. Indeed let |K_{1}| and |K_{2}| be two auxiliary systems satisfying the required conditions, of degrees m_{1} and m_{2}, and let i_{1} and i_{2} be the intersection numbers CK_{1} and CK_{2}, and j the number of variable intersections of a curve K_{1} and a K_{2}.

One can compare the degree n_{1} defined with respect to |K_{1}| and the degree n_{2} with respect to |K_{2}|, by computing the degree of the system

|(C + K_{1}) + K_{2}| = |(C + K_{2}) + K_{1}| = |C + (K_{1} + K_{2})|.

Then one has

N = (n_{1} + m_{1} + 2i_{1}) + m_{2} + (2j + 2i_{2}) = (n_{2} + m_{2} + 2i_{2}) + m_{1} + (2j + 2i_{1}),

which yields

n_{1} = n_{2}.

In a completely similar manner one can define the virtual genus π of an isolated curve C, and this character is meaningful even if C is a multiple curve, or a reducible curve containing some multiple part.

Let us add that the definition of the virtual characters (degree and genus) can be also extended to the case of a virtual curve C − K, defined as a difference, independently of its effective existence (see Section ??). It is sufficient indeed to add to C − K a curve L in such a way that |M| = |C − K + L| is an irreducible linear system, and then to compute the characters of the difference |M − L|, which turn out to be independent of |L|.

Going back to effective curves, let us notice that the notion of virtual characters of a curve C suggests some remarks about their invariance with respect to birational transformations of the surface.

Exceptional curves 1.11 31

Let us transform F into another surface \overline{F} in such a way that an exceptional curve o of \overline{F} corresponds to a point O of the curve C; then one can consider as the transform of C either the reducible curve \overline{C} + o, or simply the curve \overline{C}, taking out o which corresponds to a fundamental point of the transformation. The choice which can be made in this way corresponds to a different manner of considering the point O on the curve C: if C is considered as belonging to an irreducible linear system of dimension > 0 with the base point O, then the transform of this system will be the irreducible system |\overline{C}|, in which o does not appear as a fixed part. If by contrast |C| can be extended in such a way that it has no more the base point O, then its transform will contain in particular the system o + |\overline{C}|, where o appears as a fixed part. In other words, the exceptional curve o corresponding to O should be considered as a component of the transformed curve of C, or not, according to whether one intends to consider C as a curve of a linear system which does not have, or has, O as a base point. The circumstance that a system |C| without a base point O actually exists may be regarded as accidental with respect to the above consideration. The virtual degree of C, with regard to all possible transformations of its points to exceptional curves, will be thus defined in general assuming |C| without base points, or else by declaring which base points should be considered as assigned on the curve; apart from these, any other point of C which turns out to be a base point for an actually existing |C|, has to be considered as virtually non-existing. Let us add that if C possesses a multiple point (which is simple for the surface), of multiplicity ≥ i, then this may be assumed as a virtual i-tuple point for the curve C, even if it is in fact of higher multiplicity, which may even turn out to be the effective multiplicity of the largest system |C| having that point as an assigned i-tuple point.

So in particular, if the curve C on the surface F possesses an i-tuple point which we do not want to regard as an assigned base point for the system |C|, then this has to be considered as virtually non-existing; therefore the virtual genus of C will be obtained by adding i(i − 1)/2 to the effective genus π, and this is precisely the quantity of which the genus drops when an irreducible curve acquires an i-tuple point.

Similarly the virtual degree of C will be computed by adding i^{2} to the effective number of intersections of two C away from this point, since the number of intersections of two curves drops exactly of i^{2} when they acquire a common i-tuple point.

1.11 Exceptional curves

As we said above, the neighbourhood of a (simple) base point of the surface F, which happens to be a base point for a linear system of curves, may be regarded as an infinitesimal curve which, by means of a transformation of the surface, can be changed into an actual curve (an exceptional curve). Conversely, an exceptional curve is a curve on the surface F which can be transformed into the neighbourhood of a simple point.

In order to take a closer look at the exceptional curves, let us start by making this distinction:

An irreducible curve ω on the surface F will be said to be an exceptional curve of the 1st kind if it can be transformed into a simple point by means of a linear system |C| with no base points on ω. In contrast, a curve ω will be called an exceptional curve of the 2nd kind if it can be transformed into a point by a linear system |C| having some base point on it.

An example of exceptional curves of the first kind is given by the 27 lines lying on a general cubic surface F_{3}. Indeed, F_{3} can be represented on the plane in such a way that an arbitrarily given one among these lines corresponds to the neighbourhood of one of the 6 base points of the system of cubics representing F_{3}. By contrast, the plane does not contain any exceptional curves of the first kind. But lines, conics and more generally, plane curves that are fundamental curves for a Cremona transformation are exceptional curves of the second kind: one may transform a line into a point by means of a homaloidal net (for instance a net of conics) having two base points on the line; and one may transform a conic to a point by taking for example a homaloidal net of cubics with a double base point and four simple base points on the conic.

It follows already from these examples that the distinction between exceptional curves of the first and of the second kind only has projective meaning or is only relatively invariant under transformations without base points: indeed we can always replace an exceptional curve of the second kind ω belonging to a surface F by a exceptional curve of the first kind on a transformed surface Φ. For this, consider the transforming linear system |C| taking ω into a point; it has, by hypothesis, a certain number s of base points, with certain multiplicities i_{1}, i_{2}, …, i_{s} on ω, and the number of intersections with ω is precisely equal to the sum i_{1} + i_{2} + ⋯ + i_{s}. Now if we add to |C| a linear system |L| without base points, and for which ω is not a fundamental curve, then the system |C + L| will change ω into a fundamental curve for the transformed system of |C| (which will have no base points on it), that will thus be an exceptional curve of the first kind.

Let us try to compute the genus and the degree of an irreducible exceptional curve ω of the first kind, by referring to the simple point O whose neighbourhood corresponds to ω on a transformed surface. Consider on such a surface any linear system |C| for which O is not a base point. Imposing a base point O on |C| reduces its degree by one; moreover the C through O have one point in common with the neighbourhood of O. Thus writing n and π for the degree and genus of |C|, and ν and ρ for the degree and genus of the infinitesimal curve that corresponds to the neighbourhood of O, that is, of the exceptional curve ω, we will have

n = n − 1 + ν + 2 · 1 and π = π + ρ + 1 − 1,

and thus

ν = −1 and ρ = 0.

Actually, the formula ρ = 0 follows directly from the rationality of the neighbourhood of O.

Of course the same computation can be repeated on the transformed surface Φ of F, containing ω. Here one has a system |C| for which ω is a fundamental curve (having no moving intersections with C); then the curves C passing through a point of ω break up into ω and a residual curve C_{1}, which is unisecant to ω: writing the degree and genus of |C| = |ω + C_{1}|, we recover the above formulas.

We therefore state that:

an irreducible exceptional curve of the first kind has virtual genus ρ = 0 and virtual degree ν = −1.

Turning now to exceptional curves of the 2nd kind, it suffices to recall that such a curve ω can be transformed into an exceptional curve of the first kind, provided that one imposes certain base points to a suitable transforming system |C|; therefore the virtual genus and degree of an exceptional curve of the 2nd kind will be

ρ = 0 and ν ≥ 0.

For instance the degree of the generators of a scroll is

ν = 0;

the degree of a line in the plane is

ν = 1,

the degree of a conic is

ν = 4, etc.

Note that the constraint for the virtual genus of an exceptional curve to be ρ = 0 yields that an irreducible exceptional curve of the 1st kind ω cannot have double or multiple points. Indeed the virtual genus ρ is computed by calculating the genus of the sum system |C| = |C_{1} + ω|. Therefore a double point of the component ω, which is not a base point of |C|, has to be considered as non-existing, hence, being already ρ = 0, the curve ω would turn out to be reducible.

Anticipating some of the notions which will be introduced later on, we want to say here that the exceptional curves are defined by their virtual characters which we found above. If ω is an irreducible curve of genus ρ = 0 and degree ν = −1 on the surface F, then it may be transformed into a simple point in the following way. Let us consider on F, in a completely general way, an ∞^{3} linear system |C| without base points, of a certain genus π and a certain degree n, whose curves cut ω at i points. Adding ω to |C| one obtains a system |C + ω| of curves which are (i − 1)-secant to ω, which will be of degree n + 2i − 1 and genus π + i − 1. In this way the difference between the degree and the double of the genus n−2π increases by one. And, as we shall see in what follows, this increment has a real meaning, inasmuch as the addition of ω actually succeeds in extending the system |C|, that is, the dimension of the irreducible system |C + ω| turns out to be bigger than the dimension of |C|. After this one may proceed by adding again ω to the system thus obtained, and so on, until one arrives at a system |C+iω| without base points on F, for which ω turns out to be fundamental. By means of this system F can be transformed in a surface Φ on which the neighbourhood of a simple point O corresponds to ω. In conclusion ω is an exceptional curve of the first kind.

The transformation procedure which we used here leads to the following result: if the surface F contains a certain number s of exceptional curves which cut the C in i_{1}, i_{2}, …, i_{s} points respectively, where one has

i_{1} + i_{2} + ⋯ + i_{s} > 2π − 2 − n

then one may construct on F an irreducible linear system of degree N and genus Π for which

N > 2Π − 2.

As we shall see later on, the existence of systems whose characters satisfy such an inequality defines a particular family of surfaces. It characterizes indeed the family of surfaces which are rational or birational to scrolls; on a surface which does not belong to this particular family, for any linear system of genus π and degree n the fundamental inequality

n ≤ 2π − 2

holds.

A consequence of what we just said is that: a surface which does not belong to the family of ruled surfaces may always be transformed into another surface which has no exceptional curves at all.

Likewise one sees that

a surface which contains exceptional curves of the second kind belongs to the family of ruled surfaces.

A note on the reducible exceptional curves 1.12 35

Indeed, to start with, one may transform, as we said, an exceptional curve of the second kind into a curve ω of genus ρ = 0 and degree ν ≥ 0. Here, since ν ≥ 0, the procedure of adding ω to a system |C| never ends because the intersection number of the curves of the system with ω increases, or at least does not decrease. Therefore one will certainly arrive at a linear system of genus Π and degree N, for which

N > 2Π − 2.

1.12 A note on the reducible exceptional curves

When one transforms a surface F into another surface Φ by means of a linear system |C| having distinct base points O, O_{1}, O_{2}, … on F, the exceptional curves corresponding to these points, which are at the same time fundamental curves for the system |L| (transformed of the plane sections of F), are irreducible curves, disconnected from one another. But what happens when the points O, O_{1}, O_{2}, … become infinitely near?

Suppose that the aforementioned points O, O_{1}, O_{2}, … have multiplicities i, i_{1}, i_{2}, … for |C|, where i > i_{1} > i_{2}, … and the differences i − i_{1}, i_{1} − i_{2}, … are as large as one needs. To these points will in general correspond on Φ some exceptional curves Ω, Ω_{1}, Ω_{2}, … of orders i, i_{1}, i_{2}, … respectively. But if the point O_{1} becomes infinitely near to O, then the curve Ω of degree i becomes reducible into two parts ω and ω_{1}(= Ω_{1}): the former, of order i − i_{1}, corresponds to the points of the neighbourhood of O except O_{1}, the latter, of order i_{1}, is the limit of the exceptional curve corresponding to O_{1}.

In general if several points O_{1}O_{2} … become infinitely near to a proper point O on one or more branches, which may be either linear or superlinear, then the exceptional curve corresponding to the point O will break up into several parts, connected between them, and containing the exceptional curves which arise from the single points. To be more precise we consider the characteristic case in which O, O_{1}, O_{2}, …, O_{s} form a sequence of infinitely near points on one branch, and we start from the simplest examples. If O, O_{1}, O_{2}, …, O_{s} follow each other along a linear branch, then the exceptional curve of order i which corresponds to O becomes a composite curve of order (i − i_{1}) + (i_{1} − i_{2}) + ⋯ + (i_{s−1} − i_{s}):

Ω = ω + ω_{1} + ω_{2} + ⋯ + ω_{s},

whereas the curves coming from O_{1}, O_{2}, … become

Ω_{1} = ω_{1} + ω_{2} + ⋯ + ω_{s}

⋯  ⋯
Ω_{s−1} = ω_{s−1} + ω_{s}
Ω_{s} = ω_{s}.

And, according to the principle of continuity, these reducible curves have to be considered as exceptional curves, whereas their parts (e.g. ω_{1}, ω_{2}, …, ω_{s−1}) do not form complete exceptional curves, but only components of them. Indeed the component ω will not be in one-to-one correspondence with the (whole) neighbourhood of O, but with this neighbourhood with the point O_{1} deleted, and so ω_{1} will be in correspondence with the neighbourhood of O_{1} with the point O_{2} deleted, etc. We may pass from the surface F to the Φ by means of several subsequent transformations from F to F_{1}, from F_{1} to F_{2}, …, from F_{s−1} to F_{s} = Φ; the first of these transformations sends O to a curve ω, the second sends a point O_{1} of ω to a curve ω_{1}, the third sends a point O_{2} of ω_{1} (different from the common point of ω and ω_{1}) to a curve ω_{2}, etc. Therefore each ω_{i} (for i < s) will be connected with the following ω_{i+1} at one point and its degree (which is −1 on F_{i}) will turn out to be −1−1 = −2 on F_{i+1} and on Φ. Accordingly, the composite exceptional curve Ω = ω + ω_{1} + ω_{2} + ⋯ + ω_{s}, corresponding to a sequence of infinitely near points O, O_{1}, …, O_{s}, on a linear branch, will have genus

ρ = 0 + 0 + ⋯ + 0 + (s − 1) − (s − 1) = 0

and degree

ν = −2 − 2 − ⋯ − 2 − 1 + 2(s − 1) = −1.

At the end we find again the same characters of the irreducible exceptional curves, of which the composite curves Ω, Ω_{1}, Ω_{2}, …, Ω_{s} appear as limits.

The same values of the characters must be found also when the points O, O_{1}, O_{2}, …, O_{s} form a sequence on a superlinear branch. However in the definition of the limit curves we meet something surprising: the components of the curves Ω_{i} become in general multiple curves!

Let us consider, for the sake of simplicity, the case of three points O, O_{1}, O_{2} on a cuspidal branch: this case may be handled by looking at the cuspidal branch as a limit of a linear branch, where O_{2}, varying in the neighbourhood of O_{1}, goes to the position of its satellite point. It is apparent here that the curve of order i − i_{1}, corresponding to the neighbourhood of O (without O_{1}), breaks up into a curve ω of order i − i_{1} − i_{2} and a curve ω_{2} of order i_{2}, which corresponds to the neighbourhood of O_{2}; consequently the latter curve turns out to appear twice in

ω + ω_{1} + 2ω_{2}.

In order to understand more clearly what is going on here, let us make three successive transformations of the surface F, changing first of all O into ω, then a point O_{1} of ω into ω_{1}, and finally the common point O_{2} of ω and ω_{1} into ω_{2}: the position of the point O_{2} thus defined corresponds on the surface F to the the satellite point of O_{1}. But, according to what we said above, since O_{2} is a double point for the curve ω + ω_{1}, then the transformed exceptional curve ω_{2} has to be counted twice. The multiplicities of the components of the reducible exceptional curves corresponding to the sequences of infinitely near points on a superlinear branch have been determined by E. Barber and O. Zariski^{(13)}, who also found out how these curves are connected to one another. In this way another interpretation has been given, from a new point of view, of Enriques’ theory of singularities^{(14)}. We shall restrict ourselves here to state the following general rule: the reducible exceptional curve Ω, which corresponds to a sequence of infinitely near points O_{1}, …, O_{s} which are proximate to a proper point O on a superlinear branch, is formed by s+1 components; the ith component ω_{i}h_{i} appears with the multiplicity h_{i} which equals the minimum order of a branch issuing from O^{(15)} and containing the points O_{1}, …, O_{i}.

The same rule can be applied to the composite exceptional curves Ω_{1}, Ω_{2}, …, Ω_{s}. Furthermore the curve ω_{i} is not necessarily connected with the following curve ω_{i+1}, but – in general – to the curve ω_{i+t} which follows it after t steps; this corresponds to the fact that the points O_{i+1}, …, O_{i+t} are proximate points of O_{i} (this is also evident just by looking at the diagram of the branch O_{i+1}, …, O_{i+t}). Let us add that the curve ω_{i} turns out to be of degree −1 − t, as follows from the fact that it represents the neighbourhood of a point O_{i} from which t proximate points have been removed. Taking into account the multiplicities of the curves ω_{i} in Ω_{i}, the composite exceptional curves Ω, Ω_{1}, …, Ω_{s} always turn out to be of degree ν = −1 and genus ρ = 0, as the irreducible exceptional curve of which they are the limit.

We finally point out that the set of irreducible exceptional curves disconnected from one another, of which we consider the limit as O_{1}, O_{2}, …, O_{s} approach O, tends to a curve

Ω + Ω_{1} + ⋯ + Ω_{s}

which is composed of ω, ω_{1}, ω_{2}, …, ω_{s}, each counted with multiplicity equal to the sum of the multiplicities which it has in each Ω_{i} containing it. We will see later on the meaning of this remark.

(13)Barber and Zariski, Amer. J. of Math., 1935.

(14)See Enriques–Chisini, Lezioni etc., Libro I, Vol. II.

(15)When we talk about a branch issuing from O_{1} (or from O_{2}, etc.) we mean we have previously made a transformation which sends O_{1} to a proper point.

The virtual characteristic series 1.13 38

1.13 The virtual characteristic series

We explained how one may determine the virtual characters, and in particular the degree n, of a curve C on a surface F, even if the curve itself is an isolated curve, not contained in a system |C| of dimension r > 0. The procedure for the computation of n^{(16)}, leads also, for n > 0, to the definition of the characteristic series of C, independently of whether it belongs to a larger linear system^{(17)}.

In fact let us associate to C an irreducible linear system |K|, such that |C + K| is irreducible: the characteristic series of C will be defined by the curves of |C + K| which pass through the intersection of a C with a K. This series turns out indeed to be independent of the choice of |K|, as one sees by comparing the series defined in relation with two systems |C +K| and |C +L| and with the system |C + K + L|. Let us notice that, as we will explain later, it is always possible to choose |K| in such a way that the curves of |C + K| cut out a complete linear series on C, so that one gets on C the complete virtual characteristic series.

The importance of this series is mainly apparent in the study of continuous systems of curves C which do not consist of equivalent curves; in that case the virtual characteristic series of C is (partly or completely) cut out by the curves of the system which are infinitely near to it.

Note In this form the characteristic series of a curve C belonging to a continuous system implicitly occurred to G. Castelnuovo, as it appears from a letter to Enriques^{(18)}. Later on Castelnuovo himself explicitly pointed out its meaning, in relation with the (hyperelliptic) surface which represents the pairs of points of a curve of genus two, and drew this example to the attention of Severi; who was therefore led to the definition we have treated above, as he himself says in a footnote to the first page of his paper.

(16)See F. Enriques, Introduzione etc., 1896: no. 15 and 16.

(17)F. Severi, Osservazioni sui sistemi continui di curve, Atti Acc. Torino, 1904.

(18)See Rendic. Circ. Mat. di Palermo, dic. 25th, 1898 (n.4).

Covariant and invariant systems

2.1 Jacobian curves

Let us consider on the surface F, which we may assume to be without any singularity in a hyperspace, an irreducible linear system |C| of dimension 2, namely a net. The locus of those points of the surface that are double points for a curve of the net, is an algebraic curve C_{j}, which we shall call the Jacobian of the net. The C_{j} may be also defined as the locus of those points in which two curves of the net touch each other; in fact the pencil determined by these curves contains a curve having a double point at the contact point. In the plane the Jacobian of a net of curves of order n (> 1) is a curve of order 3n−3; and an i-tuple base point of the net has in general the multiplicity 3i − 1 for the Jacobian; thus a simple base point O is, in general, a double point for the Jacobian, but if the curves of the net have a fixed tangent line at O then the Jacobian has a triple point there, with that line as a fixed tangent line^{(1)}. These properties also hold for any net on a surface F, since the determination of the multiplicity of a base point for the Jacobian of a net |C| depends only on the differential behaviour of the curves C under consideration and of the surface in which they are contained. This may be approximated at the point O, to an arbitrarily high order, with an osculating paraboloid, which, in turn, can be set into a one-to-one correspondence with a plane. For the Jacobians of nets which are contained in a larger linear system |C| the following theorem holds:

The Jacobians of the nets contained in one and the same irreducible linear system |C| of dimension r > 2 are equivalent.

The Jacobian system 2.2 40

Let us take any two nets in |C| and let us compare their Jacobians. First we assume that the two nets have a pencil in common, and therefore both belong to the same ∞^{3} linear system^{(2)}. A generic point P of the surface F is a double point for some curve \overline{C} of the aforementioned ∞^{3} system, and \overline{C}, together with the common pencil of the two nets, determines a net of curves C whose Jacobian passes through P. This way we see that the Jacobians of the ∞^{1} nets of curves C contained in the ∞^{3} system and having a common pencil, form a pencil (since there is only one of these Jacobians passing through any point of F); but this is a linear pencil since it is rational, being in one-to-one correspondence with the pencil of the nets of an ∞^{3} linear system which have a common pencil^{(3)}. Now if we consider two nets of curves in |C| (which we assume of dimension r > 3) having a common curve \overline{C}, then we can construct an auxiliary net containing \overline{C} and having a common pencil with each one of the two nets: the Jacobian of the auxiliary net turns out to be equivalent to the Jacobians of the two nets; so, by the transitive property of equivalence, these Jacobians will be equivalent.

(1)See Enriques–Chisini, Ibidem, Libro III, Vol. II, Chapter I, §5.

(2)This system is determined by the common pencil and by two curves taken in the two

Finally, if we have in |C| two nets without a common curve (the system |C| has then dimension r > 4), we may compare their Jacobians by constructing an auxiliary net having a common curve with each one of the two nets. The Jacobians of the two nets will therefore be equivalent. Q.E.D.

2.2 The Jacobian system

From what we have just proved it follows that:

The Jacobians of the nets contained in an irreducible linear system |C| on the surface F belong to one and the same complete linear system which will be called the Jacobian linear system |C_{j}| of |C|. For the time being, we suppose the dimension of |C| to be ≥ 2.

The above definition is certainly correct when the system |C| has no base points, if it is intended that the system of Jacobians C_{j} has to be completed without imposing any base point. By contrast, if |C| has an i-tuple base point, it must be our intention to assign this as a point of multiplicity 3i − 1 to the complete system |C_{j}|, because, as we said, 3i − 1 is in general the multiplicity of the Jacobian of a net contained in |C|.

This also holds if the base points are infinitely near. So, if for instance |C| possesses a simple base point with a fixed tangent line, that is, two simple infinitely near base points O, O_{1}, then the Jacobians of the nets contained in the system will actually have a triple point at O with OO_{1} as a fixed tangent line^{(4)}. However the system of these curves will in general be contained in a larger system with two double base points O and O_{1}, which will be our complete Jacobian system |C_{j}|. And even if the enlargement of the system is not actually possible, the singularity formed by a triple point with an infinitely near simple point has to be considered as virtually consisting of two double points O and O_{1}. We supposed so far the system |C| to be irreducible, but is clear how the definition must be extended to the case in which |C| contains a fixed part K (which we assume to be simple). Here any point of K appears as a base point with a fixed tangent line for any net contained in |C|, and so the curve K will be a component of multiplicity three of the Jacobians of all the nets contained in |C|. It has to be added three times to these Jacobians and the reducible system thus defined will be contained in general in the larger irreducible complete system |C_{j}|.

nets, but not lying in the pencil.

(3)This set can be identified with a pencil of planes of S_{3}.

(4)See Section ??.

As a matter of fact one proves in this way that K appears at least with multiplicity three in the Jacobians of the nets of |C|. In order to prove that it does not actually appear with multiplicity bigger than three, one may reduce the problem from the surface to the plane, by replacing F with a suitable osculating paraboloid at a point of the curve K, or alternatively with a rational surface having a very high contact with F along the whole curve K^{(5)}. One realizes in this way that a curve K which is added as a fixed component to a net of irreducible curves, appears three times in the Jacobian of this net; it cannot appear with multiplicity bigger than 3, unless it is a part of the Jacobian of |C|.

Note One may give a direct analytic verification of the theorem we obtained. In order to do this one needs to know how a complete linear system is cut out on a surface F (with normal singularities) of ordinary space, namely the result that “the adjoint surfaces φ to f, that is, the surfaces passing simply through the double curve of f, cut out complete linear systems on f”^{(6)}. For the sake of simplicity we will suppose that f is a surface of order n, without singularities, so that the complete intersections with all surfaces of a given order m form a complete system on it.

If we are given a net of irreducible curves |C| on f, which are not complete intersections, and we add to them a fixed component K, then we may always suppose that the net |C| is the residual of a curve L, which does not have K as a part of it, so that the curves C + L are complete intersections with surfaces φ_{i} of order m. Therefore the net |K + C| is partially contained in the net represented by an equation of the form:

\begin{matrix}φ(x_{1}, x_{2}, x_{3}, x_{4}) × \\ [λ_{1}φ_{1}(x_{1}, x_{2}, x_{3}, x_{4}) + λ_{2}φ_{2}(x_{1}, x_{2}, x_{3}, x_{4}) + λ_{3}φ_{3}(x_{1}, x_{2}, x_{3}, x_{4})] = 0 \end{matrix}

According to our assumptions, φ_{1}, φ_{2}, φ_{3} do not vanish simultaneously on K, which is instead a part of φ = 0, whose order we will denote by s. Now we may write down the equation of the Jacobian of the net described above, and see that the curve φ = 0, and in particular K, which is a part of it, appears in the net as a component of multiplicity 3. The points of the Jacobian are the contact points with f of a surface of the system which cuts out the net. If we write down the conditions for contact, we find a linear system, whose compatibility condition is expressed by the vanishing of the determinant

(5)See Enriques–Chisini, Libro XV, Chapter IV, §39, Vol. II, page 634 and following.

(6)See the footnote to Section ?? and Chapter III.

D(x_{1}, x_{2}, x_{3}, x_{4}) = | \begin{matrix}\frac{∂(φφ_{1})}{∂x_{1}} & \frac{∂(φφ_{2})}{∂x_{1}} & \frac{∂(φφ_{3})}{∂x_{1}} & \frac{∂f}{∂x_{1}} \\ \frac{∂(φφ_{1})}{∂x_{2}} & \frac{∂(φφ_{2})}{∂x_{2}} & \frac{∂(φφ_{3})}{∂x_{2}} & \frac{∂f}{∂x_{2}} \\ \frac{∂(φφ_{1})}{∂x_{3}} & \frac{∂(φφ_{2})}{∂x_{3}} & \frac{∂(φφ_{3})}{∂x_{3}} & \frac{∂f}{∂x_{3}} \\ \frac{∂(φφ_{1})}{∂x_{4}} & \frac{∂(φφ_{2})}{∂x_{4}} & \frac{∂(φφ_{3})}{∂x_{4}} & \frac{∂f}{∂x_{4}} \end{matrix} | = 0

Now, once the derivatives of φφ_{1}, φφ_{2}, φφ_{3} have been computed, we may decompose the determinant D as a sum of determinants: if we neglect those determinants which are identically zero, then the others have the factor φ^{2} in common; which we factor out to leave us with four determinants, one of which contains the factor φ again. We prove now that the sum of the other 3 determinants (simply) vanishes at the points of the common curve φ = 0. This sum can be easily transformed in a single determinant. Let us write:

Φ = φ_{1} | \begin{matrix}\frac{∂φ}{∂x_{1}} & \frac{∂φ_{2}}{∂x_{1}} & \frac{∂φ_{3}}{∂x_{1}} & \frac{∂f}{∂x_{1}} \\ \frac{∂φ}{∂x_{2}} & \frac{∂φ_{2}}{∂x_{2}} & \frac{∂φ_{3}}{∂x_{2}} & \frac{∂f}{∂x_{2}} \\ \frac{∂φ}{∂x_{3}} & \frac{∂φ_{2}}{∂x_{3}} & \frac{∂φ_{3}}{∂x_{3}} & \frac{∂f}{∂x_{3}} \\ \frac{∂φ}{∂x_{4}} & \frac{∂φ_{2}}{∂x_{4}} & \frac{∂φ_{3}}{∂x_{4}} & \frac{∂f}{∂x_{4}} \end{matrix} | +

+ φ_{2} | \begin{matrix}\frac{∂φ_{1}}{∂x_{1}} & \frac{∂φ}{∂x_{1}} & \frac{∂φ_{3}}{∂x_{1}} & \frac{∂f}{∂x_{1}} \\ \frac{∂φ_{1}}{∂x_{2}} & \frac{∂φ}{∂x_{2}} & \frac{∂φ_{3}}{∂x_{2}} & \frac{∂f}{∂x_{2}} \\ \frac{∂φ_{1}}{∂x_{3}} & \frac{∂φ}{∂x_{3}} & \frac{∂φ_{3}}{∂x_{3}} & \frac{∂f}{∂x_{3}} \\ \frac{∂φ_{1}}{∂x_{4}} & \frac{∂φ}{∂x_{4}} & \frac{∂φ_{3}}{∂x_{4}} & \frac{∂f}{∂x_{4}} \end{matrix} | + φ_{3} | \begin{matrix}\frac{∂φ_{1}}{∂x_{1}} & \frac{∂φ_{2}}{∂x_{1}} & \frac{∂φ}{∂x_{1}} & \frac{∂f}{∂x_{1}} \\ \frac{∂φ_{1}}{∂x_{2}} & \frac{∂φ_{2}}{∂x_{2}} & \frac{∂φ}{∂x_{2}} & \frac{∂f}{∂x_{2}} \\ \frac{∂φ_{1}}{∂x_{3}} & \frac{∂φ_{2}}{∂x_{3}} & \frac{∂φ}{∂x_{3}} & \frac{∂f}{∂x_{3}} \\ \frac{∂φ_{1}}{∂x_{4}} & \frac{∂φ_{2}}{∂x_{4}} & \frac{∂φ}{∂x_{4}} & \frac{∂f}{∂x_{4}} \end{matrix} |;

Exchanging in the first determinant the 1st column with the 3rd and in the second determinant the 2nd column with the 3rd, and taking into account the signs, one finds the expansion of a 5th order determinant:

Φ = | \begin{matrix}φ_{1} & φ_{2} & φ_{3} & 0 & 0 \\ \frac{∂φ}{∂x_{1}} & \frac{∂φ_{1}}{∂x_{1}} & \frac{∂φ_{2}}{∂x_{1}} & \frac{∂φ_{3}}{∂x_{1}} & \frac{∂f}{∂x_{1}} \\ \frac{∂φ}{∂x_{2}} & \frac{∂φ_{1}}{∂x_{2}} & \frac{∂φ_{2}}{∂x_{2}} & \frac{∂φ_{3}}{∂x_{2}} & \frac{∂f}{∂x_{2}} \\ \frac{∂φ}{∂x_{3}} & \frac{∂φ_{1}}{∂x_{3}} & \frac{∂φ_{2}}{∂x_{3}} & \frac{∂φ_{3}}{∂x_{3}} & \frac{∂f}{∂x_{3}} \\ \frac{∂φ}{∂x_{4}} & \frac{∂φ_{1}}{∂x_{4}} & \frac{∂φ_{2}}{∂x_{4}} & \frac{∂φ_{3}}{∂x_{4}} & \frac{∂f}{∂x_{4}} \end{matrix} | .

Subtracting from the 1st row multiplied by m the 2nd row multiplied by x_{1}, the 3rd row multiplied by x_{2}, the 4th row multiplied by x_{3}, the 5th row multiplied by x_{4} and taking into account Euler’s theorem, we may replace the 1st row with

0, 0, 0, −sφ, −nf

where n, m, s denote the orders of f, the φ_{i} and φ respectively.

It is clear that the determinant in which this line appears vanishes when φ = 0 and f = 0 simultaneously occur, since, when these terms are zero, the first row identically vanishes. Q.E.D.

Let us add that K cannot appear with multiplicity bigger than three in the Jacobian of the net; or at least, if this happens, then K is a part of the Jacobian of the net from which the fixed part K has been removed.

In fact a higher multiplicity for K should be a consequence of a contact between the surfaces f and Φ along the curve K. Now let us suppose that the tangent planes to f and Φ coincide at a point P of this curve. We should then have

ρ\frac{∂f}{∂x_{i}} = \frac{∂Φ}{∂x_{i}} for i = 1, 2, 3, 4.

Let us compute ∂Φ/∂x_{i}. We may differentiate by columns; in this way the derivative will be expressed as a sum of determinants obtained by replacing a column with its derivatives with respect to x_{i}. It is clear that, since we are at a point where φ = 0, f = 0, the determinants that are obtained by differentiating the first three columns are zero. Therefore we are left with:

m (\frac{∂Φ}{∂x_{i}} )_{P} =

| \begin{matrix}0 & 0 & 0 & −s\frac{∂φ}{∂x_{i}} & 0 \\ \frac{∂φ_{1}}{∂x_{1}} & \frac{∂φ_{2}}{∂x_{1}} & \frac{∂φ_{3}}{∂x_{1}} & \frac{∂^{2}φ}{∂x_{1}∂x_{i}} & \frac{∂f}{∂x_{1}} \\ \frac{∂φ_{1}}{∂x_{2}} & \frac{∂φ_{2}}{∂x_{2}} & \frac{∂φ_{3}}{∂x_{2}} & \frac{∂^{2}φ}{∂x_{2}∂x_{i}} & \frac{∂f}{∂x_{2}} \\ \frac{∂φ_{1}}{∂x_{3}} & \frac{∂φ_{2}}{∂x_{3}} & \frac{∂φ_{3}}{∂x_{3}} & \frac{∂^{2}φ}{∂x_{3}∂x_{i}} & \frac{∂f}{∂x_{3}} \\ \frac{∂φ_{1}}{∂x_{4}} & \frac{∂φ_{2}}{∂x_{4}} & \frac{∂φ_{3}}{∂x_{4}} & \frac{∂^{2}φ}{∂x_{4}∂x_{i}} & \frac{∂f}{∂x_{4}} \end{matrix} | + | \begin{matrix}0 & 0 & 0 & −s\frac{∂φ}{∂x_{i}} & 0 \\ \frac{∂φ_{1}}{∂x_{1}} & \frac{∂φ_{2}}{∂x_{1}} & \frac{∂φ_{3}}{∂x_{1}} & \frac{∂^{2}φ}{∂x_{1}∂x_{i}} & \frac{∂f}{∂x_{1}} \\ \frac{∂φ_{1}}{∂x_{2}} & \frac{∂φ_{2}}{∂x_{2}} & \frac{∂φ_{3}}{∂x_{2}} & \frac{∂^{2}φ}{∂x_{2}∂x_{i}} & \frac{∂f}{∂x_{2}} \\ \frac{∂φ_{1}}{∂x_{3}} & \frac{∂φ_{2}}{∂x_{3}} & \frac{∂φ_{3}}{∂x_{3}} & \frac{∂^{2}φ}{∂x_{3}∂x_{i}} & \frac{∂f}{∂x_{3}} \\ \frac{∂φ_{1}}{∂x_{4}} & \frac{∂φ_{2}}{∂x_{4}} & \frac{∂φ_{3}}{∂x_{4}} & \frac{∂^{2}φ}{∂x_{4}∂x_{i}} & \frac{∂f}{∂x_{4}} \end{matrix} | .

Expanding the two determinants with respect to the first row we have:

m (\frac{∂Φ}{∂x_{i}} )_{P} = ρm (\frac{∂f}{∂x_{i}} )_{P} = s\frac{∂φ}{∂x_{i}} | \begin{matrix}\frac{∂φ_{1}}{∂x_{1}} & \frac{∂φ_{2}}{∂x_{1}} & \frac{∂φ_{3}}{∂x_{1}} & \frac{∂f}{∂x_{1}} \\ \frac{∂φ_{1}}{∂x_{2}} & \frac{∂φ_{2}}{∂x_{2}} & \frac{∂φ_{3}}{∂x_{2}} & \frac{∂f}{∂x_{2}} \\ \frac{∂φ_{1}}{∂x_{3}} & \frac{∂φ_{2}}{∂x_{3}} & \frac{∂φ_{3}}{∂x_{3}} & \frac{∂f}{∂x_{3}} \\ \frac{∂φ_{1}}{∂x_{4}} & \frac{∂φ_{2}}{∂x_{4}} & \frac{∂φ_{3}}{∂x_{4}} & \frac{∂f}{∂x_{4}} \end{matrix} | +

−n\frac{∂f}{∂x_{i}} | \begin{matrix}\frac{∂φ_{1}}{∂x_{1}} & \frac{∂φ_{2}}{∂x_{1}} & \frac{∂φ_{3}}{∂x_{1}} & \frac{∂φ}{∂x_{1}} \\ \frac{∂φ_{1}}{∂x_{2}} & \frac{∂φ_{2}}{∂x_{2}} & \frac{∂φ_{3}}{∂x_{2}} & \frac{∂φ}{∂x_{2}} \\ \frac{∂φ_{1}}{∂x_{3}} & \frac{∂φ_{2}}{∂x_{3}} & \frac{∂φ_{3}}{∂x_{3}} & \frac{∂φ}{∂x_{3}} \\ \frac{∂φ_{1}}{∂x_{4}} & \frac{∂φ_{2}}{∂x_{4}} & \frac{∂φ_{3}}{∂x_{4}} & \frac{∂φ}{∂x_{4}} \end{matrix} |

for i = 1, 2, 3, 4. If, for brevity, we denote by A the first determinant and by B the second, this equation can be written as

ρm\frac{∂f}{∂x_{i}} = s\frac{∂φ}{∂x_{i}} A − n\frac{∂f}{∂x_{i}} B, for i = 1, 2, 3, 4,

which a priori gives two possibilities:

(1) either ∂f/∂x_{i} = λ∂φ/∂x_{i}, namely the surfaces f = 0 and φ = 0 are tangent, contrary to our assumptions; or

(2) A = 0 and then, as one easily deduces from the form of the determinant A, the point P lies on the Jacobian of the net.

The fundamental theorem 2.3 45

The algebraic procedure developed here can be explained with some simple geometric argument which we only sketch, referring to the simplest case in which the surface f of order n of ordinary space has no singularities, φ_{1}, φ_{2}, φ_{3} are three linearly independent surfaces of order m > n, and φ is a surface of order s.

Let us remark that if we take the surface f together with a surface Ψ of order m − n, then it can be linearly combined with φ_{1}, φ_{2}, φ_{3}; one may then construct the Jacobian surface of the ∞^{3} linear system thus determined, which will be a surface D of order (4m − 4)^{(7)}. D cuts out on f a curve of order (4m − 4)n which consists of the Jacobian of the net of curves |C| cut out by

λ_{1}φ_{1} + λ_{2}φ_{2} + λ_{3}φ_{3} = 0

and of the intersection curve L of f and Ψ; since the latter has order n(m−n), then the Jacobian of the net |C| turns out to be of order n(3m + n − 4).

Let us add to the net

λ_{1}φ_{1} + λ_{2}φ_{2} + λ_{3}φ_{3} = 0

a surface φ of order s, cutting out a curve K of order ns on f; if we add φ also to f, then we have an ∞^{3} linear system of surfaces of order m + s in which φ appears as a fixed part. Therefore φ appears with multiplicity four in the Jacobian of the system; its intersection with f (of order ns), counted with multiplicity one is a component of the intersection curve of f with Ψ + φ; so it appears three times in the Jacobian curve of the net |K + C|, whose order is now

n[3(m + s) + n − 4].

2.3 The fundamental theorem

The above remarks yield the following result.

Theorem 1 (FUNDAMENTAL THEOREM) Let |C| and |L| be two irreducible linear systems on the surface F, which we always assume to be virtually free of base points and of dimension at least 2, then the Jacobian system |C + L|_{j} of their sum can be expressed as

|C + L|_{j} = |C_{j} + 3L| = |3C + L_{j}|.

In fact the systems |C|+L with a fixed curve L, and C +|L| with a fixed curve C, are contained in the system |C + L|; their complete Jacobian systems are

|C_{j} + 3L| and |L_{j} + 3C|

respectively, and they both coincide with the system |C + L|_{j}.

(7)This is an immediate extension of the notion of the Jacobian of a net of plane curves.

See Enriques–Chisini, Ibidem, Libro III, Chapter I, §5, Vol. II, page 31.

According to the fundamental theorem one may now define the complete Jacobian system of a linear system |C| whose dimension is less than two, namely a pencil or an isolated curve C. Indeed it suffices to add to C an auxiliary, at least ∞^{2}, system |L|, and set

|C_{j}| = |3C + L_{j} − 3L|.

The fundamental theorem can be thus extended to the Jacobian system of any linear system, however reducible. Some caution is needed only in the case when one considers linear systems |C| and |L| with assigned base points on the surface F. If |C| has an i-tuple base point O which is not a base point for |L|, and |L| has a s-tuple base point O′ which is not a base point for |C|, then the Jacobian system |C +L|_{j} has (virtual) multiplicity 3i−1 at O, and 3s−1 at O′; by contrast O has multiplicity 3i for |3C + L_{j}| and O′ multiplicity 3s for |3L + C_{j}|. Therefore the equivalence between the two systems |C_{j} + 3L| and |3C + L_{j}| only holds if the neighbourhoods of the points O and O′ are respectively added to the two systems. Namely one has

|3C + L_{j} + O| = |3L + C_{j} + O′|

According to our convention, the meaning of this equivalence is the following: if one makes a transformation of the surface F, changing O and O′ into exceptional curves, then these curves must be respectively added to the transformed systems of |C_{j} + 3L| and |L_{j} + 3C|.

Now we go back to the case we will generally consider in what follows, that is, we deal with complete systems, virtually free of base points on the surface F. The equivalence |C_{j} + 3L| = |L_{j} + 3C| may be reduced to

|C_{j} − 3C| = |L_{j} − 3L| = |K|,

which in other words means that: if the Jacobian |C_{j}| of a system |C| contains the triple of |C|, then the Jacobian |L_{j}| of any other system |L| also contains the triple of |L|, and the system |K|, obtained as the difference of |C_{j}| and |3C|, is independent of the system |C| one starts from. This system |K| is defined to be the (impure) canonical system of the surface F.

But, even if |C_{j}| does not contain |3C|, that is, if the difference system |C_{j} −3C| is only virtual, the symbolic equivalence written above is still meaningful. The meaning of the equivalence between virtual systems is nothing but a different way of writing the relation expressing the fundamental theorem on Jacobians.

It is also convenient to give a simpler form of the above-mentioned equivalence of virtual systems. Therefore we define the adjoint linear system |C′| to |C| by means of the relation

|C′| = |C_{j} − 2C|,

Canonical curves 2.4 47

and we notice that, for a system |C| on a surface F, |C′| (unlike |K|) does in general effectively exist: this is already the case for the linear systems of curves of order m and genus π > 0 in the plane, since then the C′ are the curves of order m − 3 passing i − 1 times through each i-tuple base point of |C|; and, as we will see later on, the same holds for all linear systems whose genus exceeds a certain bound.

Once the adjoint system has been introduced, the canonical system may be defined as:

|K| = |C′ − C| = |L′ − L| = ⋯,

and the meaning of this equivalence of effective or virtual systems is that

|C′ + L| = |L′ + C|

This is the fundamental relation for the adjoint systems (without base points), which in turn is just a simple translation of the relation for the Jacobian systems given above.

2.4 Canonical curves

As we saw, starting from a given linear system |C| on a surface F, one constructs the adjoint linear system |C′| and then the canonical system |K| = |C′ − C|, which may be either effective or virtual; we also realized the latter system to be independent of the choice of the system |C|, which is supposed to be base point free on F. Even when |C| has an i-tuple base point O, simple for F, the system |C′ −C| may still be considered equal to |K|, except for the infinitesimal curve corresponding to the neighbourhood of O, which must be added to the K, since O is an (i − 1)-tuple point for |C′|. The independence of |K| of the choice of |C| means that the canonical system |K| is invariant for birational transformations of the surface.

In order to explain this better, let us consider two surfaces F and Φ, of orders n and m respectively, with normal singularities in ordinary space; we suppose that there is a birational correspondence between them, and, to begin with, we make the hypothesis that this correspondence has no fundamental points on either of the two surfaces. The plane sections of the surface F form a linear system |C| whose transformed system has no base points on Φ, and the plane sections L of Φ likewise form a linear system on Φ whose transformed system has no base points on F. Now one may construct the canonical system |K| on F either starting from the plane sections system |C|, or starting from the system |L|, and one has

|K| = |C′ − C| = |L′ − L|

The same system may be constructed on Φ, either starting from the plane sections system |L|, or starting from the system |C|, and one has again

|K| = |L′ − L| = |C′ − C|.

Therefore the curves of the canonical system |K|, which are constructed on F starting from the plane sections system, are transformed into the curves of the canonical system of Φ, which in turn is constructed starting from the plane sections system |L|. We may now understand the meaning of this construction.

In order to construct the canonical system on F, one has first to construct the Jacobian system of |C|. To do so we notice that a star of plane sections of F, with its centre at a point O of the space, is a net whose Jacobian is the intersection of F with the polar surface of order n − 1 of O, away from the nodal curve.

Thus the Jacobians of the nets contained in the plane sections system |C| form an ∞^{3} linear system which is cut out on F by the polars, away from the nodal curve; but this system has the cuspidal points of the aforementioned nodal curve as simple base points. It is easy to see that these points are actually to be considered as “simple points” of the surface, and, according to our conventions, they are to be removed in order to obtain the complete Jacobian system |C_{j}|. Indeed if we consider a transformation of F into a surface on which the nodal curve is represented by the curve of neutral pairs with respect to the system |C|, then the cuspidal points give on such a curve some contact base points for a net of C, and double base points for a pencil of C: so these points are simple points for the Jacobians of all the nets contained in the ∞^{3} system of the C.

Hence the complete Jacobian system |C_{j}|, which is virtually free from base points, contains the whole system of curves cut out on F by the (adjoint) surfaces of order n − 1 passing through its double curve. If we now subtract 3 planes from these surfaces, then we have that the surfaces of order n − 4 passing through the double curve of F (if they exist) cut out canonical curves on F.

In a similar way the adjoint surfaces of order m − 4, passing through the double curve of Φ, cut out canonical curves on Φ.

Therefore the birational invariance of the canonical system means that “the section curves of F with the adjoint surfaces of order n−4, are sent, with a birational transformation, to curves which are equivalent to the sections of Φ with its adjoint surfaces of order m − 4”.

This theorem becomes even more significant if we now take for granted a result which we will prove in the next chapter, namely that “the adjoint surfaces of any order, and in particular those of order n−4, cut out a complete

Properties of canonical curves 2.5 49

linear system on the surface F of order n”, and therefore, in this case, the complete canonical system |K|.

So we have: the (canonical) curves, cut out on the surface F (away from the double curve) by its adjoint surfaces of order n−4 are sent, in a birational transformation, to the sections of Φ with its adjoint surfaces of order m − 4.

We actually supposed so far that the birational correspondence between F and Φ has no exceptions, namely that the system |L| has no base point on F, and |C| has no base point on Φ. But if, for instance, |L| has an i-tuple base point O on F, then the theorem still holds with this change: the exceptional curve ω corresponding to the point O of F has to be added to the transformed curves of |K|, since it appears as a fixed part of the impure canonical system formed by the sections of Φ with its adjoint surfaces of order m − 4.

Note We supposed F and Φ to have only normal singularities (that is, a double curve with triple points); but this restriction may be removed, provided that one defines the adjoint surfaces to a given one, according to the higher singularities it might possess. It suffices to point out here that an ordinary i-tuple curve of F has to be of multiplicity i − 1 for the adjoint surfaces, and a conic point of order s, away from the multiple curve, has to be in general of multiplicity s − 2 for the adjoint surfaces. We shall come back later to this subject.

2.5 Properties of canonical curves

The canonical curves on a surface F (more precisely the impure canonical curves, cut out on a surface F of order n, away from the double curve, by the adjoint surfaces of order n − 4) enjoy a property which we now wish to point out explicitly. As we shall see in the next chapter, this property, under mild assumptions, does in fact characterize the canonical curves.

Let us consider on F an irreducible, base point free, linear system |C|, of dimension r ≥ 2, genus π and degree n. The Jacobian of a net of curves C cuts out on a curve C the group consisting of the 2n + 2π − 2 double points of the characteristic g_{n}^{1}, and this belongs to the sum of the canonical series with the double of the g_{n}^{1}. If one subtracts 2C from |C_{j}|, then one has curves C′ adjoint to |C|, and they cut out on C canonical groups. If C is subtracted once more, then one finds that the canonical curves cut out on C residual groups to the characteristic series.

Hence we may state:

The adjoint curves cut out on the curves of the system |C| canonical divisors. The canonical curves K cut out on the curves |C| residual groups to the characteristic series.

Geometric genus and linear genus 2.6 50

Whence it follows that the adjoint curves C′ cut out 2π − 2 points on a curve C, of virtual genus π, and the canonical curves K, either effective or virtual, cut out

i = 2π − 2 − n

points on C.

This equality holds for any curve, whether irreducible, reducible, or even isolated, provided one considers its virtual characters, which are defined by assuming that C has no assigned base points on F. In fact let us add to C a suitable curve L, of genus ρ and degree ν which meets C at s points, such that the complete linear system |C + L| is irreducible of dimension > 2; it suffices to take as |L| the system cut out by those surfaces of large enough order which pass through C. Now the curves C′ + L belonging to the system |(C + L)′| cut out

2(π + ρ + s − 1) − 2

points on a C + L; by removing their intersections with L, whose number is 2ρ − 2 + s, and then the s intersections of C with L, one obtains

(CC′) = 2π − 2.

Likewise the virtual number of intersections of C and K is always (CK) = 2π − 2 − n.

Note We add that the group cut out on an irreducible curve C on F, without multiple points, by an adjoint curve C′ is always a canonical group, and the group cut out by a K is a residual group to the characteristic series (either effective or virtual). It suffices to prove the assertion for a curve C which is a part of a composite curve C + L belonging to an irreducible net. Going back to the previous argument, one finds that a curve of

|C_{j}| = |(C + L)_{j} − 3L|

cuts out on C a group which belongs to the canonical series summed with the double of the virtual characteristic series of C itself.

2.6 Geometric genus and linear genus

Since the canonical system is invariant under birational transformations of the surface, its characters likewise have an invariant meaning with respect to these transformations. However we need to distinguish between:

(1) absolute invariants, which do not change for any transformation, and (2) relative invariants which are preserved under birational transformations

without exceptions, but are not preserved under transformations having fundamental points, that change a (simple) point of one surface into an exceptional curve of the other.

Indeed the dimension of the canonical system, or rather the number of linearly independent canonical curves, gives us an absolute invariant, called the geometric genus (p or p_{g}), or simply the “genus”, of the surface.

Since the complete canonical system is cut out on a surface F of order n of ordinary space by the adjoint surfaces φ_{n−4} of order n − 4 (see Chapter III), then the genus of a general surface of order n is

p = \frac{(n − 3)(n − 2)(n − 1)}{6}

which is the number of linearly independent φ_{n−4}.

Therefore the genus of the plane, of the quadrics and of the cubic surfaces is

p = 0.

This can be directly verified for the plane by observing that the complete system |C| of plane curves of order n ≥ 3 contains the adjoint system |C′| of the curves of order n − 3; thus, since |C| is larger than |C′|, then |C′| cannot contain |C|: so C′ − C is not effective, which means p = 0.

The vanishing of the genus of the plane yields that all rational surfaces (which can be put in one-to-one birational correspondence with the plane) likewise have genus zero. Among these surfaces we find the above-mentioned quadrics and the cubics (at least those which are not cones).

Another family of surfaces having (geometric) genus zero are the scrolls, and therefore also all the surfaces birational to them. This assertion can be justified by remarking that a generator of a scroll F cannot have an adjoint system, because this system should cut out on the generator the canonical series. In general the adjoint |C′| of a linear system |C| on the scroll cannot contain |C| because, if the C cut out n points on the generators, then the C′ cut out n − 2 points on them (see Chapter III, Section ??).

Going back to surfaces of any kind of genus p ≥ 0, the genus of the canonical curves (either effective or virtual) gives a relative invariant, called the linear genus p^{(1)} or also the relative linear genus, which drops by one for any point of the surface F that is changed into an exceptional curve, and conversely increases by one for any exceptional curve that is sent to a point. In fact we saw that, when a point O of the surface is changed into an exceptional curve ω of Φ, this curve ω has to be added to the transformed \overline{K} of the canonical curves. Now such a curve is not connected with the above mentioned K, that is, its virtual number of intersections with them is zero, since the point O is – at least virtually – not a base point of |K|. In fact the system |K| has no assigned base point on F. Therefore by adding ω, of genus 0 and degree −1, to the \overline{K}, the genus of these drops by one, since ω is disconnected from them.

It is apparent here that it is more appropriate, at least for surfaces of genus p > 0 where the canonical system is effective, to consider, together with the relative linear genus p^{(1)}, the genus of the pure canonical system, from which all the exceptional fixed components have been removed. This is in turn the genus of the canonical curves on a transformed surface without exceptional curves, as we shall see more precisely in Section ??. We find in this way an absolute invariant, called the absolute linear genus of the given surface. Its value is

\overline{p}^{(1)} = p^{(1)} + σ,

where σ denotes the number of the exceptional curves belonging to the surface.

More precisely, we intend to show in the next section that, at least for p > 0, these curves are always exceptional curves of the first kind, disconnected one from another, and we will examine in particular the case of the reducible exceptional curves.

As the genus of the canonical system, its degree also gives a relative invariant of the surface, which we may denote by p^{(2)}. When, by a transformation of the surface F, an exceptional curve ω of degree −1 disconnected from the K is added to the \overline{K}, p^{(2)} also drops by 1. And here too, together with the degree of the impure canonical system, we may define the degree of the pure canonical system; this is an absolute invariant, which is defined, at least for surfaces with genus p > 0. But p^{(2)}, compared with the linear genus, gives no new invariant character. Indeed the following relation holds between the genus and the degree of the impure canonical system

p^{(2)} = p^{(1)} − 1.

In order to prove this it suffices to remark that the adjoint system to the canonical system |K| is

|K′| = |2K|.

Therefore the number of intersections of K with 2K is 2p^{(1)} − 2, and then

(KK) = p^{(1)} − 1.

In the case p > 0 (and in any other case in which the absolute characters we deal with may be defined, that is, when the surface can be transformed into another one which has no exceptional curves) the above relation between the genus p^{(1)} + σ and the degree p^{(2)} + σ of the pure canonical system still holds.

Note We will see later on that the only surfaces for which the absolute linear genus cannot be defined (by considering a transformed surface without exceptional curves) are those belonging to the family of ruled surfaces (rational or not). Nevertheless for these too one arrives at the construction of an absolute invariant, which is given – according to Castelnuovo^{(8)} – by the maximum of the relative linear genus with respect to all possible transformations of the given surface.

It is convenient here to compute the linear genus of the plane and of the irrational scrolls of genus p (> 0), for which that maximum value is achieved.

The linear genus of the plane may be computed by considering a linear system |C| of plane quartics, of genus π = 3 and degree n = 16: the genus π′ = 0 of the adjoint system, which is the net of lines, is expressed by the formula

π′ = π + p^{(1)} + 2π − 2 − n − 1.

Setting here π = 3, n = 16, π′ = 0, one finds that the linear genus of the plane is

p^{(1)} = 10.

Let us consider now a scroll of order m and genus p > 0. The double of the plane sections system (which is cut out by quadrics) has genus

π = 2p + m − 1

and degree

n = 4m.

The system adjoint to it is formed by groups of m + 2p − 2 lines^{(9)}, and therefore has genus

π′ = −(m + 2p − 2) + 1 = −m − 2p + 3.

Therefore from the formula

π′ = π + p^{(1)} + 2π − 2 − n − 1,

one deduces that: the linear genus of the scrolls of genus p > 0 is

p^{(1)} = −8(p − 1) + 1.

(8)Sul genere lineare di una superficie e sulla classificazione a cui esso da’ luogo, Rendic.

Acc. Lincei, Vol. VI, 1897. Memorie scelte; page 443.

(9)This may be verified in various ways; see Chapter III.

The exceptional curves as fixed parts of the canonical system 2.7 54

2.7 The exceptional curves as fixed parts of the canonical system

It is useful to make more precise what we said above about the exceptional curves on a surface of genus p > 0, as belonging, as fixed parts, to the canonical system |K|.

The fact that an exceptional curve ω of the first kind on a surface F belongs, as a fixed component, to the system |K| follows directly from the fact that it has genus ρ = 0 and degree ν = −1. In fact ω has to be cut by the K at

2ρ − 2 − ν = −1

points; since this (virtual) number is negative, ω has to be a part of each K. But we can say more: on the surface F of genus p > 0 we cannot have two irreducible exceptional curves of the first kind connected one to the other, namely with some point in common. In fact, assume such a pair of curves ω and ω′, of genus 0 and degree −1, with i (≥ 1) points in common, exists; they appear in all canonical curves K as fixed parts to be counted a certain number of times. But, if the former has to be counted r times and the latter s times, the number of intersections of ω and ω′ with the residual curves is respectively

(Kω) − r(ωω) − s(ωω′) = −1 + r − si
(Kω′) − s(ω′ω′) − r(ωω′) = −1 + s − ri

Now, since the sum of these two numbers is negative, at least one of them is negative, thus the corresponding curve, either ω or ω′, should appear in the K with higher multiplicity, contrary to the assumption.

With the same reasoning one proves that a surface F of genus p > 0, cannot contain an exceptional curve ω of the second kind. Indeed let us consider a transform Φ of F, on which ω becomes a curve of degree ν ≥ 0, but again with genus ρ = 0. On such a surface the virtual number of points ω has in common with a canonical curve is

2ρ − 2 − ν = −2 − ν

which is negative; ω is therefore a fixed part of the K; but its multiplicity should be infinite, which gives a contradiction. Suppose indeed its multiplicity is s; then the number of intersections with the residual curves is

(Kω) − s(ωω) = −2 − (s + 1)ν,

which is always a negative number.

We may summarize the results we obtained by stating the theorem:

On a surface of genus p > 0 one may have at most a finite number of irreducible exceptional curves of the first kind, each one disconnected from the others; there can never be exceptional curves of the second kind.

We notice that if one has only irreducible exceptional curves of the first kind, then each of them, ω, appears with multiplicity one as a component of the canonical system and is a fundamental curve for the residual system |K − ω|. Anyhow it cannot be a priori excluded that ω is again a fixed component of |K − ω|, thus appearing twice as a fixed part of |K|. This means that, by contracting ω to a point, this point turns out to be a base point for all curves of the transformed canonical system. But since this is not an assigned base point, it has to be considered as virtually nonexistent; so on the given surface, ω has still to be considered as a part of the pure canonical curves. Indeed these curves are obtained by subtracting only once the exceptional curves which are parts of the impure canonical system.

NOTE I The case in which one has reducible exceptional curves, which arise from infinitely near base points of a transformed surface, deserves particular attention. Here the multiplicity of the exceptional curves as fixed parts of the impure canonical system may be determined by using a principle of continuity; we know in fact that this case arises as a limit of the case when the base points of a transforming system become indefinitely close to each other. Taking into account the remark which concludes Section ?? of the preceding Chapter 1, we point out that it has the following meaning: Every component ω_{i} of a reducible exceptional curve

Ω = ω + ω_{1} + h_{2}ω_{2} + ⋯ + h_{i}ω_{i} + ⋯ + h_{s}ω_{s}

appears as a component of the impure canonical system with multiplicity equal to the sum of the multiplicities with which it appears in the exceptional curves containing it^{(10)}.

Suppose we have, for instance, an exceptional curve consisting of three components, each one connected with the successive, corresponding to three infinitely near points on a linear branch; then these components give rise to three reducible exceptional curves

ω + ω_{1} + ω_{2}, ω_{1} + ω_{2}, ω_{2}.

Thus ω appears as a simple fixed component of the canonical system |K|, ω_{1} appears twice, ω_{1} three times.

If by contrast the three above-mentioned components arise from proximate points on a cuspidal branch (so that ω and ω_{1} are connected to ω_{2}, but are not connected one to the other), one has three exceptional curves

(10)A. Franchetta, Sulle curve eccezionali riducibili di prima specie. Boll. Unione Mat. It., Bologna, 1940.

ω + ω_{1} + 2ω_{2}, ω_{1} + ω_{2}, ω_{2}.

So ω again appears as a simple fixed part of the canonical system |K|, ω_{1} appear twice, and ω_{2} as a 4-tuple curve.

The theorem to the effect that a surface F of genus p > 0 in general possesses only irreducible exceptional curves of the 1st kind disconnected one from another, clearly yields that F may be transformed into another surface which has no exceptional curves at all, as was pointed out by Enriques in his Introduction of 1896. But a rigorous proof of this fact, under weaker assumptions which include (as we already pointed out above) all surfaces which do not belong to the family of ruled surfaces, was achieved in the simplest way by Castelnuovo–Enriques (1901), by using the so-called Riemann–Roch theorem for surfaces. Thus we defer this development to a later time. We only point out now that the case in which one has an exceptional curve composed of s parts, gives no problem. In fact one of its irreducible components is an exceptional curve, which can be eliminated, so that one may argue by induction.

NOTE II Let us explicitly point a result which is implicit in the characteristic property of the canonical curves of cutting out on a curve C of degree n and genus π a group of 2π − 2 − n points. In view of this property the difference

2π − 2 − n

(which in general is positive, but may also be, at least virtually, negative) appears to be an additive character for the system |C| of curves on the surface; and this additive property might also be directly deduced from the formulas which give the degree and the genus of reducible curves. Now we may claim that: On a surface of genus p > 0 an irreducible curve C of degree n and genus π such that

n > 2π − 2

may only be an exceptional curve of the first kind:

π = 0, n = −1.

In fact, if 2π−2−n is smaller than zero, then C has to be a component of the canonical curves K; but, if we assume that its multiplicity as a component is s, then the number of intersections of C with the residual curves is

2π − 2 − (s + 1)n

and therefore it is always negative, unless n = −1 and, consequently π = 0.

Bicanonical and pluricanonical curves 2.8 57

This theorem can be extended to the case of reducible curves, by remarking that a sum of characters n_{i}−(2π_{i}−2) which are negative or zero can never give a positive number. But then a rather subtle analysis is needed in the case of reducible exceptional curves. One may avoid it by considering some transformed surface without exceptional curves. Here is the result: On a surface of genus p > 0 there are no curves of genus π and degree n > 2π −2 (however reducible), except for those which can be obtained by adding to curves with n ≤ 2π − 2 some exceptional curves disjoint from it (that is, such that the virtual number of points in common with them is zero).

In particular one deduces that an exceptional curve of the first kind (however reducible), belonging to a surface of genus p > 0, is characterized as a “connected curve of genus ρ = 0 and degree ν = −1”^{(11)}.

EXERCISE The reader is invited to solve the following paradox: let us start with a surface of order 4 (of genus p = 1) F_{4} containing a line a; consider a cubic C which is cut out on F_{4} by a plane through a, and make a transformation of F_{4} into a surface Φ by means of a linear system with a base point O on C. So one has on Φ an elliptic curve C with an exceptional curve ω which meets it at one point. Let us look now at the sum

C + 2ω;

this is a curve of virtual genus ρ = 0 and degree ν = −1, and yet it is not an exceptional curve!

2.8 Bicanonical and pluricanonical curves

As we saw, the virtual existence of the canonical system

|K| = |C′ − C|

constructed from a system |C| without base points on F, yields the existence of numerical characters like p^{(1)} and p^{(2)} which are defined independently of the effective existence of |K|. But the consideration of a virtual canonical system which is not effective has an even more concrete meaning when it happens that, although the system does not exist, nonetheless it gives rise to an effective double or multiple, which can be directly defined by the relation

|2K| = |2C′ − 2C|

(11)With a small restriction, that is, excluding the case when the curve belongs to a linear pencil of rational curves (with some fixed component), the theorem can be extended to all surfaces (p ≥ 0). See A. Franchetta, Sulla caratterizzazione delle curve eccezionali ecc., Boll. Unione Mat. It., 1941.

or

|iK| = |iC′ − iC|,

In fact the fundamental theorem for the adjoint systems gives in any case

|iC′ − iC| = |iL′ − iL|.

We also notice that the bicanonical system may be defined by subtracting |C| from its second adjoint system

|(C + C′)′| = |C + C′′| = |2C′|; |2K| = |C′′ − C|

The simplest example, and also the first one that occurred, of a surface of genus p = 0, which has a bicanonical curve (of order zero), is given by the surface F_{6} of the 6th order which passes doubly through the edges of a tetrahedron; we shall come back to this surface later. The plane sections of F_{6} are sextics C of genus 4; their adjoint curves are twisted sextics C′ which are cut out on the surface by the cubic surfaces passing simply through the edges of the tetrahedron; the curves C and C′ are not equivalent, but their doubles are equivalent, namely

|2C′| = |2C|;

in fact the complete system |2C| = |2C′| is cut out on F_{6} by the surfaces of order 6, which also have the edges of the tetrahedron as double curves.

The surface F_{6} has no exceptional curves, but if it is transformed into a surface Φ in such a way that one of its points O is sent to a curve ω, then the exceptional curve ω, counted with multiplicity two, is the only bicanonical curve of Φ.

The characters of the bicanonical system, as well as those of the pluricanonical ones, give invariants of the surface F. First of all the dimension of the bicanonical system, or rather this dimension increased by one, that is, the number of linearly independent bicanonical curves, is an absolute invariant which is called the bigenus P = P_{2}. The genus and the degree of the bicanonical system give, by contrast, relative invariants of the surface. But it is clear that these characters can be expressed in terms of the (relative) linear genus p^{(1)}.

In fact, denoting by |K| the virtual canonical system of genus p^{(1)} and degree p^{(2)} = p^{(1)} − 1, the genus of the bicanonical system |2K| is given by

2p^{(1)} + p^{(2)} − 1 = 3p^{(1)} − 2

and its degree by

4p^{(2)} = 4(p^{(1)} − 1).

A historical note 2.9 59

Similarly the i-canonical system of the surface F, no matter if the pluricanonical systems of lower order are effective or not, gives us: an absolute invariant which is the i-genus P_{i}, that is, the number of linearly independent i-canonical curves, and the relative invariants given by its genus and degree. But the latter depend on the linear genus p^{(1)} of the surface; precisely the genus of the i-canonical system is

\frac{i(i + 1)}{2}(p^{(1)} − 1) + 1

and its degree is

i^{2}(p^{(1)} − 1).

What we said in the previous section about the exceptional curves on a surface of genus p > 0, can be extended now to the surfaces of genus p = 0 having bigenus P > 0, or, in general, having some P_{i} > 0. In fact the effective existence of the bicanonical or i-canonical curves |iK| = |K_{i}| again implies that a curve of genus π and degree n, with n > 2π − 2, has to be a fixed part of the K_{i}. In particular the exceptional curves of the first kind, each counted in general i times, will appear as fixed parts of the K_{i}. And, as in the case p > 0, there can be on the surface only exceptional curves of the first kind not connected one to another. We shall not spend more time in discussing in detail properties of reducible exceptional curves, which do not give rise to new observations.

We shall only point out that, as a consequence of what we said, the absolute linear genus of a surface with a plurigenus P_{i} > 0 (even if p = 0), is always defined; its value is p^{(1)} + σ, where p^{(1)} denotes the relative linear genus and σ is the number of exceptional curves. However a surface with p = 0 and with some nonvanishing plurigenus can be transformed into another one which has no exceptional curves, and the linear genus of the latter gives the absolute linear genus.

2.9 A historical note

The notion of the genus of an algebraic surface F occurred for the first time to A. Clebsch^{(12)} as the “number of double integrals of the first kind (that is, everywhere defined) belonging to F ”. If one assumes F to be of order n in ordinary space, then the algebraic differentials of the first kind can be formed by means of the adjoint surfaces φ_{n−4} of order n − 4, as well as the differentials of the first kind of a plane curve of order n are formed by means of the adjoint curves of order n − 3. It is therefore apparent that these φ, or rather the (canonical) curves which they cut out, away from the double curve, have an invariant character with respect to the birational transformations of the surface. The first algebraic proof of their invariance was given by M. Noether^{(13)}. After that, Noether again noticed that the canonical system |K| gives, besides the genus p (its dimension increased by one) also another invariant of the surface, namely the genus of the K, that is, the linear genus p^{(1)}, defined at least for p > 0, which he called “Kurvengeschlecht” in contrast with the “Flächengeschlecht”. As for the degree p^{(2)} of the canonical system, he proved that it can be expressed in terms of p^{(1)} by means of the formula

(12)Comptes rendus de l’Académie des Sciences de Paris, 1868.

p^{(2)} = p^{(1)} − 1,

which is a consequence of the fundamental property of the K of cutting out on the generic curve of a linear system |C| residual groups to the characteristic series. Indeed the above formula, considered by Noether to be valid only for the effective characters, holds, with no exceptions, for the virtual characters, as defined by Enriques.

F. Enriques, took up again after twenty years the general study of algebraic surfaces from the point of view of birational transformations^{(14)}. His first aim was to understand whether the fundamental property of the canonical curves K characterizes them, if one knows that it holds for a particular system |C|: this is actually the case (with one exception) if |C| is an irreducible linear system, at least ∞^{2}, without proper fundamental curves (that is, curves such that the genus of the residual curves drop). Whence the author derives a geometric proof of the invariance of the K with respect to birational transformations: indeed a birational transformation of the surface may be decomposed as a sequence of transformations, each preserving a two dimensional linear system of plane sections. These and other subjects which are treated in the above-quoted Ricerche (which we shall mention again later on), are taken up again by F. Enriques, in the paper Introduzione alla Geometria sopra le superficie algebriche of 1896^{(15)}. Concerning the canonical system |K|, the most important issue is the extension of the invariance theorem to the case p_{g} = 0, when |K| is no longer effective. Already in Ricerche the author considered the adjoint system to a system |C|, which – for p > 0 – is the sum |C′| = |C + K|. Now he goes deeper into the study of this system independently of the existence of |K| (namely for p = 0). The adjoint curves C′ to |C| on a surface F cut out on a generic C canonical groups (as do the adjoint curves of order n − 3 on a plane curve of order n). But this property alone does not suffice to characterize the adjoint curves if |C| possesses some proper fundamental curves; however one may define them in any case, in such a way that the following functional condition

(13)Zur Theorie des eindeutigen Entsprechens algebraischer Gebilde, Math. Ann., Band 2 (1869), Band 8 (1874).

(14)Ricerche di geometria sulle superficie algebriche, Mem. R. Acc. delle Scienze di Torino, 1893.

(15)Mem. della Società Italiana delle Scienze (detta dei XL) s. III, t. X, 1896.

|C + D′| = |C′ + D|

holds with respect to any two linear systems (free from base points).

This fundamental property of adjunction yields, for p > 0, the invariance of the canonical system:

|K| = |C′ − C| = |D′ − D|,

and extends in fact the invariance theorem. Indeed this theorem is still meaningful when |K|, defined as a difference (and therefore a priori as a virtual system), is not effective. The meaning of the theorem has been pointed out by Enriques by stressing these aspects:

(1) The fundamental relation between adjoint systems leads to the definition of the virtual characters p^{(1)} and p^{(2)} of the canonical system, independently of its effective existence; as we said, these characters are related by the formula

p^{(2)} = p^{(1)} − 1.

(2) It may happen that, although |C| is not contained in |C′|, its double or a multiple |sC| of it is contained in |sC′|; this yields that a virtual canonical curve, which is not effective, may give rise to an effective multiple system.

We already pointed out examples in which this possibility occurs. We shall see later on the importance, mainly in the classification problems, of the bigenus and the plurigenera, that is, the new characters depending on the dimensions of the multiples of the canonical system.

A remarkable methodological progress in the theory of invariants outlined by Enriques was determined by the idea, due to the same author^{(16)} some years later, of defining the adjoint systems and the canonical system simply by means of the Jacobians. The idea is so simple, so easy to be carried through, so general in view of extensions to varieties of any dimension, that somebody (not only mathematicians who belonged to Enriques’ school) could have believed that it was a common patrimony of the geometers, and that it should have been known, at least for curves, since a long time. But those who read the treatments of the geometry on a curve by C. Segre and by E. Bertini of 1894^{(17)} will not find there any proof of the invariance of the canonical series on a curve other than the one by Brill and Noether based on the theorem of Riemann–Roch, according to which the canonical series is the only g_{2p−2}^{p−1} belonging to a curve of genus p^{(18)}. Were this not true, that is, had the definition of the canonical series and the proof of its invariance, based on the property of the Jacobian series of the g_{n}, been known at that time, then it would be hard to understand the more laborious development of the theory of canonical curves on a surface given by Enriques in Ricerche of 1893 and in Introduzione of 1896.

(16)It was treated in a course of lectures at the University of Bologna, in the year 1897-98.

See the Program, published in the Bollettino di Bibliografia e Storia delle Matematiche, n. 13 (1899). For the surfaces see Intorno ai fondamenti della geometria sopra le superficie algebriche, Atti Acc. Torino, 37, page 19 (1901).

Let us add that the idea of the Jacobian systems may be exploited in various ways, and this gives rise to some variants in the treatments by other authors.

The form which is adopted here, not too far from the one appearing in Fondamenti, corresponds to the development of the theory of curves as treated in the book Lezioni by Enriques–Chisini^{(19)} and – unless we are mistaken – it is of the greatest simplicity.

(17)C. Segre, Introduzione alla Geometria sopra un ente algebrico semplicemente infinito, Ann. di Mat., s. II, t. 22. E. Bertini, La geometria delle serie lineari sopra una curva piana secondo il metodo algebrico, Ann. di Mat., s. II, t. 22.

(18)See C. Segre, ibidem, n. 75; Bertini, ibidem, n. 21, b) and g).

(19)Vol. III. See also, for an analytic treatment: F. Enriques, Sul teorema di invarianza della serie canonica g_{2p−2}^{p−1}, ecc., Mem. R. Acc. di Bologna, 1914.

Adjoint surfaces

We have seen that on a surface F the curves C′ adjoint to an irreducible linear system |C| cut out canonical groups on the curves of |C|. Now we wants to investigate whether, and to what extent, this property characterizes the curves C′. This question is related to another question of projective type: how the adjoint curves to the system of plane sections C are cut out on a surface F, taken in ordinary space, by certain surfaces, that we will call adjoint to F, defined by the condition that they pass with multiplicity i − 1 through every i-tuple curve of F and behave in a suitable way at the isolated multiple points of F.

3.1 Surfaces of order n−3 adjoint to a surface of order n without proper multiple points

Let us consider a surface F of order n in ordinary space and let us denote by C the plane sections of F. F can have singularities of two kinds:

(1) one or several multiple curves, for instance a double curve that corresponds on a transform of F to a locus of pairs of points that are neutral for the transform of the linear system |C|, or an i-tuple curve corresponding to a locus of neutral groups of i points and so on;

(2) some particularly notable multiple points, both isolated multiple points and points of the multiple curves whose singularity is not due to the fact that the multiple curve passes through them.

As far as the multiple curve is concerned, we shall as a rule address, for the sake of simplicity, to the case of ordinary multiple curves and sometimes even to the case of a nodal curve; but it will be easy to understand that what

Surfaces adjoint to a surface of order n 3.1 64

we shall say holds also for an arbitrary curve, however singular, provided one takes into account the infinitely near multiple curves defining it.

As to the case of the notable isolated singularities, which have to be taken into account in the considerations of this chapter, we shall make a precise distinction by resorting to the following criterion: a multiple point O of F will be said to be improper if the genus of a plane section through O equals the genus of the sections with planes not passing through O; on the other hand, a singular point of F that lowers the genus of plane sections will be said to be a proper multiple point.

If we refer to a transform of F on which the multiple curve corresponds to a locus of points that are neutral for |C|, then an improper multiple point O of F corresponds in general to a neutral group G of points (imposing only one condition to the curves C that must contain them); so, for instance, a triple point for the surface F and for its double curve corresponds in general to a neutral triple of points to a point that is (i + 1)-tuple for F and for its i-tuple curve corresponds to a neutral group of i + 1 points, and so on.

Yet it may happen that, in a transformation of F, the point O gives rise to a fundamental curve θ for the system |C|, which in general can be composed of as many disconnected exceptional parts as the points of the group G mentioned above; and it is worth remarking that this decomposition of θ can be derived from the property, which is characteristic, of being an improper fundamental curve, not lowering the genus of the C. In fact this property yields that every connected part of θ is of genus zero and has only one common point with a curve residual to C (or to one of its multiples), and so this part is characterized as an exceptional curve.

Now let us consider a surface F that is not ruled and such that its plane sections C are of order n and of genus π > 1 and a curve L on F that cuts canonical groups on the curves C. In a generic plane the group of points cut out by L belongs to a curve φ_{n−3} of order n − 3, adjoint to the plane section C of F, consisting of all the intersection points of C and the φ_{n−3} away from the multiple points of C (where an i-tuple point of C counts for i(i − 1) intersections); the curve C cannot have any other intersection with the φ_{n−3} outside L, unless C is reducible.

Let us take a generic space line a and, in every plane through a, the curve φ_{n−3} adjoint to the plane section that is determined by the group of points where L meets the plane; as the plane through a varies, φ_{n−3} describes a surface Φ, passing i − 1 times through every i-tuple curve of F^{(1)}, and having order n − 3 + h, where h is the multiplicity with which it contains the line a.

(1)The statement can be verified analytically by considering the case of a surface described by translating a plane curve with an s-tuple point (s = i − 1) in a normal direction, in which case it is clear that the s-tuple point describes an s-tuple line.

Adjoint surfaces of arbitrary order 3.2 65

Now if h > 0, then the total intersection of F and Φ must contain all the common points of a and F, which are not on L (nor on the multiple curve). If A is one of these points, there is at least one plane, tangent to Φ at A, whose section with F meets the adjoint curve φ_{n−3} in A away from the canonical group on L and therefore turns out to be reducible. But then by a well known theorem of Kronecker–Castelnuovo^{(2)} the surface F, thus having ∞^{2} reducible plane sections, should be a Steiner surface with plane sections of genus zero, which contradicts our assumptions. One concludes that the curves L that cut out canonical groups on the plane sections C are the intersections of F, away from the multiple curve, with adjoint surfaces Φ_{n−3} of order n − 3. Therefore the curves C′ adjoint to |C| must be contained (totally or partially) in the system |L|. Now the C′ and the L are curves of the same order 2π−2, namely they have the same intersection number with the C, and therefore they can only differ by fundamental curves of |C|; this means that the linear system |C′| (if less ample than |L|) is obtained by imposing to |L| some base point occurring at some multiple point O of F; by contrast no base point of the C′ occurs at an improper multiple point O, because the adjoints to the plane sections C through O, that are again of genus π, also meet these sections at 2π − 2 points (at least virtually) away from O.

We can be make clearer what we have just said by considering a transformed surface of F, on which the point O corresponds to a neutral group G of simple points: here the system |C′_{1}| adjoint to |C_{1}| (that is obtained from |C| by imposing simple points) does not differ from |C′|. Therefore we may state the theorem:

On a surface F, of order n, of ordinary space, not ruled and without proper multiple points, the adjoint curves of the plane sections C are characterized by the property of cutting out canonical groups on the above-mentioned curves C; and the complete linear system of these adjoint curves is cut out on F by the surfaces Φ_{n−3} of order n − 3 (adjoint to it) passing i − 1 times through every i-tuple curve of F.

3.2 Adjoint surfaces of arbitrary order

This result can be extended to the curves L that cut out on the plane sections C of F groups belonging to the series which is the sum or the difference of the canonical series with a multiple, by any integer r, of the characteristic series: under the above assumptions (F not ruled and without proper multiple points) the L are cut out on F (away from the multiple curve) by surfaces Φ_{n−3+r} of order n − 3 + r, adjoint to it. To begin with, let r > 0. Then it suffices to repeat the above argument, with only one warning: when r ≥ 3

(2)See Enriques–Conforto, Ibidem, Libro II, Chapter II, §7, page 273.

Ruled surfaces 3.3 66

there are infinitely many curves φ_{n−3+r} passing through the group φ cut out by L on a plane through a, they are adjoint to the section C with the plane and form a linear system of dimension (r − 1)(r − 2)/2; and it is convenient to determine one of these by assigning, for instance, a suitable number of its points on a and then a certain number of tangents and osculating curves. If instead r < 0, then add to the curve L the curve H cut out on F by a generic surface Ψ of order |r|. Then the curve L+H cuts out on the plane sections of F groups belonging to the canonical series and therefore there exists a Φ_{n−3}, adjoint to F, that goes through it. But such a Φ_{n−3}, having the curve H of order |r|n > |r|(n − 3) in common with Ψ, breaks up as the union of Ψ itself and of a Φ_{n−3+r} still adjoint to F and passing through L.

In particular from the previous considerations it follows that:

On a surface F of order n of ordinary space, not ruled and not having any proper multiple point, the surfaces Φ_{n−3} passing i − 1 times through every ituple curve of F (namely adjoint to it) cut out the complete linear system |K| of (impure) canonical curves, as already announced in Chapter II, Section ??; and in general the Φ_{n−4+r} with r > 0 cut out on F the complete linear system adjoint to the r-tuple system of plane sections, which usually is only partially cut out by the system of all surfaces of order r.

Moreover, this system |(rC)′| = |C′+(r−1)C| = |K +rC| is characterized by the property that its curves cut out on the plane sections C of F groups belonging to the series sum of the canonical series and of the (r − 1)-tuple of the characteristic series.

3.3 Ruled surfaces

The previous theorem has been established under two restrictive assumptions:

(1) that F is neither ruled nor a Steiner surface, and that

(2) F has no proper multiple points.

We see at once that the restriction on F to be nonruled is really essential. If F is a ruled surface of order n and genus π > 0 (that, for the sake of simplicity we will assume to have only a double curve) there are neither curves C′ adjoint to the plane sections C nor adjoint surfaces Φ_{n−3}, but there are groups of 2π − 2 generators cutting out canonical groups on the C.

In fact, introducing the virtual canonical system |K|, we may write:

|C′| = |C + K|

Characteristic properties of adjoint and canonical curves 3.4 67

The C meet the generators at only one point, the K meet them at −2 points^{(3)}, therefore the C′ should cut the generators at

1 − 2 = −1

points, so they should contain them, which is a contradiction.

On the other hand, one can also see that there cannot be surfaces Φ_{n−3} of order n − 3 passing through the double curve of F, since this curve meets the generators at n − 2 points.

Let us add that, on the ruled surface F, there exist also groups of 2π −2+ (r − 1)n generators cutting out on the plane sections C groups belonging to the series sum of the canonical series and of the r-tuple of the characteristic series, but except for the case r = 2, these groups are not adjoint curves to |rC| and are not intersections of adjoint surfaces Φ_{n−4+r}, since the curves of |(rC)′| have to meet the generators of F at r − 2 (and not at zero) points.

On the contrary, for r ≥ 2, the surfaces Φ_{n−4+r} passing through the double curve cut out on the ruled surface curves that are adjoint to the r-tuple system of plane sections. This can be proved by remarking that those curves must cut out canonical groups on the curves of |rC|, at least on those that are cut out on F by general surfaces Φ_{r} of order r. Indeed on a Φ_{r} the curves intersection of the Φ_{n−4+r} = Φ_{r+n−4} are adjoint to the intersections with F.

We will prove that the system cut out on the ruled surface F by the Φ_{n−4+r} is always complete later, when we will learn how to estimate its dimension.

3.4 Characteristic properties of canonical curves and of adjoint curves to a linear system

In order to state the results obtained in an invariant form, we have to consider a surface F and an irreducible linear system |C| on it, and characterize, with respect to C, the canonical curves and the curves adjoint to the C. But before stating the result, it is convenient to get rid of a superfluous restriction. Actually we have assumed the system |C| to be not only irreducible, but also simple of dimension r ≥ 3, so that ∞^{3} C may be regarded as the plane sections of a transformed surface.

We can do without this restriction.

If |C| is (or contains) a net of irreducible curves of genus π and order n, we may assume that this net is cut out on a surface F, of order m > n, by the planes of a star with centre in an s-tuple point O (m = n + s).

(3)Since the curves of the system |2C + K| split in 2π − 2 + n generators, then they cut the generators in zero points: therefore the curves K cut the generators in −2 points.

Isolated multiple points 3.5 68

Then, starting from a curve L that cuts canonical groups on the C, and with respect to a pencil of planes whose base line goes through O, one constructs, as before, an adjoint surface Φ_{m−3} cutting out L on F, passing i − 1 times through every i-tuple curve of F, and having multiplicity s − 1 at the s-tuple point O; then one has L = C′.

However, the feasibility of this construction is based on two assumptions:

(1) there do not exist ∞^{1} reducible plane sections through O;

(2) F has neither proper multiple points (not even infinitely near O), nor multiple lines through O, that might turn out to have smaller multiplicity for the Φ_{m−3} so constructed.

Considering the matter from the point of view of the invariant theory, the two restrictions can be combined in only one requiring that: the net of the C does not contain ∞^{1} reducible curves. Actually the existence of a fundamental curve for the ∞^{r} system |C|, means that it contains ∞^{r−1} reducible curves.

Now we may state the following theorem:

If on a surface F one has an irreducible linear system |C|, of dimension r > 1, not containing ∞^{r−1} reducible curves, then the curves C′ adjoint to it are characterized by the property of cutting out canonical groups on the C. And the canonical curves of F are defined by the property of cutting out on the C groups residual to the canonical series.

Also the curves adjoint to a multiple of the system |C| can be characterized by the groups that they cut out on the C; the matter is clear and there is no need to insist on this point.

3.5 Isolated multiple points

Let us go back to the projective consideration of the plane sections of a surface in the ordinary space.

Adjoints and subadjoints 3.6 69

We want to examine to what extent the results obtained can be extended to the case when F has multiple points that lower the genus of the sections through them. For the sake of simplicity it suffices to consider the curves C′ adjoint to the system of plane sections C. So the question is to what extent these curves are characterized by the simple property of cutting out canonical groups on the plane sections C of F. On the surface F of order n considered in projective space, a curve L cutting out canonical groups on the C is again, as we have seen before, intersection with a Φ_{n−3}, of order n − 3, passing i − 1 times through every i-tuple curve of F. But the system |L|, although containing |C′| partially, may not coincide with it; in any event one has |L| = |C′ + θ|, where θ denotes a curve, or a set of proper fundamental curves of |C|. And, from the projective point of view, the surfaces Φ_{n−3} cutting out the C′ are different from the Φ_{n−3} that cut out the L (subject only to the condition of passing i − 1 times through every i-tuple curve of F) by the further condition of passing also through some multiple points of F.

To clarify the matter, let us assume that F has an isolated s-tuple point, namely away from the multiple curve. Let us consider the planes of the space passing through a point P; the sections of these planes form a net of |C|, and, from the projective point of view, the Jacobian of the net is cut out by the polar surface of P, of order n − 1, which in general passes i − 1 times through every i-tuple curve of F, and similarly with multiplicity s−1 through the s-tuple point O. However beware that the neighbourhood of the s-tuple point O corresponds (on a transformed surface) to a fundamental curve θ of |C|, which is a component (usually counted with multiplicity one) of the Jacobian C_{j}^{(4)}. As a consequence, in order to obtain a whole Jacobian curve, one has to add the neighbourhood of the point O to the above polar. So the system |C_{j}| is actually cut out by the surfaces of order n − 1 passing i − 1 times through every i-tuple curve of the surface F and with multiplicity s − 2 through every s-tuple point O. It follows that the curves C′ adjoint to the plane sections are cut out by the surfaces Φ_{n−3} subject to the condition of passing i − 1 times through every i-tuple curve of F, and of passing s − 2 times through the isolated s-tuple point.

The same argument can be repeated for a proper s-tuple point belonging to the multiple curve of F; the s-tuple point is (in general and at least) (s − 2)-tuple for the surfaces that cut out on F the adjoints C′ to the plane sections.

3.6 Adjoints and subadjoints

The previous analysis, concerning a surface F with improper multiple points, leads us to distinguish between:

(1) the surfaces passing i − 1 times through every i-tuple curve of F, and

(2) the surfaces that, besides the condition of passing i − 1 times through every i-tuple curve of F, also enjoy the particular property of passing through the proper multiple points of F in a way that (for arbitrarily large order) they give rise, by intersecting, to adjoint curves of a multiple of the plane sections.

(4)By assumption q has no variable intersections with the C. Take a generic point A on q; the C which have to pass through A split as the union of q and of the residual curves C_{1} of a pencil; among these there is a C_{1} passing through A, that together with q is a curve of the net having a double point at A.

Keeping the name of adjoint surfaces for the latter, we shall call the former subadjoint surfaces, and therefore we shall call subadjoint curves of the plane sections C the curves defined by the property of cutting out canonical groups on the C. Now one has to investigate, for every particular kind of proper multiple points, the extra conditions that they impose to the subadjoint surfaces in order that they become adjoint. In relation with this subject, it is convenient to remark that, for the adjoints of a certain order, the conditions concerning a given multiple point may turn out to depend partly on the other conditions imposed by the multiple curves or by other multiple points; but we may assume (as this will be made clearer later) that this is not the case for surfaces with large enough order. However, by computing the number of these conditions without taking into account the possible relations of dependence that we have mentioned above, we shall have a number that will always be the same^{(5)} for the adjoint surfaces of any order. If this number really corresponds to the number of adjoints Φ_{n−4} of order n − 4, then it expresses the genus drop of F due to that multiple point. However, by saying that a multiple point lets the genus of the surface drop of a certain amount, we shall mean the virtual drop of this genus, a concept which we shall explain later on, and meaning also that the virtual expression of the genus yields another invariant character of the surface (its numerical genus) that may be smaller than the (geometric) genus introduced before.

Using the language that we have introduced we may state:

A proper s-tuple point of the surface F (either belonging or not to the multiple curve) is (in general and at least) an (s − 2)-tuple point for the adjoint surfaces.

In the simplest case, when O is isolated with osculating cone of genus (s − 1)(s − 2)/2, the multiplicity of O for the adjoint surfaces is precisely s − 2, while there is no vanishing condition for the subadjoints; so one can say that a priori O imposes s(s − 1)(s − 2)/6 linear conditions on the adjoint surfaces of order n − 4, and so it makes the genus drop of s(s − 1)(s − 2)/6.

In particular a conical double point does not affect the genus of the surface.

Multiple points of the first kind 3.7 71

EXAMPLES The nonsingular surface F_{4} of order 4 has (superficial) genus p = 1, since it has one canonical curve of order zero, section with the adjoint surface of order n − 4 = 0. If F acquires a triple point, then its genus drops by one and so it becomes p = 0; according to the fact that the F_{4} with a triple point is rational.

(5)When one imposes to the surfaces f of large enough degree to have a certain singular point O, one is led to impose to the f a certain multiplicity at the point O and at a certain number of points infinitely near to O and belonging to its higher order neighbourhoods. As a consequence the f may acquire the prescribed multiplicities along infinitesimal curves of these neighbourhoods. So the number of conditions imposed on the f will be independent of the order of these.

The F_{4} with a conical double point is still of genus 1, and therefore certainly irrational. On the other hand there are rational surfaces of order 4 with (only) one double point^{(6)}: the first type of such surfaces is the F_{4} with a tacnode. From this example it is already apparent that “double points of particular type (e.g., one whose osculating cone consists of a double plane) may cause the genus of the surface to drop”: hence the need for a careful analysis.

The nonsingular surface F_{5} of order 5 (n = 5) has genus p = 4, since it has ∞^{3} canonical curves that are the sections of the planes of the space (adjoint to F_{5}). If F_{5} acquires a quadruple point (s = 4) its genus drops by s(s − 1)(s − 2)/6 = 4, according to the fact that it becomes rational.

The F_{5} with a double line a has genus p = 2, since its canonical curves are the ∞^{1} cubic sections with the (adjoint) planes passing through a. If it acquires a triple point outside a, then its genus drops to 1. In this case there is one adjoint plane, namely the plane Oa, that cuts on F the impure canonical curve consisting of three lines through O. These lines are exceptional for F_{5}; in fact one sees that the quadrics through O and a cut out on F_{5} a linear system |C| of genus 5 and order 8, which determines a transformation of F_{5} into a F_{8} of S_{5} containing a plane cubic C_{3}, the image of the point O, whereas the three lines of the impure canonical curve are transformed into three collinear points of C_{3} that are simple for the surface F_{8}: and projecting F_{8} from the line containing these three points of C_{3} one gets the F_{5} back. Since the plane sections of F_{8} are canonical curves of genus 5 in S_{4}, their system is selfadjoint, hence one verifies that the genus of the surface is equal to 1.

3.7 Multiple points of the first kind

If one wants to better understand the conditions that a singular point of F imposes to the adjoint surfaces, it is convenient, to start with, to distinguish the multiple points of the first kind from those of a higher kind. A multiple point O of F, that we assume to lower the genus of the plane sections C, will be called a “multiple point of the first kind” when near to it there is no other multiple point lowering the genus of the C through O. On a suitable transform of F the point O becomes a fundamental curve θ for the system |C|, and the property that “there is no part of it that is a fundamental curve of the residual system |C − θ|” characterizes θ as a fundamental curve of the first kind.

(6)Surfaces determined by M. Noether. See Enriques–Conforto, Ibidem, Libro I, Chapter VIII.

The conditions imposed on the adjoint surfaces Φ by a multiple point O of the first kind can be simply expressed by saying that: the Φ must be cut by the planes through O along curves that become adjoint to the sections of F if one adds to them a line through O.

The simplest way to prove this is to refer to the Φ_{n−4} of order n − 4 and to assume that the net of sections C of F with the planes through O does not contain ∞^{1} reducible curves. Then it is evident that the Φ_{n−4} mentioned above cut on the C groups residual to the characteristic series, and this property – under our assumptions – suffices to define canonical curves. The argument can be repeated in an analogous way for the Φ_{n−4+r} with r > 0, even if there are no Φ_{n−4}.

EXAMPLES Let F be a surface with a tacnode O, that is, a point O such that the plane sections through it have two infinitely near double points at O^{(7)}. Applying the previous criterion, one sees that the plane sections of the Φ_{n−4}, plus a line through O, must be adjoint curves of the sections of F. This implies that the adjoints have to pass simply through the tacnode O.

If instead one has a oscnode, that is, a point O such that the plane sections through it have three infinitely near double points at O, then one sees in a similar way that the adjoints have to pass simply through the oscnode and touch the tangent plane to F there.

EXERCISE Let F_{5} be a surface of order 5 with a double line a and a tacnode O. The adjoints of order n − 4 = 1 to this surface are reduced to the plane Oa, hence one has p = 1. The intersection of F_{5} with the plane Oa (away from a) consists of two exceptional curves: a conic and a line b, touching at O.

One can verify these assertions by transforming the F_{5} into a nonsingular F_{4} of order 4, containing a line a′: this surface (on which the system of plane sections is selfadjoint) has genus p = 1 and has a canonical curve of order zero.

Double planes 3.8 73

The above-mentioned transformation can be performed with a special quadratic transformation of the first kind of the space, defined by the homaloidal net of the quadrics passing through two concurrent lines and touching a given plane at a point of one of the lines: the lines in the base locus are a and the line b intersection of the tacnodal plane at O with the plane Oa, and the tacnodal plane at O is the fixed tangent plane that we have mentioned above. With this transformation the surface F_{5} becomes a F_{4} of order 4 and the quartic residual intersection of the tacnodal plane of F_{5}, away from the line b, becomes a line a′ of F_{4}. The exceptional conic of F_{5} is carried to a point O′ of F_{4} and the exceptional line b to a point B; the transforms of the plane sections of F_{5} are curves C_{7} of order 7, residual intersections of F_{4} with the quadrics through a′, that have a double base point at O′ and a simple base point at B.

(7)See Enriques–Chisini, Ibidem, Libro IV

Conversely, if one takes an arbitrary surface F_{4} containing a line a′, and a generic point O′ on F_{4} (belonging to a cubic K section with a plane through a), one can transform F_{4} in an F_{5} with a double line and a tacnode. In order to do this, one can use the homaloidal system of the quadrics passing through a′ and through the line b′ tangent to the cubic K at O′, and touching F_{4} at O′. One has to remark here that by imposing a double point O′ to the curves C_{4}, partial intersections of the quadrics through the line a′, gives as a consequence also the existence of a simple base point at the tangential point of O′ on K. Thus the base point free complete system of the C_{4}, which is of genus 6, dimension 6 and order 10, gives rise to a superabundant system with a double base point and a simple base point, of genus 5, order 5 and dimension 3.

3.8 Double planes

We make now a short digression in order to consider a remarkable family of surfaces, which can be explained by using the previous considerations. Let us consider the surface F defined by the equation z^{2} = f(x, y), where f is a plane curve of even order 2n: by projecting from the point at infinity O on the z-axis, F is represented on the double plane with branch curve f. An f of even order arbitrarily taken, considered up to birational transformations of the plane, defines a class of surfaces including F; if f is of odd order, one has to add to it the line at infinity of the plane as a branch line^{(8)}.

The surface F is of order 2n and, if f is nonsingular, it has only one proper multiple point at the point O. The analysis of this singularity can be reduced to the analysis of the singularity that the point at infinity on the z-axis has for the hyperelliptic curve section of F with a plane parallel to that axis, for instance the hyperelliptic curve z^{2} = f(x), with f of order 2n. It is well known^{(9)} that such a curve has a (2n − 2)-tuple point at O, and infinitely near to it and facing it there are n − 1 double points along a branch of order n − 1 tangent to the line at infinity of the plane.

So the surface F has a (2n − 2)-tuple singular point O, defined by the property that every plane section through it must have n − 1 double points near O.

(8)See Enriques–Chisini, Lezioni, Libro V, §12, Vol. III, page 90.

(9)See Enriques –chisini, Ibidem, Libro V, §12 note, Vol. III, page 93.

Let us now consider the adjoint surfaces of F of order 2n − 4. These are cones (and, more properly, cylinders) of vertex O, since, when one adds a plane to them, their sections are curves adjoint to the sections of F; furthermore these cones are subject to the condition of containing the double points of F near O, and so they have as a component the plane at infinity counted n − 1 times. We conclude that the surfaces of order 2n − 4 adjoint to z^{2} = f(x, y) consist of the plane at infinity counted n − 1 times and of the cones of order n − 3 with vertex O. The plane at infinity intersects F along 2n exceptional lines. The canonical curves of the double plane that has a general curve of order 2n as a branch curve are represented on the plane by the curves C_{n−3} of order n − 3. The genus of the double plane is:

p = \frac{n(n − 3)}{2} + 1 = \frac{(n − 1)(n − 2)}{2}

Now remark that a quadruple point on the branch curve f is a tacnode for the surface z^{2} = f(x, y), with the plane z = 0 as the tacnodal plane, whose lines through the origin have four intersections with the surface. We deduce that: a quadruple point of the branch curve lowers the genus of the double plane by one.

In a similar way a sextuple point of the branch curve gives rise to an oscnode of F and therefore it is double for the canonical C_{n−3}; in general a 2i-tuple point of the branch curve is (i − 1)-tuple for the canonical C_{n−3} and so it lowers the genus of the double plane by i(i − 1)/2.

A double point A of f, of odd multiplicity 2i + 1, must be considered, in general, as of multiplicity 2i, since the birational transformations of the plane that have O as a fundamental point transform the neighbourhood of this point into a line that is a component of the transform of the branch curve. But, for the same reason, a multiple point of f of odd multiplicity 2s + 1 occurring infinitely near A, must be regarded as a (2s + 2)-tuple point; and so, in particular, if one has two infinitely near multiple points A and A_{1}, with the same odd multiplicity 2i + 1, the canonical C_{n−3} must pass i times (instead of i − 1) through A and i − 1 times through A_{1}. One gets the first example for i = 1: the [3, 3] singularity thus defined lowers the genus of the double plane by 1, as it happens for quadruple point. In Section ?? we shall find again this result by examining the singular double point which corresponds to it for the surface F.

The genus of the fundamental curves and their influence 3.9 75

Historical note.-The notion of double planes goes back to A. Clebsch Ueber den Zusammenhang einer Klasse von Flächenabbildungen mit der Zweitheilung der Abelschen Funktionen^{(10)}, who represented on the simple plane the double planes with branch curve of order 2 and 4 (the latter arising – according to Geiser – from a projection of the cubic surface) and those with a sextic with a quadruple point as a branch curve, arising by projection from a surface of the 4th order with a double curve. After that (1878) M. Noether^{(11)} studied all the rational double points in a systematic way, reducing them to three birationally distinct types, that – via a (1, 2)-transformation of R. De Paolis – give rise to the types of involutions of E. Bertini. The third type of Noether is the double plane branched along a sextic with two infinitely near double points.

(10)Math. Ann., Band III, 1861. See also: Crelle’s J., 63.

Later on (1896) Enriques, starting the study of various classes of irrational double planes^{(12)}, showed explicitly how to obtain the images of the canonical and pluricanonical curves on the double plane (see Chapter III, Section ?? and Chapter V, Section ??).

3.9 The genus of the fundamental curves and their influence on the adjoints

We resume the consideration of multiple points of the first kind corresponding to irreducible fundamental curves of the system |C| of plane sections. Such a fundamental curve θ has (virtual) genus ρ and degree −s; as we shall see, s coincides with the multiplicity of the point O for F and ρ is the genus of the (simple or multiple) osculating cone at O (where one takes into account the branches of the double curve that go through O and regards the other possible double generators of the cone as virtually nonexistent).

We know that, because of the existence of the multiple point O, the system |C′| adjoint to the system of plane sections C is obtained from the subadjoint system |L| by imposing that its elements contain the curve θ counted a certain number of times, and we intend to determine the number x such that |L| = |C′ + xθ|.

Expressing the condition for θ to be a fundamental curve, not intersecting the curves of |C| = |C_{1} + θ|, we have:

(θC_{1}) + (θθ) = 0

and therefore

(θC_{1}) = s.

The number s, essentially positive, is the number of intersections of θ with the curves C_{1} that are residual to it with respect to |C|, that is, in general, the multiplicity of the point O for F, as we have said before. Now, in order to determine the number x defined above we may start with |C′| and try to successively extend it by adding θ, as many times as possible.

(11)Ueber die einzweideutigen Ebenentransformationen, Sitzungsberichte der Phys-med.

Societät zu Erlangen, 10.

(12)In particular in the memoire Sui piani doppi di genere 1. Mem. della Società It. delle Scienze (detta dei XL), 1896.

It is clear that these successive extensions will attain a limit, when the number of intersections of θ with the curves that one adds to it becomes zero or negative. So we shall have:

(θC′) + x(θθ) ≥ 0

We write

|C| = |C_{1} + θ|, |C′| = |C_{1} + θ′|(θC′) = (θC_{1}) + (θθ′) = s + 2ρ − 2;

the previous inequality becomes

2ρ − 2 − (x − 1)s ≥ 0,

thus determining an upper bound for the value of x:

x ≤ 1 + (2ρ − 2)/s.

The first information that we obtain from the previous inequality is that if ρ = 0, then also x = 0. So the irreducible fundamental curves of the first kind with genus zero of a linear system |C| do not affect the definition of the adjoint curves. Or, in a projective form: a multiple point of the first kind with (simple or multiple) irreducible osculating cone of (virtual) genus zero does not affect the genus of the surface.

This class of multiple points includes, as a first example, the conical double point, which we mentioned before, and then a triple point that is simple for the double curve of F, and so on. If in the previous inequality we set ρ = 1, then one has x ≤ 1. Here one can always take x = 1. In fact let us consider a surface F having a double curve on which there is an s-tuple point through which there are (s−1)(s−2)/2−1 branches of this curve. Now the osculating cone to F at O is a cone of order s that has an adjoint cone of order s − 3; this means that the subadjoint surfaces (of very high order), which have to go through the double curve, have multiplicity s − 3 at O, whereas the adjoints must have multiplicity s − 2 at O.

The case of multiple points with ρ = 1 includes also the tacnode, where the irreducible curve θ reduces to the neighbourhood of O on the tacnodal plane counted twice. In this case in fact θ is represented on the double pencil of tangents with four branch lines that correspond to the cuspidal plane sections of the tacnode. Thus one finds again the result proved in the previous section to the effect that the tacnode lowers the genus of the surface by 1.

Point singularities of a higher kind 3.10 77

In a similar way one sees that the oscnode corresponds to a fundamental curve θ of genus ρ = 2, and here the value of the number x is x = 2, according to what we have already seen: the oscnode lowers the genus of the surface by 3.

Now, referring to the case of an irreducible fundamental curve of the first kind of arbitrary genus ρ, one may wonder whether, as we have seen in the case of an isolated multiple point, the system |C′| can always be successively extended by adding θ, as far as x satisfies the previous inequality. The answer is negative: the upper bound of x is not always achieved. To see this it suffices to give an example. Let us assume that F has a double curve passing with six branches through a sextuple point O, whose osculating cone at O has therefore genus ρ = 4. In general the tangents to these branches do not belong to a quadric cone and so a subadjoint surface of F will have multiplicity 3 at O, since it has there an osculating cone of the 3rd order adjoint to the osculating cone of F. The point O will be of multiplicity 4 for the adjoint surfaces, so that the system adjoint to the plane sections is obtained from the larger subadjoint system by imposing that it contains only one curve θ corresponding to the neighbourhood of O, and the value 1 is strictly smaller than the upper bound for x indicated by the previous inequality:

6 − (x − 1)6 ≥ 0

which gives x = 2 as the maximum for x.

3.10 Point singularities of a higher kind

The analysis of point singularities of a higher kind and of the effect that they have on the genus of the surface can be reduced to the study of the multiple points that may occur infinitely near to a proper singular point and that are not defined by the fact that proper or infinitesimal singular curves go through them.

If one imagines that the point singularity that a surface F has at a point O is completely solved, then the system of plane sections of F, or equivalently one of its multiples (that we may assume of arbitrarily large dimension) gives us on the transform of F a linear system |C| having a fundamental curve corresponding to the neighbourhood of the point O. If the singularity of O is of the first kind, then this fundamental curve is also of the first kind, hence subtracting it from |C| none of its components is again fundamental for the residual system. By contrast we define fundamental curve of the second kind (corresponding to a multiple point of the second kind) a fundamental curve containing some parts that are fundamental curves of the first kind for the residual system. And so, in general, one may define by induction the fundamental curves of the sth kind, by reducing this concept to that of curves of the (s − 1)st kind.

From this definition it follows that the problem of determining the effect of a multiple point of the sth kind on the genus of the surface F can be reduced to the analogous problem for singularities of the (s − 1)st kind. Thus the determination of the conditions that a fundamental curve of the system |C| imposes on the the subadjoint curves in order that they become adjoint to |C| can be reduced to the analogous problem with respect to the system that one obtains from |C| by subtracting that curve.

For the sake of simplicity we refer to the case of a fundamental curve of the second kind of the system |C|.

On the surface F whose plane sections are the C we have a multiple point O and infinitely near to, in the first order neighbourhood, it there are some (proper) multiple points O_{1}, O_{2}, . . .. On a transformed surface, the point O corresponds to a curve θ fundamental for |C|; O_{1} to a fundamental curve θ_{1}, and so on. Now it is interesting to remark that the curves θ_{1}, θ_{2}, … must occur as parts of the fundamental curve θ, so that one has in general

θ = θ_{0} + θ_{1} + θ_{2} + ⋯

In fact the transformation applied to the surface F can be performed step by step. First one considers a transforming system of surfaces having O as a simple base point, so that the neighbourhood of O is transformed into a (simple or multiple) curve θ_{0} containing the singular points O_{1}, O_{2}, . . .. Then one transforms successively each of the points O_{1}, O_{2}, … into a curve that, in general, will be connected with θ_{0}. Because of this connectedness, when one subtracts the curve θ_{0} from |C|, as a consequence one subtracts also the curves θ_{1}, θ_{2}, … which are also fundamental curves (not intersecting the C). In other words the sum

θ = θ_{0} + θ_{1} + ⋯

is a univalent fundamental curve for |C|, that is, to subtract it which from |C| imposes only one condition. On the contrary the residual system (C − θ) still has θ_{1}, θ_{2}, … as fundamental curves, so that we may say that the curves

θ_{0} + 2θ_{1} + θ_{2} + ⋯

and

θ_{0} + θ_{1} + 2θ_{2} + ⋯

and so on are bivalent fundamental curves of |C|, that is, to subtract it from |C| imposes two conditions.

For instance, let us consider the case of an (isolated) uniplanar double point. This is a double point O whose osculating cone is reduced to a plane counted twice; the lines of this plane through O have in general a contact of order 3 with the surface, except three lines with a contact of order 4 that define three double points O_{1}, O_{2}, O_{3}, infinitely near to O^{(13)}. For the sake of simplicity we shall assume that the surface F to which the uniplanar point O belongs is a cubic surface F_{3}, and we shall elucidate what we have said by referring to its plane representation.

(13)See for instance Enriques–Chisini, Ibidem, Libro IV.

To represent F_{3} on a plane we may take the ∞^{3} linear system of the cubics C_{3} passing through three collinear base points O_{1}, O_{2}, O_{3} and through three more points O′_{1}, O′_{2}, O′_{3} infinitely near these, where the line containing O_{1}, O_{2}, O_{3} corresponds to the uniplanar point O and to the part θ_{0} of his neighbourhood. Let us transform our system of C_{3} via a Cremona transformation of order 4, taking as transforming net a net of quartics with a triple base point, three simple base points O_{1}, O_{2}, O_{3} and three more simple base points. The system |C_{3}| will be transformed in another system of curves |C_{9}|, with a sextuple base point P, three collinear triple points A_{1}, A_{2}, A_{3}, three more double points B_{1}, B_{2}, B_{3}, and three simple points O_{1}, O_{2}, O_{3}, that belong to the lines P B_{1}, P B_{2}, P B_{3} respectively. The line A_{1}A_{2}A_{3} counted twice represents the curve θ_{0}, whereas the lines P B_{1}, P B_{2}, P B_{3} are the images of θ_{1}, θ_{2}, θ_{3}. It is immediately clear that the group of lines θ_{0} + θ_{1} + θ_{2} + θ_{3} is a univalent fundamental curve for the system |C|. Indeed if one subtracts the line θ_{0}, then as a consequence the lines θ_{1}, θ_{2}, θ_{3}, each of which meets θ_{0} in a point which is not a base point, must be a base curves of the residual system, hence the above-mentioned line A_{1}A_{2}A_{3} turns out to be subtracted twice. Therefore the residual system of θ = θ_{0} + θ_{1} + θ_{2} + θ_{3} consists of the quartics C_{4} passing three times through P, simply through B_{1}, B_{2}, B_{3} and simply also through A_{1}, A_{2}, A_{3}. θ_{0} is not a fundamental line for this residual system any more; on the contrary, the lines P B_{1}, P B_{2}, P B_{3} are still fundamental.

The analysis of the singularity of the second kind that a surface F has at an ordinary uniplanar point O makes it clear that such a point does not affect neither the adjoint surfaces nor the genus.

In fact let us consider the system |C| of plane sections of F (which one may replace with one of its multiples). The plane sections C_{1} through O form a linear system that has only fundamental curves of the first kind and of genus zero θ_{1}, θ_{2}, θ_{3} corresponding to the double points O_{1}, O_{2}, O_{3}, infinitely near O. So these double points do not affect the canonical curves, that are characterized by the property of cutting out on the C_{1} residual groups of the characteristic series, that is, groups belonging to a curve of order n−3 adjoint to one of the C_{1} formed by a φ_{n−4} and a line through O. Then the curves adjoint to the C_{1} are cut out by the adjoint surfaces Φ_{n−3} that pass simply through O and, adding the curve θ that represents the neighbourhood of O, it turns out that the curves C′ adjoint to the sections C of F are cut out by Φ_{n−3} which do not have to pass through O any more.

While the ordinary uniplanar double point does not affect the genus of a surface F, there may be specializations of this singularity that do affect the genus. We have already seen that this happens for the tacnode, that is characterized by the property that the tangent lines through O have a quadruple contact with F there and that therefore makes the genus of the plane sections through it drop by two instead of one. But even more surprising is the case of a uniplanar double point that makes the genus of the plane sections through it drop by only one and has an infinitely near tacnode O_{1}, so that the genus of the plane sections through OO_{1} drops by three instead of two. In this case the system |C_{1}| of plane sections through O has a fundamental curve θ of genus 1, whose image is the tacnode O_{1}, and this implies that the surfaces Φ_{n−4} have to pass simply through O_{1}, that is, as a consequence, through O. So one deduces that a uniplanar double point, lowering the genus of the plane sections through it by only one, with an infinitely near tacnode, lowers the genus of the surface by 1.

Remark Noether pointed out two kinds of surfaces of order four that turn out to be rational as a consequence of the possess of the singularity described above^{(14)} and the rationality of these surfaces is consistent with the fact that the above-mentioned singular point lowers the genus of the surface of order four by one, and so it reduces it to zero. The two types of surfaces of Noether give rise to a plane representation respectively given by:

(1) the ∞^{3} system of curves C_{7} of order seven having a triple base point and nine double base points on a fundamental cubic; and

(2) the ∞^{3} system of curves C_{9} of order nine having eight triple base points, a double point and a simple point, on a fundamental cubic.

In both cases the fundamental cubic of the representing system is a θ of the second kind that is fundamental also for the residual system.

And it is evident from the above representation that this curve θ has genus 1 and degree −1 on the surface F_{4}, or on a transform of F on which the singularity is resolved without introducing new exceptional curves: in fact the plane cubic from which 10 base points have been removed is precisely a curve of degree −1^{(15)}.

Now the analysis of the singularity presented by Noether’s surface F_{4} can be extended to analogous singularities of an arbitrary surface: the singularity of the second kind consisting of a double point with an infinitely near tacnode corresponds in general to a fundamental curve of genus 1 and of degree −1 for the linear system |C| of the plane sections of the surface^{(16)}.

(14)See Enriques–Conforto, Le superficie razionali, Ibidem, pages 196 and 204.

(15)At this point it is necessary to warn the reader that second type of Noether’s surfaces presents a projective complication, i.e., the tacnodal plane of O_{1} passes through O, so that the plane sections through OO_{1} contain a third fixed double point collinear with OO_{1} and become reducible in the line OO_{1} and a cubic of genus 1 for which the line is a flex tangent. As a consequence the analysis of the singularity must be carried out using a system of intersections with surfaces of higher degree instead of the system of plane sections. See Enriques–Conforto, Ibidem, pages 206–7.

We can give a direct proof of the statement (independent of the possibility of approximating the given surface with an F_{4} having the same singularity). To start with one shows that a curve of genus 1 and degree −1 fundamental for a linear system |C| yields the above mentioned singularity on a surface having ∞^{3} C as plane sections; and the converse also holds, if one starts with an F having the stated singularity, resolves it and computes the characters of the fundamental curve corresponding to it.

Let us assume that θ is a fundamental curve for a system |C| of genus π and degree n, and that the curve θ has genus ρ = 1 and degree ν = −1; it is clear that the number of intersections

(C − θ)θ = Cθ − θθ = −ν = 1,

namely the curves of the residual system |C_{1}| = |C −θ| have one intersection, necessarily fixed, with θ (that is, C_{1}θ = 1). As a consequence the genus x of C_{1} is given by the equation

π = x + 1 + 1 − 1 = x + 1,

that is,

x = π − 1

and its (virtual) degree is computed by the formula

n = CC = (C_{1} + θ)(C_{1} + θ) = C_{1}C_{1} + θθ + 2

that gives

C_{1}C_{1} = n − 1

(but the effective degree of |C_{1}| is n − 2, since there is a base point on θ).

Now the image of θ on the surface F that has ∞^{3} C as plane sections is a point O that lowers the genus of the plane sections through it by 1 and therefore it is double for F; and near O there is a singular point O_{1} that lowers the genus of the (plane or superficial) sections through it by two, and is therefore a tacnode.

Conversely let us start with a surface F having a double point O with an infinitely near tacnode O_{1}. The point O corresponds on a suitable transform of F to a curve θ fundamental for the system of plane sections C, which a priori consists of two parts θ_{0} and θ_{1}, the former of genus zero corresponding to the first order neighbourhood of O (not including the tacnode), the latter of genus 1 corresponding to the neighbourhood of O_{1}: the curve

(16)See D. Pedoe, On the canonical system of certain algebraic surfaces. Proc. Cam. Phil.

Soc., Vol. 33 (1937).

θ = θ_{0} + θ_{1}

is fundamental for |C|, and necessarily connected, and represents the whole neighbourhood of the point O.

Now one can compute the genus ρ and the degree −ν of the curve. By expressing the fact that subtracting it from the system |C| of genus π lowers the genus of C by 1, we have

|C| = |C_{1} + θ|π = π − 1 + ρ + x − 1

where x ≥ 1 denotes the number of points of connection for the reducible C, that are common to θ and C_{1}; whence one deduces

x = 1 and ρ = 1.

So, computing the number of intersections θC_{1} as a function of the degree of θ, one has

C_{1}θ = (C − θ)θ = −θθ = −ν = 1

One computes next the degree −ν_{1} of θ_{1} by writing down that imposing to the sections C_{1} to pass through O lets their genus drop by 2: we have

|C_{1}| = |C_{2} + θ_{1}|π − 1 = π − 3 + 1 + y − 1

where y denotes the number of intersections

C_{2}θ_{1} = (C_{1} − θ_{1})θ_{1} = −θ_{1}θ_{1} = −ν_{1},

whence one obtains

ν_{1} = y = 2.

Now, denoting by −ν_{0} the degree of θ_{0} one has

−ν = −ν_{0} − ν_{1} + 2 = −1

thus

ν_{0} + ν_{1} = 3, ν_{0} = 1.

In conclusion the curve θ_{0} of genus zero and degree −1 turns out to be an exceptional curve that can be transformed into a simple point of the curve θ_{1} of genus 1, and therefore the latter becomes a θ of degree

ν_{0} = −1

(and still has genus ρ = 1).

The singularity consisting of a double point with infinitely near a tacnode occurs in the study of double planes, when the branch curve has two infinitely near triple points O, O_{1}, or, as somebody says, a [3, 3] point. In this case consider for instance the system of curves C having a certain genus π that is represented on the double plane by the ∞^{5} system of conics; the conics through O represent curves C_{1} of genus π − 1, but those going through O and O_{1} represent curves of genus π − 3. So the point O is a double point of the surface z^{2} = f(x, y) that makes the genus of the (plane or superficial) sections drop by only 1, whereas the point O_{1} infinitely near to it makes the genus of these sections further drop by 2, hence it is a tacnode. So we see again that: a [3, 3] point belonging to the branch curve of order 2n of a double plane must be considered as a simple base point for the curves of order n − 3 that are the images of the canonical curves (see Section ??) and so it lowers the genus of the double plane by 1.

The two rational surfaces of order four of Noether’s that we have recalled above are projected onto a double plane from their singular point. Their branch curve has order 6, and has a quadruple point, and a [3, 3] point for the two types respectively: so it is confirmed that the singularity of those surfaces lowers by one the genus of the surfaces of order 4.

EXERCISE

1. Show that the (isolated) singularity consisting of two infinitely near triple points OO_{1} imposes two simple base points to the adjoints and therefore it has the same effect as two distinct triple points on the genus of the surface.

Then compute the genus of a surface F_{5} of order 5 with a double line and two infinitely near triple points OO_{1} away from the line: one finds that

p = 0

This result agrees with the fact that F_{5} is rational, since it has a pencil of rational quartic plane sections through OO_{1}.

It is possible to obtain the plane representation of the above mentioned F_{5} by means of a quadratic transformation of the second kind, corresponding to the homaloidal system of the quadrics passing through the double line of F_{5} and through three more points of F_{5}, two of which are the infinitely near triple points O and O_{1}. In this way F_{5} will be transformed into a surface F_{4} of order 4 with a double line.

Bicanonical curves and biadjoint surfaces 3.11 84

2. Consider a nonsingular surface F_{4} of order 4, containing a line a: the sum of the linear system of plane sections and of the double of the pencil of the cubic sections K_{3} through a gives a linear system of curves C of genus 3 + 1 + 1 + (3 − 1) + (3 − 1) = 9 and it is easy to see that the degree of this system is 16 and the dimension is equal to 9. Impose that the C have a triple point at a point P of F_{4} away from a, that is a flex for the K_{3} passing through it. Thus one obtains an ∞^{3} linear system of curves of genus 6 and degree 7, that has K_{3} as a fundamental curve of genus 1 and degree −1. Via this system F_{4} can be transformed into a surface F_{7} of order 7 with a quadruple line, a triple line (that reduces to 6 the genus of the plane sections) and a double point with an infinitely near tacnode: so one finds for F_{7} the same genus of F_{4}, that is, p = 1. There is an adjoint surface of order 7 − 4 = 3, that is reduced to the plane containing the quadruple and the triple line counted twice and to the plane projecting the isolated double point from the quadruple line, which cuts F_{7} along an exceptional cubic.

Investigate the inverse transformation of F_{7} into F_{4}.

Remark In this section we have made some remarks concerning interesting examples, but we do not believe, in any way, that we have exhausted the theory of fundamental curves of linear systems and its relationships with the theory of singularities. Therefore it is necessary to examine and to meditate on a larger class of examples and to investigate the complications to which they give rise. In order to have a deeper understanding of this type of questions, we advise the scholar to consider some, even quite simple, singularities that may belong to the branch curve of a double plane: infinitely near quadruple points, series of four or six infinitely near triple points, series of triple points infinitely near to quadruple points, and so on. The question is to understand these cases by characterizing the fundamental curve of the system of plane sections of the surface F, that corresponds to the above multiple points, and the consequences of the subtraction of it from the system^{(17)}.

3.11 Bicanonical curves and biadjoint surfaces

Given a surface F of a certain order n in ordinary space, we have learned how to construct on it the canonical curves as intersections of surfaces Φ_{n−4} of order n − 4 adjoint to F, and more generally the adjoint curves of the r-tuple system of plane sections C of F, via the adjoint surfaces Φ_{n−4+r}, which always cut out on F a complete linear system. It is now easy to determine on F the bicanonical curves and also the biadjoint curves of the r-tuple system of |C|, by making use for this of biadjoint surfaces of F, ψ_{2n−8} and ψ_{2n−8+r} of order 2n − 8 and 2n − 8 + r respectively.

(17)See: A. Franchetta, Sui punti doppi isolati delle superficie algebriche, Rendic. Acc.

Lincei, 1945, Notes I and II.

For the sake of simplicity, let us refer to the case of genus p > 0, in which the adjoints Φ_{n−4} and the canonical curves that are sections with them actually exist. The behaviour of the biadjoint surfaces ψ_{2n−8} with respect to the singularities of F is defined as a rule by the behaviour of two Φ_{n−4} that, taken together, form precisely a ψ_{2n−8}. Still it is necessary to define this behaviour more precisely, as we shall do with the following remarks.

If F has a double curve and no proper multiple points, then the Φ_{n−4} pass through this curve simply and the ψ_{2n−8} go through it doubly: the linear system cut out by these ψ_{2n−8} is complete because it is deduced from the system cut out by the adjoints ψ_{2n−8} (that we already know to be complete) by subtracting the neighbourhood of the double curve.

But if F has a curve of multiplicity i > 2 then the biadjoint surfaces ψ_{2n−8} (and similarly the ψ_{2n−8+r}) should pass through this curve, not with multiplicity 2i−2, but with multiplicity i, and should have a suitable contact with the branches of the surface along it; for instance if i = 3 then the ψ should contain the triple curve of F as a triple curve and they have to touch simply the branches of the surface F along it. In fact a surface ψ that passes four times through a triple curve of F may always be replaced by a linear combination of ψ and F that passes three times through this curve and touches the branches of F. So the ψ are subject to less restrictive conditions and one can say that they cut out on F a linear system obtained from the linear system cut out by the adjoints of the same order, by subtracting a certain number of times the neighbourhood of the multiple curve. By contrast the ψ passing four times through the triple curve of F do not give in general a complete system.

Similarly an isolated s-tuple point of the surface F (s > 2), which is in general (s − 2)-tuple for the adjoint surfaces Φ, has to be a multiple point for the biadjoint surfaces ψ, and its multiplicity will not be 2s − 4 but s, and in addition the ψ have to touch F s − 4 times at that point.

Thus if O is an isolated triple point of F, then the ψ have to pass through it doubly. If instead O is a quadruple point, then the ψ must have only a quadruple point there: but if O is a quintuple point of F, then it has to be only quintuple for the ψ, that must have the same osculating cone, though

Special attention must be paid to the behaviour of the biadjoint surfaces at a tacnode of the surface F. At first sight one might think that such a point, being simple for the adjoint surfaces Φ_{n−4}, should be double for the ψ_{2n−8}, as it is for the surfaces consisting of two Φ_{n−4}. But let us think about the way the behaviour of the surfaces Φ_{n−4} at a tacnode O is determined: a plane section through O has two infinitely near double points O and O_{1}, and the section of a Φ_{n−4} is such that, together with a line through O, it must form an adjoint φ_{n−3} of the section of F. Therefore a priori Φ_{n−4} is forced to pass through the point O_{1} infinitely near O, and only as a consequence it happens to pass through the proper point O. Similarly, a plane through O will be cut out by a biadjoint surface ψ_{2n−8} along a curve that a priori should pass doubly through O_{1}, and only by an effect of transitivity will pass simply through O and O_{1}. So the conditions that a tacnode imposes on the biadjoint surfaces consist only in passing through this point and in touching the tacnodal plane there: hence the tacnode lowers the bigenus by three. These conditions can be verified also by remarking that they imply for the bicanonical curves sections of the ψ_{2n−8} the characteristic property of cutting out on the generic curve of a linear system |C| (for instance of the net of plane sections through O) groups belonging to the double of the residual series of the characteristic series.

Surfaces of genus p = 0 and bigenus P > 0 3.12 87

Let us now turn to the case of the Noetherian singularity consisting of a double point O with an infinitely near tacnode O_{1}. Also in this case the vanishing of the Φ_{n−4} at O is a consequence of the principle of transitivity of singularities: a priori the Φ_{n−4} should pass through the infinitely near point O_{1}. Therefore the ψ_{2n−8} which have to pass through O_{1} twice, are actually determined by the condition of passing through the two infinitely near points O and O_{1}. Thus: a double point with an infinitely near tacnode imposes in general to the biadjoint surfaces to pass simply through the two infinitely near points, O and O_{1}, and therefore it lowers the bigenus only by two. It is easy to check these results by referring to a double plane with branch curve f(x, y) = 0 of a certain order 2n, namely to the surface z^{2} = f(x, y). We have seen that a quadruple point of f = 0 is a tacnode for the surface F, and, in the plane, the canonical curves of F are represented by curves of order n − 3 passing simply through O. Now the curves of order 2n − 6 passing doubly through O represent bicanonical curves of F, although the system of this curves is not complete, in general^{(18)}. So one can check that a tacnode lowers the bigenus of F by 3.

(18)A complete linear system of curves |C| on the double plane does not represent on the surface F a complete system, but a system of curves belonging to the involution I whose pairs correspond to the points of the double plane; and in general this can be extended to a larger system whose curves no longer belong to the involution I. This can be seen already in the case of the double plane with branch curve f_{4} = 0 of degree 4 obtained by projecting from the point at infinity O of the z-axis the surface F_{4} of degree 4 of equation z^{2} = f_{4}(x, y), that has a tacnode at O with the plane at infinity as the tacnodal plane.

Here the ∞^{5} conics of the plane are the images of the curves cut out on F_{4} by the quadric cones through O, which are hyperelliptic curves belonging to the involution I. But the ∞^{5} system of such curves is contained in the ∞^{6} linear system cut out on F_{4} by the quadrics touching the tacnodal plane at O. Concerning the possibility of extending the system of bicanonical curves whose images on the double plane are curves of degree 2n−6, the matter

The next case is the case of the noetherian singularity that corresponds to a [3, 3] point of the branch curve f. The [3, 3] point should be considered as a double point O, with an infinitely near 4-tuple point O_{1}. The images of the bicanonical curves must then pass doubly through O_{1}, namely – by transitivity – the images of the bicanonical curves must pass simply through O and O_{1}, and so a [3, 3] point of the branch curve of the double plane lowers the bigenus of the surface by 2.

3.12 Surfaces of genus p = 0 and bigenus P > 0

The previous results allow us to construct easily surfaces of genus p = 0 and bigenus P > 0, which therefore are certainly irrational.

The first example was found by Enriques in 1896^{(19)}. It is the surface F_{6} of order 6 that passes doubly through the edges of a tetrahedron. If we consider a projective coordinate system in which the tetrahedron is fundamental, F_{6} is represented by an equation of the form:

F_{6} = φ(x_{2}x_{3}x_{4}, x_{3}x_{4}x_{1}, x_{4}x_{1}x_{2}, x_{1}x_{2}x_{3}) + x_{1}x_{2}x_{3}x_{4}f(x_{1}, x_{2}, x_{3}, x_{4}) = 0

where f and φ are quadratic forms in the variables x_{1}x_{2}x_{3}x_{4}, and respectively in the products of triples of these variables. In fact F_{6} belongs to the double of the linear system consisting of the cubic surfaces that pass simply through the edges of the tetrahedron. F_{6} has genus p = 0, since there are no quadrics adjoint to it, that is, passing through the edges of the tetrahedron. But the bigenus of F_{6} is P = 1, because there exists a surface of order 2 · 6 − 8 = 4 biadjoint to it, that is, our tetrahedron. This surface ψ_{4} cuts out on F_{6} a bicanonical curve of order 0. The linear system |C| of plane sections of F_{6} is of genus π = 4 and of degree n = 2π − 2 = 6; its characteristic series is a complete and nonspecial g_{6}, whose double is the double of the canonical series. The system |C′| adjoint to |C| is cut out on F_{6} by the cubic surfaces Φ_{3} passing simply through the edges of the tetrahedron; this is, as well as |C|, a system of sextics of genus 4 and degree 6. It is remarkable here that the double of the system |C| and the double of the system |C′| coincide:

|2C| = |2C′|

depends on the behaviour of the biadjoint surfaces at the point at infinity of the z-axis, where F has a known singularity. It suffices to say that in general the y_{2n−8} will have multiplicity 2n − 2 there and a suitable contact with the surface. Whence it is clear that, while the canonical curves of a surface that can be represented on the double plane always belong to the involution whose pairs correspond to the points of that plane, in general the same does not happen for the bicanonical curves. (See Chapter V, Section ??)

(19)Introduzione…, Ibidem

This equality just expresses the real meaning of the existence of the bicanonical system |2C − 2C′| of order zero.

Furthermore, since

|(2C)′′| = |2C′| = |C + C′′|

one deduces that

|C′′ − C| = |2C′ − 2C| = |2C − 2C|

and so

|C′′| = |C|;

in words: the system of plane sections of F_{6} is in turn the adjoint of its adjoint system, cut out by the surfaces Φ_{3} passing simply through the edges of the tetrahedron.

Let us compute the linear genus p^{(1)} of F_{6} by calculating the virtual genus of the curves of |C′ − C| according to Chapter I, Section ??. Denoting by π and n the genus and the degree of |C|, by x and y the genus and the degree of the virtual curve K = C′ − C of order 0, that has i = 0 intersections with the C, we shall write the genus and the degree of |C + K| = |C′|:

π′ = π + x + 0 − 1 = πn′ = n + y + 2 · 0 = n

whence

x = 1, y = 0 :

the linear genus of F_{6} is p^{(1)}=1.

Another example of surface of genus p = 0 and of bigenus P > 0 was pointed out by Castelnuovo, again in 1896^{(20)}. It is the surface F_{7} of order seven that passes three times through a line r, twice through a conic Γ (not intersecting r) and moreover has three tacnodes whose tacnodal planes contain r.

Such an (irreducible) surface actually exists and can be constructed as follows: first of all take a cubic surface F_{3} that passes simply through the line a and through the conic Γ and consider three planes a_{1}a_{2}a_{3} passing through the line r and tangent to F_{3} at the points A_{1}, A_{2}, A_{3} (away from r). It is easy to see that there exists an irreducible surface F_{4} of order 4 passing doubly through the conic Γ and tangent to the planes a_{1}a_{2}a_{3} at the points A_{1}, A_{2}, A_{3}: this surface taken together with the three planes a_{1}a_{2}a_{3} is a particular reducible F_{7} that passes twice through the conic Γ, three times through the line r, and has a tacnode at the points A_{1}, A_{2}, A_{3} with tacnodal planes a_{1}a_{2}a_{3}. But another reducible surface satisfying the same conditions can be obtained by adding to F_{3} counted twice a plane through r; then, taking the pencil generated by the two reducible F_{7}, one obtains an irreducible F_{7} with the singularities described above.

(20)G. Castelnuovo, Sulle superficie di genere zero. Mem. della Soc. It. delle Scienze (detta dei XL), s. III, t. 10, 1896.

There are no surfaces Φ_{3} adjoint to F_{7}: in fact a Φ_{3} which should pass through the conic Γ and doubly through the line r that does not intersect Γ, should be a priori reducible, containing the plane of Γ as a component. But the residual quadric should first of all pass doubly through r, thus it should split into two planes through r, and besides it should pass through the tacnodes A_{1}, A_{2}, A_{3}; this imposes three conditions impossible to be satisfied. One concludes that the genus of F_{7} is

p = 0.

Now we determine the surfaces ψ_{2·7−8} = ψ_{6} biadjoint to F_{7}. Among these certainly there are the surfaces consisting of the plane of Γ counted twice and of four planes through r, three of which have to pass through the tacnodes A_{1}, A_{2}, A_{3} and therefore coincide with the planes a_{1}a_{2}a_{3}; thus one obtains ∞^{1} biadjoint surfaces ψ_{6} that contain a plane through r as the variable part and cut out ∞^{1} bicanonical quartics of genus 1 on F_{7}, forming a base point free pencil. It is clear that this bicanonical system of order zero cannot be extended, and therefore the condition that the biadjoint surfaces ψ_{6} pass three times through the line r and touch the branches of F_{7} through it, implies here, as a consequence, that the line r is quadruple for ψ_{6}.

Summing up we shall say that: the surface F_{7} of Castelnuovo has genus p = 0 and bigenus P=2, since it contains a pencil of plane bicanonical quartics of genus 1 and degree 0.

It is easy to determine the linear genus of F_{7}, that is the (virtual) genus p^{(1)} of its virtual canonical curves K; in fact, since the degree of |K| is equal to p^{(1)} − 1, the genus and the degree of |2K| are respectively given by

3p^{(1)} − 2 and 4p^{(1)} − 4.

From either of of the relations 3p^{(1)} − 2 = 1 and 4p^{(1)} − 4 = 0 one deduces that the linear genus is p^{(1)} = 1.

No other example of surfaces of genus p = 0 and of bigenus P > 0 was found for many years, and in particular, it remained an unanswered question whether among such surfaces there are any for which the linear genus is p^{(1)} > 1, having an irreducible canonical system of dimension > 1. This question is now answered by the examples actually constructed, simultaneously and independently of one another, by L. Godeaux^{(21)} and by L. Campedelli^{(22)} in 1931–32.

(21)See L. Godeaux, Sur une surface algébrique de genre zéro et de bigenre deux, Rendic.

Acc. Lincei, Vol. XIV, 1931 (22)L. Campedelli, Sui piani doppi con curva di diramazione del decimo ordine, Ibidem,

We shall only describe the example constructed by Campedelli that refers to a double plane with branch curve of order ten, having six [3, 3] points (pairs of infinitely near triple points). It is possible to actually construct such a curve f_{10}, at least as a curve formed by three conics two by two bitangent and by a quartic f_{4} that touches them at their six contact points. Indeed one obtains three mutually bitangent conics by taking for instance two concentric circles (that are tangent at the cyclic points of the plane), and an ellipse that touches the smaller circle at the ends of a diameter and the larger one at the ends of an orthogonal diameter. The 6 contact points of the conics thus obtained yield 12 linear conditions for a quartic. Taking one of these quartics together with the 3 conics as the branch curve of a double plane, we shall have that:

(1) on the double plane there is no canonical curve, which should be a conic passing through the six singular points of f_{10}, whereas one sees that in the example we constructed those points do not belong to a conic (circle); hence the genus of the double plane is

p = 0;

(2) there is a net of quartics that, like the quartic f_{4} which is a component of the branch curve f_{10}, pass through its twelve triple points: they represent ∞^{2} bicanonical curves of the surface, belonging to the involution whose pairs of points correspond to points of the plane, and one may see that they form the complete bicanonical system, so that the bigenus of the surface is

P = 3.

Remark The proof that P = 3, and not P > 3 will follow later on from the expression of P via the linear genus, that here gives P = p + p^{(1)} = p^{(1)} = 3, where the value of p^{(1)} stems directly by remarking that the virtual double conic forming the canonical curve would have 8 branch points: 6 at the triple points of f_{10} and two more outside. If one does not want to work with the virtual canonical curve, then it also suffices to remark that the degree of the bicanonical system is the double of the degree of the system of quartics, so that one has

4(p^{(1)} − 1) = 2 · 4 = 8, p^{(1)} = 3.

Equivalence criteria 3.13 91

If one does not want to use the relationship between the bigenus and the linear genus, then one can nevertheless prove directly that, for our surface, the net of double quartics that are the images of the bicanonical curves is complete. In order to do this it suffices to remark that the component f_{4} of the branch curve is a part of one and only one bicanonical curve, consisting of the curve itself counted twice, and that the ∞^{2} quartics through the triple points of f_{10} cut out on f_{4} a complete and nonspecial g_{4}: in fact if this series were the canonical series of f_{4} then the 12 triple points of f_{10} should lie on a cubic, and it is easy to see that this gives a contradiction.

Vol. XV, 1932.

3.13 Equivalence criteria

The construction of the subadjoint and adjoint surfaces of a nonruled surface F of ordinary space contains an equivalence criterion: two curves L and M that cut out on the plane sections of F groups belonging to the sum (or difference) series of the canonical series of C and of a multiple of the characteristic series, are equivalent, up to fundamental curves of the system |C|; they are certainly equivalent if |C| does not have any proper fundamental curve. And one can remark that, when the system |C|, as we have said, has no proper fundamental curve, the equivalence of the curve L and M can be deduced just by the condition that they cut out on the C equivalent groups, even if the series containing them is not the sum of the canonical series and of a multiple of the characteristic series.

In fact, taking an adjoint surface of F of large enough order containing L, one obtains a curve D as the residual intersection. Now L + D and M + D cut on a curve C groups equivalent to the sum of the canonical series and of a multiple of the characteristic series; therefore they are equivalent, and so L and M also turn out to be equivalent.

The equivalence criteria can be developed independently of the theory of adjoint and subadjoint surfaces, as F. Severi^{(23)} did, in an interesting way. We shall briefly explain these developments. To start with we consider on a surface F a linear pencil of curves C with at least one simple base point O and without reducible curves. Then if the two curves K_{0} and K_{∞} not passing through O cut out equivalent groups of points on the C, then they are equivalent.

In fact on any curve C of the pencil we can assume that the groups cut out by K_{0} and K_{∞} are the groups of zeros and poles of a rational function φ that is determined up to a nonzero constant, and we can determine this constant too, by imposing that φ takes the value 1 at the point O.

As C varies in its pencil, φ gives us a rational function of the points of the surface that has K_{0} as the curve of zeros and K_{∞} as the curve of poles, so that these curves turn out to be equivalent, Q.E.D. But, in order that the argument be rigorous, one has to prove that K_{0} is the whole curve of zeros of our function and K_{∞} is the whole curve of poles. To do this remark that in case the opposite were true one should add to K_{0} a curve that, having no variable intersection with the irreducible C of the pencil, must be composed with one or more of the C; but then, on such a C, the point O would be an indeterminacy point of the function, where the function would take the values zero and 1 at the same time, and this is impossible, since O belongs neither to the group of zeros nor to the group of poles that determine the corresponding rational function on C.

(23)See F. Severi, Il teorema di Abel sulle superficie algebriche, Ann. di Mat., (3), 12, 1905.

Osservazioni varie di geometria sopra una superficie algebrica, Atti Ist. Veneto, 65, parte II, 1906. Alcune relazioni di equivalenza ecc., Ibidem, 70, 1911.

The previous equivalence criterion can be generalized. Let |C| be a pencil of curves, rational or irrational, without reducible curves: two curves K_{0} and K_{∞} that cut out equivalent groups on the C are equivalent up to curves of the pencil, that is, one has

K_{0} + ∑ C ≡ K_{∞} + ∑′ C.

To prove this second equivalence criterion, we may proceed as before to construct a rational function having K_{0} and K_{∞} as the curves of zeros and the curve of poles respectively. We know that φ is defined up to a multiplicative constant on every C of the pencil, and (denoting by t the parameter of the C of this pencil) this way we obtain a function of the points of the surface, λ(t)φ, where λ has to be determined. In order to do this it is not necessary to determine a point on every C, as we did in the proof of the first criterion, but it suffices to determine a group of a certain number m of points, consisting of the intersections that C has with an auxiliary curve L lying on the surface; then one may prescribe the sum of the values

λφ_{1} + λφ_{2} + ⋯ + λφ_{m}

that φ takes at the points of this group. Even more simply it suffices to take instead of φ the function

f = φ/(φ_{1} + φ_{2} + ⋯ + φ_{m})

that is a rational function of the points of the surface, independent of λ.

It is obvious that K_{0} is contained in the curve of zeros and K_{∞} is contained in the curve of poles, but these are not the entire curves of zeros and poles: one has to add to K_{0} a group ∑ C consisting of the C that pass through the intersection points of L and K_{∞}; in fact on these C the denominator of f becomes ∞ and so f is everywhere zero. Similarly, one has to add to K_{∞} a group ∑′ C consisting of the C that pass through the intersections of L and K_{0}, as one sees by interchanging the roles of K_{0} and K_{∞}, that is, by replacing f by 1/f.

Then we shall have

K_{0} + ∑ C ≡ K_{∞} + ∑^{′} C.

That K_{0} + ∑ C is actually the whole curve of zeros of f can be proved by remarking that this curve of zeros can contain, besides K_{0}, only a certain number of curves having no variable intersections with the C, and therefore a group of (irreducible) C; but f cannot be identically zero on a C unless the sum φ_{1} + φ_{2} + ⋯ + φ_{m} becomes infinite and so the same happens for one of the φ_{i}, which means that C itself has to go through a common point of L and K_{∞}.

Let us add two remarks. The first is that when the pencil of curves C is rational, the C are equivalent, and so there exists a difference ∑ C − ∑′ C consisting of a group of a certain number s of curves that has to be taken with a plus or minus sign, so that one has

K_{0} ≡ K_{∞} − sC or K_{∞} ≡ K_{0} + sC.

And if in particular the pencil of the C has base points, then one certainly has the equivalence

K_{0} ≡ K_{∞}

when the equivalence of the two groups (CK_{0}) and (CK_{∞}) holds, also taking into account the points that occur at the base points.

In fact, if, for instance, one has

K_{0} ≡ K_{∞} − sC

then it follows that

(CK_{0}) ≡ (CK_{∞}) − s(CC);

but one has also

(CK_{0}) ≡ (CK_{∞}),

and so, if the group (CC) is nonzero, one should have s = 0.

Let us now turn to the third equivalence criterion: if on a surface F it is given an ∞^{1} system of index^{(24)} ν > 1 of curves all of which are irreducible, then two curves K_{0} and K_{∞} cutting out equivalent groups on the C are equivalent.

In order to prove this, one constructs as before on every curve C a rational function φ having its zeros at the intersections with K_{0} and its poles at the intersections with K_{∞}, and this function is determined up to a multiplicative constant λ. Then one take an auxiliary curve L on F having a certain number m of intersections with the C, and – for every C – one considers the sum of the values that φ takes at these m points:

(24)This means that there are n curves of the system passing through a generic point of the surface.

φ_{1} + φ_{2} + ⋯ + φ_{m}.

In this way, the rational function

f = \frac{φ}{φ_{1} + φ_{2} + ⋯ + φ_{m}}

is determined on every C. After this, we determine a rational function ψ of the points of the surface by assigning as its value at a point P the sum of the values f_{1}, f_{2}, … f_{ν} that f takes on the ν curves C passing through P:

ψ = f_{1} + f_{2} + ⋯ + f_{ν}.

We may denote the above mentioned values of the f_{i} by means of an upper index; and then we shall write

f_{i} = \frac{φ^{(i)}}{φ_{1}^{(i)} + ⋯ + φ_{m}^{(i)}} .

The rational function ψ clearly vanishes on the curve K_{0}, where all the φ^{(i)} are equal to zero, and its poles must necessarily occur on K_{∞}; but it is not evident a priori that all the points of K_{∞} are poles of the rational function ψ, since a sum of infinities is not always infinite. What we can say is that the curve \overline{K}_{0} of zeros of ψ is of the form \overline{K}_{0} = K_{0} + A, whereas, if \overline{K}_{∞} is the curve of poles, then one has K_{∞} = \overline{K}_{∞} + B. Intersecting the curves \overline{K}_{0} and \overline{K}_{∞} with a generic C, one has:

(K_{0}, C) + (A, C) ≡ (K_{∞}, C) − (B, C)

But one has (K_{0}, C) ≡ (K_{∞}, C) by assumption, thus the curves A and B would have no intersections with the generic curve C, whereas our assumptions exclude the existence of such curves. Concluding, one has

K_{0} = K_{0}, \overline{K}_{∞} = K_{∞}

and so the equivalence of the curves K_{0} and K_{∞} is proved.

Remark In the above discussion we have considered criteria that enable us to say when two (isolated) curves on a surface F are equivalent. There exist other more significant criteria, concerning a continuous (at least ∞^{1}) series of curves |C| given on F. The first criterion of this kind was established by Enriques in 1895. A rational series of curves |C| always consists of equivalent curves, that is, it is contained in a linear system. The property can be shown as for the rational series of groups of points on a curve^{(25)}. But, if one does not want to reduce to that case, it can also be proved using the criteria given before. Another remarkable criterion was pointed out by Severi (1905)^{(26)}. The result is that, in general, an ∞^{1} (at least) series of curves consists of equivalent curves if the C cut out equivalent groups on only one generic curve of a continuous system {D} of index ν > 1.

In fact the condition for an ∞^{1} series of groups of points on a curve to consist of equivalent groups (that is, to be contained in a linear series) can be expressed, with Castelnuovo, via a numerical criterion, in which appears the number of double points of the series (that has to attain the maximum value)^{(27)}. Now, if this criterion is satisfied by the groups that are sections of one of the D with the C, by continuity it will be satisfied also with respect to the other D, and so one is reduced to the third criterion, given before.

We add that the equivalence criterion of the ∞^{1} series of groups of points on a curve, that, as we recalled above, was established by Castelnuovo, can be extended also to surfaces, as R. Torelli showed. He writes an inequality that is satisfied by the number d of curves with a double point that belong to an ∞^{1} series (where the Zeuthen–Segre invariant of the surface appears; see Chapter V); the maximum of this number d corresponds to the case when the series consists of equivalent curves^{(28)}.

(25)See Enriques–Chisini, Ibidem, Libro V, Chapter I, note to §10, Vol. III, page 78 (and page 485).

(26)Osservazioni ecc., Ibidem.

(27)See Enriques–Chisini, Ibidem, Libro V, Chapter IV, 42 (Vol. II, page 482).

(28)See R.!Torelli, Sui sistemi algebrici di curve appartenenti ad una superficie algebrica.

Atti Acc. Torino, 1907.

The numerical genus and the Riemann–Roch theorem for surfaces

4.1 Introduction

In order to understand the meaning of what follows it is worth recalling the problems in the theory of curves that have their natural extension in this setting^{(1)}.

After having defined the linear series on a curve and the elementary operations concerning them, and after having proved the invariance of the canonical series, one is faced with the question of computing the dimension r of the complete series determined on a curve of genus p by a group of n points. The first answer that one obtains here, that is, r ≥ n − p, follows by computing the dimension of the system of adjoint curves of large enough order to the plane curve C of order m and genus p. Since the system of these adjoints φ_{m−3+s} always cuts out on C a complete series (for any s, positive or negative), and since for large enough s the double points of C impose

d = \frac{(m − 1)(m − 2)}{2} − p.

independent linear conditions on the curves of order m − 3 + s that must contain them, the series of degree n = 2p − 2 + sm cut out on C by these adjoints turns out to be of dimension

n − p.

Conditions imposed on a surface: postulation formulas 4.2 97

Therefore it is clear that any series cut out on C by the φ_{m−3+s} passing through a group G of points imposing independent conditions on them has dimension equal to the degree minus the genus p. These results are then made more precise by proving that the canonical series, of degree 2p − 2, cut out on C by the adjoints φ_{m−3}, always has dimension p − 1: this follows by computing the number of conditions that the double points of C impose on the curves of order m − 3 that have to contain them, these conditions being independent. Finally, the theorem of Riemann–Roch actually answers in the most general setting the question of computing the dimension of the complete linear series determined on C by a group of n points: denoting by i (≥ 0) the index of speciality of the group, that is, the number of linearly independent canonical groups that contain it, the dimension of the stated series is precisely

(1)See, e.g., Enriques–Chisini, Ibidem, Libro V, Vol. III.

r = n − p + i.

Now, as we want to extend this type of questions to surfaces, we shall first of all be led to investigate what is, in the case of a surface F of order n of ordinary space, the dimension of the linear system cut out by the adjoint surfaces Φ_{m−3+s} for s large enough, and to relate the result obtained with the computation of the superficial genus of F, for which we shall thus obtain a virtual expression. Following this pattern, we shall be led to compute the dimension of the linear systems adjoint to some other linear systems, thus establishing a “theorem of Riemann–Roch for adjoint systems”, and then to examine the case of an arbitrary linear system, not given as the adjoint of another one.

In order to compute the dimension of the system of Φ_{m−3+s} adjoint to a surface F of order n of ordinary space, it is necessary to recall the socalled postulation formulas of Cayley (1870) and Noether (1871), expressing the number of linearly independent conditions that a curve imposes on the surfaces of large enough order that have to contain it one or more times.

4.2 Conditions imposed on a surface that contains a curve: postulation formulas

Let us consider an (irreducible) twisted curve C of a certain order m and genus p, without multiple points, in ordinary space; the surfaces Φ of arbitrary order n cut out on C a linear series g_{mn}^{r} of degree mn, whose dimension we denote by r. If the series is complete and nonspecial, one can write

r = mn − p.

Under these assumptions one immediately finds the postulation of the curve C with respect to the surfaces of order n, that is, the number of independent linear conditions that C imposes on the Φ that have to contain it; in fact the number of these conditions will be

r + 1 = mn − p + 1.

Our question would thus be answered in the simplest way. But two causes may modify the result mentioned above in opposite directions: the series g_{mn}^{r} may be special, and therefore of dimension r > mn − p, and it may be deficient, that is, it may have smaller dimension than the complete series. Because of the former circumstance the postulation of C for the Φ may turn out to be > mn − p + 1, and because of the latter it may be smaller. But one of the two perturbing causes can be certainly removed by referring to surfaces Φ of large enough order n; indeed it suffices to choose n so that

mn > 2p − 2.

Therefore one can say that for the surfaces Φ of large enough order n, the twisted curve C of order m and genus p, has postulation ≤ mn − p + 1; this means that there are at least

\frac{(n + 3)(n + 2)(n + 1)}{6} − mn + p − 1

linearly independent surfaces Φ passing through C.

One can make the result thus obtained more precise, by proving that for n large enough, n ≥ m − 2, the above series g_{mn}^{r} turns out to be complete^{(2)}. To do this, one projects the curve C from a generic point O of the space to a plane: one thus has a curve of order m with d = (m − 1)(m − 2)/2 − p double points, on which the adjoint curves of order ≥ m − 2 cut out a complete and nonspecial series. Whence it follows that the cones of order m − 2 (or larger) having the vertex at O and containing the d chords of C passing through O cut out on C a complete series, away from the intersection points of these chords with the curve. A fortiori the surfaces of order n ≥ m − 2 passing through the same points cut out a complete series on C away from these points. This series becomes larger if one extends the system of surfaces mentioned before by considering all surfaces Φ_{n} of order n of the space. Now, in order to show that these cut out on C a complete series it suffices to see that the group G of the 2d points of intersection of C with its chords through O imposes independent linear conditions on the Φ_{n}, so that adding the points of G leads from a complete (nonspecial) series of dimension

mn − 2d − p

(2)The proof that follows is due to G. Castelnuovo, Sui multipli di una serie lineare di gruppi di punti appartenente ad una curva algebrica. Rendic. Circ. Mat. di Palermo, t. VII, 1893. Reprinted in Memorie scelte, page 95.

to a complete series of dimension

mn − p.

To see this, we refer for the sake of simplicity to the case n = m−2; it suffices to construct a surface of order m − 2 that passes through 2d − 1 points of G and not through the remaining one; one such a surface will be obtained by adding to a cone of order m−3 that passes through d−1 chords of C passing through O, and not through the remaining one, a plane that passes through one of the two intersection points of the remaining chord, but not through the other. The existence of the aforementioned cone of order m − 3 follows from the well known theorem that the double points of a plane curve of order m (the projection of C) impose independent conditions on the curves of order m − 3 that have to be adjoint to it.

Therefore we conclude: an irreducible twisted curve of order m and genus p, without multiple points, imposes

mn − p + 1

linear conditions on the surfaces of order n ≥ m − 2 that must contain it.

The computation of the postulation of a curve described above, can be extended with slight modifications to the case of twisted curves with multiple points, and even to reducible curves. Let us start by considering the case, most relevant to us, of a curve C (of order m and genus p) having a certain number t of given triple points, with tangents not lying in the same plane, and that we still wish to assume irreducible. Here one shows that the t stated points impose 4t linearly independent conditions on the surfaces Φ of large enough order n for which they must be double, and then that the Φ, passing doubly through the t triple points of C, cut out on C, away from these, a (nonspecial) complete series, so that the postulation of the curve C for the Φ of large enough order n turns out to be

4t + nm − 6t − p + 1 = nm − 2t − p + 1.

The first assertion is self-evident, since to prescribe a double point at a fixed position gives four linear conditions that the surface must satisfy; the latter assertion can be shown by the same argument as for the curves C without multiple points.

In a similar way one obtains the postulation of C in the case when C has a certain number τ of (nodal) double points: in this case it is better to impose first that Φ pass also through these τ points, which gives τ linear conditions, and lowers the number of intersections with C by 2τ; so the postulation of C for the F of order n turns out to be

τ + nm − 2t − 2τ − p + 1 = nm − 2t − τ − p + 1.

Finally we wish to extend this formula to the case in which the curve C splits as the union of several parts, of order m_{1}, m_{2}, … and genus p_{1}, p_{2}, … respectively.

For the sake of simplicity, we refer to the case in which one has only two components C_{1} and C_{2}, that are only allowed to have normal crossings, that is, double points of the reducible curve C = C_{1} + C_{2}, or may also be connected at some of the triple points of the curve C that are double for one of the components and simple for the other. Now, we remark that the surfaces Φ of large enough order n, passing doubly through the triple points of C and simply through its double points, cut out a (nonspecial) complete linear series on both C_{1} and C_{2}; indeed, let us drop the conditions imposed on the Φ requiring that they pass through the double or triple points of C_{2}; the \overline{Φ} thus obtained cut out a complete series on C_{1}; but we may always assume that there is a system of surfaces \overline{Φ} passing through a group of this series, and so large that it is always possible to satisfy the omitted conditions and therefore to find a Φ that passes through that group. But we can say more: not only is each of the linear series cut out by the Φ on C_{1} and C_{2} complete, but also the series that one has on the reducible curve turns out to be complete, allowing us to associate to an arbitrary group of the series on C_{1} a group on C_{2}; in other words the Φ passing through a group of the series on C_{1} and through another group of the series on C_{2}, both taken arbitrarily, do not have as a consequence to contain one of the two curves. To see this, by the same argument we made in the case of an irreducible C, we can reduce to the plane, and invoke the result that “on a reducible plane curve, the adjoint curves of sufficiently large order always cut out a complete series”.

We have thus established that the postulation of the curve C for the surfaces Φ of large enough order, passing as we said through the intersection points of the components of C, is equal to the sum of the postulations of its components. It remains only to compute these postulations. In order to do this we denote by m_{1} and m_{2} the orders and by p_{1} and p_{2} the genera of the above components. If one does not take into account the assigned fixed points of the Φ, the degrees of the two series cut out on C_{1} and C_{2} would be nm_{1} and nm_{2} respectively and the dimensions would be nm_{1} − p_{1} and nm_{2} − p_{2}. But a double point of C = C_{1} + C_{2} lowers the dimension of the system of the Φ by 1, and it lowers the sum of the degrees of the two series cut out on C_{1} and C_{2} by 2 in total, irrespective of whether this point is a double point for one of the irreducible components of C or a simple point for both. Similarly, requiring a triple point of the reducible curve C to be double for the Φ lowers the dimension of the system of the Φ by 4, and it lowers the sum of the degrees of the two series cut out on C_{1} and C_{2} by 6 in total, whether it is a double point for one and a simple point for the other, or a triple point for one of them. In conclusion, one sees in this way that the t triple points of the curve C and its τ double points have the same influence as in the case of an irreducible curve. So the postulation of C will be expressed by the same formula written before, involving the t triple points of the curve C and its τ double points, provided that one makes the modifications that we now indicate. Instead of the series of degree x = mn − 6t − 2τ cut out on the irreducible curve C by the Φ, one has to consider the two series cut out on C_{1} and C_{2}, whose degrees we denote by x_{1} and x_{2}; and, instead of the dimension x − p of that series, one has the sum of the dimensions of the two given series on C_{1} and C_{2}, that is,

x_{1} − p_{1} + x_{2} − p_{2}.

But, while in the previous case the postulation of C was given by x − p + 1, now this postulation is obtained by adding the postulations of C_{1} and C_{2} and is therefore

x_{1} − p_{1} + 1 + x_{2} − p_{2} + 1 = x − (p_{1} + p_{2}) + 2.

In conclusion the postulation of the curve C = C_{1} + C_{2} for the surfaces of large enough order n is expressed by

mn − 2t − τ − (p_{1} + p_{2}) + 2

This formula can be reduced to the one we have written for the irreducible C if, instead of the genera p_{1} and p_{2} of the components C_{1} and C_{2} of C, one introduces the genus p of the reducible curve, computed, according to Noether, under the assumption that the components have no intersection points^{(3)}:

p = p_{1} + p_{2} − 1.

Let us state the general result that one obtains in the case of a curve of order m consisting of h irreducible curves of genera p_{1}, p_{2}, …, p_{h}, which has t triple points (with generic tangents) and τ double points: the postulation of the curve for the surfaces of large enough order n is

nm − 2t − τ − p + 1,

where

p = p_{1} + p_{2} + ⋯ + p_{h} − h + 1

denotes the genus of the reducible curve under the assumption that its parts are not connected.

(3)These points have been removed by imposing them as base points of the Φ.

Dimension of adjoint systems 4.3 102

4.3 Dimension of the systems of adjoint surfaces to a given one

Let us consider a surface F of a certain order n in ordinary space, with normal singularities: that is, a nodal double curve with triple points with noncoplanar tangents, and, if one likes, also with double points^{(4)}. We compute the number N_{m} of linearly independent surfaces adjoint to F, of large enough order m = n − 4 + s. We still denote these surfaces by Φ_{m} and by d the order of the double curve of F, by ρ its genus, by t the number of triple points and by τ the number of double points. We have that the number N_{m} of the Φ_{m} will be

N_{m} = (\begin{matrix}m + 3 \\ 3 \end{matrix}) − (dm − 2t − τ − ρ + 1);

thus the difference N_{m} − N_{m−1} will be given by

N_{m} − N_{m−1} = (\begin{matrix}m + 2 \\ 2 \end{matrix}) − d.

The meaning of this relation is that the adjoint surfaces Φ_{m} of large enough order cut out on a plane the complete linear system of the adjoint curves φ_{m} of the section of F.

In fact the dimension N_{m−1} −1 of the system |Φ_{m−1}| can be deduced from the dimension N_{m−1} of the system of the Φ_{m}, by subtracting a plane from these; and splitting off the plane yields precisely r+1 linear conditions, where r is the dimension of the system cut out by the Φ_{m}. Then in this case one finds

r + 1 = (\begin{matrix}m + 2 \\ 2 \end{matrix}) − d, that is, r = \frac{m(m + 3)}{2} − d,

which is the dimension of the complete system of the adjoints φ_{m} to the plane section of F.

In what follows we shall say that the number N_{m} defined by the above formula expresses in any case the virtual number of surfaces Φ_{m} adjoint to F: we know that for m large enough the virtual number is equal to the actual number of Φ_{m}, whereas for smaller values of m there may be a difference. Still it is important to remark that: for m ≥ n − 4 the actual number of adjoint surfaces of F is always ≥ N_{m}.

In order to prove this, we start with a value m of the order such that the actual number of Φ_{m} is equal to the virtual number N_{m}, and such that m ≥ n − 3.

(4)A priori one may assume that the given surface F has a double curve without selfcrossings: τ = 0 (see Introduction, etc., Section ??). Still it may be useful to have the formulas also in the case τ > 0.

The number of Φ_{m−1} adjoint to F can be deduced from the number of Φ_{m} by subtracting one plus the dimension r of the system cut out on a plane by the Φ_{m}. If this system is the complete system of the φ_{m} adjoint to the plane section, then the actual number of the Φ_{m−1} is again equal to the virtual number; however, if there is a difference, then the system of the φ_{m} sections of the Φ_{m} on a plane can no longer be complete, and then

r < \frac{m(m + 3)}{2} − d;

hence the actual number of the Φ_{m−1} turns out to be larger than N_{m−1}. This argument can be applied as long as m ≥ n − 3, since under this assumption the complete system of the adjoint curves of the section of F, of order n, is regular, that is, it has dimension m(m + 3)/2 − d and not larger, since the double points of the plane curve section of F impose independent conditions on the curves φ_{m} that have to pass through them. On the other hand if one takes m = n − 4, the system of the φ_{n−4} adjoint to the section of F turns out to be superabundant, so that the previous argument is no longer applicable.

We shall call numerical or arithmetic genus p_{a}, of the surface F, the virtual number of linearly independent canonical curves, that is the number N_{n−4} of adjoint surfaces Φ_{n−4}; the superficial genus of the same surface F, that we have defined in the previous chapter, will be distinguished, where it may be necessary, by denoting it by p_{g} as the geometric genus.

Because of what we have seen above one can say that: the numerical genus of the surface F is in any case smaller than or equal to the geometric genus:

p_{a} ≤ p_{g}.

We shall soon prove the invariance of the numerical genus with respect to birational transformations of the surface F. In the meantime, we write the value of this character by means of the postulation formulas established above:

p_{a} = (\begin{matrix}n − 1 \\ 3 \end{matrix}) − (n − 4)d + 2t + τ + ρ − 1.

As a consequence the number of adjoint surfaces Φ_{n−4+s} of F that are linearly independent can be expressed as follows:

N_{n−4+s} = (\begin{matrix}n − 1 + s \\ 3 \end{matrix}) − (n − 4 + s)d + 2t + τ + ρ − 1 =
= p_{a} + (\begin{matrix}n − 1 + s \\ 3 \end{matrix}) − (\begin{matrix}n − 1 \\ 3 \end{matrix}) − sd − 1.

Therefore the linear system consisting of the intersections of these adjoint Φ_{n−4+s} with F will in general have dimension:

N_{n−4+s} − 1 − (\begin{matrix}s − 1 \\ 3 \end{matrix}) = p_{a} + (\begin{matrix}n − 1 + s \\ 3 \end{matrix}) − (\begin{matrix}n − 1 \\ 3 \end{matrix}) − (\begin{matrix}s − 1 \\ 3 \end{matrix}) − sd − 1,

The numerical genus 4.4 104

where (\begin{matrix}s − 1 \\ 3 \end{matrix}) (note that this vanishes for s = 1, 2, 3) is the number of linearly

independent surfaces of order s − 4 (≥ 0), or the dimension of the system of surfaces Φ_{n−4+s} having the same intersection with the surface F of order n. The dimension of the system of sections of the Φ_{n−4+s} written above can be expressed in a very simple form, by introducing the genus π_{s} of the curves cut out on F by the surfaces of order s. First of all we compute the π_{s} in terms of the genus π (= π_{1}) of the plane sections of F; we have

π_{s} = sπ + \frac{s(s − 1)}{2}n − s + 1,

and, substituting the value of π

π = \frac{(n − 1)(n − 2)}{2} − d,

gives

π_{s} = s [\frac{(n − 1)(n − 2)}{2} − d] + \frac{s(s − 1)}{2}n − s + 1 =
= \frac{s(n − 1)(n − 2)}{2} + \frac{s(s − 1)}{2}n − s + 1 − sd.

But we have

\frac{s(n − 1)(n − 2)}{2} − \frac{s(s − 1)}{2}n − s + 1 = (\begin{matrix}n − 1 + s \\ 3 \end{matrix}) − (\begin{matrix}n − 1 \\ 3 \end{matrix}) − (\begin{matrix}s − 1 \\ 3 \end{matrix}) ,

and therefore

N_{n−4+s} − 1 − (\begin{matrix}s − 1 \\ 3 \end{matrix}) = p_{a} + π_{s} − 1.

We conclude that the linear system cut out on F by the surfaces Φ_{n−4+s} of order n − 4 + s, adjoint to the system |C_{s}| cut out by the surfaces of order s has dimension

p_{a} + π_{s} − 1

for large enough s, where π_{s} denotes the genus of |C_{s}|, the s-tuple of the system of plane sections.

4.4 The numerical genus

We consider always a surface F of ordinary space, having a certain order n and normal singularities. We propose to compute the dimension of the adjoint system |C′| of an irreducible system |C| belonging to F. We start by recalling that the adjoint curves C′ cut out canonical groups on a curve C; subtracting C from |C′| one gets the canonical system |K| of dimension p_{g} − 1, and there will thus be ∞^{p_{g}} curves C′ passing through the canonical group formed by the intersection of an adjoint C′ with a curve C. From this it follows that in all cases the dimension of the system |C′| is given by

r = p_{g} + π − 1 − δ(C),

where δ(C) denotes the deficiency of the linear series cut out by the C′ on one of the C; in particular, if |C′| cuts out on C the complete canonical series g_{2π−2}^{π−1}, then δ(C) = 0 and

r = p_{g} + π − 1.

The deficiency δ(C) depends a priori on the choice of the system |C|; but we can prove a fundamental inequality obtained by considering the sum of the irreducible (both at least ∞^{1})^{(5)} linear systems |C| and |L|.

Indeed one has

δ(C) ≤ δ(C + L).

Assume for simplicity that the systems |C| and |L| have no base points on the surface F. We start with the basic relation

|(L + C)′| = |L + C′| = |L′ + C|.

By this relation the system |C′| is obtained from |(L + C)′|, by subtracting the curve L, which yields a number of linear conditions, given by one plus the dimension of the series cut out on L by |(L + C)′|.

Now this series on L is the sum of the canonical series (LL′) and of the series cut out by |C|, whose degree is certainly greater than zero, since both systems |C| and |L| are of dimension greater or equal to one.

In this setting we denote by π the genus of |C|, by ρ the genus of |L|, and by i the number of intersections (LC). Then the genus of |L + C| is

π′ = π + ρ + i − 1,

and the dimension of |(L + C)′| is thus

p_{g} + π′ − 1 − δ(L + C) = p_{g} + π + ρ + i − 2 − δ(L + C).

On the other hand, the dimension of |C′| is

p_{g} + π − 1 − δ(C).

(5)In the case of two pencils it is understood that they are distinct, so that also |C + L| turns out to be irreducible.

As we have said, the difference between the two dimensions is one plus the dimension of the nonspecial series g cut out on L by |(L + C)′|, plus one. If the series g is complete and thus of dimension

2ρ − 2 + i − ρ = ρ + i − 2,

we certainly have

δ(C) = δ(L + C);

if, by contrast, g is deficient, its deficiency expresses the difference δ(L+C)− δ(C). In any case one has:

δ(C) ≤ δ(L + C).

The argument still works if the two systems |C| and |L| have base points on F. A priori the base points that are common to the two systems, or the base points of |C| not belonging to |L|, have no influence. If there is a base point of |L| that does not belong to |C| and therefore, at least virtually, not to |C′| either, then the neighbourhood of the point (regarded as an infinitesimal curve) ought to be added to |C′|, but this does not increase the dimension of the system; so the previous conclusion also extends to this case.

We remark in addition that the hypothesis that we have made, that is, that |L| belongs to an irreducible system, at least ∞^{1}, is stronger than necessary: actually, L could also be an isolated curve, made up of several parts L_{1}, L_{2}, …, provided that none of these is fundamental for the system |C| and that when one adds L_{1}, L_{2}, … successively to |C| one obtains irreducible systems of larger dimension each time. In fact one can subtract L from |(L + C)′| by subtracting in turn its parts L_{1}, L_{2}, …, and if these are not fundamental for |C|, then the series cut out on them, whose dimensions we need to know, are always nonspecial, since they contain the canonical series.

Now we recall that we have computed the dimension of the system cut out on F by the adjoint surfaces for s large enough; here it will be convenient to denote by |L_{s}| this system, which is the adjoint of the s-tuple system of plane sections. Denoting by π_{s} the genus of |L_{s}|, we have found that the dimension of the system |L′_{s}| is

p_{a} + π_{s} − 1.

On the other hand, this dimension can be expressed in terms of the geometric genus p_{g} and the deficiency of the series cut out on L_{s} by L′_{s}, and is

p_{g} + π_{s} − 1 − δ(L_{s}).

Whence one deduces that

δ(L_{s}) = p_{g} − p_{a},

where as we have said, s is a large enough number.

Now, consider on F an arbitrary irreducible linear system |C| at least ∞^{1}, of genus π, that we shall, as usual, suppose to be base point free. It is clear that for s large enough there are surfaces of order s passing through a curve C, and therefore |C| is contained in |L_{s}|, which can thus be viewed as the sum of |C| and of another irreducible system |M| = |L_{s} − C|.

We thus deduce that

δ(C) ≤ δ(L_{s}).

This conclusion can be easily extended to the case in which |C| has base points, with certain multiplicities i_{1}, i_{2}, … In fact, referring to a large enough value of s, we may always impose that the system |L_{s}| has the same base points as |C| with the same multiplicities, thus constructing a system |\overline{L}_{s}| that contains |C|, and even such that the residual system |M| = |\overline{L}_{s} − C| is an irreducible linear system. Now one can assume that the base points of |\overline{L}_{s}| impose i(i − 1)/2 independent linear conditions on the adjoint curves L′_{s}, that have to pass through a i-tuple point with multiplicity i − 1. Thus we shall have a linear system |\overline{L}_{s}| with the same base points as |C|, for which one has again

δ(\overline{L}_{s}) ≤ p_{g} − p_{a}.

Whence one deduces as before that

δ(C) ≤ δ(\overline{L}s).

In this way we have proved that: on the surface F, the deficiency of the (canonical) series cut out by the adjoint system to an irreducible linear system |C| on a generic C has a maximum value. This value is an intrinsic character of the surface, equal to the difference p_{g} − p_{a} between its geometric and numerical genus. This maximum is attained by a class of linear systems belonging to F, including in any case the large enough systems, that is, the systems such that they contain a multiple of large enough order of another irreducible system. Indeed this class enjoys, by definition, the property that “any irreducible linear system larger than another system belonging to the class (that is, partially containing it), belongs again to the class”.

A priori one can have only another restricted class of linear systems |C| for which the deficiency of the (canonical) series cut out by |C′| on C is strictly smaller than the maximum.

From what we have said it follows that the numerical genus p_{a} of a surface, defined directly by means of the postulation formulas (either on the given surface or on one of its transforms with normal singularities) is a character which is invariant for birational transformations of the surface, not only in the case in which one has p_{a} = p_{g}, but also when p_{a} < p_{g}.

Complements 4.5 108

Nevertheless, it will be convenient to distinguish between the two cases by calling regular surfaces those for which p_{a} = p_{g}, and irregular (of irregularity p_{g} − p_{a}) those for which p_{a} < p_{g}.

For either type of surface, the result obtained contains the extension (from the series on a curve to the systems on a surface) of the theorem of Riemann– Roch, but so far only for those linear systems |C′| that are adjoint to another irreducible linear systems |C| (of genus π and degree n). The dimension of |C′|, or at least a lower bound for it, is computed in terms of the genus of C by the formula:

r′ ≥ p_{a} + π − 1.

But it is easy to express r′ also in terms of the characters of the system |C′|, that is, its genus π′ and degree n′. In fact, denoting by p^{(1)} the (relative) linear genus of the surface F, and assuming, for the sake of simplicity, that |C| and |C′| = |C + K| are base point free on F, we have

π′ = π + p^{(1)} + 2π − 2 − n − 1 = 3π − 3 − n + p^{(1)},
n′ = n + (p^{(1)} − 1) + 4π − 4 − 2n = 4π − 5 − n + p^{(1)},

and thus

π − 1 = n′ − π′ + 1.

Thus the dimension of the system |C′| will be

r′ ≥ p_{a} + n′ − π′ + 1;

and similarly, if also |C| is the adjoint of another irreducible linear system, then its dimension will be

r ≥ p_{a} + n − π + 1.

Later on we shall see how this fundamental inequality, which is the theorem of Riemann–Roch for adjoint linear systems, can be extended, with a slight modification, to all linear systems belonging to a surface of numerical genus p_{a}. Then we shall call regular any complete nonspecial linear system whose dimension is expressed by the previous formula in which the equality holds, and superabundant a system for which the strict inequality holds.

4.5 Complements

In the above sections we have referred to a surface F of ordinary space, with normal singularities: under this assumption we have determined the postulation of the adjoint surfaces, which are required to pass simply through the double curve of F. But there is no difficulty in extending our considerations, considering also surfaces with higher order singularities. If one wants to follow the pattern of the previous sections, then one is led, as a first step, to extend the postulation formula, determining (according to Noether) the number of conditions that a curve, of order m and genus p, imposes on the surfaces of large enough order n that have to contain it with multiplicity s, that is,^{(6)}:

\frac{s(s + 1)}{6}[(3n − 4s + 4)m − (2s + 1)(p − 1)].^{(7)}

And similarly, we must compute the number of conditions imposed on the surfaces of given order by requiring that they have an isolated multiple point, that is, for an s-tuple point

(\begin{matrix}s + 2 \\ 3 \end{matrix}) .

But, as the singularities of F become more complicated, the computation of the postulation, on which the dimension of the systems of adjoint surfaces depends, becomes more difficult, as is already apparent from what we have said concerning the virtual influence on the genus of the proper multiple points of F. It is thus useful to point out a faster way, independent of the nature of the singularities of the surface F. We shall always denote by n the order of F, by π the genus of its plane sections, by n_{s} and π_{s} the characters of the s-tuple system, cut out on F by the surfaces of order s.

We start with the following remark: if one assumes that the arithmetic genus p_{a} has been determined (for a transform of F with normal singularities), we know a priori the dimension of the systems of adjoint surfaces of F and we may say that the adjoint surfaces Φ_{m−3+s}, for s large enough, cut out on F a complete linear system of dimension

p_{a} + π_{s} − 1,

and that this in turn cuts out a complete series of degree 2π − 2 + sn on the plane section C of F. Therefore the Φ_{m−3+s} cut out on every plane the complete system of adjoint curves of order n − 3 + s of the plane section C.

Now one sees that the dimensions of the linear systems of the adjoint surfaces Φ_{m−3+s} to F, for s large enough, form an arithmetic progression of the third order whose ratio is

π + sn + \frac{s(s − 3)}{2}.

This progression, extended for the smaller values of s, defines in any case the virtual expression of the number of linearly independent adjoint surfaces of given order and in particular the numerical genus p_{a}.

(6)See also L. Campedelli, Sulla postulazione di una curva i-pla, Rendic. Circ. Mat. di Palermo, 55, 1931.

(7)To be modified in case of presence of multiple points of the curve.

In this sense one can say that the above arithmetic progression, which is by the way a corollary of the postulation formulas, replaces the actual knowledge of these formulas^{(8)}.

We resume the consideration of the systems of adjoint surfaces Φ_{m−3+s} of F.

We denote by h ≥ 0 the smallest number such that for s ≥ h these systems cut out on a plane all the adjoint curves of the same order; this means that, provided that h ≠ 0, the system of Φ_{n−3+h−1} (which is of order greater than or equal to n − 3) cuts out on a plane an incomplete system of adjoint curves, with deficiency δ_{h−1} > 0. For s < h−1 the systems of Φ_{m−3+s} (for s = 0, 1, …, h − 2) cut out on the same plane complete or incomplete systems of adjoint curves, that a priori we may assume to have deficiencies

d_{0} ≥ 0, d_{1} ≥ 0, …, δ_{h−2} ≥ 0.

Now we have seen that, corresponding to the deficiency δ_{h−1}, and in contrast to its virtual expression, the dimension of the system of adjoint surfaces of order n − 3 + h − 2 increases precisely by δ_{h−1}, and the same holds for the dimension of the system of curves cut out by these on F, which will be

p_{a} + π_{h−2} − 1 + δ_{h−1}.

Similarly if δ_{h−2} > 0, then the dimension of the system cut out by the Φ_{n−3+h−3} increases in the same way by δ_{h−2}, and is equal to

p_{a} + π_{h−3} + δ_{h−2} + δ_{h−1}.

Finally, the dimension of the canonical system, cut out by the Φ_{n−4}, will turn out to be

p_{g} − 1 = p_{a} − 1 + δ_{0} + δ_{1} + ⋯ + δ_{h−1}.

In other words: the irregularity p_{g} − p_{a} of the surface equals the sum of the deficiencies of the systems of curves cut out on a plane by the adjoint surfaces of F, of order

n − 3 + s for s = 1, 2, …

(8)Independently of the fact that the base group assigned to the Φ consists of the singularities of a surface F with respect to which the Φ must be adjoint, one can show that the numbers expressing the dimension of the system of surfaces Φ of large enough order required to pass with certain multiplicities through the points and the curves of an assigned base group, always give rise to this type of third order arithmetic progression. The knowledge of this progression suffices, in many cases, to replace the actual knowledge of the postulation formula. See G. Castelnuovo, Alcuni risultati sui sistemi lineari di curve appartenenti ad una superficie algebrica. Mem. Soc. It. delle Scienze (detta dei XL), s. III, t. 10, 1896. Reprinted in Memorie scelte, page 335.

We can make the previous statements slightly more precise, by proving that if the adjoint surfaces of order greater than or equal to n − 2 cut out on a plane the complete linear system of adjoint curves of the same order, then the same happens for all the linear systems of adjoint surfaces of larger order. (We shall explain later how the statement extends for s = 0, that is, to the case that the adjoint surfaces of order n − 3 cut out on a plane the complete system of adjoints of that order).

To prove the theorem we have stated, we can pass from the systems of surfaces Φ to the systems of curves they cut out on F, and then to the series cut out by these systems on a plane section C of F. And it suffices to prove that: “if the curves of the system |sC + C′| cut out on C the complete series for s ≥ 1, then also the series cut out on C by the curves of |(s + 1)C + C′| is certainly complete”.

We start by assuming that s > 1 and – for the sake of simplicity – we set s = 2. The assumption is thus that the curves of |2C + C′| cut out on C the complete series g of degree 2π − 2 + 2n determined by the adjoint curves φ_{n−1}. And obviously it suffices to prove that the minimal sum of this series g and of that cut out by the lines is necessarily complete.

In conclusion, the question is to prove the following result (following Castelnuovo):

Lemma on the completeness of the sum of two series belonging to a curve. On a curve of genus π, to a complete and nonspecial series g_{m} of a certain degree m, we add a base point free series g_{n}, at least ∞^{1}, which is contained in g_{m}, and with respect to which the residual series is nonspecial; then the minimal sum of g_{m} and g_{n} is necessarily complete.

In fact let us add to g_{m}, which is of dimension m−π, two groups G and G′ in g_{n}, that we fix throughout: thus we shall have two linear series of degree m + n and dimension m − π, that have in common the series consisting of 2n fixed points given by the group G + G′ and the groups of |g_{m} − g_{n}|, which is a series of degree m − n, assumed to be nonspecial, and thus of dimension

m − n − π.

So our two series, that have the groups G and G′ as fixed, will be contained in a series of dimension at least

m + n − π.

which therefore turns out to be complete. Q.E.D.

Remarks on the regularity conditions for a surface 4.6 112

This lemma can be applied immediately to the case h ≥ 2. But, with a slight modification, the argument can be extended to h = 1 (but not to h = 0). Indeed one has to prove that if one adds to the series g_{2π−2+n} cut out on C by the adjoint curves φ_{n−2} the series cut out by the lines, then one always obtains, as the minimal sum, the complete series cut out by the φ_{n−1}.

In fact let us add to g_{2π−2+n} two fixed groups G and G′ cut out by two lines a and a′ of the plane; we shall have two linear series of dimension π − 2 + n, that have in common the canonical g_{2π−2}^{π−1} (not of dimension π − 2, but of dimension π − 1): the minimal sum of the two series with fixed groups G and G′ is now a series of degree 2π − 2 + 2n and of dimension π − 2 + n − 1 instead of π − 2 + n. This series, which is incomplete, is cut out on C by the system of all φ_{n−1} that pass through the common point of the lines a and a′. Now the minimal linear system that contains two systems of φ_{n−1} with a different base point is the complete system of all the φ_{n−1}. So it follows that the minimal sum of the series g_{2π−2+n} and of the series cut out by the lines of the plane is necessarily the complete series cut out by the φ_{n−1}.

4.6 Remarks on regularity conditions for a surface

Let F be a surface of order n of ordinary space, that for the sake of simplicity we wish to assume with normal singularities. We have seen that, if the Φ_{n−2} of order n − 2 adjoint to it cut out on a plane the complete system of the adjoint curves of the section of F, then the same happens for the adjoint surfaces Φ_{n−2+r} of order > n − 2; thus in this case the irregularity p_{g} − p_{a} of the surface is equal to the deficiency of the (canonical) series cut out on the plane sections C of F by the Φ_{n−3}. In particular, if the Φ_{n−3} cut out the complete canonical series on the above C, then the irregularity is p_{g} − p_{a} = 0, that is, the surface F is regular.

But the regularity criterion, thus established by Enriques, can be reduced to a simpler one.

The surface F of order n of ordinary space is regular (p_{g} = p_{a}) if the surfaces Φ_{n−3} of order n − 3 adjoint to it cut out the complete canonical series on a plane section.

We shall briefly sketch the proof of this theorem given by Castelnuovo^{(9)}. To start with it is useful to prove the following result:

Lemma 2 If a group of m < n points of our surface F, chosen among the n points cut out by a line a, belongs to a special series g_{m}^{h} of dimension h ≥ 1, on a plane section C through a, then the same happens for every other section of F with any plane through a.

(9)Alcune proprietà fondamentali dei sistemi lineari di curve tracciati sopra una superficie algebrica, Ann. di Mat., II, 25 (1897), reprinted in Memorie scelte, XXIII.

This is an immediate consequence of the assumption that the Φ_{n−3} cut out the complete canonical series on the plane sections C of F. In fact, by the theorem of Riemann–Roch, a group of the aforementioned g_{m}^{h} on C imposes precisely m−h conditions on the canonical groups of C, that is, also on the C′ that have to contain it: but the number of such conditions is independent of the choice of a particular C of the pencil cut out on F by the planes through a.

Using the above lemma we are able to see that if the Φ_{n−3} cut out the complete canonical series on the plane sections C of F, then the characteristic series of the complete system |C| is complete.

We know that the complete system |C| can be constructed on F via the linear system of all the adjoints Φ_{n−2} that pass through the section C′ of a Φ_{n−3} (the residue theorem). So it suffices to show that these Φ_{n−2} through a C′ cut out on a plane the complete series of the curves φ_{n−2} of order n − 2 adjoint to the section of F.

To this purpose one should try to construct a surface Φ_{n−2} adjoint to F, as the locus of a curve φ_{n−2} of the same order n − 2 adjoint to the section C of a plane a varying in a pencil, having a generic line a of space, not meeting C′, as its axis. One will impose the following conditions on φ_{n−2}:

(1) that is passes through the 2π − 2 points of a canonical group, section of C′;

(2) that is passes through s − 1 points A_{1}, A_{2}, …, A_{s−1} chosen among the n common points of C and of the line a, where s denotes the dimension of the characteristic series of C when completed, that is, the dimension of the complete series to which the g_{n} cut out on C by the lines of the plane a belongs;

(3) finally, touches a plane γ, passing through a point A_{1} of this group of s − 1 points and not containing a.

Conditions 1–2 determine, on a generic plane through a, a pencil of curves φ_{n−2}, containing the curve consisting of the line a and of the φ_{n−3} passing through the canonical group of C section of C′. If one adds Condition 3, then a single curve φ_{n−2} in the plane is determined; and, as this plane varies, the φ_{n−2} thus constructed describes a surface Φ_{n−2+h}, adjoint to F, that a priori will be of order n − 2 + h > n − 2, passing h (> 0) times through the line a. For our purposes, we need to show that h = 0. Therefore we shall assume that h > 0 and we shall show that this assumption leads to a contradiction.

In order to do that, let us remark that, since h > 0, one has h tangent planes to Φ_{n−2+h} at a point A_{s} of a. Any one of these planes \overline{a} cuts this surface along a curve consisting of the line a counted h times and a φ_{n−2} passing through the s points A_{1}, A_{2}, …, A_{s−1}, A_{s}. Now one can take as A_{s} one of the n − (s − 1) intersections with a of the surface F, or of the curve \overline{C} section of \overline{a}, away from A_{1}, A_{2}, …, A_{s−1}: since the \overline{φ}_{n−3} section with \overline{a} does not pass through A_{1} (as a does not intersect C′), the group of points A_{1}, A_{2}, …, A_{s} must belong, not to one curve, but to a pencil of curves φ_{n−2} adjoint to the section \overline{C} of F with \overline{a}, and thus this group has to belong to a special g_{s}^{1}, of dimension (at least) equal to 1. But this cannot happen on a particular plane \overline{a} containing a, unless it happens for all the planes a. Therefore one deduces that h = 0, and so we succeed in constructing a surface Φ_{n−2} adjoint to F satisfying the conditions of:

(1) passing through a curve C′ section of F with an adjoint Φ_{n−3};

(2) passing through s − 1 among the n points of intersection of F with a line a;

(3) touching a plane g through one of these points.

By varying the points appearing in Condition 2, we can obtain ∞^{s+1} adjoints Φ_{n−2} through C′, which cut out ∞^{s} curves φ_{n−2} on a generic plane, that is, all the adjoint curves φ_{n−2} passing through the canonical group G_{2π−2} section with C′. This means that the Φ_{n−2} through C′ cut out on F a linear system |C| of dimension s+1, whose characteristic series is complete, Q.E.D.

To establish Castelnuovo’s theorem it suffices now to remark that: “on a curve C of genus π (> 0), the minimal sum of the canonical series g_{2π−2}^{π−1} and of another, base point free, complete series g_{n}^{s}, not composed with an involution, is always complete”.

This follows by remarking that a group G_{n} of the latter series imposes s independent conditions on a group of the same series g_{n}^{s} that must contain it and n − s conditions on a group of the canonical series, therefore^{(10)} at least

s + n − s − 1 = n − 1

conditions on a group of the minimal sum series g_{2π−2+n} = g_{2π−2}^{π−1} + g_{n}^{s}; so the dimension of this series exceeds the dimension of the canonical series by (at least) n − 1, hence it is equal to

(n − 1) + π − 1 = n + π − 2.

This means that the minimal series g_{2π−2}^{π−1} + g_{n}^{s} has the same dimension as the complete series g_{2π−2+n}^{π−2+n}, thus coincides with it.

(10)See Castelnuovo, Sui multipli di una serie lineare di gruppi di punti appartenente ad una curva algebrica, Rendic. Circ. Mat. di Palermo, 1893, reprinted in Memorie scelte,

Examples of irregular surfaces 4.7 115

Note Castelnuovo’s theorem that expresses the regularity conditions for a surface, with respect to the system |C| of its plane sections, extends easily also to the irreducible linear systems |C| that are not simple, as the author himself pointed out. Thus we have:

The necessary and sufficient condition in order that a surface be regular (p_{a} = p_{g}) is that the adjoint system |C′| of an irreducible linear system |C|, of genus π > 0, at least ∞^{2}, cuts out the complete canonical series on the generic curve C, that is, its dimension is p_{g} + π − 1.

Every irreducible complete linear system belonging to a regular surface has complete characteristic series.

Later we shall again obtain this theorem as a particular case of a property of the deficiency of the characteristic series of the complete linear systems belonging to an irregular surface.

4.7 Examples of irregular surfaces

We now illustrate the previous results by some examples. For the general surface of order n, without singularities, the adjoint surfaces Φ_{n−4} are all the surfaces of order n − 4; the actual number of these, linearly independent, is equal to the virtual number^{(11)}

p_{g} = p_{a} = (\begin{matrix}n − 1 \\ 3 \end{matrix}) = (n − 1)(n − 2)(n − 3)/6

Thus, the general surfaces of any order are regular; in particular the planes, the quadrics and the cubic surfaces that are not cones, therefore also all the rational surfaces have superficial genus

p_{g} = p_{a} = 0.

Now if one imposes that a surface of order n has a double curve of relatively small order, one also obtains regular surfaces; if instead one requires that the surface has a double curve of quite large order, then a priori one finds a number of conditions that exceeds the number of parameters on which the surface of given order depends; therefore it is difficult to obtain examples of irregular surfaces this way.

But let us consider a cone F of order n > 2, and of genus p > 0, and let O be its vertex (a n-tuple point); here there are certainly no adjoint surfaces of order n − 4, that would have to pass through the point O with multiplicity n − 2 > n − 4, so that it follows that

p_{g} = 0

(11)If one wants to apply the general postulation formula, one has to regard the double curve as having order n = 0 as genus ρ = 1.

(as we already know). For the same reason there are no adjoint surfaces Φ_{n−3} of F of order n − 3, that should cut out on F the p_{a} + p linearly independent adjoint curves of the plane sections, so that

p_{a} + p ≤ 0;

that is,

p_{a} ≤ −p.

But on the other hand there exist adjoint cones Φ_{n−2} (with O as an (n − 2)- tuple point) that cut out on a plane the complete system of adjoint curves of order n − 2 to the section of F. Thus by our results we can say that the numerical genus is

p_{a} = −p.

In conclusion: the ruled surfaces of genus p have numerical genus −p and irregularity

p_{g} − p_{a} = p.

This theorem is proved by transforming the ruled surface into a cone; but even without performing this transformation one can repeat the previous argument, based on the following facts:

1. there do not exist any adjoint curves of the plane sections C; and

2. the curves of |C + C′| (sections with the adjoints Φ_{n−2}) cut out the complete series on a C.

The example of the ruled surfaces has been known since the first studies of A. Cayley^{(12)} (1871) and of M. Noether, who proved that the virtual expression for the genus becomes negative in this case. Later on the first example of irregular surface, for which p_{g} > 0, was pointed out by G. Castelnuovo^{(13)}.

Let us impose on a surface F_{6} of order 6 the condition of having 8 triple points A_{1}, A_{2}, …, A_{8} with arbitrarily fixed position in the space. The F_{6} depend on 83 parameters, imposing a triple point yields ten linear conditions, so the F_{6} that pass triply through A_{1}, A_{2}, …, A_{8} are ∞^{3}, and split as triples of quadrics (belonging to a pencil) that pass through those 8 points. But instead of taking A_{1}, A_{2}, …, A_{8} in arbitrary position, let us assume that they are the base group of a net of quadrics Φ_{2}. In this case, by taking linear combinations of the triples of Φ_{2}, one obtains a linear system of irreducible F_{6}, of dimension 9^{(14)}.

(12)On the deficiency of certain surfaces…, Math. Ann., Band III.

(13)See G. Castelnuovo, Osservazioni intorno alla geometria sopra una superficie, Rendic.

Ist. Lombardo, 1891. Note I. Memorie scelte, n. XVII.

(14)One can show that the system of F_{6} with the 8 triple points cannot have dimension r > 9, by computing the dimension of the system that the F_{6} cut out on a quadric of the net.

Now the F_{6} thus constructed turn out to be of genus p_{g} = 3, since they have the quadrics Φ_{2} of the net that we are considering as adjoints of order 6 − 4 = 2. But the eight base points A_{1}, A_{2}, …, A_{8} impose on the Φ_{2} seven linearly independent conditions instead of eight, which is the virtual number of these conditions. So the virtual genus of F_{6} is

p_{a} = p_{g} − 1 = 2.

It is interesting to remark that we have thus constructed the first instance of a series of analogous examples. In general it is possible to construct irreducible surfaces of order 2n (n ≥ 3) that have 8 n-tuple points at the base points of a net of quadrics: these F_{2n} turn out to be irregular of genus

p_{g} = \frac{n(n − 1)}{2}

and

p_{a} = n − 1.

It is important to remark now that the quadrics of the net cut out on one of the surfaces F_{2n} a linear system composed with quartic elliptic curves, and that these quartics form on F_{2n} an irrational pencil of genus

p_{g} − p_{a} = \frac{(n − 1)(n − 2)}{2};

in fact the genus of the pencil equals the genus of the osculating cone to F_{2n} at one of the points A, which is a base point for the pencil of these quartics.

Let us add that the above series of examples, pointed out by Castelnuovo, can be generalized as L. Campedelli (1936)^{(15)} has shown. The construction of an irreducible F_{2n} with eight n-tuple points actually depends on the construction of a linear pencil of elliptic curves of some order m dividing n (n = ms), having eight s-tuple base points on a quadric: such a group of base points always belongs to an elliptic quartic, and it gives on it a group resulting from the division by n of the series cut out on the curve by all the surfaces of order 2n. Such a pencil of elliptic curves can be projected into the plane from one of its base points to a Halphen pencil of curves C_{3m} with 9m-tuple base points.

Starting with the pencil that we have described on a quadric, one obtains irregular surfaces F_{2n}, with eight n-tuple points, that have always an irrational pencil of genus p_{g} − p_{a}.

(15)L. Campedelli, in Atti Acc. Torino, 1936.

Riemann–Roch Theorem: systems larger than the canonical 4.8 118

Remark Using the theorem of Chapter VI to the effect that the characteristic series of a complete linear system on a regular surface is complete, one sees in general that the existence of an irrational pencil (of genus π > 0) implies that the surface is irregular.

More precisely we shall find that the surface has irregularity p_{g} − p_{a} ≥ π. But more of this later.

4.8 The theorem of Riemann–Roch for surfaces: systems larger than the canonical

We have seen that a linear system of degree n and genus π, which is the adjoint of another irreducible system on a surface of numerical genus p_{a}, has dimension

r ≥ p_{a} + n − π + 1,

and this fundamental inequality, even under the restrictive assumptions that we have made here, is an extension to surfaces of the well-known theorem of Riemann–Roch for curves. It is convenient now to understand the meaning of the restriction that |C| is an irreducible system. For the sake of simplicity we assume that |C| is base point free on the surface F, and that the same happens for the system |L| to which it is adjoint. Denoting by |K| the canonical system we have

|C| = |L + K|.

So the condition imposed on |C| of being the adjoint of another system can be translated in the condition of containing the canonical system, with a residual system |L| of degree greater than zero, or, as we shall say, of being more ample than the canonical system.

A system |C| more ample than the canonical system |K| can always be regarded as the adjoint of the system

|L| = |C − K|,

but it is not possible to say a priori that |L| is irreducible. On the contrary there are cases in which, even though one can increase the dimension of |C| by multiplying it by an arbitrarily large number, the system |L| always turns out to be reducible, containing for instance some fixed components, not connected with the variable components. Indeed this happens if |C| is the system of plane sections of a surface F that has a multiple point that affects the genus, for instance an isolated multiple point, of multiplicity n ≥ 3, that corresponds to a fundamental curve θ of |C|. Nonetheless it is easy to see that in this case the above inequality (dimension dim |C| ≥ p_{a} + n − π + 1) a fortiori holds true. In fact, denoting by \overline{n} and \overline{π} the characters of the system \overline{C} = |C − q|, we shall have that |\overline{C}| is the adjoint of the irreducible system |\overline{L}| = |L − q|:

|\overline{C}| = |\overline{L}′|,

and so the dimension of |\overline{C}| is

\overline{r} ≥ p_{a} + \overline{n} − \overline{π} + 1;

now the degree of θ is −n and its genus r = (n − 1)(n − 2)/2 ≥ 1, thus we have

n = \overline{n} − nπ = \overline{π} + r + n − 1

and accordingly the dimension of |C|, r > \overline{r}, will be

r > p_{a} + n − π + 1.

But, in order to proceed rigorously with the greatest generality, we shall prove that any linear system |C| more ample than the canonical system has dimension r ≥ p_{a} + n − π + 1, making use of another argument that holds equally well, both for reducible and irreducible systems |C|.

Given an irreducible or reducible curve C on the surface F, we may always assume that the adjoint surfaces Φ_{n−4+s} (of large enough order n − 4 + s) of F passing through C, do not pass as a consequence through another curve or through points away from C or the double curve^{(16)}. In this setting, the curve C is contained in the system |L′_{s}| cut out on F by the Φ_{n−4+s}, and gives with respect to it an irreducible (base point free) residual system

|D| = |L′_{s} − C|.

Now, denoting by r′_{s} the dimension, by π′_{s} the genus and n′_{s} the degree of |L′_{s}|, we have found that

r′_{s} ≥ p_{a} + n′_{s} − π′_{s} + 1

(indeed the equality actually holds in this case); subtracting a curve D, one finds that the residual system |C| = |L′_{s} − D| has to have dimension

r ≥ p_{a} + n − π + 1,

provided that one knows that the series cut out by |L′_{s}| on D is nonspecial; and this is the case when |C| is more ample than the canonical system |K|, the curves of which cut out on D residual groups of the characteristic series of D.

(16)This follows by remarking that in the postulation formulas for a reducible curve (or for a curve and a group of points) each component actually yields a certain number of conditions.

Let us carry out the computation introducing the characters, genus ρ and the degree n of D and the number t of the intersections of C and D. The series cut out on D by |L′_{s}| has degree n + t, and (if it is complete) it has dimension n + t − r, so that if one subtracts D then the dimension r′_{s} of |L′_{s}| decreases at most by n + t − r + 1; then we have

r ≥ r′_{s} − (n + t − r + 1).

On the other hand

n′_{s} = n + n + 2t π′_{s} = π + r + t − 1

hence

n′_{s} − π′_{s} = n − π + n − r + t + 1;

as a consequence we have

r ≥ p_{a} + n − π + 1.

Thus we have proved that: for the irreducible or reducible linear systems of degree n and genus π that are more ample than the canonical system, the value of the dimension is always

r ≥ p_{a} + n − π + 1.

This inequality is the so-called theorem of Riemann–Roch for surfaces, and it will be extended later-as we shall see-to arbitrary linear systems.

Meanwhile, referring to the systems more ample than the canonical system, we may distinguish:

(1) the regular systems, for which the equality holds

r = p_{a} + n − π + 1

(2) and the superabundant systems for which instead

r > p_{a} + n − π + 1.

We have already seen that some regular systems do exist, since the adjoints of the multiples of quite large degree of an irreducible system |L| are such.

The result that we have obtained yields at once some remarkable consequences.

Elimination of the exceptional curves 4.9 121

Let |C| be an irreducible linear system of genus π and degree n, more ample than the canonical system and regular on the surface F, and let us add to it an irreducible curve D of genus r and degree n, which intersects the curves C at a certain number m (> 0) of points.

The degree and the genus of |D + C| are respectively given by

n + n + 2m and π + r + m − 1,

hence their difference is

(n − π) + (n − r) + m + 1 > n − π

provided that it is

m ≥ r − n.

One deduces that under this assumption the system |C + D| has dimension greater than

p_{a} + n − π + 1

which is the dimension of |C|, and therefore it is certainly irreducible.

Moreover the system |C + D| will be regular and it will have to cut out a (nonspecial) complete series on D, otherwise subtracting D one would obtain a superabundant system |C|.

We notice that this extension of |C| would be impossible for m = 0, that is, if the curve D were fundamental for |C|; in this case therefore one has n − r < 0, as it follows a priori by remarking that the fundamental curve D of genus r ≥ 0 must have negative degree: n < 0, where −n is the number of intersections of D with its residual curves (of |C − D|).

If one interchanges the roles of |C| and |D| in the previous remark, one is led to state that: given a linear system |C| (virtually free from base point), however reducible on the surface F, it is always possible to add to it an irreducible regular linear system |D| so that the sum is an irreducible linear system |C + D| which is again regular.

It suffices to take as the system |D| the multiple |L_{s}| = |sL| of a regular system |L| with no fundamental curves, whose number of intersections with the C (or with their components) thus grows with s. In the case in which the curves C are reducible, e.g., C = C_{1}+C_{2}+⋯, one firstly adds to C_{1} a multiple of |L|, then adds to C_{2} the irreducible system (as ample as one wishes) thus obtained and so on, until one obtains an irreducible system |sL + C|.

4.9 Elimination of the exceptional curves

We continue the analysis of the previous section, about the addition of a curve to a regular linear system |C| on a surface F. And we take as the curve to be added to |C| an exceptional curve ω of genus r = 0 and degree n > −1. If this is not fundamental for |C|, and so it meets the C at m > 0 points, adding it to |C| one always obtains a linear system of greater dimension, irreducible and regular as |C|. And the curves of this sum system have m + n intersections with ω. Therefore ω can be successively added to |C|, thus obtaining an infinite sequence of systems of increasing dimension, provided that it is n > 0, that is, in the case in which ω is an exceptional curve of the second kind. Conversely, if ω is of the first kind (n = −1), then the sequence of systems thus obtained stops at the system |C + mω| for which ω becomes fundamental.

Now let us consider, for every linear system |C| of degree n and genus π, belonging to the surface F, the difference 2π−2−n (number of intersections of its curves with the virtual canonical curves) that yields an additive character a(C) of the systems on F. This difference decreases at least by one every time that one adds ω to |C| or to one of the systems thus obtained, hence, if the family of these is infinite, then it will certainly end up to be negative, so that one will have systems of genus π and degree n > 2π − 2 on F.

Now, we wish to assume that the surface F does not contain any system for which the difference −a(C) = n − (2π − 2) > 0. Under this assumption it does not contain any exceptional curve of the second kind and it contains at most a finite number s of exceptional curves of the first kind.

For the sake of simplicity we shall assume that these are irreducible exceptional curves, but we shall explain later how the argument works in the other case (which corresponds to exceptional curves that are the images of infinitely near points).

Let us assume that the (irreducible, regular) system |C| is, not only virtually, but also actually free from base point on F. If, as we have said earlier, the C intersect the exceptional curve ω at m points, then ω is a fundamental curve (of degree-1, thus decreasing by one the degree of the residual curves) for the system |C +mω|. And transforming the surface F into another surface F′ via this system, ω is transformed into a simple point O of F′.

Now each of the other s−1 exceptional curves is not connected with ω, and therefore it still gives an exceptional curve of the first kind (or a simple point of the surface) on F′, otherwise one would have on F′ exceptional curves of degree ≥ 0 (that is, of the second kind), hence one would be able to construct on the surface F′, and also on F, systems |C| for which the character

−a(C) > 0.

So by transforming F into F′ one succeeds in decreasing the number of exceptional curves belonging to the surface, so that after s transformations we shall obtain a surface F completely free of exceptional curves.

Completeness of the series cut out by a small linear system 4.10 123

Actually we have simplified our discussion by assuming that the exceptional curves of F are all irreducible. But if there on F is an exceptional curve of the first kind consisting of h parts (given in a certain order ω, ω′, …, ω^{(h−1)}), the last one of these components is always an exceptional curve of degree n = −1, and when one reduces this curve to a point with the transformation that takes F into F′, the preceding component becomes an exceptional curve (of the first kind) of degree n = −1. So the exceptional curve consisting of h parts (which form h exceptional curves of F, one irreducible and h − 1 reducible) becomes an exceptional curve on F′ consisting of h − 1 parts. And eventually it disappears after h transformations, since it is taken (on F) to a group of h infinitely near simple points.

Therefore we conclude: In general a surface F can be transformed into another one free of exceptional curves. This transformation fails to exist only for the surfaces (of genus p_{g} = 0, bigenus P = 0, and so on) on which there exist linear systems |C| for which the character −a(C) = n − (2π − 2) > 0, that is, as we shall see, for the surfaces that belong to the family of ruled surfaces.

4.10 Completeness of the series cut out by a small linear system on the curves of a large system

The extension of the theorem of Riemann–Roch to the case of arbitrary linear systems (even to the “special” ones, that are less ample than the canonical system) will be given in the simplest way by exploiting the completeness of the series that a “small” linear system cuts out on the generic curve of a “normally large” system. “Small” and “large” are relative terms, that are defined as follows: one regards as “large” with respect to a given linear system |C| (virtually free from base point on the surface F) a linear system |D| that contains |C| in such a way that the difference |D − C| equals the multiple |sL|, with s an arbitrarily large integer, of an irreducible base point free linear system |L|, that is, it is a system “more ample” than |sL|. And one says that |D| is normally large, with respect to |C|, if the previous condition is satisfied with a system |L| free of fundamental curves, so that picking s large enough one can achieve that the system |B| = |D − C| is irreducible^{(17)}.

We wish to prove the Main Lemma.

A complete, reducible or irreducible, linear system |C| cuts out a complete series on the irreducible curve of a linear system |D|, which is normally large with respect to it.

(17)This is no longer true if |L| and so also |L_{1}| = |D| has a fundamental curve θ that is not fundamental for for |C|, since in that case θ is a fixed part for |D−C|. So also the main lemma does not hold any more for |C| with respect to this (not normally) large system |D|: for instance the canonical system |K| does not cut out any more on D, in general, the complete series (cut out by |K + θ|).

To prove this lemma, let us take on a curve D an arbitrary group G of the complete series CD, cut out by |C|, and associate with it a group G′, whose existence will be shown right away, satisfying the two following properties:

(1) G+G’ is a group of the (usually incomplete) characteristic series cut out on D by the other curves of |D|;

(2) G′ is a group of the series cut out on D by an irreducible curve B of |D − C|.

Then G + G′ is the base group of a pencil of curves D and there is an irreducible fundamental curve B with respect to it passing through G′; accordingly the aforesaid pencil contains a curve D consisting of B and of a C through G; therefore every group G of the complete series CD on D is cut out by a C.

It is only necessary to see that, when |D| is quite large with respect to |C|, then for every group G of the complete series CD actually there is a group G′ satisfying properties (1) and (2) that can be associated to it. In order to do this, we consider the two, a priori incomplete, linear series DD − G and BD, cut out on D by the D through G and by the B respectively; they are contained in the same complete series, and we should prove that they certainly have an effective linear series in common, which is described by the groups G’. Thus it suffices to prove that the deficiencies of these two series are smaller than their dimensions, because then, as one easily sees, the sum of the dimensions of the two series exceeds the dimension of the complete series.

Now the last condition is certainly satisfied because one sees that, for large s going to infinity, the dimension of the characteristic series of |D| on D, hence also the dimension of the series DD −G, becomes goes to infinity to the second order as \frac{s^{2}}{2} n (where n is the degree of |L|), and the same happens for the dimension of the series BD; on the other hand the deficiency of the characteristic series, and thus also of the series DD − G, may very well grow indefinitely with s, but we know a priori it cannot become infinity of order greater than one.

We shall limit ourselves to carry out this simple computation in the case in which

|D| = |L_{s}| |C| = |L|.

Denoting by n and π the degree and the genus of |C| and by n_{s} and π_{s} the degree and the genus of |L_{s}|, the dimension r_{s} of |L_{s}| is given by the theorem of Riemann–Roch:

r_{s} ≥ p_{a} + n_{s} − π_{s} + 1,

where

n_{s} = s^{2}n, π_{s} = sπ + \frac{s(s − 1)}{2}n − s + 1,

and so

r_{s} ≥ \frac{s^{2}}{2} n − sπ − \frac{s}{2}n + s − 1;

thus the actual dimension of the characteristic series |DD| = (L_{s}L_{s}) is equal to

r_{s} − 1 ≥ \frac{s^{2}}{2} n + ⋯

disregarding the terms of order s. And, since the group G = CL_{s} is made up of sn points, the dimension of the effective series DD − G will be

\frac{s^{2}}{2} n + ⋯,

disregarding the terms of order smaller than two.

On the other hand the dimension of the series cut out on D = L_{s} by the curves B of |L_{s−1}| = |L_{s} − C| is also

\frac{s^{2}}{2} n + ⋯,

It is now possible to compute the deficiency of these series, or at least an upper bound for it, depending on the deficiency of the characteristic series DD and thus on its index of speciality. If this series is complete (of dimension r_{s}−1 = p_{a}+n_{s}−π_{s}) then its index of speciality is p_{a}, if instead it has deficiency d, then its index of speciality is

i = d + p_{a}.

In any case this index i is smaller than the degree of the residual series, that is, than the number of intersections of D = L_{s} with a canonical curve K:

DK = s(2π − 2 − n),

and accordingly the deficiency of the aforementioned series DD, and thus also the deficiencies of DD − G and of BD, cannot become infinite with s of order greater than one. Q.E.D.

The same computation can be extended, with slight modifications, to the case in which one has again |D| = |L_{s}| = |sL| and |L| = |B + C|, where |L| is a system free of fundamental curves, that is, to the case in which one has

|D| = |L_{s} + E|,

and |C| coincides with |L| or it is partially contained in it:

|L| = |B + C|.

We arrive again at the conclusion that the complete system |C| cuts out a complete series on the curve D of a normally large system.

The theorem of Riemann–Roch for arbitrary linear systems 4.11 126

4.11 The theorem of Riemann–Roch for arbitrary linear systems

Let us take now, instead of the linear system |C| considered in the previous section, the canonical system |K|, thus we are assuming p_{g} > 0, and let us construct an irreducible linear system |D| that is normally large with respect to |K|. We may say that |K| cuts out the complete residual series of the characteristic series on a generic curve D of |D|.

This statement holds also for p_{g} = 0, meaning with it that “on a surface F of genus p_{g} = 0, the characteristic series of a normally large system |D| is a nonspecial series”. To prove this, take an auxiliary linear system |C| and consider, together with it, its adjoint system |C′|. If |D| is large enough with respect to |C| and to |C′|, then both these systems cut out complete series on a generic D: say g and g′. If the characteristic series DD is special and thus contained in the canonical series DD′, where |D′| = |D + C′ − C|, then the series g has to be contained in g′, and precisely a group G of g will be part of a group G′ = G + G_{1} of g′. We can see that, under this assumption, |C′| contains |C|, too, and so p_{g} > 0, contrary to the assumption.

In order to do this it suffices to remark that the curve C passing through G (and, in the reducible case, each of its components) has a number of intersections greater than CC′ with the (normally large) curve D, and so it is part of the C′ through G + G_{1}.

We have thus established that on a surface of genus p_{g} > 0 the characteristic series of a normally large linear system (with respect to the canonical system) has index of speciality p_{g}.

Let us now consider on the surface F a system |C|, however special, of index i (≥ 0), and a system |D| normally large with respect to it. We want to see that the index of speciality of the series cut out on a D by the system |C + D| is equal to the index of speciality of |C|, that is, to the number i of linearly independent canonical curves which contain a given C as a part.

Indeed, since the series cut out by |K| on D is complete, the linearly independent canonical curves K passing through a group G of the series CD on a D are as many as the index of speciality of the series cut out by |D + C| on D; each one of these K has a number CD > CK of intersections with C (or, in the reducible case, with any component of C) and therefore it contains C entirely.

Now we can deduce the dimension r of the system |C| of degree n, genus π and index of speciality i, that one obtains by subtracting a D from the system |D + C|; in order to do this, we have to subtract from the dimension of the latter system, which is regular, the dimension of the series cut out on D by |D + C|, of which we know the index of speciality, but it may be not complete. In this way we find a lower bound for r: precisely we find, as for the systems more ample than the canonical system: r ≥ p_{a} + n − π + 1 if i = 0, that is, for all the nonspecial systems |C|, and in general

r ≥ p_{a} + n − π + 1 − i

for the special systems of index i.

This inequality is the most general form of the theorem of Riemann–Roch for surfaces.

Since it has been proved for the irreducible or reducible base point free linear systems |C| on the surface F, one deduces that it holds true for any linear system, with arbitrarily assigned base points: in fact an i-tuple base point imposed on the C lowers their degree by i^{2}, their genus by i(i − 1)/2, and their dimension by at most

\frac{i(i + 1)}{2} = i^{2} − \frac{i(i − 1)}{2}.

The linear systems |C| for which the dimension r is equal to r = p_{a}+n−π+1−i are said to be regular; they are said to be superabundant if instead

r > p_{a} + n − π + 1 − i.

The theorem of Riemann–Roch for curves of genus p gives us the dimension of the complete special series g_{n}^{r} expressed in terms of the dimension of its residual series with respect to the canonical g_{2p−2}^{p−1}. The obvious complement of that theorem is the remark that the residual series of the g_{n}^{r} has degree 2p − 2 − n. It is also possible to complete the theorem of Riemann–Roch for special linear systems on a surface, by expressing characters as genus and degree of the residual system by means of those of the given system. Let in fact |C_{1}| and |C_{2}| be two linear systems, each residual to the other with respect to the canonical system |K|. Let n_{1} and π_{1} be the characters, genus and degree, of the former system; n_{2} and π_{2} the characters of the latter, and let us denote by m the number of intersections of |C_{1}| and |C_{2}|. Introducing the linear genus p^{(1)} of the surface, we may express the genus and the degree of |K| by means of the formulas

p^{(1)} = π_{1} + π_{2} + m − 1, p^{(1)} − 1 = n_{1} + n_{2} + 2m.

But one can compute m by taking into account that |K| cuts out on C_{1} a group of

2π_{1} − 2 − n_{1} = n_{1} + m

points, so that

m = 2π_{1} − 2 − 2n_{1}.

The deficiency of the characteristic series 4.12 128

Therefore we have

p^{(1)} = π_{1} + π_{2} + 2π_{1} − 2 − 2n_{1} − 1 = 3π_{1} + π_{2} − 2n_{1} − 3p^{(1)} − 1
= n_{1} + n_{2} + 4π_{1} − 4 − 4n_{1} = 4π_{1} − 3n_{1} + n_{2} − 4,

whence we deduce

π_{2} = p^{(1)} − 3π_{1} + 2n_{1} + 3
n_{2} = p^{(1)} + 3n_{1} − 4π_{1} + 3.

These formulas express the characters of the system |C_{1}| residual of the system |C_{2}| with respect to the canonical system by means of the characters of |C_{1}|.

We notice that the dimensions of the two systems |C_{1}| and |C_{2}|, each residual to the other with respect to the canonical system, are given by

r_{1} = p_{a} + n_{1} − π_{1} − r_{2} + ω_{1}
r_{2} = p_{a} + n_{2} − π_{2} − r_{1} + ω_{2},

where ω_{1} and ω_{2} denote their respective superabundances. So by the previous formulas one obtains

ω_{1} = ω_{2}

that is: two special systems one residual to the other with respect to the canonical system have the same superabundance.

4.12 The deficiency of the characteristic series

Consider on the surface F of numerical genus p_{a} and geometric genus p_{g} ≥ p_{a} an irreducible linear system of genus π, of degree n and dimension r, that for the sake of simplicity we assume to be nonspecial. On the generic curve C we have two linear series, one residual to the other with respect to the canonical series: that is, the characteristic series g_{n}^{r−1}, cut out by the other curves C, and the series of degree 2π − 2 − n and dimension p_{g} − 1 cut out by the canonical curves K. If these two series have deficiency d and d′ respectively, then they are contained in two complete series of dimension

r − 1 + dandp_{g} − 1 + d′.

Applying the theorem of Riemann–Roch on the curve, one has

r − 1 + d = n − π + p_{g} − 1 + d′ + 1

that is,

r = n − π + p_{g} + d′ − d + 1.

So if d′ = 0 we have

r = p_{g} − d + n − π + 1.

Now, if |C| is a regular linear system, normally large with respect to the canonical system, then its dimension is

r = p_{a} + n − π + 1

and on the other hand, we know that the canonical curves K cut out the complete series on C, so that one has d′ = 0.

Whence we deduce that

d = p_{g} − p_{a}.

In words: the characteristic series of a complete, normally large, irreducible linear system, which is regular^{(18)}, has deficiency

d = p_{g} − p_{a}.

With this in mind we wish to prove that the characteristic series of an arbitrary irreducible linear system |C| belonging to the surface F has always deficiency

d ≤ p_{g} − p_{a}.

In order to obtain this result we shall establish the following:

Lemma: If a linear system

|C| = |C_{1} + C_{2}|,

sum of two irreducible linear systems, cuts out the complete series on a curve of the system |C_{1}|, then the deficiency d_{1} of the characteristic series of |C_{1}| is smaller than or equal to the deficiency d_{2} of |C_{2}|:

d_{1} ≤ d_{2}.

Let us denote by n_{1}, π_{1}, n_{2}, π_{2}, the characters of |C_{1}| and |C_{2}|, by r_{1} and r_{2} their respective dimensions, and by s the number of intersections of a C_{1} with a C_{2}; finally let r be the dimension of |C|. The series cut out by |C| on a C_{1} is complete by assumption. Therefore the C passing through the group G of the s common points of a C_{1} and a C_{2} also cut out a complete series of dimension r_{1} − 1 + d_{1} on C_{1}.

On the other hand the dimension of the complete series cut out on C_{1} by C is

r − r_{2} − 1

(18)Later we shall see that “every normally large linear system is regular”.

hence the dimension of the series cut out on C_{1} by the C through G is

r − r_{2} − 1 − (s − e),

where s − e denotes the number of linearly independent conditions that the group G of s points imposes on the C that have to contain it. Thus we have

r_{1} − 1 + d_{1} = r − r_{2} − 1 − (s − e),

that is,

d_{1} = r − (r_{1} + r_{2}) − (s − e).

This equality holds also independently of the assumption that |C| cuts out the complete series on C_{1}, provided that one replaces d_{1} with the deficiency d′_{1} of the characteristic series of |C_{1}| with respect to the series cut out by the C through G (d′_{1} ≤ d_{1}); by applying the same argument to the system |C_{2}| one finds:

d_{2} ≥ d′_{2} = r − (r_{1} + r_{2}) − (s − e).

Comparing the expressions of d_{1} and d_{2} it follows in any case

d_{1} ≤ d_{2} Q.E.D.

Now, given an arbitrary irreducible, at least ∞^{1}, linear system |C| of degree n and genus π on the surface F, let us construct on F another irreducible linear system |D|, which is regular and large enough so that deficiency of its characteristic series is p_{g} − p_{a}.

We know (Section ??) that (since the number of intersections (CD) is greater than n − π) by adding |C| to |D| one obtains again a regular linear system |C +D| cutting out the complete series on C. Therefore the deficiency d of the characteristic series of |C| has to be

d ≤ p_{g} − p_{a}.

In words: the characteristic series of an irreducible system on a surface of genera p_{a} and p_{g} ≥ p_{a} has deficiency

d ≤ p_{g} − p_{a}.

So the irregularity p_{g} − p_{a} of a surface is also defined as the maximum value of the deficiency of the characteristic series for the irreducible linear systems that belong to it.

On a regular surface (p_{g} = p_{a}) every irreducible complete linear system has complete characteristic series. Or, from the projective point of view: a regular normal surface, in some space, has normal curves as its hyperplane sections. If the sections of a normal surface are not normal, then one deduces that

p_{a} < p_{g}.

We may elucidate what we have said by means of the example of the ruled surfaces.

We know that the ruled surfaces (with sections) of genus p ‘have geometric genus p_{g} = 0 and numerical genus p_{a} = −p. Therefore in general the normal ruled surfaces of order n belong to spaces of dimension r = n − 2p + 1, and actually C. Segre^{(19)}, already in 1890, has proved that a nonspecial ruled surface of order n and genus p that is not a cone, belongs precisely to a space S_{n−2p+1}; by contrast a special ruled surface, whose plane or hyperplane sections are special curves of index i, is normal in a space of dimension n − 2p+1+i (we always exclude the case of a cone in which the system of sections turns out to be superabundant); so the characteristic series of the system of hyperplane sections has deficiency p in any case.

Remark The theorem on the deficiency of the characteristic series of the complete linear systems has led G. Castelnuovo to recognize that: every surface containing an irrational pencil of genus p (> 0) is irregular with irregularity

p_{g} − p_{a} ≥ p.

This theorem is a consequence of a more general one proved by F. Enriques^{(20)}: a surface that contains an irreducible algebraic system {C} of inequivalent curves is irregular; more precisely if {C} contains ∞^{d} inequivalent curves then the irregularity of the surface is

p_{g} − p_{a} ≥ d.

Historical note 4.13 132

Enriques computed indeed the deficiency of the (canonical) series cut out on C by the adjoint system |C′|. On that occasion Castelnuovo – whom he had informed of the result – pointed out the deficiency of the characteristic system of the complete system to which a curve C belongs. The proof takes a simpler form in the treatment of F. Severi^{(21)}. Indeed it suffices to remark that the complete continuous system {C} consists of ∞^{d} linear systems |C| having, in general, some dimension r: so the characteristic series of |C| on a generic curve C has dimension r − 1 but it is contained in the characteristic series of the continuous system of dimension r − 1 + d, that is, in the series cut out on C by the curves of {C} infinitely near to C.

(19)C. Segre, Courbes et surfaces reglées, Math. Ann., 1892.

(20)Una proprietà delle serie continue di curve appartenenti ad una superficie algebrica regolare, Rendic. Circ. Mat. di Palermo, t. XIII, 1899.

(21)Osservazioni sui sistemi continui di curve appartenenti ad una superficie algebrica regolare. Atti Acc. Torino, 1904.

We explicitly notice that the theorem about the irrational pencil is a consequence of the theorem thus stated: it suffices to remark that, by subtracting the groups of p curves of a pencil of genus p from a large enough system, one obtains a continuous system consisting of ∞^{p} inequivalent linear systems.

4.13 Historical note

The numerical genus of a surface has been considered since the first studies of A. Cayley, H.G. Zeuthen and M. Noether. After that Noether had defined the (geometric) genus p of a surface F_{n} of order n as the number of linearly independent adjoint surfaces Φ_{n−4}, it was natural to look for an expression of p in terms of the characters of the double curve of F_{n}; and Cayley’s postulation formulas were useful to this purpose. Cayley himself^{(22)} remarked that, in the case of the ruled surfaces of genus p, the aforementioned computation leads to the negative value −p, whereas the geometrically defined genus cannot, by definition, become negative and, in this case, it is precisely zero.

Meanwhile Zeuthen was led, by the study of correspondences between algebraic surfaces, to prove (although with some restriction on the singularities) that “the arithmetic (virtual) expression of the genus, according to Cayley, is invariant under birational transformations of the surface, and this invariance is independent of its (supposed) geometric meaning and of the proof of the invariance of the canonical system provided by Noether”.

Noether’s interpretation of these results^{(23)} was that the genus of a surface F_{n} of order n can be defined in two ways: geometrically as the actual number of linearly independent adjoint surfaces Φ_{n−4}, and numerically by means of the postulation formulas; so one introduces the geometric genus and the numerical genus of the surface, both a priori invariant for birational transformations; they are supposed to coincide in general, except that the numerical genus can become negative (for the ruled surfaces) and therefore it seems to include the other, as a more meaningful character.

But Noether’s conjecture, that one should have p_{a} = p_{g} for nonnegative p_{a}, was disproved by the new examples of irregular surfaces that were added to the example of the ruled surfaces. As we have said before the first such example was provided by G. Castelnuovo in a Note of the Ist. Lombardo in 1891. Other examples came from the investigations of E. Picard and G. Humbert on the surfaces having integrals of total differentials of the first kind, and in particular on the hyperelliptic surfaces, for which p_{g} = 1 and p_{a} = −1.

(22)A. Cayley, On the deficiency of certain surfaces. Math. Ann., Band III (1871).

(23)Math. Ann., Band VIII (1875).

Subsequent studies have brought to light large classes of irregular surfaces, as we showed in the Remark of Section ??.

In the meantime the investigations begun in 1893 and continued in 1896 (in the Introduzione quoted earlier) have led F. Enriques to recognize the geometric meaning of the difference p_{g} − p_{a} with respect to the deficiency of the (canonical) series that the system |C′| adjoint to an irreducible linear system |C| cuts out on the generic curve of |C|; whence it follows the completely general proof of the invariance of p_{a}, while the invariance of p_{g} had been already established earlier. According to Enriques the irregularity p_{g} −p_{a} is the maximum value of the aforesaid deficiency, but a priori it is still possible that there exist systems |C| for which the aforementioned deficiency is < p_{g} − p_{a}. At this point E. Picard completed the theory, by proving (by transcendental methods) that the adjoint system |C′| of a simple irreducible system |C| (at least ∞^{3}) is always regular, and therefore p_{g} − p_{a} can be simply defined as the deficiency of the series cut out by the adjoint surfaces Φ_{n−3} on the plane section of a surface F_{n}^{(24)}. This theorem includes for p_{g} = p_{a} Castelnuovo’s theorem stated in Section ??; more about it will be said in next section.

The extension to surfaces of the theorem of Riemann–Roch was sketched for the first time by M. Noether in 1886^{(25)}. The author considers a complete irreducible linear system of dimension r, of degree n and genus π, having a certain index of speciality i ≥ 0, on a surface of (geometric) genus p. The characteristic series of |C|, cut out by |C| on a generic C, is a g_{n}^{r−1} residual of the series g^{p−i}2π − 2 − n cut out by the canonical system |K|; and Noether, without stating it explicitly, assumes that the two series, cut out by complete systems, are complete; under this assumption the theorem of Riemann–Roch applied to the curve C yields the equality

r = p + n − π + 1.

The matter is reconsidered by Enriques since the 1893 in the paper Ricerche, where he succeeds in proving that for the regular surfaces of genus p > 1 the characteristic series of a complete linear system is complete, provided that the same happens for the characteristic series of the canonical system; under this assumption (that later will be seen to be superfluous) the following inequality is thus proved

r ≥ p + n − π + 1,

(24)E. Picard, Sur quelques questions se rattachant á la connexion linéaire dans la théorie des fonctions algébriques de deux variables indépendantes. v. Crelle, Band 129, 1905.

See Picard and Simart, Traité t. II, page 437. An algebro-geometric proof (under milder assumptions) of the theorem, has been given by Severi, in the Note Sulla regolarità del sistema aggiunto (Lincei, 8 nov. 1908); however this proof uses the theorem of completeness of the characteristic series of a continuous system.

(25)Extension du théorème de Riemann–Roch aux surfaces algébriques, Comptes Rendus de l’Académie des Sciences, Paris, t. 103.

and the author points out that this cannot be changed into an equality, that is, that superabundant systems do exist, since it suffices for the existence of such systems the presence of fundamental curves of genus r > 0.

In the Introduzione of 1896, with the new definition of the numerical genus, Enriques gave the extension of the theorem of Riemann–Roch for all surfaces, regular and irregular, to the adjoint systems, which means– substantially –to the systems more ample than the canonical system.

In order to extend the theorem to arbitrary linear systems, it was necessary to find the maximum value of the deficiency of the characteristic series proving that it is given by p_{g} − p_{a}. This was precisely the result obtained by G. Castelnuovo in two memoirs^{(26)} in the years 1896 and 1897.

The method used by Castelnuovo is based on a suitable extension of the idea that leads to find the simplest conditions for the regularity of a surface, as we have explained in Section ??. These are methods that still seem to be usefully applicable, but that turn out to be quite cumbersome.

A remarkable simplification in this type of questions was given by F. Severi, in two Notes of 1903 and 1905^{(27)}; in particular in the latter Note the author is able to reverse the order of the deductions, proving first of all the theorem of Riemann–Roch and then deriving from it the maximum of the deficiency of the characteristic series of the irreducible systems.

The starting point of these developments is the well known Noether’s theorem of Af + Bf extended to surfaces in the following form:

Every surface f_{m}(xyz) = 0, of order m, not containing the plane z = 0, passing through the points of the group G common to the plane z = 0 and to the curve D_{s} of order ns, complete intersection of the surface f_{n} of order n, without proper multiple points, with the surface f_{s} of order s, has an equation of the form

f_{m} = Af_{n} + Bfs + zf_{m−1} = 0,

where f_{m−1} is a polynomial of degree m − 1 and A and B are polynomials in x, y of degree m − n and m − s respectively.

Assuming directly that the surface f_{n} have normal singularities, from this theorem it follows that the surfaces f_{m} of any order, passing through the double points of a D_{s}, cut out a complete series on D_{s}: actually one proves that if the proposition is true for m then it is true for m − 1.

(26)Alcuni risultati sui sistemi lineari di curve appartenenti ad una superficie algebrica.

Mem. della Società Italiana delle Scienze, detta dei XL (1896, Vol. X). Alcune proprietà fondamentali dei sistemi lineari tracciati sopra una superficie algebrica, Ann. di Mat., s. II, Vol. XXV. Reprinted in Memorie scelte, XXII and XIII.

(27)Sulla serie caratteristica di un sistema lineare di curve appartenente ad una superficie algebrica, Rendic. Acc. Lincei, s. V, Vol. XII (1903). Sul teorema di Riemann–Roch e sulle serie continue di curve appartenenti ad una superficie algebrica. Atti Acc. Torino, Vol. XL

Whence Severi derives the:

Lemma.-For s > n − 4 the surfaces Φ_{n−4} adjoint to f_{n} cut out a complete series on the curve D_{s} (complete intersection of f_{n} and f_{s}).

Indeed the surfaces of order n−4 passing through the double points of D_{s} have to contain the double curve of f_{n} entirely and therefore they are adjoint to it.

As a consequence of the previous lemma one has that “the canonical curves K, and then also the curves of a (special) complete system |C| contained in |K|, cut out the complete series on the curves D_{s}”. Thus, subtracting D_{s} from the complete system |Ds + sC|, one obtains the inequality for the dimension of |C| that expresses the theorem of Riemann–Roch.

On the other hand, once one has verified that the complete system |D_{s}| = |sD| (free of proper fundamental curves) is regular (see Section ??), the above lemma again implies that the characteristic series of a D_{s} has deficiency p_{g} − p_{a}.

Whence one deduces (as we did in the previous section) that the characteristic series of an arbitrary irreducible linear system |C|, at least ∞^{1}, has deficiency ≤ p_{g} − p_{a}.

The argument that leads to such a result is the same used by Enriques in the Ricerche, in order to pass from the characteristic series of the canonical system to the characteristic series of another arbitrary linear system, on a regular surface. The same argument was taken up again in the most general form, but as a quite secondary tool, by Castelnuovo to prove the theorem on the deficiency of the characteristic series in the special cases that escaped its direct proof (nonsimple systems).

The new proof of the theorem of Riemann–Roch for arbitrary linear systems, as given in this treatise^{(28)} is conceptually at least as simple as Severi’s. In our opinion it has the advantage of showing the deep reason of the fact that the canonical curves K (or the curves of a special system) cut out a complete series on the D_{s}: this is not because the D_{s} are complete intersections of the given surface F_{n} with a surface F_{s}, but because |K| is quite small in comparison with the system |D_{s}|. Moreover our method yields the most direct proof of the theorem in consideration, by keeping the same point of view from which one treats the preliminary case of the adjoint systems or of the systems more ample than the canonical system, that is, one does not have to appeal to some new argument, as it appears to be necessary following the ways of Castelnuovo and Severi.

Virtual curves 4.14 136

And moreover, since the theorem of Riemann–Roch is obtained by considering the series that a system |L| = |C + D| cuts out on a curve D, that has to be subtracted from |L|, then it is proved for the system |C| = |L + D|, however reducible this is.

(28)See F. Enriques, Sur l’extension du théorème de Riemann–Roch aux systèmes linéaires de courbes appartenant à une surface algébrique. Bull. Sci. Math., 1940 (see in particular n. 5).

(29) Now the argument involving subtracting a curve D from |L| already appears in Enriques’ Ricerche (of 1893)^{(30)}. Therefore one might think that Enriques himself, and Castelnuovo along with him, knew at a very early stage, at least implicitly, about the possibility of extending the theorem of Riemann–Roch to reducible systems. According to their inductive attitude of mind the result occurred to them by starting from concrete examples. The extension to reducible systems is formulated in their joint memoir of 1900^{(31)}.

The theorem of Riemann–Roch also yields some notable criteria for virtual curves to exist as effective curves, as we shall see in the next section.

4.14 Virtual curves

Let us bear in mind that the computation of the dimension of a linear system |C| = |A − B|, residual of a curve B with respect to a regular and nonspecial system |A|, has been obtained by computing the dimension of the series cut out by |A| on the curve B; from this it would seem that the theorem of Riemann–Roch, expressing the result of this computation, should also yield a criterion for the effective existence of the virtual curves of |A − B|. First of all this is evident in the case in which |A| cuts out a nonspecial series on B.

The formula that gives the dimension of the adjoint system

r ≥ p_{a} + π − 1

to another system of genus π is a first application of this criterion, which has already been encountered in Enriques’ Introduzione of 1896. The formula that gives the dimension of the i-canonical system on a surface of linear genus p^{(1)} is another application:

P_{i} − 1 ≥ p_{a} + \frac{i(i − 1)}{2}(p^{(1)} − 1) (i > 1).

This appears in a note by Castelnuovo of 1897^{(32)} and in the paper by Castelnuovo–Enriques of 1900^{(33)}: actually the i-canonical system |iK| can be obtained from a regular i-tuple adjoint system |A| = |B + iK| of a regular system |B|, by subtracting |B| from it, and B + iK cuts out precisely a nonspecial series on B.

(29)The following paragraph appears in the main text of the Italian edition, but from the context, is presumably written by Castelnuovo or a subsequent editor.

(30)Ibidem, IV, 4.

(31)Castelnuovo–Enriques, Sopra alcune questioni fondamentali nella teoria delle superficie algebriche, Ann. di Mat., t. VI, s. III, n. 4.

(32)Sul genere lineare di una superficie ecc., Rendic. Acc. Lincei, 1897: reprinted in Memorie scelte, XXIV.

(33)Sopra alcune questioni fondamentali della teoria delle superficie, Ann. di Mat., 1900.

A third application of the theorem of Riemann–Roch as a criterion for the effective existence of virtual curves is given by Severi (1905) and concerns the general case in which |A| cuts out an arbitrarily special series on the curve B: we shall say more about this later on.

If one wants to examine the question of the effective existence of the curves of a difference system |C| = |A−B|, one may assume that the two systems |A| and |B| defining the virtual curves C are both irreducible, regular and free of fundamental curves, and also free of base points on a nonsingular surface F; this is because one can always add to the given systems |A| and |B| another regular system free of fundamental curves.

Now we remark that:

(1) The effective existence of the virtual curves of |C| = |A − B|, requires first of all that the degree of the system from which one subtracts is greater than or equal to the degree of the system which is subtracted

A_{2} ≥ B_{2}.

Indeed if A_{2} < B_{2}, the curves C have a negative intersection number with the curves of the system |A + B|, since (A + B)(A − B) = A_{2} − B_{2}, and therefore they should be fixed components of this system, which is irreducible by the assumptions we have made.

(2) Secondly, in order that there exist curves of |A − B|, it is necessary that the intersection number of the curves A and B be in between the degrees of the two systems |A| and |B|:

B_{2} ≤ AB ≤ A_{2}.

Otherwise the curves C would have negative intersection number with the curves B or with the curves A, since one has respectively: for AB < B_{2}

B(A + B) < 0

and for AB > A_{2}

A(A − B) < 0.

The conditions (1) and (2) are necessary both for the effective existence of |C| = |A − B| and for the existence of a multiple |i(A − B)|. Indeed if

A_{2} > B_{2},

we must have

B_{2} < AB < A_{2},

otherwise the C (or their C) – that are not actually curves of order zero – would have zero intersections with the B or with the A and therefore they would be fundamental curves for the corresponding system (that we have assumed to possess no such curves).

To the two necessary conditions stated above there corresponds the sufficient condition for the existence of a multiple of |C|:

If the virtual degree of |C| = |A − B| is positive and A_{2} > AB > B_{2}, then the curves of the multiple system |iC| is always effective for large enough i.

Indeed let us consider the series cut out by the system |iA − (i − 1)B| on a curve B (assuming that this system actually exists); this series is of degree

AB + (i − 1)(AB − B_{2}),

that for i large enough becomes greater than the order of the canonical groups of B; therefore the aforesaid series is nonspecial, and it is possible to compute the dimension of the residual system

|iA − (i − 1)B − B| = |iA − iB|

with the formula

r_{i} ≥ p_{a} + n_{i} − π_{i} + 1,

where

ni = i2n π_{i} = iπ + ni(i − 1)/2 − i + 1

(n = (A − B)^{2} = virtual degree of |C|: π =virtual genus of |C|).

From the previous formula it follows r_{i} > 0 for large enough values of i, hence the effective existence of the curves of |iC| also follows.

Nevertheless in the previous argument we have assumed a priori that the system |iA − (i − 1)B| existed; this assumption can be justified for the values of i that make r_{i} > 0, since under such conditions it is possible to compute the virtual dimensions of |iA − B|, |iA − B − B| = |iA − 2B|, …, up to |iA − (i − 1)B|, that a fortiori all turn out to be greater than r_{i}: indeed for the aforementioned values of i, the dimension of the series cut out on B by |iA − (i − 2)B| is a fortiori greater than the dimension of the canonical series, and therefore it is nonspecial, and so on.

The criterion for the existence of virtual curves that we have thus proved can be applied to the pluricanonical system of a surface for which the linear genus is p^{(1)} > 1, and so the virtual degree of the canonical system is

p^{(2)} = p^{(1)} − 1 > 0.

Thus one finds the formula already written above

P i ≥ p_{a} + (p^{(1)} − 1)i(i − 1)/2 + 1.

Let us notice that the criterion does not lead to any conclusive answer about the existence of the multiple |iC| of a virtual system of degree n = 0, hence it does not enable us to claim a priori the existence of pluricanonical curves for p^{(1)} = 1; although this existence is then proved for surfaces free of exceptional curves (not belonging to the family of ruled surfaces) by means of other considerations.

The case in which the degree of |C| is n ¡0 remains a fortiori unsettled; one cannot say whether or not the two systems |A| and |B|, or their multiples |iA| and |iB|, for i large enough, give rise to an effective difference.

In order to complete this discussion we still want to consider the case in which the systems |A| and |B| have the same degree

A_{2} = B_{2}.

In this case the difference |A − B| or one of its multiples cannot be effective if the intersection number of A and B is different from A_{2}; then one has the necessary condition for the existence

AB = A_{2} = B_{2},

and the virtual degree of |C| = |A − B| is

n = (A − B)^{2} = 0.

Now the curve C cannot be effective unless is a curve of order zero, that is, when A ≡ B. Analogously the curve of |iC| will be effective, again as a curve of order zero, when

iA ≡ iB.

We notice here that if one does not have iA ≡ iB for some value of i, then one may construct on the surface F (that we may assume to have the curves B as plane or hyperplane sections) an infinite family of systems |iA − (i − 1)B|, all consisting of curves of the same order and therefore contained in one or several continuous series of inequivalent curves. This implies that the surface is irregular:

p_{g} > p_{a}.

Thus we can say that: On a regular surface two linear systems of curves |A| and |B| of the same degree, that meet at

AB = A_{2} = B_{2}

points, are parts of a same multiple system:

|iA| = |iB|,

so that the multiple according to i of their difference is the effective curve of order zero: iA − iA.

Leaving the regularity of the surface aside, one can say in general that: “if two curves A and B of the same degree n meet at n points, then two of their multiples belong to the same continuous series of curves, or as one also says, they are algebraically equivalent”^{(34)}.

The criterion for the effective existence of |C| = |A−B|, that we mentioned above and attributed to Severi^{(35)}, just consists in the formal extension to virtual curves of the theorem of Riemann–Roch, which one obtains by defining the index of speciality i of |C| as “the number of linearly independent curves of the residual system of |C| itself with respect to the (impure) canonical system |K|”, that is, as the dimension of the system

|K − C| = |B + K − A|

plus one. Denoting by n and π the virtual characters of |C|, the inequality

r ≥ p_{a} + n − π + 1 − i,

yields a lower bound for the dimension r of |C| and thus ensures the effective existence of the system whenever one has

p_{a} + n − π + 1 − i ≥ 0.

In order to justify in our circle of ideas what we have just said (and restricting ourselves, for the sake of simplicity, to the assumption i > 0) it suffices to keep in mind the lemma that has led us to the theorem of Riemann–Roch for special systems, and to remark that the system |B + K − A| can be regarded as small with respect to the normally large system |sB| for quite large s. Therefore |B + K − A| cuts out a complete series of dimension i − 1 on a curve of the aforesaid system and so the index of speciality of the series cut out by |A + (s − 1)B| on the curve itself has precisely index of speciality i: whence one deduces the above-written formula, that gives a lower bound for the dimension r of

|A + (s − 1)B − sB| = |A − B|.

(34)See F. Severi, Math. Ann., 62 (1906), page 194. A. Todd, Proc. Camb. Phil. Soc., march 23, 1939. About the relations with a theorem on the base of W. V. Hodge, J.

London Math. Soc., 12, 1937, see B. Segre, Ann. di Mat.,1937 and J. Bronowski, J. London Math. Soc., 1938.

(35)Sulle curve algebriche virtuali appartenenti ad una superficie algebrica. Rendic. Ist.

Lombardo II, 38 (1905).

Regular and superabundant systems 4.15 141

At this point it is convenient to add the remark that the index of speciality i of a virtual curve may turn out to be arbitrarily large, unlike the index of an effective curve which is always

i ≤ p_{g}.

Indeed the residual system of |C| with respect to |K|, that is,

|B + K − A|,

is not a priori contained in |K|: actually the fact that it is contained in |K| expresses in a different way the necessary and sufficient condition for a virtual curve to have effective existence.

4.15 Regular and superabundant systems

Let us consider an irreducible complete linear system of genus π, degree n and index of speciality i on a surface of arithmetic genus p_{a}: the dimension of this system satisfies the inequality expressing the theorem of Riemann–Roch

r ≥ p_{a} + n − π + 1 − i,

that is,

r = p_{a} + n − π + 1 − i + w where w ≥ 0.

The system is said to be regular if w = 0, and superabundant if w > 0; in the latter case ω denotes the superabundance of the system.

It is convenient to see whether there is any criterion that enables one to compute the superabundance of a system |C|, or at least to decide whether or not |C| is regular.

The most important result, in this line of thought, concerns the regularity of the adjoint system of an irreducible linear system |C|. We have proved this for a large enough multiple of |C| (at least ∞^{3} or ∞^{2}) and we have said that it has been firstly recognized by Picard (Section ??) for the system adjoint to |C| itself.

Here we can establish in a very simple way the theorem of Picard provided that we add a unnecessary restriction:

The system |C′|, adjoint to an irreducible linear system |C| of dimension r ≥ p_{g} − p_{a}, is certainly regular (of dimension p_{a} + π − 1).^{(36)}

The proof is based on a:

Lemma.-Let |C| and |D| be two irreducible linear systems, that for the sake of simplicity we shall assume of dimension greater than one. One can prove that “if the dimension of |C + D| is greater than p_{g} − p_{a}, then |C| cuts out on a curve of the sum system |C + D| a complete series”. Indeed consider on the curve L of |L| = |C + D| a group G of the complete series containing the groups of the intersections with the C. Since the characteristic series of |L| has deficiency d ≤ p_{g} − p_{a}, among the ∞^{r} groups LD there are ∞^{r−(p_{g}−p_{a})} groups that added to G form groups of the characteristic series of |L|: let G′ be one of these groups. Now there is a pencil of curves L passing through G + G′, and the curve D through G′ is fundamental with respect to this pencil: subtracting it, one obtains a curve of |C| passing through G. Thus the complete series of the groups G is cut out on L by the system |C|.

(36)See Enriques, Bull. de Sci. Math., Ibidem 1940.

Now, let us consider on a surface an irreducible linear system |C|, of a certain genus π and of dimension greater than p_{g} − p_{a}; we wish to prove that the adjoint |C + K| of this system (where K denotes the canonical system) is regular, of dimension p_{a} + π − 1.

In order to do this, associate with |C| an irreducible linear system |D| of dimension greater than p_{g} − p_{a} and such that the adjoint of this system is regular; and construct the system

|L| = |C + D + K|.

Both the system |C + K| and the system |D + K|, cut out on a curve L complete series, which must be one residual to the other with respect to the canonical series. Since one knows the dimension of the series cut out by |D + K|, which is equal to the dimension of the system |D + K|, one deduces the dimension of the series cut out by |C + K|, which equals the dimension of this system.

Indeed denote by r and m the genus and the degree of |D + K|, and by s the intersection number of a C and a D + K: the dimension of the series cut out on L by |D + K| is

p_{a} + m − r + 1

and its degree is m + s, while the genus of L is

π′ = π + r + s − 1.

So the theorem of Riemann–Roch tells us that the dimension of the series cut out on L by |C + K| is

p_{a} + m − r + 1 − (m + s − π′) − 1 = p_{a} + π − 1.

Another regularity criterion (that is useful in finding the postulation of a surface with respect to the varieties V_{n−1} of a space S_{n} of dimension n > 3) concerns the multiples of a linear system: the multiples of very large order of an irreducible, at least ∞^{3}, linear system |C| without fundamental curves are regular.

Consider some integer h such that the system |hC| contains the canonical system |K| as a part, so that one has

|hC − K| = |D|.

We know that for s large enough the adjoint system of |sC|, namely |sC +K|, is regular. Now we have

|(s + h)C| = |sC + K + D|,

where D, in general, is a curve consisting of several simple or multiple components, of either positive or negative degree. In this setting one has to show that when one successively adds the irreducible components of D to the regular system |sC + K| one obtains larger and larger regular systems (according to the criterion of Section ??).

Indeed let us denote by L an irreducible component of D, with characters r and n; since one has always r ≥ 0, then one sees that the intersection number of L with the curves of |sC + K| that have to be added to it (with s arbitrarily large) is

m = st + 2r − 2 − r > r − n,

where t > 0, since L is not a fundamental curve for |C|. The analogous inequality always holds for the intersection number of the curves of |sC + K + L| with a component of D, and so successively for the sum of |sC + K| with all the irreducible components of |D|.

Finally the system |sC + K + D|, that is a multiple of |(s + h)C|, turns out to be regular, for s + h large enough, Q.E.D.

It suffices to explain the argument by considering to the first relevant case, where one has to add |sC + K + L| to L itself. Here we assume that L is a double component of a curve D and is of genus r ≥ 0 and of negative degree n = −n′. Then we claim that in this case the intersection number of L with the curves of |sC + K + L| is

st + 2r − 2 + n′ − n′ = st + 2r − 2 > r − n′,

for s large enough compared to n′ = −n.

We may state the theorem that we have just proved by saying that: Any linear system which is normally large (with respect to the canonical system) is regular.

More on the regularity of the adjoint on a regular surface 4.16 144

It is useful to remark explicitly that the property does not hold for large systems that are not normally large. The simplest example is provided by the linear systems that have proper fundamental curves: If the linear system |C| has a proper fundamental curve θ of genus r > 0, the arbitrarily large multiples of |C| have superabundance^{(37)}

ω ≥ r

Indeed the subtraction of a curve q, of degree −n from |sC|, decreases by one the dimension of the system, by n its degree and by n + r − 1 its genus, and thus by r − 1 its virtual dimension; this means that the superabundance ω of |sC| exceeds by r the superabundance w′ of |sC − θ|:

ω = ω′ + r.

We notice that, in general, (for r > 1) one has also ω′ > 0, and thus the genus r of the fundamental curve is only a lower bound for the superabundance of |sC|. It is instructive to recall here the simple projective example of a surface F of order n (with plane sections C), having an isolated multiple point O of order i > 3, on which the surfaces φ_{s} (of large enough order s) cut out the linear system |sC|.

One sees at once that not only the general φ_{s} cut out on F a superabundant system (for which the curve θ represented by the neighbourhood of O is a fundamental curve), but also that the systems cut out on F by the φ_{s} passing through O with multiplicities 1, 2, …, i − 3 are superabundant, whereas the system cut out by the adjoints φ_{s} that pass i − 2 times through O is regular. Whence one easily deduces the computation of the superabundance ω of |sC|.

4.16 More on the regularity of the adjoint system of a curve on a regular surface

We wish to go deeper into the question of the regularity or superabundance of the linear systems more ample than the canonical system, referring to the simple case of the regular surfaces, with p_{g} = p_{a} = p > 0.

If |C| is a linear system more ample than the canonical system |K|, then one has

|C − K| = |L|,

where |L| denotes an effective system, and so

|C| = |L + K| = |L′|.

Thus |C| can be considered as the adjoint system of an effective curve L. Now if L is irreducible, of genus π¿0, then its adjoint system is certainly regular, of dimension p + π − 1, since it cuts out on L a (complete) series that cannot be larger than the canonical series g_{2π−2}^{π−1}.

(37)See Enriques, Ricerche…, 1893.

The same conclusion – regularity of the adjoint system of L – can be extended to the case in which the curve L is an arbitrarily reducible, connected curve, consisting of simple parts.

Connected curve consisting of simple components means “curve such that whenever it is divided in two parts in any possible way, always gives rise to curves having some (connection) point in common”, that is, topologically, a “curve or Riemann surface on which it is always possible to give a continuous path joining two arbitrary points”.

For connected curves, consisting of simple components, one has the:

Lemma of geometric composition of connected curves. A connected curve L consisting of s simple components can always be obtained by adding to a connected part consisting of s − 1 components L_{1} + L_{2} + ⋯ + L_{s−1} a further sth component L_{s} with some common (connection) point with it.

This is proved by induction: assuming that the given L contains a connected part, consisting of h simple irreducible components with h < s, this part must be connected with the rest of the curve and therefore with at least one of its components; adding such a component to it, one obtains a connected component of L, consisting of h + 1 components.

As a Corollary:: given a connected curve L consisting of s simple components, it is always possible to take these in some order L_{1}, L_{2}, …, L_{s} so that in the curve

L = L_{1} + L_{2} + ⋯ + L_{s}

each part L_{1} + L_{2} + ⋯ + Lh (h = 1, 2, …, s) is connected.

Using this property it is easy to recognize that “the adjoint system |L′| is regular”, as we have stated above. Indeed denote by π_{h} the genus of L_{1} + L_{2} + ⋯ + L_{h}, by π′_{h} the genus of L_{h}, and by i_{h} the intersection number (L_{1} + L_{2} + ⋯ + L_{h})L_{h}. We have that the genus of

L = L_{1} + L_{2} + ⋯ + L_{s} = (L_{1} + L_{2} + ⋯ + L_{s−1}) + L_{s}

is

π_{s} = π′_{s} + π_{s} − 1 + is − 1,

and the series cut out on L_{s} by |L′| has degree

n_{s} = 2π′_{s} − 2 + is

and dimension

r_{s} ≤ π′_{s} − 2 + is;

thus the subtraction of L_{s} from |L′| yields

r_{s} + 1 ≤ π′_{s} + is − 1

The adjoint system of a connected curve (continued) 4.17 146

conditions.

Thus: if the system |(L_{1}+L_{2}+⋯+L_{s−1})′| adjoint to |L_{1}+L_{2}+⋯+L_{s−1}| is regular then one has precisely

r_{s} + 1 = π′_{s} + is − 1

and the system |L′| adjoint to |L| = |L_{1} + L_{2} + ⋯ + L_{s}| is also regular. The theorem thus stated is verified for a curve L consisting of s components (that are simple and connected between them) if it is verified for a curve consisting of s − 1 components. Since |L′_{1}| is regular, the theorem is proved in general.

Contrary to what happens for connected curves, the adjoint systems of disconnected reducible curves are not regular, but superabundant. More precisely:

The system |L′| adjoint to a curve L consisting of s simple irreducible curves without common points has superabundance equal to s − 1.

This is proved with the same inductive argument used for connected curves. Consider for instance a curve L (of genus π) made of two parts L_{1} and L_{2} of genera π_{1} and π_{2}, without common points, (π = π_{1} + π_{2} − 1). Then the adjoint system |L′| = |(L_{1} + L_{2})′| cuts out the complete canonical series g_{2π_{1}−2}^{π_{1}−1} on L_{1}, and the residual system of L_{1} with respect to it – that is, |L′_{2}| – cuts out the canonical series g_{2π_{2}−2}^{π_{2}−1} on L_{2}; the latter system |L′_{2}| is regular, thus |L′| is superabundant, of dimension

p + π = p + (π_{1} + π_{2} − 1) − 1 + w

where

w = 1.

Now it is easy to compute the superabundance of the system |(L + ∑ s_{i}q_{i})′| adjoint to a disconnected curve L + ∑ s_{i}q_{i}, consisting of an irreducible curve L of degree > 0, that, for the sake of simplicity, we may assume to vary in an ∞^{2} linear system, and of several irreducible curves q_{i} of degree n_{i} < 0, fundamental for this system |L|, each counted a certain number s_{i} of times.

4.17 The adjoint system of a connected curve consisting of multiple components on a regular surface (continued)

The above results do not settle the question of the regularity or superabundance of the adjoint system of a curve L on a regular surface, when L is however formed by simple and multiple irreducible components. The difficulty that one has to face here is actually related to the concept of “connected curve”, that has to be precisely extended to an arbitrary L, of the form described above.

We define a curve L, arbitrarily made of simple or multiple irreducible components, to be (arithmetically) connected when any two parts, L_{1} and L_{2}, in which it is decomposed, always have a (virtual) number of common points

L_{1}L2 > 0 :

it is not excluded that some part of L_{1} has a negative number −n of intersections with L_{2} (thus containing at least one multiple component of L_{2}, of negative degree), provided that then the remaining part has more then n points in common with L_{2}.

The consideration of connected curves, however formed of multiple parts, gives rise to difficulties that have been explained by A. Franchetta^{(38)}, and that are worth to be pointed out here.

Contrary to what intuition suggests in the case of simple parts (having a nonnegative number of common points with other components) one cannot say that in general, by adding to a connected curve L a component C that meets it at n > 0 points, one always obtains a connected curve L+C.

Indeed Franchetta points out examples of disconnected curves consisting of s components, that can be split into a connected curve made of s − 1 components and one component that meets it at a positive number of points.

As a consequence of this, also the lemma (of geometric composition) that we have proved in the previous section for connected curves made of simple components, does not hold any more: in general one cannot say that an arithmetically connected curve made of s − 1 components can always be obtained by adding an irreducible component to a connected curve made of s − 1 components.

Nevertheless some properties still hold for arithmetically connected curves; they have been pointed out by Franchetta^{(39)} and take the place of the analogous properties in the case of simple components. We state them as the following theorems:

Theorem I.-A curve L (however formed by simple or multiple irreducible components) is arithmetically connected if each of its connected parts is connected with the remaining part (that is, it has a positive virtual number of points in common with it).

Indeed if L = A + B is not connected, then we have AB ≤ 0. So either A is connected and then there is a connected part of L that intersects the rest in a nonpositive number of points, or else it is possible to decompose A as the sum of two parts A_{1} and A_{2} disconnected one from the other:

A = A_{1} + A_{2}, A_{1}A_{2} ≤ 0;

(38)Sulla caratterizzazione delle curve eccezionali riducibili di prima specie, in Boll.

dell’Unione Mat., Bologna, 1941.

(39)Ibidem.

from the fact that AB is nonpositive it follows that at least one of the two curves, say A_{1}, has a nonpositive number of intersections with B; but in this case one has

L = A_{1} + (A_{2} + B)

with

A_{1}(A_{2} + B) = A_{1}A_{2} + A_{1}B ≤ 0,

that is, there is a part A_{1} of A, contained in L, disconnected from the remaining part. Now either A_{1} is connected or it contains a part A_{11} disconnected from the remaining part (L − A_{11}) and so on. But the sequence of parts A, A_{1}, A_{11}, …, whose number of components decreases, ends with a connected curve, consisting of one or several irreducible components: the last part thus obtained would be a connected curve disconnected from the rest.

Theorem II.-Given an (arithmetically) connected curve L, it is possible to remove from it a connected part B (consisting of one or several components) in such a way that the remaining part L − B is again connected.

Among all the possible connected parts of L, choose one, call it B, having the smallest (positive) number of common points with the remaining part A = L − B; we claim that A is also connected. If this were not the case, one would have a decomposition into disconnected parts

A = A_{1} + A_{2};

where one may assume that A_{1} itself is connected (Theorem I). But, in this case, B must have a positive number of intersections with each of the parts A_{1} and A_{2} of A, since otherwise one would have either (B + A_{1})A_{2} ≤ 0 or (B + A_{2})A_{1} ≤ 0 and thus L would be disconnected; so we have

BA_{1} > 0 and BA_{2} > 0,

thus

BA_{1} < BA = B(A_{1} + A_{2}),

and a fortiori

(B + A_{2})A_{1} < BA;

so A_{1}, a connected part of L, would have with the rest a number of intersections smaller than B, contradicting our assumption.

Franchetta’s theorems replace, to a certain extent, the intuitive properties that characterize connected curves made of simple components, as we shall see in what follows.

However it does not seem easy to deduce from these properties, without exceptions, the regularity of the system |L′| adjoint to an arbitrary connected curve L (with multiple components of negative degree). One may try to follow a quite indirect way, strictly related to our approach to the Riemann–Roch theorem, but subject to a working hypothesis that will be clarified later on.

Consider on the surface F a regular linear system |C|, which is a multiple of large enough order of an irreducible system,, at least ∞^{2}, free of fundamental curves. Hence, to start with, |D| = |C + L| is irreducible. One may assume that both |L| and its adjoint system |L′| = |L + K| cut out complete series on a curve \overline{D} of the normally large system |C + L|. Now one can prove that also |C| cuts out a complete series on this curve \overline{D}. Indeed let G be a group of the complete series CD′ and let G be a residual group of G with respect to the characteristic series of |D|, section of D′ with L; G + G is the base group of a pencil of curves D, that have no intersection with the components of L away from \overline{D}.

Now, since L is connected, if one imposes that a D of this pencil contains a point of L away from \overline{D} then the whole connected curve will be contained in D and the remaining part of D will be a C through G. Q.E.D.

The above argument is based on the assumption that, since L is arithmetically connected, when one imposes that a curve D of the above-said pencil contains a point of L, away from G, then the whole of L becomes a component of D, that is, each of its components is contained in D with the same multiplicity with which it appears in L.

We believe this property to be true, but we have no satisfactory proof of it. We shall introduce it as a working hypothesis, thus it has to be clear that the above result and the consequences that we shall deduce from it are subject to the validity of that assumption.

Now, we have seen that the regular system |C| cuts out on the curve D of |D| a complete series g, which is residual of the series cut out by |L′| = |L+K|. Hence we are able to compute the dimension of the latter series (and so of |L′|) that is equal to the index of speciality of g minus one. Denoting by n the degree of |C|, by ρ its genus, and by m the intersection number CL, the degree of the series g is

n + m

and its dimension is

r = p + n − ρ + 1,

whereas the genus of D is

π + ρ + m − 1.

Thus the index of speciality i of g is given by:

n + m − (π + ρ + m − 1) + i = p + n − ρ + 1,

whence it follows that the dimension of the series (CL′) residual of CD is

i − 1 = p + π − 1.

On the reducibility of the canonical system 4.18 150

So |L′| has exactly this dimension, and it is regular.

In conclusion: On a regular surface of genus p > 0, the adjoint system of an arithmetically connected curve is always regular.

4.18 On the reducibility of the canonical system

The theorem proved in the previous section leads to remarkable consequences concerning the bicanonical system and the pluricanonical systems. In order to understand the meaning of these, it is convenient to go back to the question of the reducibility of the canonical system.

Noether believed he was able to prove that the canonical system of a surface of genus p > 0 can be reducible only in two cases:

(1) because of the existence of fundamental curves (of genus ρ = 0 and degree n = −1) that are fixed parts of the impure canonical system;

(2) because the (absolute) linear genus is p^{(1)} = 1, in which case the pure canonical curves are in general composed with the elliptic curves of a pencil.

So for p^{(1)} > 1 the pure canonical system should always be irreducible. But Noether’s argument is wrong, and there are actually examples of (pure) reducible canonical curves K, belonging to a surface F of genus p > 0 and linear genus p^{(1)} > 1:

(1) the first example is provided by the double plane with branch curve C_{2n} of order 2n, having a (2n − 6)-tuple point O and two [3, 3] points A_{1} and A_{2}, lying together with O on a line a

(p = 2n − 7, p^{(1)} = 4n − 15).

The (double) images of the canonical curves K are the curves C_{n−3} that have O as a (n − 4)-tuple point and pass simply through A_{1} and A_{2}, and therefore they are the sum of a C_{n−4} with O as a (n−5)-tuple point and of the fixed part a. Now this line represents an (irreducible) curve θ, a fixed component of the K, which is not exceptional, since it has genus ρ = 0 and degree n = −2: as it appears also from the fact that it has two intersections with the residual components of the K, since the number of these intersections is:

2ρ − 2 − 2n = 2, (r = 0, n = −2).

(2) A second example is provided by the double plane with branch curve C_{10}, of order 10, with three [3, 3] points on a line a, that is not a component of C_{10}. A simple count of dimensions tells us that the linear system of C_{10} with three assigned triple points on a and three triple points infinitely near to these in assigned directions has dimension 29, and therefore it is larger than the ∞^{26} system consisting of a counted twice and of the curve C_{8} with three tacnodes and given tacnodal tangents. Now, on the double plane that we have constructed the images of the canonical curves are the conics that split as the sum of the fixed line a and of a variable line. The corresponding fixed part θ of the canonical system |K| is a curve of genus r = 1 and degree n = −1; the genera of the surface F are

p = 3, p^{(1)} = 6.

One may compute the degree n = −1 of θ by remarking that θ – which is a fixed part of |K| – has two intersections with the residual curves, so that

2r − 2 − 2n = −2n = 2, n = −1.

On the other hand the complete curve corresponding to a on F is a curve of genus four consisting of three elliptic curves of degree −1, corresponding to the [3, 3] points, and of our curve θ, which has a point in common with each one of them: the total degree of this complete curve is

−3 − 1 + 2.3 = 2.

(3) A third example is provided by the double plane with branch curve C_{12}, of order 12, with four [3, 3] points belonging to a line a (which is not a component of C_{12}); counting dimensions one shows the existence of irreducible C_{12}, that do not contain a. Now the canonical curves K are given on the double plane by the cubics that split as the sum of the line a and of a variable conic. The line a corresponds to a fixed component of the K that has genus r = 1 and degree n = −2; indeed θ has 4 intersections with the residual curves, so that^{(40)}

2r − 2 − 2n = −2n = 4, n = −2.

(40)We also give the example of an irregular surface with p_{a} = 2, p_{g} = 3, p^{(1)} = 11, represented on the double plane with a branch curve C_{12} consisting of 3 quartics C_{4} (belonging to a pencil) that touch each other at 8 points lying on a conic C_{2}; the images of the canonical curves split as the sum of the conic C_{2}, which corresponds to a fixed curve θ of genus 3, and of a variable line. The surface contains an irrational (elliptic) pencil of curves, represented by the pencil of the C_{4}, according to a theorem of M. De Franchis’ on the irregular double planes.

The complete curve corresponding to a on F is a curve of genus five consisting of four elliptic curves of degree −1, corresponding to the [3, 3] points, and of our curve θ, that has a point in common with each one of them: the total degree of this curve is

−4 − 2 + 2 · 4 = 2.

Note It is not known at the present moment whether there exist surfaces with p^{(1)} > 1 whose canonical system is reducible in such a way that its variable parts are composed with the curves k of a pencil^{(41)}. About this possibility we can only prove the following proposition:

If the pure canonical curves of a surface of genus p > 1 (apart from the fixed components) are composed with the curves k of a pencil of genus π and degree n, then one has in any case

π ≤ p^{(1)}, n ≤ p^{(2)} = p^{(1)} − 1.

Denote by θ the fixed components, if any, of the K:

|K| = ∑ θ + |sk|,

and let ρ and n be the characters of one of the θ. We express the degree of |K| as the sum of the intersection numbers of its components (irreducible or not) k and of θ; we have

s(2π − 2 − n) + ∑(2ρ − 2 − n) = p^{(1)} − 1.

Now, on a transformed surface without exceptional curves, we have for each θ

2ρ − 2 − n ≥ 0

therefore

2π − 2 − n ≤ p^{(1)} − 1.

On the other hand a curve k intersects the residual curve (s − 1)k + ∑ θ in the virtual number of points

2π − 2 − 2n ≥ 0.

Whence one deduces

n ≤ π − 1,

and

π ≤ p^{(1)}, n ≤ p^{(1)} − 1. q.e.d.

(41)Examples have actually been constructed recently by G. Pompilj: Alcuni esempi di superficie algebriche a sistema canonico puro degenere, Rendic. Acc. Lincei, s. 8, v. IV,

On the regularity of the bicanonical system 4.19 153

4.19 On the regularity of the bicanonical system

The possibility that the pure canonical system |K| of a surface with p > 0 and p^{(1)} > 1 may be reducible forces us to examine more closely the question of “whether the bicanonical system and the pluricanonical systems are regular”. In order to do this, one needs to prove that, although the K may be arbitrarily reducible into simple or multiple components, they are always connected (for p^{(1)} > 1); so that the adjoint system

|2K| = |K′|

is regular, and similarly, since the curves of |2K| are also connected, the tricanonical system

|3K| = |(2K)′|

is also regular, and so on.

In words: on a regular surface of genus p > 0 and of linear genus p^{(1)} > 1, the bigenus is exactly

P = p + p^{(1)}

and the ith genus is

P_{i} = p + \frac{i(i − 1)}{2}p^{(1)}.

In order to prove this important theorem it is convenient to show that, for p > 0 and p^{(1)} > 1:

(1) the variable parts of the canonical system are irreducible;

(2) the canonical curves K, however reducible, are always connected, with order of connection ≥ 2 (that is, any part of K has at least two intersections with the residual part);

(3) as a consequence the pure bicanonical system is regular of dimension p + p^{(1)} − 1 and not greater.

These results can be extended a fortiori to the pluricanonical systems of order > 2.

(1) The first proposition is easily proved. First of all if p ≥ 2 and p^{(1)} > 1, then the variable parts of the canonical curves are either irreducible or composed with the irreducible curves C, of genus π > 1, of a pencil; therefore the curves C′ adjoint to a curve C form a linear system of dimension (at least) p + π − 1 that cuts out the complete canonical series g_{2π−2}^{π−1} on C; two of the C′ have at least 2π − 2 variable intersections and thus the C′ are not composed with the curves of a pencil; a fortiori the bicanonical system |2K|, that contains |C′| as a part, does not have reducible variable part composed with the curves of a pencil.

The theorem extends to the case p = 1. In this case, if the variable parts of the bicanonical curves are composed with the irreducible curves C of a pencil, first of all – since p^{(1)} > 1 – the genus of these curves has to be π > 1^{(42)}. Now, if the number of the components C that are parts of the curves of |2K| is equal to s (P = s + 1 is the bigenus of the surface), since p = 1, there will be s curves C_{1/2} such that their doubles belong to the pencil |C|:

K = θ + sC_{1/2}.

Now the adjoint system |C′_{12}| of a curve C_{1/2}, which is contained in |2K|, should have the curves of |C| as variable parts; but this is impossible because the dimension of this system is not one, but (at least) p + π − 1 = π > 1^{(43)}.

(42)Actually we shall see later that if a surface has a pencil of elliptic curves and its geometric genus is positive, then its linear genus has to be p^{(1)} = 1. (43)Proposition 1 can be also proved in another way, first showing that π = 2 and therefore necessarily p > 1. If the variable parts of the bicanonical curves 2K are composed with s (> 1) curves of a complete linear pencil, then, denoting by θ the fixed parts of |2K| we shall write

|2K| = ∑ θ + |sC|.

Now denote by n and k the degree and the genus of |C|, and by ν and ρ the degree and the genus of a θ, and let us compute the additive character of the 2K expressing their intersection number with a K; we have

2K · K = 2p^{(1)} − 2 = ∑(2ρ − 2 − ν) + s(2π − 2 − n).

But, referring, as we may, to a surface without exceptional curves:

2ρ − 2 − ν ≥ 0,

and therefore

s(2π − 2 − n) ≤ 2p^{(1)} − 2.

Thus, in order that the pencil |C| be complete, its dimension has to be r = 1; therefore, if |C| is not contained in the canonical system |K|, we have:

p + n − π + 1 ≤ 1 and n < π − 1;

and this inequality holds a fortiori if |C| is contained in |K|, because in that case |2C| is contained in the adjoint system

|C′| = |C + K|.

In this set up, if π > 2, one has

s(π − 1) ≤ 2p^{(1)} − 2, π − 1 ≥ 2, p^{(1)} − 1 ≥ s,

(2) To prove that the canonical curves K are connected, it suffices to show that the bicanonical curves of |2K| are connected, and therefore one has to contradict the assumption that they are disconnected. Now the bicanonical system |2K| consists of a fixed part L (arbitrarily reducible) and of irreducible variable parts C:

|2K| = L + |C|.

If the curves of this system are disconnected, then they can be decomposed in two parts having a number of common points which is nonpositive: one of these parts will contain a C and a (fixed) part E of L, whereas the other will be a fixed D:

L = D + E, (C + E)D ≤ 0.

Now we may assume that D is connected; if this were not the case then D could be decomposed into two parts D_{1} and D_{2} with a nonpositive number of common points, and one of these parts – say D_{1} – would have

D_{1}(C + E + D_{2}) ≤ 0

intersections with the residual one; continuing in this way, in any case one would eventually find a connected part of L, and thus of C + L, having a nonpositive number of intersections with the residual curve.

Therefore we assume D to be connected and denote by r its genus and by n its degree: in order that D be connected we should have^{(44)}

ρ ≥ 0.

and as a consequence the dimension of |2K| is

≥ p + p^{(1)} − 1 > s,

contradicting the hypothesis that |2K| is formed by the groups of s curves of the pencil |C| and has therefore dimension s.

After having proved in this way our proposition in the case π > 2, one has to discuss the case π = 2.

In this case the presence of the pencil of curves C of genus π = 2 makes it possible to represent the surface on a double plane with branch curve f_{2m} of some even order 2m, with a (2m−6)-tuple or (2m−5)-tuple point O and other elementary singularities: 4-tuple points and [3, 3] points.

The bicanonical curves have as their images in this plane curves Φ_{2m−6} adjoint to f_{2m}, that are made of a fixed Φ_{2m−6−s} and of s lines through O. Since among the bicanonical curves there is at least one canonical curve counted twice, the Φ_{2m−6−s} must be the double of a curve Ψ_{m−3−h} (s = 2h) that, together with h lines through O, is the image of a canonical curve. Since h ≥ 1, one deduces that p > 1, and this case is already settled by the first remarks in the text. (44)This proposition is proved by induction, for the curves D consisting of s irreducible components, assuming that the proposition holds for the curves consisting of less than s components. Indeed D can be decomposed into two parts, D_{1} and D_{2}, each one connected and connected to the other, with n common points; thus, denoting by ρ_{1} and ρ_{2} the genera

Let us write the intersection number of D with the residual curve with respect to the bicanonical system |2K|:

D(2K − D) = D(C + E) = 4ρ − 4 − 3ν ≤ 0.

Whence we deduce

ν ≥ \frac{4}{3}(ρ − 1).

Now, if ρ = 0, since n is an integer, one has

ν ≥ −1 :

so that either ν ≥ 0, hence p = 0, or D is an exceptional curve of the surface (ρ = 0, n = −1). One may assume that the surface is transformed in such a way that it does not contain any such curve, and therefore one may exclude the case ρ = 0, and take

ρ ≥ 1.

But if ρ = 1, since it is not possible to have ν > 0 (otherwise p = 0), then one has n = 0, and so the bicanonical and pluricanonical curves of |iK| have zero intersections with D, whereas D has always zero intersections with the residual curves of |iK − D|, |iK − 2D| and so on; whence it follows that the curves of |iK| are composed with the elliptic curves of a pencil and p^{(1)} = 1, contradicting our assumption (p^{(1)} > 1).

In conclusion, if the bicanonical curves are disconnected, then there is one of their parts D, which is connected and has a nonpositive number of intersections with the residual curves, of genus ρ > 1, and degree

ν ≥ \frac{4}{3}(ρ − 1) > ρ − 1.

But this is impossible, since it contradicts the hypothesis that D is contained in the fixed part L of the curves of |2K|; indeed the dimension of the complete system |D| of index of speciality p is, according to Riemann–Roch:

p + ν − (ρ − 1) − p > 0.

Thus the curves of the bicanonical system |2K|, and thus also the curves K of the canonical system, are connected: any part of a K, of genus ρ and degree ν, has to have a positive even number of common points with the residual curve:

2ρ − 2 − 2ν ≥ 2,

and so the order of connection of K is ≥ 2.

of these curves, one has

r = ρ_{1} + ρ_{2} + n − 1, ρ_{1} ≥ 0, ρ_{2} ≥ 0, r ≥ 0. Q.E.D.

Remark on the irreducibility of the bicanonical system 4.20 157

(3) From the fact that the pure canonical curves are connected (for p > 0, p^{(1)} > 1) one deduces that the bicanonical system |2K| = |K′| is regular. In a similar way, all pluricanonical curves turn out to be connected, and so the corresponding systems

|K_{i} = |{(i − 1)K}′|

are also regular.

4.20 Remark on the irreducibility of the bicanonical system

After having proved that on a regular surface with p > 0 and p^{(1)} > 1 the pure bicanonical system is regular, we may wonder whether it is actually always irreducible. Since we have excluded that its variable parts may be reducible, one has to show that it has no fixed part (except for the exceptional curves).

But the argument fails because of a rather subtle problem arising in case the canonical curves are however reducible into simple and multiple components. We shall restrict ourselves to prove the theorem in the case in which the (pure) canonical curves K consist of simple components.

In that case we know that a curve K is still connected when one subtracts from it an irreducible component K_{1} having the least possible number n (≥ 2) of common points with the residual curve K_{2}.

Now, denoting by π_{1} and π_{2} the genera of K_{1} and K_{2} respectively, we have the genus of K given by:

π = π_{1} + π_{2} + n − 1.

So when one subtracts K_{1} from |K|, the dimension of the regular system adjoint to |K| drops by r_{1} + 1 = π_{1} + n − 1 > 0, where r_{1} denotes the dimension of the series cut out by |K′| on K_{1}. This means that K_{1} cannot be a fixed component of |K′| = |K_{1} + K′_{2}|, since otherwise subtracting that part one would obtain the system |K′_{2}| having the same dimension.

Now – once the component K_{1} has been subtracted – the curves of |K_{2}| are still connected, but in general no longer has order of connection ≥ 2: they may have handles, that is, some components having a single point in common with the residual curve; these components – when they are of genus zero – will be fixed parts of |K′_{2}|. Nevertheless these handles are not – in any case – fixed parts of |K′|; this is because otherwise the series g_{2π_{1}−2+n} cut out by |K′| on K_{1} would have fixed points and would not be complete, whereas it must have dimension

r_{1} = π_{1} − 2 + n.

Genera of a reducible surface 4.21 158

Thus we have seen that, since the K are connected curves with order of connection ≥ 2, the adjoint system |K′| cannot contain either K_{1} or the handles of the residual curve K_{2} as fixed parts. Now, removing these handles from K_{2}, one has a connected curve \overline{K}_{2}, with order of connection ≥ 2, and one proves in a similar way that neither a component that can be subtracted without disconnecting the curve, nor the handles of the residual curve, are fixed parts of |K′_{2}|. Proceeding in this way, one sees that no component of K can be a fixed part of |2K| = |K′|. Q.E.D.

4.21 Genera of a reducible surface

We want to add to the topics treated in this chapter some information about reducible surfaces. Suppose that surface F of some order m, that we wish to assume with normal singularities, splits into two surfaces F_{1} and F_{2} of orders m_{1} and m_{2} respectively. Then we want to determine the numerical characters of F in terms of those of F_{1} and F_{2} and of the curve in which they intersect.

We should point out that the limit surface F_{1} +F_{2} has a double curve that is the limit of the double curve of F and that consists of three parts: a part of certain order d_{1}, double for F_{1} and not contained in F_{2}, a second part, the order of which we shall denote by d_{2}, double for F_{2} and not contained in F_{1}, and finally a third part – of order d, say – contained in F_{1} and F_{2}. Besides this curve one has a curve C of a certain genus π, intersection of F_{1} and F_{2}, that connects the two surfaces F_{1} and F_{2}, the sum of which is regarded as the limit of F.

Now we have to determine the numerical characters of the reducible surface F_{1} + F_{2} in terms of the analogous characters of the two components and of the characters of the common curve C connecting them^{(45)}.

Let us start by discussing the arithmetic genus. We denote simply by p (equal to p_{a}) the arithmetic genus of F, by p_{1} and p_{2} the arithmetic genera of F_{1} and F_{2} respectively, while we have denoted by π the genus of the curve C.

It is clear that one can compute p by means of the postulation formulas for the double curves of F_{1} and F_{2}, or rather for the curves of order d_{1}, d_{2} and d that are the limit of the double curve of F when F degenerates as the sum of F_{1} and F_{2}. In this way one can find the formula

p = p_{1} + p_{2} + π,

(45)From the degeneration principle applied to the plane curves sections of F it follows that there must be a curve of connection of order greater than zero. See Enriques–Chisini, ibidem.

which is the extension to surfaces of Noether’s formula that gives the genus of a reducible curve^{(46)}.

But we can justify the formula without performing the computation mentioned above.

Actually the surfaces F_{m_{1}+m_{2}−4} of order m_{1}+m_{2}−4 adjoint to F_{1}+F_{2} (for which C must not be regarded as a double curve) cut out on F_{1} the adjoint system of |C| that has virtual dimension:

p_{1} + π − 1,

and for each one of these curves there are in general infinitely many surfaces F_{m_{1}+m_{2}−4}; the virtual dimension of this system of F_{m_{1}+m_{2}−4} will be p_{2}, since there are ∞^{p_{2}−1} surfaces F_{m_{1}+m_{2}−4} that split into F_{1} and a surface F_{m_{2}−4} adjoint to F_{2}. In this way one finds the dimension of the system of F_{m_{1}+m_{2}−4} adjoint to F_{1} + F_{2}:

p − 1 = p_{1} + p_{2} + π − 1.

One may object to this argument that since the surfaces F adjoint to F_{1} + F_{2} are subject to pass through a curve of order d_{2} not belonging to F_{1}, they might cut out on F_{1} a system of dimension smaller than the dimension of the system |C′| adjoint to |C|. But, since we deal here with virtual dimensions, this circumstance does not affect our computation, in which the given adjoint surfaces might be replaced by adjoint surfaces of higher order.

Therefore we conclude that the numerical genus of a reducible surfaces whose components of genera p_{1} and p_{2} are connected by a curve of genus π is

p_{1} + p_{2} + π.

Now we want to compute the linear genus p^{(1)} of F_{1} + F_{2}.

In order to do this one needs to know the linear genera p_{1}^{(1)} and p_{2}^{(1)} of F_{1} and F_{2}, the genus of the curve of connection C, that we have already denoted by π, and the degrees n_{1} and n_{2} that this curve has on F_{1} and F_{2} respectively.

The linear genus of the reducible surface is

p_{1}^{(1)} + p_{2}^{(1)}(1) + 8π − n_{1} − n_{2} − 9.

Indeed, as we have remarked, the surfaces of order m_{1} + m_{2} − 4 adjoint to F_{1} + F_{2} cut out on F_{1} curves C′ adjoint to C that have genus

p_{1}^{(1)} + 3π − 3 − n_{1};

similarly the curves that the same F cut out on F_{2} have genus

p_{2}^{(1)} + 3π − 3 − n_{2}.

(46)In general, for varieties of any dimension: see F. Severi, Fondamenti per la geometria sulle varietà algebriche, Rendic. Circ. Mat. di Palermo, 1909.

Now a curve of the first system on F_{1} and a curve of the second system on F_{2} taken together form the intersection of F_{1} + F_{2} with a F_{m_{1}+m_{2}−4} whenever they have in common a canonical group of 2π − 2 points, which are their intersections with C. So the genus of the curves sum of these two is

p^{(1)} = p_{1}^{(1)} + p_{2}^{(1)} + 8π − n_{1} − n_{2} − 9. Q.E.D.

Numerical invariants and multiple planes

5.1 The Zeuthen–Segre invariant

In the previous chapters we have introduced the invariants of surfaces that are defined in relation with the canonical system. In doing this we followed a pattern that appears as a natural extension of the procedure that leads to the definition of the genus of a curve. But the extension can be done in different ways, and so it is possible to define other numerical characters of surfaces, that also have an invariant meaning, and that can be usefully exploited in the further developments of the theory. Nevertheless these characters are not essentially new, since they can be expressed in terms of the numerical genus p_{a} and the linear genus p^{(1)}, as we shall explain later.

Consider on the surface F a linear pencil of irreducible curves of genus π ≥ 0, having some number n of (simple or multiple) effective base points, and denote by δ the number of curves of the pencil with a double point away from the base points.

The expression

I = δ − n − 4π,

involving the number of base points, the genus, and the number of curves with a double point of a linear pencil, does not depend on the arbitrary choice of the pencil and therefore it is a (relative) invariant of the surface.

The Zeuthen–Segre invariant 5.1 162

The invariant meaning of this character was recognized for the first time by Zeuthen^{(1)} by considering a pencil of plane sections of a surface. C. Segre^{(2)} has proved directly the equality of the characters equally defined in relations with two arbitrary pencils of the surface. And so I is usually referred to as to the Zeuthen–Segre invariant.

(1)H. G. Zeuthen, Etudes géométriques etc. Math. Ann., IV, 1871.

(2)C. Segre, Intorno ad un carattere delle superficie ecc., Atti Acc. Torino, 1896.

We explain Segre’s proof, which consists in comparing two pencils |C_{1}| and |C_{2}| on the surface, in no special relation one to the other.

The first pencil has a certain number n_{1} of base points, that we assume to be simple and distinct one from another; the second pencil also has a certain number n_{2} of base points, different from the previous ones and also simple and distinct one from another. We shall denote by π_{1} and π_{2} respectively the genera of the generic curves of the two pencils, and by s the number of intersections (C_{1}C_{2}). And we shall consider the curve L which is the locus of the contact points of C_{1} and C_{2}, namely the Jacobian of the two pencils.

A curve C_{1} meets L at the points of the Jacobian group of the series g_{s}^{1} cut out on C_{1} by |C_{2}|, thus at 2s + 2π_{1} − 2 points away from the n_{1} base points that also belong to L. Similarly a C_{2} cuts L at 2s + 2π_{2} − 2 variable points.

Now on the curve L, whose genus we denote by π, we have two ∞^{1} series, that we shall denote by g_{1} and g_{2}, of degrees 2s + 2π_{1} − 2 and 2s + 2π_{2} − 2 respectively. Let us consider the Jacobian group of g_{1}, consisting of

4s + 4(π_{1} − 1) + 2π − 2

points.

Let us remark that the double points of the series g_{1} on L are given by:

(1) the δ_{1} double points of the pencil |C_{1}|; indeed each of these points is double for the group cut out on L by the corresponding C_{1};

(2) the τ points where a C_{1} and a C_{2} have a triple contact, because at such points there are two infinitely near points of a C_{1} in common with L;

(3) and finally, the n_{2} base points of the pencil C_{2}; indeed each of these points appears as a fixed point for the series cut out by the C_{2} on the curve C_{1} which passes through it, hence it has multiplicity two in the Jacobian of this series, and therefore it is a point of simple contact of C_{1} and L.

Now we have

4s + 4(π_{1} − 1) + 2π − 2 = δ_{1} + τ + n_{2}.

Similarly, interchanging the role of |C_{1}| and |C_{2}| one finds

4s + 4(π_{2} − 1) + 2π − 2 = δ_{2} + τ + n_{1}.

Subtracting one deduces

δ_{1} − n_{1} − 4π_{1} = δ_{2} − n_{2} − 4π_{2}. Q.E.D.

Remark We have assumed the base points of each of the two pencils |C_{1}| and |C_{2}| to be simple and distinct, but we can get rid of these restrictions.

Assume for instance that a base point O of |C_{2}| is a multiple point of order r greater than one, with variable tangents. Then it is easy to show that this point has multiplicity 2r − 1 for the Jacobian L; indeed one checks that L completed with two curves \overline{C}_{1} and \overline{C}_{2} of the two pencils is the Jacobian of the net containing the two pencils

\overline{C}_{2} + |C_{1}| and \overline{C}_{1} + |C_{2}|,

thus it has multiplicity 3r − 1 at O; subtracting C_{2}, it follows that L has multiplicity 2r − 1 at O. Now when one looks, in the previous argument, for the multiplicity of O in the group of points section of L with the C_{1} passing through O, one finds that this multiplicity is 2r, thus that O belongs 2r times to the Jacobian group of the g_{s}^{1} on L. The point O accordingly always appears with multiplicity one in the computation of the degree of the Jacobian series of g_{1}.

Concerning the case in which one of the two pencils has some infinitely near base points, we can say, arguing by continuity, that our formula always holds, provided that one keeps track of the curves of the pencil that acquire a double point (lowering the genus π) at a base point of the pencil, or at a point infinitely near to it. It suffices to refer to the case in which the pencil |C_{1}| has a simple base point where the curves of the pencil touch each other: then the pencil contains a curve with a double point at O and another curve, infinitely near to it, that has a double point infinitely near O: these two curves must appear in the computation of δ. Therefore δ will be equal to the number of curves of the pencil with a double point awayfrom the base points plus 2.

After this, we can compute the equivalence of a curve C_{1} with an r-tuple point, awayfrom the base points, by showing that in general it gives the same contribution as (r −1)^{2} curves with a double point. Let us compare the pencil |C_{1}| containing the curve with the r-tuple point with another generic pencil |C_{2}|. First of all, as we have already remarked, the r-tuple point P will be (r − 1)-tuple for the Jacobian of the two pencils |C_{1}| and |C_{2}|. And on every branch of this Jacobian the point P appears as a (r − 1)-tuple point for the series cut out by the C_{2}; so that, repeating the previous argument, P will be counted (r − 1)^{2} times in the computation of the invariant I.

It is possible that, instead of a curve with a multiple point, there is in a pencil |C| a curve containing a whole component which is double or multiple: for instance a curve

C^{(0)} = 2χ + ψ,

where χ has some genus ρ ≥ 0 and the residual curve ψ (which may also be missing) meets χ at a certain number i ≥ 0 of points. It is not difficult to compute the equivalence of this curve 2χ + ψ, that is, the number of curves C with a double point that are absorbed by it. This computation is worked out in a paper by Castelnuovo and Enriques^{(3)}. Here we shall explain it in the simplest case, in which there no complication arising for the presence of infinitely near base points. Thus we make the assumption that the pencil has no base point. Actually this is the case in which later on it will occur to us to apply the formula to the irrational pencils.

To start with, by comparing the linear pencil |C| = |C_{1}| with another generic pencil |C_{2}|, one remarks that χ is a component of the Jacobian L of the two pencils, and that the residual curve L_{1} passes through the i common points of χ and ψ that are triple for C^{(0)}. With an argument analogous to the one used in general case, one has to compute the degree of the series cut out on L_{1} by the curves of the pencils |C_{1}| and |C_{2}|.

The degree of the series g_{m_{1}}^{1} cut out on L_{1} by the curves C_{1} is

m_{1} = 2s + 2π_{1} − 2 with s = (C_{1}C_{2}),

since that is the number of the C_{2} that touch a generic C_{1}.

In order to determine the degree of the series g_{m_{2}}^{1} cut out on L_{1} by the C_{2}, one has to keep in mind that the Jacobian group of the series g_{s}^{1} cut out on a generic C_{2} by the C_{1} contains also the group of the points of intersection of C_{2} with χ. Denoting the number of these points (C_{2}χ) by σ, one has that the curves C_{1} that touch properly a generic C_{2} are

m_{2} = 2s + 2π_{2} − 2 − σ,

and this is also the intersection number (C_{2}L). Now, denoting by P the genus of the Jacobian L_{1}, the Jacobian group of the series g_{m_{1}}^{1} on L_{1} consists of 2m_{1} + 2P − 2 points, and this group consists of:

(1) the \overline{δ} double points of the remaining nodal curves that belong to the pencil |C_{1}|;

(2) the τ points of contact of order 3 of a C_{1} and a C_{2};

(3) the n_{2} base points of the pencil |C_{2}|;

(4) the i common points of χ and ψ, each to be counted twice, since it is triple for the group cut out by C^{(0)} = 2χ + ψ on L_{1};

(5) the 2σ + 2ρ − 2 double points of the series cut out on χ by the pencil |C_{2}|, since L_{1} passes through these points, that are double for the group cut out by C^{(0)}.

(3)Castelnuovo–Enriques, Sopra alcune questioni fondamentali ecc., Ann. di Mat., 1901.

Thus we have

2m_{1} + 2P − 2 = \overline{δ} + τ + n_{2} + 2i + 2σ + 2ρ − 2.

On the other hand, for the series cut out on L_{1} by the curves C_{2} one has simply

2m_{2} + 2P − 2 = δ_{2} + τ,

since the pencil |C_{1}| has no base points.

Eliminating 2P −τ −2 from these two relations and substituting the values of m_{1} and m_{2} one obtains

\overline{δ} − 4π_{1} + 2ρ − 2 + 2i = δ_{2} − n_{2} − 4π_{2}.

But if the pencil |C_{1}| had only curves with isolated double points, whose number we denote by δ_{1}, we should have

I = δ_{1} − 4π_{1}.

Comparing expressions it follows that a curve C^{(0)} with a double component of genus ρ intersecting the residual curve at i points must be counted as

δ_{0} = 2ρ − 2 + 2i

curves with a double point in the computation of the Zeuthen–Segre invariant, at least under the assumption that the pencil has no base points.

The previous argument can be extended to the case in which the pencil |C| = |C_{1}| contains a curve

C^{(0)} = hχ + ψ,

containing a multiple component with multiplicity h (greater than 1), without double points, meeting the residual curve ψ at i points.

In this case one has

δ_{0} = (h − 1)(2r − 2) + hi.

More generally, assume that the pencil |C| contains a reducible curve

C^{(0)} = h_{1}C_{1} + h_{2}C_{2} + ⋯ + h_{t}C_{t}

consisting of t simple or multiple components χ (h_{r} ≥ 1), and that the curves χ_{r} and χ_{s} have i_{rs} points in common. The equivalence of this curve in the

Irrational pencil 5.2 166

computation of the Zeuthen–Segre invariant is given by the Godeaux formula^{(4)}

δ_{0} = ∑(h_{r} − 1)(2ρ_{r} − 2) + ∑ i_{rs}(h_{r} + h_{s} − 1).

This formula can be applied to all cases, provided that one takes into account two modifications:

(1) If one of the curves, χ_{1}, has a double or multiple point, then δ_{0} must be increased accordingly: in the case of a double point, taking ρ_{1} to be the effective genus, one has to add 2h_{1} − 1 to δ_{0}; on the other hand, if ρ_{1} represents the virtual genus of χ_{1}, then δ_{0} has to be increased only by one.

(2) In the case in which the pencil |C| has infinitely near base points on a multiple component χ_{1} of C^{(0)}, it is possible to eliminate these points with a transformation of the surface that sends them into exceptional curves: this is equivalent to adding one or several neighbourhoods of base points to the χ_{r}. With this convention the Godeaux formula holds also in the case of infinitely near base points.

5.2 Irrational pencil

The Zeuthen–Segre invariant of a surface F can be computed not only for the linear pencils, but also for the irrational pencils belonging to F.

Precisely, the invariant I can be expressed in relation with a pencil of genus ρ > 0 consisting of curves of genus π and containing Δ curves with a double point by the formula

I = Δ + 4(ρ − 1)(π − 1) − 4^{(5)}.

This formula can be proved directly, by comparing again two pencils |C_{1}| and |C_{2}| and replacing one of them with the irrational pencil {C} that we are considering here. But it is also possible to reduce the proof to the case of two linear pencils, by replacing {C} with a linear pencil consisting of the groups of n curves C of a g_{n}^{1}, where n is a large enough integer, equal for instance to ρ + 1. Indeed the fact that one of the two pencils consists of reducible curves does not affect the arguments developed in the previous section.

(4)L. Godeaux, Sur le calcul de l’invariant de Zeuthen–Segre. Mémoires de la Société des Sciences du Hainaut, t. 66, 1920. See Bull. de l’Académie Royale de Belgique, 13 oct.

1934. See also Campedelli, Sul computo dell’invariante di Zeuthen–Segre per una superficie algebrica; Ancora sul computo ecc., Rendic. Acc. Lincei, s. 6, t. XIX, 1934.

(5)See Castelnuovo–Enriques, Ibidem.

The Zeuthen–Segre invariant in terms of the genera 5.3 167

Thus the groups of n curves C of the aforementioned g_{n}^{1} form a linear pencil of curves, of genus:

Π = nπ − n + 1

without base points; and in this pencil there will be:

(1) Δ curves containing a C with a double point and

(2) 2n + 2ρ − 2 curves containing a C of genus π counted twice, each equivalent to 2π − 2 curves with a double point; thus we have

I = Δ + (π − 2)(2n + 2ρ − 2) − 4Π = Δ + 4(ρ − 1)(π − 1) − 4. Q.E.D.

5.3 Expressing the Zeuthen–Segre invariant in terms of the genera

We have said that the Zeuthen–Segre invariant I is a relative invariant of the surface F; in fact it increases by one when a point O of F is transformed into an exceptional curve. This statement can be verified in two ways: if one takes a point O of F that is not a base point and is simple for a curve of the pencil |C|, then transforming O into an exceptional curve increases by one the number δ of curves with a double point; on the other hand if O is a base point of the pencil |C|, then the transformation decreases the number of base points by one.

Let us remark now that a transformation of the surface F that replaces a point O with an exceptional curve affects the invariant I and the linear genus p^{(1)} in opposite ways, since we know that the latter decreases by one. Therefore the sum

I + p^{(1)}

is an absolute invariant of the surface. But this is not a new invariant, since Noether has shown that it can be expressed in terms of the numerical genus by of the formula^{(6)}

I + p^{(1)} = 12p_{a} + 9.

This expression is related to a set of formulas involving several invariant numerical characters of the surfaces. We shall prove it here following the way indicated by T. Bonnesen^{(7)}.

Let us refer to a simple irreducible ∞^{3} linear system of curves |C|, that we may regard as plane sections of a surface F with a double curve D; for the sake of simplicity, we shall assume that d is irreducible and has only t triple points that are also triple for F. The invariant I of the surface F can be computed with respect to a pencil of plane sections C, and it is given by:

(6)Noether, Ibidem, Math. Ann., VIII.

(7)P. Bonnesen, Sur les séries linéaires triplement infinies de courbes algébriques sur une surface algébrique. Bull. de l’Académie Royale des Sciences et des Lettres de Danemarque, 1906, n. 4.

I = δ − n − 4π,  (5.1)

where n is the order of F, π the genus of the C and δ denotes the class of the surface F. The class of F is also given by the effective degree of the system |C_{j}| cut out on F by the polar surfaces Φ_{n−2}; this system has the τ cuspidal points of F on D as simple base points, so that δ + τ equals the degree of

|C_{j}| = |3C + K|

computed without taking into account the τ stated base points. Thus introducing the linear genus p^{(1)} (genus of the impure canonical curves K) we have

δ + τ = 3n + 12(π − 1) + p^{(1)} − 1.  (5.2)

Now there are some characters (projective characters of the surface F) that are related to the ∞^{3} system |C| and in particular the following:

(α) the degree d of the double curve D, that depends on the degree n and on the genus π of |C|:

d = \frac{(n − 1)(n − 2)}{2} − π;  (5.3)

(β) the genus ρ of the double curve D;

(γ) the number t of triple points of D, that are also triple for the surface F; moreover, denoting by Δ the curve that corresponds to the double curve D on a nonsingular model F′ of the surface F:

(δ) the (effective) genus Π of the curve of the neutral pairs of |C|, that is, the curve Δ which is represented on the double curve D with τ branch points at the cuspidal points. Note that the curve Δ has t triples of double points corresponding to the t triple points of D, so that the virtual genus of Δ is

Π + 3τ.

In addition to (??) and (??), there are other relations involving the aforementioned characters. From these relations it follows that besides the two characters n and π of the system |C|, only two characters of the surface assume independent values, and are thus up to a point arbitrary.

First of all we remark that the covariant curve Δ of the ∞^{3} system |C| is a partial curve of the system |(n − 4)C| on the surface F′ (a transform of F where the double curve D is resolved); more precisely, adding a canonical curve K to Δ one obtains

|Δ + K| = |(n − 4)C|.

Therefore the intersections of Δ with the curves of the adjoint system |Δ+K| system give a canonical group on the curve Δ of virtual genus Π+3τ, and thus we have

2(Π + 3t) − 2 = 2(n − 4)d,  (5.4)

where d is expressed by the formula (??).

On the other hand, since the double curve D is in 2-to-1 correspondence with Δ and there are τ branch points on it, by a well known formula of Zeuthen’s we have

2(2ρ − 2) + τ = 2Π − 2.  (5.5)

Finally, we may obtain a sixth relation, by computing the number of intersections of a C_{j}, the Jacobian of a net in |C|, with Δ, or, what is the same, with D. To do this, we remark that

|C_{j}| = |3C + K| and |Δ + K| = |(n − 4)C|;

as a consequence the number of the intersections (C_{j}D) = (C_{j}D) is given by

(3C + K)[(n − 4)C − K] = 2n^{2} − 7n + 15 + 2π(n − 7).

But this same number (C_{j}D) can be also computed directly on the surface F, by looking for the number a of the intersections of D with the second polar Φ_{n−2} of a generic point O; indeed the C_{j} cut by the polar of O meets the double curve D: at the τ cuspidal points and at the a common points of D and Φ_{n−2} (away from the triple points of D); at each of these points the tangent plane to a branch of F passes through O. Thus we have

(C_{j}D) = a + t, where a = (n − 2)d − 3t.

Therefore we may write

2n^{2} + n + 14 + 2π(n − 7) − p^{(1)} = (n − 2)d − 3t.  (5.6)

It is convenient to add another relation involving p_{a} to the six relations that we have just written: one has

p_{a} = \frac{(n − 1)(n − 2)(n − 3)}{6}

−(n − 4) [\frac{(n − 1)(n − 2)}{2} − π] + 2t + ρ − 1

= \frac{1}{6}(n − 1)(n − 2)(9 − 2n) + π(n − 4) + 2t + ρ − 1.

We have found seven equations involving the eleven characters: n, π, δ, I, d, ρ, Π, τ, t, p_{a} and p^{(1)}.

Of these characters, n and π can take arbitrary values, up to a point, since they depend on the choice of a pencil |C| on the surface F; therefore it is a priori clear that the seven characters

δ, I, ρ, Π, τ, t,

can be expressed as functions of π and n and of the invariants p_{a} and p^{(1)} of the surface. In particular one has to express I as a function of p_{a} and p^{(1)}: a priori one may think that this function also depends on π and n, but in fact it has to be independent of them, since it is an invariant character of F.

Thus one only has to perform the computation, and since (??) and (??) already contain an explicit expression of I and d, one has to solve five linear equations, in which n, π, p_{a} and p^{(1)} appear as parameters. Thus one obtains:

δ = n + 4π − p^{(1)} + 12p_{a} + 9
ρ = \frac{1}{2}(n^{2} − 7n + 48) + (n − 12)π + 9p_{a} − 2p^{(1)}
Π = n^{2} − 6n + 2(n − 10)π + 12p_{a} − 3p^{(1)} + 36
t = \frac{1}{6}(n^{3} − 6n^{2} + 26n − 78) − (n − 8)π − 4p_{a} + p^{(1)}
τ = 2n + 8π − 12p_{a} + 2p^{(1)} − 22.

From the first formula one deduces Noether’s expression of the Zeuthen– Segre invariant:

I = 12p_{a} − p^{(1)} + 9.

Note The four formulas for ρ, Π, t and τ express four relative invariants constructed with the characters of an ∞^{3} system |C| belonging to the surface. The last formula in particular is worthy of note, since it involves n and π linearly: it defines the invariant

2n + 8π − τ = 12p_{a} − 2p^{(1)} + 22.

Combining this formula with the first one, one obtains another very simple relative invariant^{(8)}, that is,

2δ − τ = 36p_{a} − 4p^{(1)} + 40.

This invariant is constructed with two characters of an ∞^{3} linear system |C| on the surface; δ denotes, in general, the number of curves C of a pencil that have a double point (away from the base points) and t denotes the number of pencils contained in the ∞^{3} system |C| that have a double base point (away from the base points of |C|).

(8)See Bonnesen, Ibidem.

Remark The expression of the Zeuthen–Segre invariant in terms of the numerical genus and of the linear genus suggests another way of finding the linear genus of the plane and of the ruled surfaces, that we computed in Chapter II, Section ??.

To start with one may compute the invariant I of the plane, by considering a pencil of conics, for which one has:

• the genus π=0,

• the number of base points n = 4,

• and the number of curves with a double point δ = 3.

We have

I = δ − n − 4π = −1;

therefore from the formula

I = 12p_{a} − p^{(1)} + 9

and since p_{a} = 0, one obtains

p^{(1)} = 10.

Now we turn to the irrational ruled surfaces of genus p_{a} = −p (with p > 0). Here one may compute the value of I in relation to the pencil of generating lines, which is of genus p and for which

π = 0, Δ = 0, n = 0.

We have

I = Δ + 4(p − 1)(π − 1) − 4 = −4p.

Now from Noether’s formula one deduces that

12p_{a} − p^{(1)} + 9 = −4p

and since p_{a} = −p, then

p^{(1)} = −8(p − 1) + 1.

The same result can be found by considering a linear pencil of plane sections instead of the irrational pencils of the generating lines.

Cuspidal curves of a net 5.4 172

5.4 Cuspidal curves of a net

We turn again to the consideration of the ∞^{3} linear system |C| and to the characters of it that we have described in the previous section. Using the formulas we have proved we may determine the characters of a net of curves |C|, and in particular the number χ of the curves of a net having a cusp.

To do this, let us consider the surface F with the curves |C| as plane sections. The locus of points of F for which the tangent plane section has a cusp is the parabolic curve C_{h} of F, the section of F with the Hessian surface H. Thus the χ plane sections with a cusp passing through a point O of the space correspond to the intersections of C_{h} with the curve C_{j}, the Jacobian of the net cut out by the planes through O, awayfrom the cuspidal points of F. These are simple points of C_{j}, and double for the parabolic curve C_{h}, as we shall see.

Therefore in order to compute χ we have to:

(1) determine the behaviour of the Hessian at the nodal double curve of the surface F;

(2) then find the multiplicity that curve C_{h} has at the τ cuspidal points of F.

The first computation can be performed by simply replacing the given surface with another one, that we still temporarily denote by F, that has a biplanar double point at the origin, with x and y as tangent planes; in homogeneous coordinates x, y, z, u, we write:

F = u^{n−2}xy + u^{n−3}φ_{3}(x, y, z) + ⋯ + φ_{n}(x, y, z).

The Hessian of F is

H = | \begin{matrix}\frac{∂^{2}F}{∂x^{2}} & \frac{∂^{2}F}{∂x∂y} & \frac{∂^{2}F}{∂x∂z} & \frac{∂^{2}F}{∂x∂u} \\ \frac{∂^{2}F}{∂y∂x} & \frac{∂^{2}F}{∂y^{2}} & \frac{∂^{2}F}{∂y∂z} & \frac{∂^{2}F}{∂y∂u} \\ \frac{∂^{2}F}{∂z∂x} & \frac{∂^{2}F}{∂z∂y} & \frac{∂^{2}F}{∂z^{2}} & \frac{∂2F}{∂z∂u} \\ \frac{∂^{2}F}{∂u∂x} & \frac{∂^{2}F}{∂u∂y} & \frac{∂^{2}F}{∂u∂z} & \frac{∂^{2}F}{∂u^{2}} \end{matrix} | = 0

Making the computation, one finds that the behaviour of the Hessian at the origin of coordinates (x = y = z = 0, u = 1) is determined by the Hessian of the approximating surface of order three, so H has a quadruple point at a biplanar point of F. Now, going back to our surface F with a nodal curve, one deduces that its Hessian H of degree 4n − 8 passes through the stated nodal curve four times. Therefore the parabolic curve C_{h} belongs to the linear system

|4K + 8C|,

where, as usual, K denotes the canonical curves of the surface.

Now we see that the τ cuspidal points of F (simple for the C_{j}) are double for the parabolic curve C_{h}. Indeed, considering a transformed surface on which the nodal curve of F is resolved, on such a surface a cuspidal point of F gives rise to a double base point of a pencil of |C|. Hence in that pencil there are two curves with a cusp.

Now the number of cuspidal curves of a net contained in |C| is given by

χ = (C_{j} · C_{h}) − 2τ.

But, since

|C_{j}| = |3C + K|, |C_{h}| = |8C + 4K|,

one has

(C_{j} · C_{h}) = 24n + 20(2π − 2 − n) + 4p^{(1)} − 4,

and recalling the expression for τ

τ = 2n + 8π − 12p_{a} + 2p^{(1)} − 22

from the previous section, it follows that

χ = 24(p_{a} + π).

Thus we arrive at the relation discovered by Zeuthen: the number of cuspidal curves in a net of genus π is in general 24(p_{a} + π).

To be precise, we have proved this formula for a net of curves C that is contained in an ∞^{3} system. But it is easy to extend the above argument considering an ∞^{3} system |L + C| that contains |C|. And so the formula can be verified in every case^{(9)}.

(9)The proof of the formula given in the text can also be explained in a slightly different form, without using the properties of the Hessian. To do this, it is convenient to associate to the surface F its dual F′, which is the transform of F obtained by taking the Jacobians C_{j} of the nets of |C| as plane sections. Now if one constructs the dual surface of F′ one find F back. This means that the Jacobian curves of the nets of C_{j} on F, say the C_{jj}, coincide with the plane sections C, apart from a fixed component, that one proves to be the parabolic curve C_{h}.

Indeed consider the polars with respect to F of the points of a plane α, which form a net.

5.5 Historical note

The relation between the number of cuspidal curves of a net and the numerical genus was discovered by H.G. Zeuthen^{(10)}. Assuming the invariance of the numerical genus (a fact known by Zeuthen and Noether, although under some restriction on the singularities of the surface), Zeuthen proved that the number c − 24π, computed for a net (of plane sections), is independent of the net and is therefore an intrinsic character of the surface. F. Severi gave a direct proof of this theorem in 1902, with a very elementary argument^{(11)}.

The proof described in this text can be found in Bonnesen, Ibidem^{(12)}. We cannot conclude this historical note, that completes the information

given in the previous sections, without recalling a circle of ideas in which one looks for the functional geometric interpretation of the enumerative formulas that we have found. It is known that this point of view was introduced in the theory of curves with the theorem of Painlevé–Castelnuovo (giving the interpretation of Zeuthen’s formula for [1, n] correspondences) and with the consideration of the Jacobian series of a linear series, that Enriques recognized as the simplest way of defining the canonical series and of proving its invariance.

In order to find an analogous interpretation for the enumerative geometry on surfaces, Severi tried, in a series of memoirs since 1932, to construct a theory of the series of groups of points, that he called of series of equivalence, that should be a natural extension of the theory of linear series on a curve.

Characters of a net and multiple planes 5.6 175

For a large number of references on the subject see F. Severi’s book: Serie, sistemi di equivalenza e corrispondenze algebriche sulle varietà algebriche, Roma, ed. Cremonese, 1942.

It is clear that the plane section C of α belongs to the Jacobian C_{jj} of this net, because the polar of a point P of C intersects F in a curve that passes doubly through P. Moreover, if one takes a point A on the parabolic curve C_{h}, the tangent plane at A meets α along a line a, and the polars of the points of α intersect F along the curves of a pencil that has a base point with fixed tangent at A. Among the curves of this pencil there is one with a double point at A; so the curve C_{h} is a part of the second Jacobian C_{jj}.

Thus from the relation

|C_{j}j| = |C_{h} + C| it follows that

|C_{h}| = |8C + 4K|.

(10)H.G. Zeuthen, Etudes géométriques etc., Math. Ann., Band IV, 1871. See the memoir of Noether’s, Ibidem, in Math. Ann., VIII.

(11)F. Severi, Il genere aritmetico ed il genere lineare in relazione alle reti di curve tracciate sopra una superficie algebrica. Atti Acc. Torino, Vol. XXXVII, 1902.

(12)See also M. PANNELLI, Sui sistemi lineari triplamente infiniti di curve tracciati sopra una superficie algebrica. Rendic. Circ. Mat. di Palermo, t. XX, 1905. The remark about the Jacobian of a net of Jacobians, that we have made in one of the previous footnotes, was also made by E. Caporali.

5.6 Characters of a net and multiple planes

With the formulas we proved in the previous sections we may also express the characters of a net of curves |C| in terms of two of characters, e.g., the genus π and the degree n, and of the invariants of the surface: p_{a} and p^{(1)}.

Let |C| be a base point free net of irreducible curves (of genus π and degree n) on the surface F; in general we may attach to it the following numerical characters:

(1) the number N of intersections of the curves C with the Jacobian C_{j} of

the net:

N = 2n + 2π − 2;

(2) the number δ of curves C of a pencil having a double point; this number, as we know, is expressed by

δ = n + 4π + 12p_{a} − p^{(1)} + 9;

(3) the genus of the above Jacobian C_{j} (= 3C + K)

P = p^{(1)} + 9π − 9;

(4) the number k of contact points of order 3 of the C of the net (base points of a pencil of osculating C);

(5) the number i of curves C of the net |C| having a cusp;

(6) the number d of the double contact points of the C of the net (base points of a pencil of bitangent C);

(7) the number t of the C of the net having two double points.

We may assume that the net of the C is cut out on F by the star of planes through a point O, which in general will be a multiple point of the surface and will correspond to an irreducible curve L (adding which it to |C| one obtains the plane sections system).

Then F is projected from O onto a n-tuple multiple plane, the points of which correspond to groups of n points of F, belonging to the involution I defined by the net |C|. And the above characters acquire the following meaning:

(1) N is the order of the branch curve D of the n-tuple plane (curve of points of the plane corresponding to groups of I with two coincident points);

(2) P is the genus of the branch curve D;

(3) δ is the class of the branch curve D;

(4) k is the number of lines through O having a contact of order three with the surface F, and therefore it is the number of cusps of D;

(5) i is the number of cuspidal plane sections of F through O, and therefore it is the number of flexes of the branch curve D;

(6) d is the number of lines through O that are bitangent to F and therefore it is the number of nodes of D;

(7) t is the number of planes through O bitangent to F, and so it is the number of double tangents (with distinct contact points) of D.

These characters are related by the Plücker formulas for the curve D, that in general is irreducible and has no other singular points and no other exceptional tangents.

In this way one can prove the following relations:

(A) \{ \begin{matrix}N & = & 2n + 2π − 2 \\ P & = & p^{(1)} + 9π − 9 \\ δ & = & n + 4π + 12p_{a} − p^{(1)} + 9 \\ k & = & 3(n + 6π + p^{(1)} − 4p_{a} − 11) \\ i & = & 24(π + p_{a}) \\ d & = & 2[(n + π)^{2} − 5n − 17π − 2p^{(1)} + 6p_{a} + 24] \\ t & = & \frac{1}{2}[(n + 4π + 12p_{a} − p^{(1)})^{2} \\ & & +15n − 6π + 132p_{a} − 17p^{(1)} + 74] \end{matrix}

These formulas in general express the characters of an irreducible base point free net |C| on the surface, or else the Plücker characters of the branch curve D of the corresponding multiple plane, in terms of the genus π and of the degree n of |C| and of the invariants p_{a} and p^{(1)} of the surface.

(1, n) correspondence between two surfaces: double planes 5.7 177

Conversely these invariants are expressed in terms of the characters of D by the formulas:

(B) \{ \begin{matrix}p^{(1)} & = & P − 9π + 9 \\ p_{a} & = & \frac{i}{24} − π \\ & or else \\ 4p_{a} & = & n + P − 3π − \frac{k}{3} − 2 \\ (π & = & \frac{N}{2} − n + 1). \end{matrix}

Remark The fourth and fifth formulas among the (A) are especially important, as they compute two invariant characters of the net |C| that depend linearly on n and π, namely 3(n + 6π − k) and 24π − i.

But the latter of these two formulas is particularly important, as it involves the absolute invariant p_{a}. As we have said, these relation between p_{a} and the number of cuspidal curves of a net was discovered by G.H. Zeuthen. And F. Severi showed in a direct and elementary way that the expression 24π − i is equal for all base point free nets of the surface F. This gives a new proof of the invariance of p_{a}.

5.7 (1, n) correspondence between two surfaces: double planes

Let us consider a (1, n) correspondence between two surfaces f and F: a given point of F corresponds to one point of f, whereas the points of f correspond to the groups of an involution I of degree n on F (every point of F belongs to one group of I). For the sake of simplicity we assume also that there are no fundamental points, that is, points of f or F which in the correspondence are associated to a curve on the other surface.

Given an irreducible linear system |c| on f, there is a linear system |C| on F corresponding to it and belonging to the involution I. The Jacobian |c_{j}| of a net contained in |c| is related in the correspondence with a curve of F contained in the Jacobian of the corresponding net of the homologous |C|. But in order to obtain the whole Jacobian of this net it is necessary to add the branch curve D, that corresponds to the branch curve on f; indeed a point of d is a base point of a pencil of |C| contained in the aforementioned net, which has a fixed tangent, and therefore contains a C with a double point. Now recalling the definition of the canonical curves (|K| = |C_{j} − 3C|) one deduces the theorem:

Let us consider a (1, n) correspondence between the surfaces f and F; then canonical curves of F are obtained by adding the curve homologous to the branch curve to the curves of F corresponding to the canonical curves of f^{(13)}.

This statement has to be modified when there is a fundamental point on f, corresponding to a curve on F (or to a curve and a group of points); this fundamental curve must be added to the images of the canonical curves of f and to the branch curve, in order to make a canonical curve of F.

The theorem we have just stated holds even if the canonical curves of f are virtual. We may apply it to the problem of determining the canonical and pluricanonical curves of a double plane.

Let F be a surface with a rational involution I of order two, such that F is represented on the double plane

z^{2} = f_{2n}(x, y)

where the branch curve f_{2n} of degree 2n may have singular points, the multiplicity of which has to be computed following the conventions of Chapter III, Section ??. Consider the curve φ_{m} of the double plane belonging to a linear system with no base points away from the singular points of f_{2n}; the adjoints of the corresponding curve of φ_{m} on F are obtained by adding the branch curve homologous to f_{2n} to the transforms of the φ_{m−3} adjoint to f_{m} in the plane. If one considers a biadjoint φ_{m−6} instead of an adjoint φ_{m−3}, then one has to add the branch curve counted twice to the transform of this curve, and so on.

The virtual canonical curves K on the plane are defined as differences

|K| = |φ_{m−3} − fm|,

hence one finds canonical curves on F by adding the K to the branch curve homologous to f_{2n}; thus adding these curves to the curves homologous to the φ_{m} one obtains the adjoints of the φ_{m}.

Let us apply this result by taking, instead of the φ_{m}, the branch curve f_{2n} and its biadjoints f_{2n−6}. In order to have the biadjoints on F of the branch curve counted twice, that corresponds to f_{2n}, it suffices to add this curve counted twice to the transforms of the f_{2n−6}. Subtracting the branch curve counted twice from the system thus obtained, one has the bicanonical curves of F; among these there are those belonging to the involution I, having the f_{2n−6} as images.

Thus: the biadjoint curves of the branch curve of the double plane are images of bicanonical curves belonging to the involution I. This result agrees with what we saw in Chapter III, Section ??: the canonical curves of the surface are represented by the f_{n−3} belonging to the half of the biadjoint system of the branch curve.

(13)See Enriques, Ricerche etc.

Correspondence formulas 5.8 179

The argument can be extended to pluricanonical curves: the tricanonical curves of the double plane are obtained by adding to the branch curve the third adjoints of this curve, and so on.

But let us go back to the bicanonical curves, in order to remark that, we can see also in another way that one obtains bicanonical curves of the double plane by adding the branch curve to the adjoints of the curves f_{n−3}, images of the canonical curves; it suffices to recall that the bicanonical system cuts out groups of the canonical series on the curves of the canonical system.

Now remark that the bicanonical system is preserved by the involution I, whereas its curves in general do not belong to this involution. Thus we must regard this system as a linear space of dimension P − 1 (where P denotes the bigenus of F) on which I induces a projective involution with two subspaces of fixed points S_{r} and S_{t}, where

P − 1 = r + t + 1.

One of the two spaces, say S_{r}, corresponds to the system of f_{2n−6}, biadjoints of f_{2n}, while the other space S_{t} corresponds to the system consisting of the branch curve f_{2n} taken together with the adjoints f_{n−6} of the images of the canonical curves. Thus, denoting by π = t + 1 the genus of the f_{n−3}, one has

P − 1 = r + π + 1.

This formula expresses the bigenus of the double plane in terms of the genus r + 1 of the adjoint curves f_{2n−3} of the branch curve and of the genus π of the curves that are one half of the biadjoints f_{2n−6}, that is, of the genus of the involution of order two that I induces on a canonical curve.

These results can be easily extended to the plurigenera. We shall just remark that if all the plurigenera vanish, then there are no adjoints of the branch curve of any order, and so the curve is birationally equivalent either to a curve of degree 2n with a (2n − 2)-tuple point or to a quartic or to a sextic with two infinitely near triple points, or to a specialization of one of these^{(14)}.

Therefore we can state: A double plane having all the genera equal to zero either is rational or it can be transformed into a hyperelliptic ruled surface.

5.8 Correspondence formulas

A first consequence of the theorem given in the previous section, about the transformation of the canonical curves in a (1, n) correspondence, is that the genus P_{g} of F cannot be, in any case, smaller than the genus of f:

(14)Theorem of Castelnuovo–Enriques, 1900. See Enriques–Conforto, Le superficie razionali, §27, page 489.

P_{g} ≥ p_{g}.

Furthermore the same theorem immediately leads to a relation between the linear genera P^{(1)} and p^{(1)} of the two surfaces.

It is convenient to make first a remark concerning the branch curve D of the correspondence on f. In general this curve will have a certain number of nodes that correspond on F to groups of n points having two distinct coincident pairs. In addition, D will have a certain number of cusps; each of these corresponds to a group of n points, of which three coincide. Now according to a well-known theorem of Zeuthen, the curve corresponding to a n-tuple canonical curve on f that meets the branch curve D at s points has genus

n(p^{(1)} − 1) + \frac{s}{2} + 1.

This curve meets the branch curve D at s points, so that denoting by ρ the effective genus^{(15)} of D, one has

P^{(1)} = n(p^{(1)} − 1) + ρ + 3s/2.

This relation connects the linear genera of the two surfaces in (1, n) correspondence^{(16)}. One has to set ρ = 1 and s = 0 in it when there is no branch curve (that is, it is of degree zero on a nonsingular f).

Together with the previous formula, one can write another relation involving the numerical genera P_{a} and p_{a} of F and f^{(17)}.

To do that, we take a pencil |c| of curves on f and the corresponding pencil |C| of curves on F. We compute the Zeuthen–Segre invariants i and I of the two surfaces using the characters of these two pencils and write the relation that connects them.

Thus denote by m the number of base points of |c|, by π the genus of the c, and by δ the number of c having a double point; we have

i = δ − m − 4π.

Now the number M of base points of the pencil |C|, and the genus Π of the C are given respectively by

M = mn and P = n(π − 1) + 1 + \frac{t}{2},

(15)We assume that the only singularities of d are the nodes and the cusps that have the meaning explained earlier with respect to our correspondence.

(16)Enriques, Ricerche etc., 1893.

(17)See Severi, Sulle relazioni che legano i caratteri invarianti di due superficie algebriche.

Rendic. Ist. Lombardo, 1903.

where t denotes the number of intersections of a curve c with the branch curve D on f. As for the number Δ of curves C with a double point, a first set of these is given by the δ curves corresponding to the c with a double point, each of which has be counted n times, since it has n double points. But in general there will be other Δ − nδ curves obtained from the c that touch the curve D, except those having an improper contact with D, that is, those passing through a cusp of d, which corresponds to a point of F where three of the n corresponding points come together.

Thus, denoting by ρ the effective genus of the branch curve D, and by χ the number of its cusps, one has

Δ = nδ + 2t + 2ρ − 2 − χ,

thus the Zeuthen–Segre invariant of F is expressed by

I = Δ − M − 4Π = nδ + 2ρ − 2 − χ − nm − 4n(π − 1) − 4.

Now recalling Noether’s relation

i = 12p_{a} − p^{(1)} + 9
I = 12P_{a} − P^{(1)} + 9,

one deduces

24(P_{a} + 1) = 24n(p_{a} + 1) + 6ρ − 6 + 3s − 2χ.

This relation connects the numerical genera of the two surfaces f and F in (1, n) correspondence, under the simplifying assumption that there are no fundamental points of the correspondence on f. Here again, if there is no branch curve, one has to set

ρ = 1, s = 0.

The formulas relating the numerical genera and the linear genera of two surfaces f and F in (1, n) correspondence can also be written in a different form, introducing the degree ν of the curve D instead of the number s that denotes the number of intersections of the branch curve D of f with the canonical curves k: this is a virtual degree expressed in terms of s and of the virtual genus ρ + ε + χ of d, where ε denotes the number of nodes of d and χ the number of cusps. We have

s = 2(ρ + ε + χ) − 2 − ν

and so the formulas relating the characters of f and F become

P^{(1)} = n(p^{(1)} − 1) + 4ρ + 3(ε + χ) − \frac{3}{2}ν − 3
24(P_{a} + 1) = 24n(p_{a} + 1) + 12ρ − 12 + 6ε + 4χ − 3.

The first formula can be also deduced by comparing the degrees of the canonical system on f and F. We recall that one obtains a canonical curve K of F by adding the branch curve D to the transform of a k of f that meets it at s points; denoting by x the degree of D, we have

P^{(1)} − 1 = n(p^{(1)} − 1) + x + 2s.

This formula agrees with the previous one, as one sees by computing the value of x which is

x = \frac{ν}{2} − (ε + χ).

In particular, if on f there is a branch curve or one of its components of degree n, which is completely free of double points, the branch curve corresponding to it on F has degree ν/2. This especially is clear in the case of a (1, 2) correspondence, n = 2, where in general the branch curve has no double points: this curve of degree ν corresponds to a double curve of degree 2ν, the component of which has therefore degree ν/2.

It is interesting to compare the formulas we obtained here with the formulas yielding the numerical genus and the linear genus of multiple planes given in Section ??. We should find again the same formulas, bearing in mind that the (relative) linear genus of the plane is p^{(1)} = 10^{(18)}, while its geometric genus is p_{a} = 0.

To do that, consider again the notation of formulas (B) of Section ??, thus replacing ρ with P, χ with k, ε with d and ν with N^{2}, where

N = 2n + 2π − 2.

With this substitutions, our formulas yield

P^{(1)} = 9n + 4P + 3(d + k) − \frac{3}{2}N^{2} − 3

and since

P = \frac{(N − 1)(N − 2)}{2} − (d + k),

one deduces

P^{(1)} = P − 9π + 9;

which is the first formula of (B).

We have thus found again the formulas that give the characters of a multiple plane.

(18)See Chapter II, Section ??; Chapter V, Section ??.

EXAMPLES We use again the notation of this section and apply the formulas thus obtained to some interesting examples.

Consider the Kummer surface, that is, the singular surface of the quadric complex of lines in space, that is defined as a surface f_{4} of degree 4 with 16 double points^{(19)}. It is known that f_{4} coincides with its dual surface and has 16 bitangent planes; each of these touches the surface along a conic containing 6 double points. This follows immediately from the representation of f_{4} on a double plane with reducible branch sextic consisting of 6 lines; such a representation is obtained by projecting f_{4} from one of its double points. Now consider 3 bitangent planes of f_{4} passing through a double point A; the pairwise intersections of these planes are three lines, joining A with three double points B, C, D; hence it follows that the tetrahedron ABCD, whose vertices are double for f_{4}, contains simply the remaining 12 double points of f_{4}, three on each face.

Now denoting by T (x, y, z) = 0 the equation of this tetrahedron, we construct in the space S_{4}(xyzu) the surface F of equation

u^{2} = T (x, y, z), (f_{4}(x, y, z) = 0),

which is represented on the double surface f_{4}.

It is important to remark that there is no branch curve of order greater than zero on this double surface. But there are 16 infinitesimal branch curves, consisting of the neighbourhoods of the 16 double points, through which T passes an odd number of times. From this it follows that our double f_{4} is irreducible; on the other hand this also follows from Comessatti’s criterion^{(20)}, by remarking that on every quartic plane section of f_{4} there are 8 apparent critical points occurring at the contact points of a quadrilateral, which do not belong to a conic. In the (1, 2) correspondence between f_{4} and F, the system of plane sections of f_{4}, that has genus π = 3 and degree n = 4, corresponds to a linear system of genus 5 and degree 8 = (2 · 5 − 2) on F. One deduces that F has, as well as the double f_{4}, genus

p_{g} = 1

and has, as the double f_{4}, a canonical curve of degree zero, hence all its plurigenera are equal to one.

Thus we a priori know that the (absolute) linear genus of F is P^{(1)} = 1. But let us compute it by means of our formula:

P^{(1)} = n(p^{(1)} − 1) + 4ρ + 3(ε + χ) − \frac{3}{2}ν − 3.

(19)See Enriques–Chisini, Lezioni etc., Libro VI, §40, Vol. IV, page 246.

(20)We recall that according to Comessatti’s criterion the necessary and sufficient condition for an irreducible curve C, tangent to the branch curve D_{2m} at all the intersection points, to be the image of a reducible curve is that the group of the points of contact belong to the series cut out on C by the curves D_{m} that are halves of the branch curve.

To do this, remark that the branch curve D on f_{4} consists of the 16 neighbourhoods of the 16 double points; each of these is an infinitesimal curve of genus zero and degree −2 (with ε = χ = 0). Thus we find

P^{(1)} = −4 · 5 + \frac{3}{2} · 32 − 3 = −15.

This paradoxical conclusion is clarified if one considers that a priori each branch curve of genus zero and degree −2 corresponds to a curve of genus zero and degree −1, which is exceptional, and which in our case occurs as the neighbourhood of a simple point of F. To construct the canonical curve of F, we have added the transform of the branch curve to the transform of the canonical curve of degree zero of f_{4}; as a consequence we have computed P^{(1)} regarding the neighbourhoods of the 16 simple points as exceptional curves. In other words, we have calculated the linear genus of the transformed surface of F with 16 exceptional curves. Eliminating these curves we find, as we should,

P^{(1)} = 1.

Then computing the numerical genus of F, we have

24(P_{a} + 1) = 96 − 12 · 15 − 12 + 3 · 32 = 0

that is,

P_{a} = −1.

Remark As F. Klein has shown^{(21)}, the Kummer surface f_{4} can be represented as an involution on the surface Φ whose elements are the pairs of points of a curve C of genus 2; precisely, regard as conjugated on C the pairs of points that are mutually residual with respect to a g_{4}^{2}. In this way one defines an involution I on F that has 16 points of coincidence, corresponding to the 16 pairs of C that are obtained by bisecting the g_{4}^{2}. Now, since I is represented by f_{4}, one has that Φ is represented by this double Kummer surface, with 16 branch points, that occur at the 16 double points. Therefore one deduces that the surface Φ is equivalent to the surface F whose equation we have written above. And thus the hyperelliptic surface representing the set of pairs of points of a curve of genus 2 has genera

p_{g} = 1, p_{a} = −1

and all plurigenera equal to 1. We shall come back again to this surface later on^{(22)}.

(21)See Enriques–Chisini, Ibidem (22)For the construction of the hyperelliptic surface as a double Kummer surface with branch curve defined by the tetrahedron T, as we have explained before, see Enriques– Severi, Mémoire sur les surfaces hyperelliptiques. Acta Math., 1909.

Another interesting example for our theory is the surface of genus p_{a} = p_{g} = 0 and of bigenus P = 1, consisting of the sextic f_{6} that passes doubly through the edges of a tetrahedron T (see Chapter II, Section ??). We construct the surface F defined by the equations

u^{2} = T (x, y, z), f_{6}(x, y, z) = 0,

which is represented on the double f_{6}. Since T and f_{6} have the 6 edges of the tetrahedron T counted twice in common, there is no branch curve of order greater than zero on f_{6}. But there are no infinitesimal branch curves either, namely no branch points; a priori these might occur only at the vertices of T; but one of these vertices on f_{6} comes from the coalescence of three simple points, whose infinitesimal neighbourhoods belong to the plane faces of the tetrahedron passing through A. Now since A is 4-tuple for the section of f_{6} with a face a, the points on a that are infinitely near to A will not be branch points of our double f_{6}. Moreover we remark that the curves C plane sections of f_{6} are irreducible plane curves (Comessatti’s criterion), hence they represent curves C of genus π = 7 and degree N = 2 · 6 = 2π − 2 on F. Since the double of |C| coincides with the double of the adjoint system:

|2C′| = |2C|,

one deduces that on F,

|C′| = |C|

hence the geometric genus of F is

P_{g} = 1.

F has a canonical curve of degree zero, so its linear genus is

P^{(1)} = 1

and all its plurigenera P_{i} = 1.

We compare the characters of F with those of the double surface f_{6} that has

p_{a} = p_{g} = 0, p^{(1)} = 1

(see Chapter II, 8). Our formulas give us

\begin{matrix}n = 2, ρ = 1, ν = 0, ε = χ = 0, \\ P^{(1)} = 2(p^{(1)} − 1) + 4ρ − 3 = 1 \end{matrix}

and

24(P_{a} + 1) = 2 · 24(p_{a} + 1) = 48,

The existence theorem 5.9 186

whence

P_{a} = 1.

The sextic f_{6} passing doubly through the edges of a tetrahedron – once desingularized – is a regular surface of genus zero and bigenus one with a bicanonical curve of degree zero, can be regarded as a double surface with all the genera equal to 1 and with canonical curve of degree zero^{(23)}.

5.9 The existence theorem

Let us consider again the multiple planes. Whenever a surface F with given genera p_{a} and p^{(1)} is represented on a multiple plane of some order n by means of a net of curves of genus π and degree n,, we have learned how to compute the characters of the branch curve of the n-tuple plane in terms of n, π, p_{a} and p^{(1)} (see Section ??).

The formulas we obtained already show that the branch curve of a n-tuple plane, for n > 2, cannot be arbitrarily taken. Actually, given any plane curve

f(x, y) = 0

one can construct the (cyclic) n-tuple plane

z^{n} = f(x, y)

whose branch curve is precisely f = 0.

But the branch curve of this n-tuple plane is not only the irreducible curve f, but the curve f counted n−1 times plus the line at infinity, counted one or several times, if the degree m of f is not a multiple of n^{(24)}. Thus the branch curve of a cyclic n-tuple plane is not at all an arbitrarily given irreducible curve.

Since the branch curve of a n-tuple plane cannot be given arbitrarily for n > 2, the problem arises of determining the conditions for a plane curve f(x, y) = 0 to be the branch curve of a n-tuple plane. These conditions lead to an existence theorem for multiple planes, or for the algebraic functions of two variables. This theorem has a meaning which is analogous, but under a certain respect different, from the existence theorem for the n-tuple lines or for the algebraic functions of one variable, given by Riemann^{(25)}. Indeed in Riemann’s case the group of the branch points can be given arbitrarily, and in correspondence with it one has a finite number of classes of algebraic functions (or of n-tuple lines), each depending on the assigned substitutions of the branches with respect to a system of loops surrounding the branch points; here instead, as we have said, the question is to determine the particular conditions that the branch curve has to satisfy for the existence of the n-tuple plane. When these conditions are satisfied, there will be, at least in general, only one class^{(26)} of surfaces represented on the plane, namely a unique ntuple plane, corresponding to one, well determined, system of substitutions with respect to the loops surrounding the branch curve, that we shall describe later.

(23)See F. Enriques, Un’osservazione relativa alle superficie di bigenere uno. Rendic. Acc.

Bologna, 1908.

(24)To see this it suffices to consider an n-tuple line or better to a n-sheeted Riemann surface, that is constructed by taking m branch points in the complex plane and associating to each of these points the cyclic substitution (123 ⋯ n) of the branches. A loop surrounding all the m branch points gives rise to a permutation of the branches expressed by (123 ⋯ n)^{m}, and this is the substitution corresponding to a loop around infinity, that therefore turns out to be a branch point, unless the above said substitution is the identity, i.e., unless m is a multiple of n.

Nevertheless, the algebraic meaning of the question is analogous to case of the construction of an algebraic function of one variable with prescribed branch points. When the surface F (x, y, z) = 0 is projected from the point at infinity Z of the z-axis onto the plane z = 0, in general one has in this plane a curve which is the locus of the critical points of z(x, y), defined by the vanishing of the discriminant of this function

Δ(x, y) = 0

that is the resultant of the two equations

F (x, y, z) = 0, \frac{∂F}{∂z} = 0.

But this curve Δ = 0 in general breaks up into two parts,

Δ = Δ_{1} · (Δ_{2})^{2},

one of which, Δ_{2} = 0, is an apparent critical curve, projection of the double curve of F, while the other component, Δ_{1} = 0 (the so-called principal part of the discriminant) is the actual branch curve of the algebraic function z(x, y). Our problem consists to assign the conditions in order that a curve be the branch curve of an algebraic function of two variables and thus to construct this algebraic function with given branch curve. The formulation of this problem can be made more precise by assuming that the plane curve f, that one wants to be the branch curve of a n-tuple plane, is a curve of even degree 2m^{(27)}, projection from a point O of the branch curve C of a surface F; so f has a certain number d of nodes and a certain number of cusps, corresponding to the apparent double points of the aforementioned curve C: in addition to these double points, f can have nodes and cusps corresponding to isolated double points of F (conical points and biplanar points respectively); but these double points of f can be left out of consideration by regarding them as virtually nonexistent.

(25)See, for instance, Enriques–Chisini, Lezioni, Libro V, Chapter III, §33 (Vol. III, page 355).

(26)See: O. Chisini, Sulla identità birazionale delle funzioni algebriche di due variabili dotate di una medesima curva di diramazione. Rendic. Ist. Lombardo di Scienze e Lettere,

Now let us consider in the plane of the given curve f a pencil of lines through a point that, for the sake of simplicity, we may assume not to be on f, say for instance the pencil y = tx through the origin O. For a generic value of t, one has an n-tuple line on which there are 2m simple branch points; in the representing plane of the complex variable x one takes 2m loops issuing from the point O and surrounding these 2m points; one can assign arbitrarily 2m transpositions of the branches, corresponding to the above-mentioned loops, with the condition that the group generated by them is transitive and that their product is the identity; in this way one finds a finite number of irreducible curves, in general birationally distinct, that are represented on the n-tuple line y = tx and are all of genus

π = \frac{m}{2} − n + 1;

these curves are defined up to birational transformations, that is, they correspond to well determined, and in general distinct, classes of curves or algebraic functions.

The number r of the above curves C_{1}, C_{2}, …, C_{r} has been determined by Hurwitz, but it is not necessary to know it here. It is important instead to remember that, when we let t vary along a closed path in the plane τ of the complex variable t, the curves C_{1}, C_{2}, …, C_{r} (or the classes of curves to which they belong) will in general be permuted: this happens first of all because the 2m branch points intersections of the line y = tx with f are permuted, and also because the system of loops, by means of which the nsheeted Riemann surface C_{1} is constructed, can be continuously transformed in a new system of loops, that can be reduced to the first one by suitably changing the corresponding transpositions. Now if f is the branch curve of a n-tuple plane, projection of a rationally determined surface F, one of the above curves, say C_{1}, must be a section of F. Hence it must be rationally determined and therefore its Riemann surface must not be interchanged with those of C_{2}, …, C_{r} but must be transformed again into C_{1} whenever t varies along a closed loop in the plane of this complex variable.

The main question is to establish the conditions for the invariance of (the class determined by) C_{1}, or rather of its Riemann surface, with respect to all the possible closed paths along which t can vary in the plane τ. These conditions will certainly be necessary conditions for the existence of a n-tuple plane with f as branch curve.

(27)If the degree of f were odd, one would necessarily have to add the line at infinity to it.

Let us narrow down the meaning of our research. We start by choosing an initial value of t, say t = 0, and assigning an ordered system of loops in the plane of the complex variable x, issuing from the point O and going around the branch points A_{1}, A_{2}, …, A_{2m} (intersections of f with the line y = 0); then, according to a given labelling 1, 2, …, n of the branches of the algebraic function z(x), we keep track of the transpositions that correspond to the above loops OA_{i}, thus defining the n-sheeted Riemann surface that represents our curve C_{1} (the one for which we are investigating the invariance conditions with respect to the closed paths along which t varies in the plane τ).

If we fix the initial Riemann surface, among those representing the multiple lines y = tx, we let t vary continuously (starting at the point t = 0) along a closed path in the plane τ of this variable; as t varies in this way, C_{1} also varies continuously, and its Riemann surface can be still constructed with sheets covering the same plane of the complex variable x, since (in the language of algebraic geometry) the multiple line y = tx – having the intersections with f as branch points – can be replaced in general with its projection on the x-axis from the point at infinity of the y-axis.

Now, following a continuous variation of t in the plane τ, which avoids the positions of t where two points A_{r} and A_{s} come together, we may see the points A_{1}, A_{2}, …, A_{2m} move continuously in the plane of the complex variable x without ever coming together, and describe arbitrary (closed or open) paths. Moreover one may let the loops OA_{i} around them vary, still continuously, so that they keep their ordering and are simple open paths, not crossing each other.

Then consider, in the plane τ, a closed path along which t moves, starting at the origin t = 0. If this loop does not surround any critical point corresponding to the fact that two points A_{i} come together, it can be continuously deformed into the constant loop, therefore the ordered system of loops and the corresponding substitutions that define the Riemann surface C_{1} have to be preserved along this loop: thus the above cycle does not give rise to any condition for the invariance of C_{1}.

Therefore the conditions for the invariance of C_{1} only depend on the elementary loops of the plane τ going along a critical point t, corresponding to which two of the points A_{i} come together.

Now these loops may be of three different types:

(1) loops surrounding a point T, corresponding to a simple tangent of the curve f which passes through the point O that does not lie on f, (2) loops surrounding a point D, corresponding to a line joining O with a

node of f; and

(3) loops surrounding a point Q, corresponding to a line joining O with a cusp of f.

In each of these three cases, it is easy to determine the invariance conditions for the Riemann surface C_{1}, if one starts from the simple situation in which the two points A_{1}A_{2} that come together at the critical point (T or D or Q) belong to loops OA_{1} and OA_{2} contiguous and honestly near: by this we mean that in the limit (when A_{1} and A_{2} come together) the two paths OA_{1} and OA_{2}, having the common endpoint A_{1} = A_{2}, bound a surface which does not contain any other point A_{i}. So these two paths can get infinitely near without crossing any other loop of the variable Riemann surface, hence without changing the substitutions of the branches of z(x) corresponding to them.

Let us examine the three cases defined above:

(1) If A_{1} and A_{2} come together at a point T, then on the limit multiple line OT the two branch points A_{1} and A_{2} disappear, giving rise to one apparent critical point that is the image of a double point of C_{1} (and when t goes around the loop they are interchanged); this means that the two transpositions corresponding to the two (honestly near) points A_{1} and A_{2} must be the same. Conversely, if this happens, e.g., if the same transposition (12) corresponds to both OA_{1} and OA_{2}, then the Riemann surface C_{1} is invariant with respect to the loop surrounding T in the plane τ.

(2) If A_{1} and A_{2} come together at a point D, then on the limit multiple line OD the two branch points A_{1} and A_{2} coincide at a point that is the image of two distinct points of C_{1}. This means that the transpositions corresponding to the two (honestly near) loops OA_{1} and OA_{2} must act on two distinct pairs of branches, that is, they must commute and be in general disjoint like (12) and (34).^{(28)} Conversely, if this happens, then the Riemann surface C_{1} is invariant with respect to the loop surrounding D in the plane τ.

(3) Finally if A_{1} and A_{2} come together at a point Q, then on the limit multiple line OQ the two branch points A_{1} and A_{2} coincide at the point Q and they are interchanged when t goes around the loop; but the point Q becomes the image of a flex of C_{1}, so that the transpositions corresponding to A_{1} and A_{2} are such that their product has to be a cyclic substitution of order three, hence they act on two pairs of branches that have a point in common, namely that are linked like (12) and (23).

(28)Instead, they will be the same, and therefore they will again commute, if the point D is a virtually nonexistent node of f.

Conditions (1), (2), and (3) ensure that the Riemann surface C_{1} is invariant with respect to the loops surrounding the points T, D and Q in the plane τ when the two points A_{r}, A_{s} that come together are the end points of loops OA_{r} and OA_{s} that become honestly near. The case in which OA_{r} and OA_{s} do not become honestly near can be reduced to the previous one: indeed in this case it happens that the two aforementioned paths having the common endpoint A_{r} = A_{s} form a closed path that bounds a surface containing some other points A_{i}; but it is always possible to let the line OA_{r} become infinitely near to the line OA_{s} by letting it vary continuously and cross the above-said points A_{i}: this means that, as a consequence of this reduction, also the transposition corresponding to the loop OA_{i} is modified^{(29)}.

We can summarize the result obtained by saying that: the conditions for the invariance of the Riemann surface C_{1},when one considers it as varying in the pencil of n-tuple lines y = tx with branch points on the curve f, correspond to the loops, described by t in the plane τ, respectively surrounding the critical points T, D and Q (points that correspond to the proper tangents and to the lines joining O with the nodes and the cusps of the branch curve f); these conditions are expressed by saying that the transpositions corresponding to the branch points A_{1} and A_{2} that come together at one of those critical points must be respectively the same or must commute (and be in general disjoint) or must be linked, when the system of loops that defines the Riemann surface is transformed in such a way that the loops surrounding the above mentioned points, OA_{1} and OA_{2}, become honestly near.

The conditions stated above are necessary conditions in order that the curve f, with nodes and cusps, is the branch curve of an n-tuple plane (n > 2). We want to show that they are also sufficient for the existence of such an n-tuple plane.

Complements: limit of the branch curve according to Chisini 5.10 192

To do this, consider the pencil of planes through the z-axis, whose intersections with the plane (z = 0) of the curve f are the lines y = tz. We want to construct in each of these planes a well defined curve, that, by projection from the point at infinity on the z-axis, call it Z, gives rise to the n-tuple line C_{1} with branch points at the intersections A_{1}, A_{2}, …, A_{2m} with the curve f. C_{1}, as we have said before, is defined as a Riemann surface in an invariant way with respect to the closed paths of t in the plane; thus one can construct an irreducible projective image of C_{1}, that is, a curve C^{(n+h)}, of some large enough order n + h, passing h times through the point Z and touching the lines ZA_{i}. One can also impose that such a curve C^{(n+h)} passes through n distinct points O_{1}, O_{2}, …, O_{n} of the line OZ and that moreover it satisfies a certain number of linear conditions, depending on h. Thus one finds a finite number of irreducible curves C^{(n+h)}, and among these a finite number s of curves represented by the Riemann Surface C_{1}, corresponding to s algebraic functions z_{1}(x), z_{2}(x), …, z_{s}(x) branched in the same way^{(30)}; then the sum

(29)See, e.g., Enriques–Chisini, Lezioni, Libro V, §3, Vol. III, page 18.

z_{(}x) = z_{1}(x) + z_{2}(x) + ⋯ + z_{s}(x),

also corresponds to an irreducible curve C represented on the Riemann surface C_{1} and rationally defined. Letting this C vary with the parameter t, we shall obtain a surface F represented n-tuply on the plane z = 0 and with f as branch curve. It is obvious that f is contained in the branch curve of the n-tuple plane obtained by projection of the surface F from the point Z; nevertheless one may doubt that a certain number of branch lines through O should be added to it: but if it happened, then some plane through the line OZ should touch F along a curve, and this is impossible if the points O_{1}, O_{2}, …, O_{n} are distinct, as we have assumed.

In conclusion: The conditions for the existence of an n-tuple plane (n > 2) with assigned branch curve f(x, y) = 0, with nodes and cusps (and not passing through the origin O = [0, 0]) are expressed by the invariance conditions of one of the Riemann surfaces represented on the multiple line y = tx, with respect to the closed paths of the parameter t in the plane of its complex variable, surrounding the points corresponding to the tangents of f through O or to the lines joining O with the nodes or the cusps of f.

It is essential to remark that the conditions required for the existence of a multiple plane with some branch curve f are qualitative conditions, that cannot vary when f varies continuously. Thus the existence theorem that we have established gives rise to the following:

Corollary 1 Given a continuous system of curves f with nodes and cusps the plane, if one of these curves is the branch curve of an n-tuple plane, then the same holds for all the curves of the system.

5.10 Complements: limit forms of the branch curve according to Chisini

The existence theorem for multiple planes proved in the previous section with its corollary, has been established by F. Enriques in the paper Sulla costruzione delle funzioni algebriche di due variabili possedenti una data curva di diramazione, in Ann. di Mat., s. IV, t. I (1923).

(30)See Enriques–Chisini, Lezioni. Libro V, §33, Vol. III, page 355.

There are also some interesting developments related to this theorem. First of all O. Zariski^{(31)} has put Enriques’ existence conditions in a topological group theoretic form, from which he deduces in particular that a curve without cusps cannot be the branch curve of a multiple plane of degree n > 2, except in the case of a curve counted n − 1 times that corresponds to a cyclic n-tuple plane.

But, both in Enriques’ form and in Zariski’s form, the existence theorem for multiple planes gives rise to great difficulties in its concrete application, since it relies on the solution of the further problem of determining the branch curves satisfying the given conditions. An important contribution to this problem has been given by O. Chisini^{(32)}, with the investigation of the limit forms contained in a continuous system of branch curves with a certain number of nodes and cusps.

To understand Chisini’s research, consider a reducible surface F of degree m + h, with an h-tuple point Z, that is the union of two parts f_{1} and f_{2} meeting the lines through Z at n and s points respectively (with m = n + s). Here the branch curve of the m-tuple plane, on which F is projected from Z breaks up as the union of the two branch curves of the n-tuple plane and of the s-tuple plane corresponding to f_{1} and f_{2}, and of the double curve projection of the common curve of f_{1} and f_{2}, which becomes an apparent branch curve. This remark leads to the question whether it is possible to construct the branch curve of a m-tuple plane, with m = n + s, by adding to the branch curves of the n-tuple plane and of the s-tuple plane a double curve that has to have a suitable behaviour with respect to them, so that the curve sum of the three, constructed as we have said, represents the limit curve of a continuous system of branch curves of m-tuple planes, having a certain number number of nodes and cusps.

What we have said explains the origin of Chisini’s idea. Still in general the search for the double curves to be added to the branch curves of a n-tuple plane and of a s-tuple plane gives rise again to great difficulties. To overcome them, Chisini restricts himself to the case in which one has, for instance, s = 1, that is, one has only the branch curve of a n-tuple plane, from which, by adding a suitable double curve, one wants to obtain the branch curve of a (n + 1)-tuple plane. It will suffice to explain Chisini’s construction by considering a simple example.

(31)O. Zariski, On the problem of existence of algebraic functions of two variables possessing a given branch curve. Amer. J. Math., Vol. 51 (1929). On the linear connection index of the algebraic surface z^{n} = f(x, y). Proc. Mat. Acad. Sci. U.S.A., Vol. 15 (1929). On the irregularity of cyclic multiple planes, Ann. of Math., s. II, Vol. 32 (1931). See Algebraic surfaces, Berlin, Springer, 1935.

(32)O. Chisini, Un teorema di esistenza dei piani multipli. Notes I and II, Atti Acc. Lincei, VI, 19 (1934). Un piu’ generale teorema di esistenza dei piani multipli, Atti Acc. Lincei, VI, 27, (1938). Altre curve di diramazione dei piani n-pli. Atti Acc. Lincei, VI, 29 (1939).

We want to construct the system of the branch curves of the quadruple planes that arise by projecting the general surface F_{4} of degree 4. These branch curves are C_{12} of order 12 with d = 12 nodes and k = 24 cusps.

If we let the surface F_{4} pass through the centre of projection Z, then the branch curve C_{12} of the quadruple plane degenerates as the union of the branch curve C_{10} of a triple plane, having k′ = 18 cusps (and no more nodes), and of a line C_{1} doubly tangent to C_{10}, counted twice. In turn C_{10} can degenerate as the union of a completely general sextic C_{6}, branch curve of a double plane, and a conic tangent to it at six points, counted twice. This happens when one imposes that F_{4} has a double point at Z. Finally if the double point Z of F_{4} becomes a triple point, then the branch curve C_{6} of the aforesaid double plane degenerates as a cubic counted twice.

Now it is useful to examine what the singularities of the branch curve become like, following the subsequent degenerations shown above.

To start with, when the aforementioned C_{12} degenerates as C_{10} + 2C_{1}, each of the two contact points P_{1} and P_{2} of C_{1} and C_{2} appears as the limit of three cusps of C_{12}, and each of the remaining 6 common points of C_{1} and C_{10} turns out to be the limit of two nodes of C_{12}. The statement can be justified by referring to the surface F_{4} passing through the centre of projection Z; one has to take into account that the cusps of C_{12} are obtained by projecting from Z the intersections of F_{4} with the first and the second polar of the point Z, and that the nodes of C_{12} come from the bitangents of F_{4} passing through Z, which now become the six lines passing through Z and tangent at some other point to the quartic curve intersection of F_{4} with the tangent plane at Z, each counted twice.

What we have remarked about the degeneration of the singularities of C_{12} when it becomes a C_{10} + 2C_{1} still holds for C_{10} that degenerates as C_{6} + 2C_{2}; this means that each of the 6 points of contact of C_{2} and C_{6} absorbs 3 cusps of C_{10}.

We have noticed that it is possible to let the branch curve C_{12} of a quadruple plane representing a surface F_{4} degenerate first as a C_{10} + 2C_{1}, then as a C_{6} + 2C_{2} + 2C_{1}, and finally also, if one wants, as a 2C_{3} + 2C_{2} + 2C_{1}, where C_{10} is the branch curve of a triple plane, bitangent to the line C_{1}, and C_{6} (branch curve of a double plane) is tangent to the conic C_{2} at 6 points. Now this analytic method of reduction can be reversed, thus leading to a synthetic method of construction of the multiple planes, that is precisely Chisini’s existence theorem. Indeed the author goes back from the limit forms to more general forms of multiple planes by recovering the triples of cusps and the pairs of double points that are absorbed by the limit singularities. Thus, starting with the double plane with branch sextic C_{6} and specializing C_{6} so that it admits a conic C_{2} tangent to it at six points, one will consider C_{6}+2C_{2} as a special case of C_{10}, for which the conditions of Enriques’ explained in the previous section are satisfied, therefore it turns out to be the branch curve of a triple plane. And then, adding again a bitangent C_{1} to C_{10}, one proves in a similar way that C_{10} + 2C_{1} (or C_{6} + 2C_{2} + 2C_{1}) is the limit form of C_{12} which is the branch curve of a quadruple plane, representing a surface F_{4} with plane sections of genus 3. It is needless to say that the case that we have described is just a special example, but rather significant, of a method that enables the author to prove the existence conditions for a larger family of multiple planes. Here we do not intend to describe Chisini’s more general developments, but we shall restrict ourselves to explain his construction method, and also the subtle questions which it poses, by considering a notable family of surfaces that includes the quartics F_{4}, and that we shall have to consider again later on in these lectures: we are talking about the surfaces with genera p_{a} = p_{g} = 1 and p^{(1)} = 1 with a canonical curve of degree zero.

According to the formulas of §6, one obtains surfaces belonging to the aforementioned family by starting with the n-tuple multiple planes with n = 2π − 2, defined by a branch curve C_{3n} = C_{6π−6} of degree

2n + 2π − 2 = 3n = 6π − 6,

with k(π) = 24(π − 2) cusps and d(π) = 18π_{2} − 78π + 84 nodes: actually such a n-tuple plane (if it exists) has genera p_{a} = p^{(1)} = 1 and also p_{g} = 1, since it has a canonical curve of order zero; and the images of the lines in the plane belong to a linear system of genus π and degree n = 2π − 2, and thus of dimension π. Thus the n-tuple plane arises as a projection, from points not on the surface, of a surface F_{n} of order n = 2π − 2 of S_{π}, with canonical hyperplane sections of genus π.

Now we shall try to prove here the existence of n-tuple planes corresponding to the branch curves C_{3n} mentioned above, and thus of the surfaces F_{n} = F_{2π−2} of S_{π} with canonical sections of genus π, for all the values of π = 3, 4, 5,… We have already shown in the previous discussion how one can construct the branch curve C_{12} of a quadruple plane (representing a surface F_{4}) corresponding to π = 3, starting with the double plane with a sextic C_{6} branch curve, tangent to a conic C_{2} at six points. Now we shall proceed in a similar way to construct the sextuple plane corresponding to π = 4, starting with a C_{12} tangent to a conic C_{2} at six points: we shall investigate whether C_{12}+2C_{2} is the degenerate form of a C_{16}, a branch curve of a quintuple plane, and whether one obtains a C_{18}, a branch curve of a sextuple plane, by adding to it a bitangent line C_{1} counted twice. If this happens, the aforementioned C_{12} has k(π) = k(3) = 24 cusps and d(π) = d(3) = 12 nodes, whereas the C_{16} has 18 more cusps (3 cusps corresponding to each of the contact points of C_{12} and C_{2}) and 24 more nodes (2 nodes arising from each of the 12 simple intersections of C_{12} and C_{2}), hence it has 42 cusps and 36 nodes in total; then adding to C_{16} the bitangent C_{1} counted twice, one obtains a C_{18} with 6 more cusps and 24 more nodes, and so

k(4) = 48 cusps and d(4) = 60 nodes.

To continue this construction in general for the higher values of π, it is convenient to use a fact that will be proved in the following section, namely that the surfaces F_{n} (of genera p_{a} = p_{g} = p = 1, p^{(1)} = 1) depend on at least 9p − 2p^{(1)} + 12 = 19 essential parameters or moduli, with respect to birational transformations. Therefore given a multiple plane defined by a branch curve C_{3n} = C_{6π−6} with the above characters, since we have a free parameter, it is possible to specialize the C_{3n} in such a way that it has a conic C_{2} tangent at six points. In order to follow Chisini’s method we should ask whether C_{3n} + 2C_{2} can be regarded as the limit form of a C_{3n+4}, having, besides the cusps and nodes of C_{3n}, 18 cusps and 12n − 24 = 24(π − 2) nodes corresponding respectively to the 6 contact points of C_{2} and C_{3n} and to the 6n − 12 further intersections of these curves; and we should also ask whether the C_{3n+4} thus constructed are branch curves of (n + 1)-tuple planes.

If the answer is affirmative (so that the invariance conditions of the previous section are satisfied) the C_{3n+4} thus constructed turn out to be branch curves of (n + 1)-tuple planes. And, as we shall see, by adding a suitable bitangent line C_{1} to them, one obtains the branch curve of a (n + 2)-tuple plane, representing a surface F_{n+2} of the space S^{π+1}, with canonical sections of genus π + 1.

But the question formulated above does not always has a positive answer. In order that C_{3n} + 2C_{2} be actually a limit form of a C_{3n+4} satisfying our conditions, it is necessary that its singularities can be deformed to nodes and cusps in the way indicated before, and that the curve C_{3n+4} thus constructed satisfies the invariance conditions for a branch curve of a (n + 1)-tuple plane. Now these conditions are not satisfied for an arbitrary conic tangent to C_{3n} at six points; but considering, for instance, to the case n = 4, they can be satisfied by reducing C_{12} to Chisini’s limit form C_{12} = 2C_{2} + 2C_{1} + C_{6} and assuming then that there is another conic, different from C_{2}, tangent to C_{6} at six points. Thus it is clear that the difficulty can be overcome for small values of n, but it is not easy to extend the argument to larger values of n.

The moduli of a class of algebraic surfaces 5.11 197

However, if one could show that the curve C_{3n} + 2C_{2} is the limit form of the branch curve of a (n + 1)-tuple plane, next step is to add again a bitangent line C_{1} to it, chosen in such a way that the curve C_{3n} + 2C_{2} + 2C_{1} is a limit form of the branch curve of a (n + 2)-tuple plane. But here we do not need to repeat the argument we made the case of n = 4 for the choice of the C_{2} tangent to C_{12} at six points. The difficulty in the choice of C_{1} can be overcome at once thanks to the following simple remark: the genera of the (n + 1)-tuple plane defined by the branch curve C_{3n+4} are equal to one and therefore such a multiple plane represents a F_{n+2} (with n = 2π − 2) with canonical sections of genus π + 1, projected from one of its simple points; this point corresponds to an exceptional line, bitangent to C_{3n+4}, which is the required C_{1}.

What we have said suffices to explain Chisini’s constructive method, and to test somehow its meaning by examining a concrete example. About this problem – that is, the existence and the classification of surfaces F_{n}, with all genera equal to one – we shall give a direct solution of it in §3 of Chapter VII, where there is also a historical note on the subject.

We add some more references to those already mentioned. In the quotations we made above we referred to n-tuple planes with n > 2, but there are some special researches in the case of double planes: F. Enriques, Sui piani doppi di genere 1, Mem. della Società It. delle Scienze, detta dei XL, Roma, 1896: Sui piani doppi di genere lineare p^{(1)} = 1, Rendic. Acc. Lincei, 1898. L. Campedelli, Sopra i piani doppi con tutti i generi uguali all’unità, Rendic. del Sem. Mat. di Padova, Vol. XI, 1940; Le superficie con i generi uguali all’unità, rappresentabili in infiniti modi sul piano doppio, Rendic. Sem. Mat. R. Università di Roma, s. V, Vol. I, 1940 and La classificazione dei piani doppi con tutti i generi uguali all’unità, Atti del 2^{o} Congresso dell’Unione Mat. Ital., Bologna, 1940; Sui piani doppi con curva di diramazione dell’ottavo ordine, Rendic. Acc. Lincei, 1932; Sui piani doppi con curva di diramazione del 10^{o} ordine, ibidem, 1932; Sopra alcuni piani doppi notevoli con curva di diramazione del 10^{o} ordine, ibidem, 1932.

The case of double planes is somehow close to the study of those particular quadruple planes that depend on Abelian equations, namely on extracting two square roots: P. Libois, Rendic. Acc. Lincei, 1934.

The following papers deal with notable classes of n-tuple multiple planes, with n > 2: B. Segre, Sulla caratterizzazione delle curve di diramazione dei piani multipli generali, Rendic. R. Acc. d’Italia, Vol. I, 1930, G. Pompilj, Sulla rappresentazione algebrica dei piani tripli, Rendic. Sem. Mat. R. Università di Roma, 1939; Osservazioni sui piani tripli, Rendic. Ist. Lombardo, 1941; Sui piani multipli diedrici, Rendic. Sem. Mat. R. Università di Roma, 1939.

5.11 The moduli of a class of algebraic surfaces

We have considered so far multiple planes with simple branch curve, on which in general it is possible to represent an algebraic surface, and we explained in the previous sections the corresponding existence theorem. Using these concepts we can now say something about the number of moduli or parameters on which depend the surfaces of a given class varying in a continuous family corresponding to certain given values of the characters p and p^{(1)}. For the sake of simplicity we assume that the surfaces are regular p = p_{g} = p_{a}, and that they have some plurigenus P_{i} > 0, so that they can be transformed in such a way that the exceptional curves can be eliminated and p^{(1)} becomes the absolute linear genus.

On a surface F satisfying the conditions stated above, let us choose arbitrarily an irreducible, base point free, linear system |C|, of genus π and degree n ≤ 2π − 2, that we assume regular and nonspecial, thus of dimension

r = p + n − π + 1.

By means of a generic net contained in |C|, F can be represented on a multiple plane of order n, with branch curve D of order 2n + 2π − 2, that has a certain number d of nodes and a certain number k of cusps. These numbers can be expressed in terms of p and p^{(1)} using the formulas of §6. Now we have seen that, if we let the curve D, of order 2n + 2π − 2, vary in a continuous system of curves still having d nodes and k cusps, then this is again the branch curve of a n-tuple plane, corresponding to a surface with the same characters p and p^{(1)} as F. But not all the curves of this system, of dimension

s ≥ (n + π − 1)(2n + 2π + 1) − d − 2k,

correspond to surfaces birationally distinct from F. First, the ∞^{8} homographic transforms of D give rise to the same F, and to the same net of curves |C| on it; secondly, the ∞^{r} system |C| contains infinitely may nets, depending on 3r − 6 parameters. Thus we have at least

(n + π − 1)(2n + 2π + 1) − d − 2k − 8 − (3r − 6)

parameters for the surfaces, that must be regarded as birationally distinct from F. Indeed if ∞^{1} curves D not corresponding to nets contained in |C| give rise to surfaces that are birationally equivalent, this means that F has a continuous series of inequivalent nets of curves, containing a net of |C|, against the assumption that the surface is regular (p_{a} = p_{g}).

To be precise, one might object to this argument that a net of C, say |C|^{2}, may appear as the limit of a net of curves |L|2, of higher order and with base points, as well as the net of the lines in the plane appears as the limit of a net of conics with three base points. But this doubt can be easily excluded for the surfaces F we are considering, on which the degree of a linear system cannot exceed twice the genus minus two, and that therefore can be regarded as free from exceptional curves. Actually if a net |L|_{2}, of effective genus π and effective degree n, has some base point of order i ≥ 1, then the virtual intersection number of the L with the canonical curves K is 2π − 2 − n − ∑ i^{2}, and in the limit they have to become reducible as curves C plus a fixed component θ, that must intersect the K in a negative number of points, − ∑ i^{2}. Thus this curve θ has a certain genus ρ and degree

ν = 2ρ − 2 + ∑ i^{2} > 2ρ − 2;

and the presence of such a curve on the surface F contradicts the assumption we made, that F can be transformed in such a way that it has no exceptional curves. In order to show the correctness of our computation (that is, the statement that there are ∞^{3r−6+8} birationally equivalent multiple planes), it is convenient to add here that F cannot admit a continuous series of birational transformations into itself; actually this would imply that: either the complete system |C| belongs to a larger series of inequivalent curves, against the assumption p_{a} = p_{g}, or |C| would be transformed into itself, against the assumption that F does not belong to the family of ruled surfaces^{(33)}.

We are now able to write down the number M of moduli on which the class of surfaces F depends, or at least a lower bound for it. Indeed we have

M ≥ (n + π − 1)(2n + 2π + 1) − d − 2k − 8 − (3r − 6)

that is,

M ≥ (n + π − 1)(2n + 2π + 1) − d − 2k − 3r − 2.

Let us replace in this formula d and k with their expressions in terms of p and p^{(1)} given by the formulas of §6:

d = 2[(n + π)^{2} − 5n − 17π − 2p^{(1)} + 6p_{a} + 24]
k = 3(n + 6π + p^{(1)} − 4p_{a} − 11);

moreover we replace r with its value

r = p + n − π + 1.

Thus we find

M ≥ 9p − 2p^{(1)} + 12,

or, more precisely,

M = 9p − 2p^{(1)} + 12 + ω,

where ω ≥ 0 denotes the superabundance of the system of plane curves D; this means that in the continuous system of the curves d of degree 2n+2π−2, with d + k double points, the conditions that k of the double points become cusps are not k independent conditions but only k − ω conditions.

(33)Indeed the surfaces that have a continuous family of projective transformations are rational or ruled. Theorem of Enriques–Fano. See Enriques–Conforto, Le superficie razionali, quoted work, Chapter VI, §37, page 496.

Digression on the completeness of the characteristic series 5.12 200

For the time being we may conclude by asserting that: The regular surfaces of some plurigenus P_{i} > 0, of geometric genus p and absolute linear genus p^{(1)}, depend, in general, on

M = 9p − 2p^{(1)} + 12 + ω

moduli, where ω ≥ 0.

For surfaces with p_{a} < p_{g}, the previous computation must be modified in view of the characteristic property of these surfaces, that we shall establish later on, that is, the property of containing ∞^{p_{g}−p_{a}} series of inequivalent linear systems. Thus for irregular surfaces, the number of moduli will be

M = 10p_{a} − p_{g} − 2p^{(1)} + 12 + ω,

with ω ≥ 0^{(34)}.

Let us go back to regular surfaces (p_{a} = p_{g} = p). For the surfaces with canonical or bicanonical curve of order zero (p = 1, p^{(1)} = 1, and p = 0, p^{(1)} = 1), one has respectively

9p − 2p^{(1)} + 12 = 19

and

9p − 2p^{(1)} + 12 = 10,

and indeed we shall verify that the number of moduli is precisely, for the former surfaces

M = 19,

and for the latter

M = 10;

thus

ω = 0.

But in general one has

ω > 0.

5.12 Digression on the completeness of the characteristic series of a complete system of plane curves with nodes and cusps

To express the computation of the moduli of a surface in a more meaningful formula, it is convenient to study the meaning of the character ω appearing in the formula of the previous section. It is easy to recognize that the superabundance ω of a continuous system of plane curves {D}, of order n with d nodes and k cusps, is smaller than or equal to the index of speciality of its characteristic series on a generic D, that is, of the series cut out on D by the curves of the same order that pass simply through the d nodes and the k cusps, and touch the cuspidal tangents at those points.

(34)With possibly a different meaning of ω, the formula a fortiori holds for surfaces admitting a continuous group of transformations in themselves.

Indeed let us recall that, given a continuous series {D} of plane curves D, of order n with d nodes and k cusps, the curves of {D} infinitely near to a given D = D_{0} have 2 intersections with it absorbed by each node and 3 intersections absorbed by each cusp^{(35)}. This means that the characteristic series, cut out on D_{0} by the infinitely near D, is contained in the complete series cut out on D_{0} by the curves of the same order passing through its d nodes and its k cusps and tangent to the cuspidal tangents at these points. Now, if the continuous system {D} has superabundance ω, then its dimension is not equal to

r = \frac{n(n + 3)}{2} − d − 2k,

but rather

r + ω = \frac{n(n + 3)}{2} − d − 2k + ω,

Thus the dimension of the series g_{m}, of degree m = n − 2d − 3k, cut out on the curve D_{0} of genus

π = \frac{n(n − 3)}{2} + 1 − d − k,

by the infinitely near d is ≥ r + ω = m − π + ω, that is, this characteristic series is special, with index of speciality not smaller than ω. At first sight, one would be tempted to say that ω is exactly equal to the abovementioned index of speciality, i.e., that the the characteristic series of the complete system {D}, formed by the plane curves of order n with d nodes and k cusps, is complete. Indeed it is easy to verify that the following differential property is satisfied: given a curve D of the system, the curve infinitely near to d in the pencil determined by d and by one of its adjoint curves φ_{n} touching the cuspidal tangents at the k cusps, has d nodes and k cusps infinitely near to those of D.

The moduli of the regular surfaces of genus p > 3 5.13 202

But this differential property does not yield that each of the aforementioned curves with d nodes and k cusps infinitely near to D, belongs to a continuous series of curves having the same number of nodes and cusps. Indeed let us consider the variety V representing the elements (curves) of the system D. This variety V appears as the intersection of a variety W of the space S_{R} of dimension R = n(n + 3)/2 representing the system of all plane curves of order n with d + k nodes, with k hypersurfaces or linear branches of hypersurfaces of the same space S_{R}. And, for instance, it might happen that these k branches have the same tangent space of dimension R − (k − 1) at each common point although they intersect along a variety of dimension R − k and not of dimension R − k + 1. In this case a point of V (representing a curve of {D}) would not have ∞^{r−1} but ∞^{r} infinitely near points belonging to V (that is, images of curves of {D}).

(35)Remark of De Jonquières and Beck (Math. Ann., Band 14, 1878). See Enriques–Chisini, Libro II, Chapter II (Vol. I, page 331, see Vol. II, page 286).

Although the examples that we know suggest to exclude this possibility, we have not yet succeeded in proving what we believe to be true, therefore we have to restrict ourselves to make the following:

Working hypothesis: The complete continuous system of the plane curves d with d nodes and k cusps has a complete characteristic series; therefore the superabundance ω of this system {D} equals the index of speciality of the characteristic series, cut out on d by the infinitely near D.

This assumption yields a few remarkable consequences and has some heuristic value, in relation with the developments that follow.

5.13 The moduli of the regular surfaces of genus p > 3

We have seen how to express in general the number of moduli on which a class of algebraic surfaces depends (assuming that some plurigenus is greater than zero, that is, for surfaces not belonging to the family of ruled surfaces), for surfaces represented as multiple planes with branch curves d of a certain order n, with a certain number of nodes and cusps. The formula we found in §11 contains, besides characters such as superficial genera p_{a}, p_{g} and absolute linear genus p^{(1)}, also a numerical, nonnegative, character ω, that is, the superabundance of the continuous system of the D, or, according to our working hypothesis (§12), the index of speciality of the characteristic series of {D}.

Now the meaning we have given for ω admits a different interpretation which suggests a more significant formula for the number of moduli of regular surfaces of genus p > 3.

To do this, let us consider a surface F = F_{n} of ordinary space, of a certain order n, with normal singularities, and consider the representation of F on a n-tuple plane obtained by projecting F from a generic point O external to F. In this representation the branch curve D arises as the projection of the Jacobian curve C_{j} of the net of plane sections C with planes through O. Now the complete characteristic series on the plane curve D of the continuous system {D}, consisting of the curves of the same order with the same number of nodes and cusps, has the index of speciality precisely equal to ω. We wish to read this series, with speciality ω, on the Jacobian curve C_{j}. To do this, remark that C_{j} is a section of F with the polar of the point O, that is, (denoting by K the canonical curves of F) it is a curve of the linear system |3C + K| cut out on F by its adjoint surfaces φ_{n−1} of order n − 1, passing through the τ cuspidal points of F; we will denote by G the group of these points, belonging to the double curve. The other curves C_{j}, passing through the τ cuspidal points, cut out on C_{j}, away from these points, a linear series that, as we shall see in a moment, is projected from O into the series cut out on the branch curve D by the first polars. Indeed, denoted by \overline{C}_{j} the section of F with the first polar of a point \overline{O}, the tangent plane to F at a point P common to C_{j} and \overline{C}_{j} contains the line O\overline{O}; thus the tangent to the curve D at the point P′, projection of P from O, passes through the intersection of the line O\overline{O} with the multiple plane.

It follows from this that the characteristic series of {D} on the curve D projection of C_{j} is the projection of the series cut out on C_{j} by the curves of the linear system |L| = |4C + K| passing through G (that is, by the adjoint surfaces φ_{n} passing through the τ points of the group G). Thus the character ω, that is, the index of speciality of the aforementioned characteristic series of {D}, may also be interpreted as the index of speciality of the series cut out on the curve C_{j}, which is a curve of |3C + K| passing through G, by the curves of the linear system |L| = |4C + K| also passing through G.

This projective interpretation of the character ω for the surface F, holds for both regular and irregular surfaces. And, according to it, the computation of the number of moduli on which a class of algebraic surfaces depends when it varies in the larger continuous family of surfaces containing it, acquires a meaning that agrees very well with intuition.

Indeed, consider the continuous system of all surfaces F, having the same order n and a double curve with the same characters, on which there is the same number τ of cuspidal points: the surfaces of such a system {F_{n}} containing F determine on one of them, and in particular on the given F, a characteristic linear system, consisting of the intersections of F with the surfaces of the system that are infinitely near to it. This system (that may a priori be complete or not) is cut out on F by the adjoint surfaces φ_{n} of order n, which are tangent to the cuspidal planes of F at the points of the group G.

Our existence theorem together with the working hypothesis of §12 yields that the continuous system F_{n} cuts out the complete characteristic linear system on a surface F; and conversely, this is equivalent to the above working hypothesis, and it leads to the computation of the number of moduli of the class of F as varying inside the system {F_{n}}. Of course in order to obtain a surface with general moduli, it is convenient to start with a projective model F which one a priori knows to be contained in a continuous family of surfaces with given characters. For instance one may consider a surface F corresponding to a regular linear system of large enough degree n and genus π, for which the inequality n > π − 1 holds; or one may also consider a canonical surface, that is, a surface having ∞^{3} canonical curves K as plane sections, except the case of small genus or some other analogous exception. Let us see what is the outcome of the computation of the moduli, referring to the two choices of F explained above and assuming, for the sake of simplicity, the surface to be regular.

If we start with the first choice, we find again the formula of §11. Indeed the question is of computing the dimension R of the characteristic system of {F_{n}} and then the number S that denotes the number of parameters of the F_{n} of the above continuous system that are birationally equivalent to a given F; then one has

M = R − S.

In order to compute R, we start by calculating the dimension of the base point free system |L| = |4C + K|, that we may assume to cut out a complete and nonspecial series on a C_{j}. Then we subtract from this dimension τ − ω, that is, the number of independent conditions that the group of the τ cuspidal points of F imposes on the curves L that contain it. The dimension of |L| is

p + 4π + 6n − 4,

and therefore the dimension of the characteristic system of {F_{n}} is

R − 1 = p + 4π + 6n − 4 − τ + ω,

where

τ = 2n + 8π − 12p + 2p^{(1)} − 22,

so that

R = 13p + 4n − 4p − 2p^{(1)} + 19 + ω.

On the other hand, S is computed by adding 15 (number of parameters of a space homography) to the number of parameters defining an ∞^{3} system inside the complete linear system |C| of dimension r. Thus one obtains

S = 4r − 12 + 15 = 4r + 3

that is,

S = 4p + 4n − 4π + 7.

Thus one finds

M = R − S = 9p − 2p^{(1)} + 12 + ω,

which is the formula given in §11.

But, if we take into account the meaning of ω (according to our working hypothesis of §12), it is possible to give a more meaningful formula, showing that for p > 3 one has in general

ω ≥ p, and M = 10p − 2p^{(1)} + 12 + ω′ (with ω′ ≥ 0).

In order to see this, we go back to our previous computation, considering a surface F_{n} of degree n = p^{(1)} − 1 with canonical curves K as plane sections. The following modifications are necessary here:

(1) the curves L of |L| = |5K| cut out the complete canonical series on a C_{j} = K_{j}, thus, if we still denote by ω the index of speciality of the series cut out on K_{j} by the curves of |5K| that pass through G, the computation of R gives us again

R − 1 = p + 4π + 6n − 4 − τ + ω

whence it follows that the τ cuspidal points do not impose τ − ω but τ − (ω − 1) independent conditions on the L to contain them;

(2) the system |K|, unlike the nonspecial regular system |C| considered before, has dimension

p − 1 = p + n − π (with n = p^{(1)} − 1, and π = p^{(1)}),

instead of

p + n − π + 1;

thus the number of surfaces F_{n} birationally equal to a given one turns out to be

4p + 4n − 4π + 3

instead of

4p + 4n − 4π + 7.

Therefore one finds that the number of moduli of F_{n} is given by

M = 9p − 2p^{(1)} + 16 + θ

where θ = ω − 4 denotes the index of speciality of the series cut out on a K_{j} by the curves of |5K| passing through the group G of cuspidal points lying on the surface F, image of a linear system chosen inside |K|.

Now we can compute ω, or at least give a lower bound for it, by remarking that the group G of cuspidal points is a group of points lying on the Jacobian curve K_{j} of a net of curves K, and is a partial intersection with a curve L of the system

|L| = |5K|

Historical note and complements 5.14 206

that is, the adjoint of

|K_{j}| = |4K|.

Therefore the group G_{τ} is a special group, imposing precisely τ − θ conditions on the curves of |5K| that must contain it, where θ is the dimension of the complete series to which G belongs on K_{j} (theorem of Riemann–Roch).

Now a lower bound for θ can be found by remarking that the net whose Jacobian is K_{j} belongs to ∞^{p−4} linear systems of dimension 3 of curves K contained in the canonical system. Each of these ∞^{3} systems defines a group G_{τ} on K_{j}, that is, the group of double points for ∞^{1} curves of the system. Thus we have

θ ≥ p − 4

and finally

M = 10p − 2p^{(1)} + 12 + ω′ (with ω′ ≥ 0).

Let us state the result thus obtained, but keeping in mind that it is based on the working hypothesis of §12, a proof of which is still missing:

The number of moduli on which depends a class of regular surfaces of genus p > 3, with simple irreducible canonical system (p^{(1)} > 5), is most likely to be

M ≥ 10p − 2p^{(1)} + 12.

In the next section we shall say how this formula can be extended to the case of irregular surfaces.

5.14 Historical note and complements

The first attempt to determine the number M of moduli on which a class of algebraic surfaces depends is due to M. Noether^{(36)} (1888) who considered the formula

M = 10p − 2p^{(1)} + 12, where (p = p_{a} = p_{g}),

to be true under very mild assumptions; this is precisely the formula that, according to our working hypothesis of §12, gives in general a lower bound for M for regular surfaces with p > 3, corresponding to ω′ = 0. Noether arrived at this type of divination by the intuition of genius. He started with the formula that gives the postulation of a twisted curve C (with triple points) for the surfaces F_{n} of order n that have to contain it as a double curve. If the number of conditions thus imposed on F_{n} is negative, −R, he conjectures that R coincides with the number of conditions in order that the curve be double for some surface F_{n}; thus one finds the number of moduli corresponding to the F_{n} having C as a double curve, by computing the number N of parameters on which C depends:

(36)M. Noether, Anzahl der Moduln einer Classe algebraischer Flächen etc., Sitzungsberichte der Akademie, Berlin, 1888.

M = N − R.

The question of the moduli of the surfaces was reconsidered by Enriques in two papers, of 1908 and 1912^{(37)}. The computation is performed by considering the continuous system of the surfaces F_{n} of order n – in ordinary space – having a double curve with given characters, and assuming that the characteristic system, cut out on F_{n} by the infinitely near F_{n}, is complete: the author did not realize at that moment that this hypothesis contains implicitly an assumption that is not satisfactorily proved by showing the corresponding differential property; the gap was pointed out only by Zariski^{(38)} (1935) after that a similar gap had been discovered by Severi (1921) in the proof of the fundamental theorem expressing the characteristic property of irregular surfaces (see Chapter IX, §6).

However Enriques, implicitly relying on the hypothesis mentioned above, found the formula of §11:

M = 10pa − p_{g} − 2p^{(1)} + 12 + ω (with ω ≥ 0).

which holds for all regular and irregular surfaces that do not belong to the family of ruled surfaces (that is, with some plurigenus P_{i} > 0); then he found the more meaningful formula that should hold, in general, for the regular surfaces (with simple irreducible canonical system) of genus p > 3 (p^{(1)} ≥ 6):

M = 10p − 2p^{(1)} + 12 + ω′ (with ω′ ≥ 0);

as we have said, this formula reduces to Noether’s for ω = 0. In the second note of 1912, Enriques started from the formula of the moduli to make some remark on the existence theorem for the multiple planes. Indeed he pointed out that the conditions in order that a plane curve with some number of nodes and cusps be the branch curve of a multiple plane are purely arithmetic in nature, so that one can say that if a curve D is the branch curve of a n-tuple plane, then the same must happen for all the curves of the largest continuous system having the same number of nodes and cusps. The author claimed, on that occasion, that it should be possible to determine such arithmetic conditions by studying the monodromy group, as he did later on in the paper in Ann. di Mat., 1923, as we have explained here in §9.

(37)F. Enriques, Sui moduli delle superficie algebriche. Rendic. Acc. Lincei, June 1908; Sui moduli di una classe di superficie algebriche e sul teorema d’esistenza per le funzioni algebriche di due variabili. Atti Acc. Torino, 1912. See Ann. di Mat., 1923–24.

(38)O. Zariski, Algebraic Surfaces. Berlin, 1935 (see V, 8).

It is now clear that, starting – as we did before in §11 – from the existence theorem for the multiple planes, one can deduce the first formula of the moduli that (except for the family of ruled surfaces) is completely general, and it is therefore immune to Zariski’s criticism. Which affects instead the second more meaningful formula (for p > 3), that in the present treatment we prove under the working hypothesis of §12.

The question of moduli of the algebraic surfaces was considered again by B. Segre^{(39)} in 1934, especially with the aim of finding a more meaningful formula for the irregular surfaces. But in this research he still makes use of the hypothesis (that the characteristic system of a continuous system of surfaces F_{n} is complete) that we have reduced to the working hypothesis of §12; actually Segre was aware of the difficulty brought to light subsequently by Zariski’s criticism.

Here we remark that, on the basis of the stated hypothesis, the argument we developed above for the general case extends immediately to the case of irregular surfaces (if one keeps in mind the characteristic property of these surfaces that we have already mentioned). In this way one obtains the formula

M = 12p_{a} − 2p_{g} − 2p^{(1)} + 12 + ω (with ω ≥ 0).

But Segre follows another way that leads to another more meaningful formula, so that already in the case of the regular surfaces he can write ω ≥ p − 1, and thus

M ≥ 10p − 2p^{(1)} + 11.

This formula (apart from the restrictions mentioned above) should hold for all surfaces with some nonzero plurigenus, with no need of assuming p > 3, irreducible canonical curves, and so on.

On the other hand, this result is included in the more general formula that gives the moduli of the irregular surfaces. Indeed B. Segre finds that ω ≥ 2p_{g} − p_{a} − 1, hence

M = 9p_{a} + p_{g} − 2p^{(1)} + 11 + θ.

We close by stating a formula that has been communicated to us by Prof. G. Castelnuovo, who has found the following inequality for the character ω:

ω ≥ 2p_{a} + p_{g} + p^{(1)} − I − 3.

(39)B. Segre, Sui moduli delle superficie algebriche irregolari. Rendic. Acc. Lincei, Vol. XIX

Regular surfaces: the minimum for the genera and rationality conditions

In this chapter we want to consider for simplicity regular surfaces with genus

p_{g} = p_{a} = p,

although many results concerning these surfaces may be extended without difficulties to irregular surfaces, as we will often point out.

6.1 The lower bound for the linear genus

Let us consider a surface F of genus p > 0, which we may assume to have no exceptional curves (Chapter IV, Section ??). In view of this hypothesis, F possesses some canonical curves, which, to begin with, we will suppose to be of order greater than zero. We want to prove that the linear genus of F (which here is just the absolute linear genus) is

p^{(1)} > 0.

We need to show that the hypothesis p^{(1)} ≤ 0 leads to a contradiction. Therefore let p^{(1)} ≤ 0, hence let the degree of the canonical system |K| be

p^{(2)} = p^{(1)} − 1 < 0.

So we will have (at least) one canonical curve K with negative degree, which may consist of several, simple or multiple, parts:

K = s_{1}K_{1} + s_{2}K_{2} + ⋯ + s_{r}K_{r},

The lower bound for the linear genus 6.1 210

where the K_{i} cannot be curves with virtual genus ρ_{i} = 0 and degree ν_{i} = −1, which would be exceptional curves of the first kind on F. Let us intersect one of the components of K, for instance K_{1}, with K. The number of intersections will be

2ρ_{1} − 2 − ν_{1}

This number cannot be negative. In fact, since ρ_{1} ≥ 0,

2ρ_{1} − 2 − ν_{1} < 0,

yields

ν_{1} ≥ −1

and thus

ν_{1} ≥ 0.

(since the values ν_{1} = −1 and ρ_{1} = 0 are excluded).

But this is a contradiction, because a component K_{1} of degree ν_{1} ≥ 0 cannot have a negative number of intersections with K = s_{1}K_{1} + s_{2}K_{2} + · · ·. Now, since we have proved that every component K_{i} of K must have a nonnegative number of intersections with K, one easily deduces that K itself must have degree p^{(2)} ≥ 0.

Indeed, denoting by n_{ij} the number of the intersections of two components K_{i} and K_{j} of K, the degree of K is given by

p^{(2)} = ∑ s_{i}^{2}ν_{i} + 2 ∑ s_{i}s_{j}n_{ij}.

The expression of p^{(2)} appears now as a quadratic form in the s_{i}, and, taking into account Euler’s theorem on homogeneous functions, one sees that it has to be essentially nonnegative, since its partial derivatives, which represent the double of the number of intersections (K_{i}K_{j}), are

\frac{∂p^{(2)}}{∂s_{i}} = 2(s_{i}ν_{i} + ∑ s_{j}n_{ij}) ≥ 0.

The inequality p^{(1)} > 0 still holds when the surface F has a single canonical curve of order zero (p = 1). Consider in fact on the surface F, which has no exceptional curves, an irreducible linear system |C| of genus π and degree n, with no base points; we have in this case

n = 2π − 2,

since the number of intersections is (CK) = 2π − 2 − n = 0. And, denoting by |C′| the adjoint system of |C|, of genus π′ and degree n′, we have

|C′| = |C|.

Now, by computing the genus π′ = π of |C′|, we find

π′ = π = π + p^{(1)} + 2π − 2 − n − 1

and so

p^{(1)} = 1.

The theorem that we have proved (p^{(1)} ≥ 1) holds also for all surfaces of genus p = 0 which have a plurigenus P_{i} > 0. In fact one may repeat the above argument, where the canonical curve is replaced by an ith canonical curve iK; were p^{(1)} ≤ 0, then this curve would have a nonnegative number of intersections with each one of its components (of genus ρ and degree ν). This number would be given by

i(2ρ − 2 − ν).

Also in the case (P_{i} = 1) in which the ith canonical curve is of order zero, and thus a system |C| coincides with its ith adjoint system, one may conclude, as in the case p = 1, that

p^{(1)} = 1.

Finally we notice that the proof of our theorem is completely independent of the hypothesis that F is regular. Therefore we can state, without any restriction, that:

A surface with genus p_{g} > 0, or even with p_{g} = 0 but with a plurigenus P_{i} > 0, has the absolute linear genus

p^{(1)} ≥ 1.

We want to present another proof of the theorem we just stated; this proof relies on a lemma which will have important applications later on. Let F be a surface, which we assume to have no exceptional curves, with linear genus p^{(1)} ≤ 0, and let us consider an irreducible, regular linear system on it, of genus π and degree n ≤ 2π − 2. The C meet a virtual canonical curve K at

2π − 2 − n ≥ 0

points. The curves C′ of the adjoint system (|C′| = |C + K|) meet K at

2π − 2 − n + p^{(1)} − 1 < 2π − 2 − n

The bigenus and the plurigenera 6.2 212

points. So the number of intersections of the curves of the system with the canonical curve decreases in passing from a linear system to its adjoint. But this implies that either the adjunction process must have an end, or that one eventually obtains linear systems of genus π and degree N > 2π − 2. Now again the existence of such a system |L| necessarily implies that the sequence of the adjoint systems |L′|, |L′′|, etc. must have an end, since the numbers of intersections are (L · L) > (L′ · L) > (L′′ · L) > · · ·.

Therefore the assumption p^{(1)} ≤ 0 yields that, starting from a linear system |C|, one obtains only a finite sequence of successive adjoint systems; thus a system |C| cannot be contained in any one of its adjoint systems |C^{(i)}|, namely all the plurigenera of the surface are P_{i} = 0.

Here two remarks are in order. The first one is that the assumption F without exceptional curves is introduced only in order to find a contradiction; if this assumption is removed then one is led at once to the conclusion that F has a system with n > 2π − 2, and that the adjunction process terminates as soon as one obtains that system.

The second remark concerns the possibility that the adjoint system |C′| of the irreducible system |C| from which we started is reducible. If the surface F is regular and |C| has no fundamental curves, then it is clear that |C′| cannot be reducible, and so, in particular, it cannot be reduced to the system |C| plus a fixed canonical curve. But, independently of the irreducibility of the successive adjoint systems of |C|, it is sufficient to obtain a system |L|, however reducible, of genus π and degree N > 2π − 2; any multiple of this again gives a linear system a fortiori satisfying the same inequality. Therefore it is easy to construct an irreducible system which again satisfies an analogous inequality; this can be done just by adding |sL| to a suitable irreducible linear system. We do not believe it necessary to spend more time on this.

Meanwhile we may conclude by confirming the above stated result.

6.2 The bigenus and the plurigenera

Let us consider a surface F such that for any linear system of genus π the degree is n ≤ 2π −2; as we saw (Chapter IV, Section ??), F can be transformed in such a way that its exceptional curves are eliminated. Starting from an irreducible linear system |C|, which can be the hyperplane section system of a surface F without singularities, let us try to construct the bicanonical system of F. For this, we need to subtract a curve C from its second adjoint system |C′′|. Now if we assume that

d = 2π − 2 − n ≥ 0,

then the C′′ cut out on C a series of order 2π − 2 + d which is certainly nonspecial; therefore one has to impose at most π − 1 + d conditions on the curves of |C′′| in order that they contain C. Thus by computing the dimension of |C′′| by the Riemann–Roch theorem, one finds that the dimension of the bicanonical system |C′′ − C| is given by

P − 1 ≥ p_{a} + p^{(1)} − 1;

thus we have

P ≥ p_{a} + p^{(1)}.

This result does not depend on the effective existence of the canonical curves (of order d ≥ 0), that is, it holds for p_{g} ≥ 0. In particular, for regular surfaces (p = p_{a} = p_{g}), without exceptional curves, with virtual canonical curves of order d (≥ 0), the bigenus is always

P ≥ p + p^{(1)}.  (6.1)

If one has irreducible canonical curves of genus p^{(1)} > 1, on which the bicanonical system cuts out the complete canonical series, then one has exactly:

P = p + p^{(1)}.  (6.2)

Instead of equality (??) one has in general the inequality

P ≥ p + 1 for p^{(1)} = 1,  (6.3)

since the canonical system of degree p^{(1)} − 1 = 0 is then composed with the elliptic curves k of a pencil (which is linear, since F is regular): if a canonical curve is composed of p − 1 curves k of such a pencil, then the bicanonical curves are formed by p − 2 curves k and therefore they form a linear system of dimension

P − 1 = 2p − 2.

If a fixed curve has to be added to the p − 1 curves k, then the double of such a fixed curve may be a curve k susceptible of varying in the pencil; thus it may happen that

P − 1 > 2p − 2.

The above results have to be modified if p = 1, when there is an effective canonical curve of order d = 0: then, instead of the inequality (??) one has:

P = 1

and

p^{(1)} = 1 (that is, P < p + p^{(1)}).

In fact, if d = 0, then the system |C| that we considered on the surface F has degree

n = 2π − 2,

and, since p = 1, one has

|C| = |C′| = |C′′|.

Thus the bicanonical system coincides with the system of order zero which gives the canonical system and has dimension

p − 1 = P − 1 = 0;

the genus of such a system |K| is

p^{(1)} = 1,

since, by adding a K of order zero to a C one has a reducible curve of genus

π = π + p^{(1)} + 0 − 1.

So the regular surfaces of genus p = 1 with a canonical curve of order zero are different from the surfaces with canonical curve of order greater than zero in this respect: the former have

P = 1,

whereas the latter have

P ≥ p + p^{(1)} ≥ 2.

For a regular surface of genus p = 0, without exceptional curves (n ≤ 2π − 2), with a virtual canonical curve of order d = 0 (d = 2π − 2 − n), the inequality (??) still holds, as in the case d > 0, namely

P ≥ p^{(1)},

since

P = 1, p^{(1)} = 1.

In fact, since |C′| is different from |C|, |C′′| is distinct from |C′|, too, and it cuts out on a C a nonspecial series g_{2π−2}; accordingly, subtracting C from |C′′| imposes at most π − 1 linear conditions: whence we deduce that it imposes exactly π − 1 conditions, and we have

|C′′| = |C|, P = 1, p^{(1)} = 1.

Remark It is important to notice that the above results, especially the fundamental inequality

P ≥ p + p^{(1)},  (6.4)

fail absolutely for surfaces which contain linear systems |C| of genus π and degree n > 2π − 2, that is, having virtual canonical curves of negative order

−d = n − (2π − 2).

Indeed the system |C′′| will now cut out on a C a series of order 2π −2−d that may be special.

As an example, look at the case of the plane, where we may consider the complete linear system |C_{7}| formed by the curves of the 7th order. The adjoint curves are the quartics C_{4}, and the second adjoint curves are the lines C_{1}, which cut out on a C_{7} a series g_{2}^{7}, which is special with index of speciality

i = 10.

Here we can compute the (linear) genus p^{(1)} of the virtual canonical system

|K| = |C_{1} − C_{4}|.

Since the intersection number of the K with the C_{1} is −d = −3 and with the C_{4} (of genus 3) is −12, we have

0 = p^{(1)} + 3 − 12 − 1

whence we find again that the (relative) linear genus of the plane is:

p^{(1)} = 10.

Meanwhile we note that, since it is not possible to subtract |C| from |C_{1}|, there is no bicanonical system and therefore

P = 0.

The formulas which express the bigenus in terms of the linear and geometric genera can be extended to the plurigenera. If a surface F, without exceptional curves, has virtual canonical curves of order d > 0 (d = 2π−2−n), then one has

P_{i} ≥ p_{a} + \frac{i(i − 1)}{2}(p^{(1)} − 1) + 1

and for a regular surface (p_{a} = p_{g} = p) with linear genus p^{(1)} > 1 and with irreducible (i − 1)th canonical curves, one has

P_{i} ≥ p_{a} + \frac{i(i − 1)}{2}(p^{(1)} − 1) + 1.

Examples: double planes with linear genus p^{(1)} = 1 6.3 216

The equality becomes an inequality in the case p^{(1)} = 1, where the pluricanonical curves of degree i^{2}(p^{(1)} − 1) = 0, are all composed with the elliptic curves of one and the same pencil:

P_{i} ≥ p + 1.

An exception to the fundamental inequality

P_{i} ≥ p_{a} + \frac{i(i − 1)}{2}(p^{(1)} − 1) + 1.

is given by the surfaces with bigenus P = P_{2} = 1 which possess a virtual canonical curve of order zero and an effective bicanonical curve also of order zero. In this case the series cut out by the third adjoint system |C′′′| = |C′| on a C is the canonical series (of dimension π − 1); then there is no curve of |C′′′| containing C, since C imposes π conditions on |C′′′|, hence

P_{3} = 0.

Therefore the regular surfaces of genus zero and bigenus one with a bicanonical curve of order zero differ from the surfaces having a bicanonical curve of order greater than zero in this respect: that the former have trigenus

P_{3} = 0,

whereas the latter have

P_{3} ≥ 1.

Furthermore the former have

P_{2i} = 1, P_{2i+1} = 0,

whereas for the latter all plurigenera are essentially positive. And moreover they are greater than one even in the case p^{(1)} = 1. In fact, by combining a bicanonical curve counted three times with a tricanonical one counted twice, one obtains a pencil of sexticanonical curves, so that

P_{6} ≥ 2.

6.3 Examples: double planes with linear genus p^{(1)} = 1

An interesting illustration of the formulas of the above section is given by the double planes with linear genus p^{(1)} = 1^{(1)}. One proves here that, with the exception of the double plane branched on a completely general sextic

z^{2} = D_{6}(x, y)

(1)See F. Enriques, Sui piani doppi di genere lineare p^{(1)=1}, Rendic. Acc. Lincei, April and May 1898.

which has a canonical curve of order zero (genus and plurigenera p = P_{i} = 1), all the other double planes for which p^{(1)} = 1 and some plurigenus P_{i} > 0, contain a pencil (rational or not) of elliptic curves L, which is transformed into itself by the involution I whose pairs correspond to the points of the plane. Therefore one is led to two families of surfaces which are representable as a double plane of this sort:

(I) surfaces with a linear pencil of (irreducible) elliptic curves L fixed by the involution I which induces a linear involution g_{2}^{1} on them; these surfaces are represented as double planes with branch curve D_{2n} of order 2n (n > 3) having a point O of multiplicity 2n − 4 or 2n − 3, and possibly other singularities which let the genus drop: quadruple points and [3, 3] points; the lines through O are the images of the curves L on the double plane;

(II) surfaces with a pencil of irreducible elliptic curves L, which may either be the curves of a rational pencil, fixed by the involution I which induces an elliptic involution γ_{2}^{1} on them, or the curves of a pencil (rational or not), pairwise associated by the involution I which induces a rational involution in the pencil; these surfaces are represented as double planes in such a way that the L correspond to the elliptic curves of a pencil of Halphen^{(2)} (curves C_{3s} with nine s-tuple points); therefore (provided they have no exceptional curves) the branch curves of these double planes are composed with the curves of the above-mentioned pencil and possibly also with the cubic curve which passes through its nine base points.

Among the double planes of class I, it is worth mentioning those having a branch curve D_{2n} with pairs of infinitely near triple points ([3, 3] points) collinear with the (2n − 4)-tuple point O.

For n = 4 we have the double plane with a branch curve D_{8} having a 4-tuple point O, and a pair of infinitely near triple points A and A_{1}, collinear with O; D_{8} is composed of a D_{7} with O^{3}A^{2}A_{1}^{2} and of the line r = OA. The genus of this double plane is

p_{g} = p_{a} = p = 1,

since there is one canonical curve represented by the line r = OA (which is subject to the condition of passing simply through O and A). Less easy is the determination of the linear genus p^{(1)}, since the double line r = OA seems to correspond to a disconnected curve formed by a rational curve counted twice. This difficulty can be overcome is one understands that the surface F, represented as a double plane by projection, possesses an impure canonical curve, formed by two infinitely near exceptional curves whose points correspond to the generic points of r, and by an infinitesimal curve represented by the neighbourhood of the point A_{1}; the latter curve may be transformed into a curve of order greater than zero and genus p^{(1)} = 1, by means of a transformation of F into a surface without singularities. Accordingly, F has the bigenus

(2)See, for instance, Enriques–Conforto, Le superficie razionali, Zanichelli, Bologna.

P = p + p^{(1)} = 2,

and there is in fact a pencil of elliptic bicanonical curves represented on the double plane by the lines through O: if we add the exceptional curves, then we have, as images of the impure bicanonical curves, the pairs of lines formed by r and by a variable line through O. What we have just said is made clear by considering the ∞^{3} linear system of the conics through AA_{1}, which determines the transformation of F into a double quadric cone without exceptional curves: the images of these double conics with 10 branch points are hyperelliptic curves of genus 4, forming a linear system of degree 4 (< 2 · 4 − 2); they are met at two (conjugate) points by the canonical curve which corresponds to the neighbourhood of A_{1} and at 4 points, consisting of two pairs of points that are conjugate in the g_{2}^{1}, by the bicanonical curves; therefore the latter are represented on the cone by the plane sections of a pencil, and on the plane by the lines through O. Let us notice that the quadric cone has a branch curve D_{10} of the 10th order; the D_{7} may be regarded as a projection of this curve from a triple point P which has another infinitely near triple point P_{1} whose projection is O. The line P P_{1} lies in the tangent plane to the cone along the generator k through P, and the planes containing k cut out on the cone the images of the bicanonical curves.

Beside the above-mentioned double plane, with branch curve D_{8} = rD_{7}, with O^{4}A^{3}A_{1}^{3}, one can also consider the double plane with branch curve D_{8} passing through O^{4} and through two pairs of infinitely near triple points A^{3}A_{1}^{3} and B^{3}B_{1}^{3} collinear with O. This D_{8} breaks up into the two lines r = OA and s = OB and into a D_{6}(O^{2}A^{2}A_{1}^{2}B^{2}B_{1}^{2}). The genus of such a double plane is

p = 0,

and its bigenus

P = 1,

since there is an impure bicanonical curve, represented by the pair of lines r and s. This curve consists of a pair of exceptional curves each counted twice. Computing the trigenus, one finds P_{3} = 0. This also indirectly stems from the following remarks.

If one transforms the double plane into a surface F without exceptional curves, then the bicanonical curve appears to be of order zero: one may construct F by means of the ∞^{4} linear system of the cubic curves through OAA_{1}BB_{1}, which are the images of curves of genus 4 and degree 6; one is therefore led to a double cubic surface (normal in S_{4}) with branch curve of the 10th order.

We will see later on that F may also be transformed into the surface F_{6} of the 6th order which passes doubly through the edges of a tetrahedron.

Let us consider now the double planes (again in class I) with branch curve D_{10} of the 10th order and two pairs of double points AA_{1} and BB_{1}, collinear with O: D_{10} is composed of the lines r = OA and s = OB and of a D_{8}(O^{4}A^{2}A_{1}^{2}B^{2}B_{1}^{2}). We have here an impure canonical curve, represented by the two exceptional lines r and s, and by a residual curve (which becomes of order > 0 on a transformed surface without singularities): this is composed of two elliptic components corresponding to the neighbourhoods of the points A_{1} and B_{1}, and thus

p_{g} = p_{a} = p = 1, p^{(1)} = 1.

The bigenus P has to be greater than one, and in fact there is a net of bicanonical curves whose images on the double plane are the pairs of lines through O; so we may conclude that

P = 3 (> p + p^{(1)}).

Beside the aforesaid example, consider the double plane branched on a curve D_{10} with a 6-tuple point O and 3 pairs of infinitely near triple points AA_{1}, BB_{1}, EE_{1}, collinear with O. The surface F thus defined has no canonical curves, since there is no conic passing doubly through O and simply through A, B, E; therefore

p_{a} = p_{g} = p = 0.

We have instead elliptic bicanonical curves, represented by the lines through O, plus the exceptional curves which are the images of the branch lines

r = OA, s = OB, t = OE;

so

p^{(1)} = 1

and

P = 2 (> p^{(1)}).

Among these bicanonical curves there are three double curves (which correspond to the three lines r, s, t, or rather to the neighbourhoods of A_{1}, B_{1} and E_{1}), which, taken together and with multiplicity one, form a tricanonical curve of the surface F:

P_{3} = 1 (= p^{(1)}).

The double of this tricanonical curve belongs to the triple of the bicanonical system; in this way one obtains the linear system of sexticanonical curves of F, whose images are the triples of lines through O. One has therefore:

P_{6} = 4.

We also find interesting examples for our theory in Class II. Let us consider a double plane with a branch curve D_{6n} (n > 1) formed by an irreducible elliptic curve with 9 points of multiplicity 2n, belonging – as it is well known – to a pencil of Halphen^{(3)}. The surface thus defined possesses a canonical curve whose image is a curve of order 3n − 3 subject to the condition of passing n − 1 times through the 9 base points of the pencil; so it turns out to be the cubic curve φ_{3} through the 9 points, counted n − 1 times, or, as we shall say, the φ_{3}^{n−1}. We have therefore

p_{a} = p_{g} = p = 1, p^{(1)} = 1.

Now the branch curve D_{6n}, together with the φ_{3}^{n−2}, adjoint to φ_{3}^{n−1}, (to be counted twice) gives a bicanonical curve of F (see Chapter V, Section ??), whereas another bicanonical curve is given by the double φ_{3}^{2n−2}. Whence we infer that the curves of the pencil of Halphen defined by the D_{6n}, taken together with the fixed φ_{3}^{n−2}, represent a pencil of bicanonical curves; so these are formed by irreducible elliptic curves L^{(4)} (which do not belong to the involution I) and by a fixed part, which is a (n − 2)-tuple elliptic curve, whose component belongs to the involution I and is equivalent to an (n−1)st submultiple of L. So we have

P = 2(= p + p^{(1)}).

Finally we easily find that the tricanonical curves are given by the pairs of curves L, so that

P_{3} = 3.

Together with the above example we also find in class II the double plane with branch curve D_{6n} = D_{6n−3} + D_{3}, which is formed by an irreducible Halphen curve D_{6n−3}, with 9 points of multiplicity 2n − 1, and by the cubic curve D_{3} through these 9 points.

(3)Any (double) curve of the pencil of Halphen represents two distinct curves L which do not belong to the involution I, since this is the case for D_{6n}; however this agrees with Comessatti’s criterion, since the D_{6n} may be reduced by continuity to a double φ_{n}^{3}. See for instance Enriques–Chisini, Libro V, Chapter IV, §38 (Vol. III, page 445).

(4)According with Comessatti’s criterion, each curve L of this pencil is the image of two distinct elliptic curves, because the group of apparent critical points cut out by D_{6n} on L is also cut out by the D_{3}^{2n}, the double of the D_{3n}. See Enriques–Chisini, Ibidem

The surface thus defined possesses one canonical curve represented by the D_{3} counted n − 1 times, or, as we shall say, by the D_{3}^{n−1}, so that

p_{a} = p_{g} = p = 1, p^{(1)} = 1.

The bicanonical curves which are fixed by the involution I are: the double D_{3}^{n−1}, counted twice, and the branch curve D_{6n−3} + D_{3} together with the adjoint curve to the image of the canonical curve, which is the double D_{3}^{n−2}. Therefore one has a pencil of elliptic bicanonical curves, represented by the simple^{(5)} curves of order 6n − 3 of the pencil of Halphen, taken together with the multiple curve, represented by the double D_{3}^{n−2}:

P = 2.

We will not insist on the double planes of class II with branch curve however composed with the curves of a pencil of Halphen: these lead in general to irregular surfaces with

p_{a} = 1, p_{g} > 1, p^{(1)} = 1.

Still we want to point out a double plane with p^{(1)} = 1, p = 0, P = 1 and with a bicanonical curve of order greater than zero, possessing a pencil of elliptic curves (not selfconjugate in I), represented by the curves of a pencil of Halphen. Nevertheless this plane does not belong to class II because there exists a line, the image of an exceptional curve, which is a component of the branch curve. This is a double plane which already occurred to Enriques, whose branch curve D_{10} has order 10, and it has been met by L. Campedelli^{(6)} in the classification of these double planes: the D_{10} is composed of a D_{9} + r, where the D_{9} is an (irreducible) Halphen elliptic curve with 9 triple points, six of which AA_{1}, BB_{1}, EE_{1} are pairwise infinitely near, while the other (distinct) three lie on the line r.

The surface represented on this double plane has no canonical curves, since a conic passing through the three triple points of D_{9} and through the three points A, B, and E, would be reducible into the exceptional curve corresponding to the line r^{(7)} and into a line through A, B and E, which, by contrast, are not collinear.

(5)See the previous note.

(6)L. Campedelli, Sui piani doppi con curva di diramazione del 10^{o} ordine and Sopra alcuni piani doppi notevoli con curva di diramazione del 10^{o} ordine, Rendic. Acc. Lincei, 1932.

(7)It is easy to see that the curve corresponding to this branch line of genus 0 and degree −2 has genus 0 and degree −1 on F.

Rationality conditions for a surface 6.4 222

We thus have

p = 0.

On the other hand if a double line through A, B and E existed, then it would have 4 branch points (three at the aforementioned three points and one on r), thus the absolute linear genus of F (the genus of the canonical curves from which the exceptional curve has been subtracted) would be

p^{(1)} = 1.

Now we have on F one irreducible^{(8)} bicanonical curve, represented on the double plane by the cubic curve D_{3} passing through the 9 triple points of D_{9} plus the exceptional curve represented by the line r. Therefore the bigenus of F is

P = 1.

According to our theorem one must have the trigenus P_{3} ≥ 1 and the sextic genus P_{6} ≥ 2.

In order to verify this we observe that (since D_{3} possesses an adjoint curve of order 0) a tricanonical curve of F is given by the branch curve D_{9} plus the exceptional curve r. This tricanonical curve is isolated, that is, it does not vary in a pencil. In fact, if this were not the case, the bicanonical curve represented by D_{3} would be contained in it, since it has no common points with this tricanonical curve, and we would have p_{g} > 0. This means that the double elliptic curves of the 9th order of the pencil of Halphen defined by the D_{9} are the images of irreducible curves (which are selfconjugate in the involution I defined on F by the double plane). This conclusion may also be directly confirmed by the criterion of Comessatti. Finally we have one tricanonical curve D_{3} and a pencil of sexticanonical irreducible curves represented by curves of the pencil of Halphen, which contains the tricanonical curve counted twice and the bicanonical curve counted three times. Then one has:

P_{3} = 1, P_{6} = 2.

6.4 Rationality conditions for a surface

A rational surface, that is, a surface which can be put in one-to-one correspondence with a plane, must have, like the plane, the genus p = p_{a} = p_{g} = 0, and all its plurigenera must vanish together with the genus. The examples that we have considered (in particular the surface of order 6 which passes through the edges of a tetrahedron with multiplicity two) show that a regular surface of genus p = 0 may have the bigenus P > 0, hence the vanishing of the genus does not give a sufficient condition for the rationality of a surface. Now it is extremely interesting the result that this condition can be expressed by imposing that, beside the genus, also the bigenus vanishes, namely by writing

(8)The irreducibility of the double cubic D_{3} stems from Comessatti’s criterion, taking into account the construction of the pencil of Halphen to which the D_{9} belongs.

p_{a} = P = 0

(whence also p_{g} = 0 necessarily follows). This theorem is due to G. Castelnuovo who wrote it in a memoir of 1896^{(9)}.

The proof of our theorem can be reduced to the rationality criteria of Noether and Clebsch–Noether, concerning the surfaces with a linear pencil of rational curves and the types of double planes which they determine. With this in mind, one starts from an irreducible linear system |C| on the given surface and one successively constructs its adjoint system |C′|, the second adjoint |C′′| and so on. Then one realizes that, since one has p_{a} = P = 0, the sequence of these successive adjoint systems terminates, and the last adjoint system that is at least ∞^{1} is composed of either rational or elliptic curves; therefore it leads to one of the cases of rational surfaces which are contemplated in the above mentioned criteria of Noether and of Noether-Clebsch.

The discussion can be carried out in a very simple way by distinguishing between two assumptions:

(1) the given surface F contains an irreducible linear system |C| of genus π and degree n > 2π − 2, which may be assumed to be the plane or hyperplane section system of a surface F having no singularities at all;

(2) for any given irreducible linear system on the surface F the inequality n ≤ 2π − 2 holds, where n and π denote the degree and the genus of the system.

Hypothesis I. It is clear that, starting from a given system |C| on F and constructing the successive adjoint systems |C′|, |C′′|, etc. one obtains a finite number of systems; in other words the adjunction process terminates. In fact the curves C′, C′′, … have decreasing order

2π − 2, 2π − 2 − d, 2π − 2 − 2d, …

where

d = n − (2π − 2).

(9)G. Castelnuovo, Sulle superficie di genere zero. Soc. It. delle Scienze (detta dei XL), see memorie scelte, page 307.

Therefore we want to consider the last adjoint system |C^{(i)}| of dimension r_{i} ≥ 1.

First we assume the adjoint systems |C′|, |C′′|, …, |C^{(i)}| of the above sequence to be irreducible. The system |C^{(i)}|, the first one of the sequence which has not an adjoint system of dimension greater than zero, is formed by curves whose genus is either zero or one. If the system |C^{(i)}| is formed by curves of genus zero, then the theorem of Noether^{(10)} about surfaces containing a linear pencil of rational curves enables us to deduce immediately that F is rational.

If by contrast the system |C^{(i)}| is of genus 1, we need to distinguish between the assumptions that its dimension is r_{i} ≥ 2, or r_{i} = 1. If r_{i} ≥ 2 then the complete system of elliptic curves |C^{(i)}|, which, by the regularity of the surface, has a complete characteristic series, has degree n_{i} = r_{i}, and therefore it contains a net of degree 2; by means of this net the surface F can be represented on the double plane with a quartic branch curve: whence it follows^{(11)} that our surface is rational.

On the other hand, if r_{i} = 1, then the system |C^{(i−1)}| which appears before |C^{(i)}| in the sequence of the adjoints has to be of genus two, and may be supposed of dimension r_{i−1} ≥ 2, since otherwise we may replace it with one of the preceding systems, again of genus two. Now we can see that the system |C^{(i−1)}| is not simple: every C^{(i−1)} passing through a point P of the surface F passes as a consequence through the point P′ which is the conjugate of P in the g_{2}^{1} on the curve; in fact the conjugate of P on a variable C^{(i−1)} through P has to be fixed, otherwise it would describe a curve C^{(i)}, or a component of C^{(i)}, of genus zero, contrary to our assumptions.

Now, since F is a regular surface, the adjoint system C^{(i−1)} to C^{(i−2)} is a regular system, of dimension p_{i−2}−1, where p_{i−2} denotes the genus of |C^{(i−2)}|, and its characteristic series is complete and nonspecial. Therefore this series, which, as we have seen, is composed with the g_{2}^{1}, is precisely the double of this g_{2}^{1}. Then the system |C^{(i−1)}| is an ∞^{3} system of curves of genus 2, which belongs to an involution of pairs of points, and our surface F is represented by means of this system on a double quadric Q, whose branch curve D_{6} is of order six. Since the generators of Q cannot have an odd number of branch points, we deduce that Q is a cone, whose vertex is a branch point that has to be added to D_{6}, which cuts out three points on the generators. The double cone thus defined may be represented on Noether’s double plane with a sextic branch curve having two infinitely near triple points, whence it follows that the double plane is rational^{(12)}.

We have assumed that the sequence of successive adjoint systems |C′|, |C′′|, … is formed by irreducible systems. If in this series one finds a system |C^{(h)}| having a fixed curve θ and an irreducible variable part, which belongs to a system of dimension r_{h} ≥ 2, then one may in general go on in the construction of our sequence of adjoint systems replacing |C^{(h)}| with the irreducible system |C^{(h)} −θ|, from which the fixed component θ has been subtracted. And if this system has genus 0 or 1, therefore (by the completeness of the characteristic series) has degree n_{h} = r_{h} −1 or n_{h} = r_{h}, then one may conclude (like before) that the surface F is rational.

(10)See for instance Enriques–Conforto, Ibidem (11)Theorem of Clebsch, See Enriques–Conforto, Ibidem (12)Theorem of Noether, See Enriques–Conforto, Ibidem

We still have to investigate carefully the case in which the system |C^{(h)}−θ|, without the fixed part θ, is a pencil or is composed with the curves L of a pencil that, by the regularity of the surface, has to be linear.

To start with, if the curves L of the pencil are rational then the aforementioned theorem of Noether enables us to say that the surface is rational.

Let us assume by contrast that the curves L have the genus ρ > 0. Then we notice that θ, the fixed part of |C^{(h)}|, has to be a fundamental curve for the irreducible system |C^{(h−1)}|, since |C^{(h)}| cuts out on a C^{(h−1)} the complete canonical series, which has no base points.

Then the curves L meet the C^{(h−1)} at ν > 1 variable points (since from the hypothesis ν = 1 we would deduce that the L are rational), and the complete canonical series cut out by |C^{(h)} − θ| on a C^{(h−1)} is composed with the groups of the involution of order ν cut out by the pencil of the L; whence we deduce that the C^{(h−1)} are hyperelliptic curves and that ν = 2: the pencil of the L, with which the curves of |C^{(h)} − θ| are composed, is a linear pencil of bisecants.

Now we recall that the complete system |C^{(h−1)}|, adjoint to the system |C^{(h−2)}| (on the regular surface F of genus p = 0), is a regular system (of dimension π_{h−2} − 1) and therefore its characteristic series is complete and nonspecial. This series is composed with the g_{2}^{1}, since otherwise the conjugate of a point P on a C^{(h−1)} through P would describe a rational curve, which would be the curve L through P or one of its components. Accordingly the system |C^{(h−1)}| is not simple, but composed with an involution of order 2 and the degree n_{h−1} of this system of hyperelliptic curves is equal to the double of the dimension minus one:

n_{h−1} = 2(r_{h−1} − 1);

whereas its genus π_{h−1} is

π_{h−1} = r_{h−1} − 1.

We can achieve a more precise determination of these characters with the following argument. If we subtract from the system |C^{(h−1)}| a curve L (on which the C^{(h−1)} cut out the pairs of a g_{2}^{1}) then we obtain a linear system of dimension r_{h−1} − 2, whose adjoint system must be the adjoint system of |C^{(h−1)}|, minus the curve L, and therefore the system formed by all the groups of π_{h−1} − 2 curves L; so the subtraction of L from |C^{(h−1)}| lets its genus drop by one only. On the other hand the C^{(h−1)} which are reducible into the curve L and a residual curve must be connected, thus L has to meet the residual curves at i ≥ 0 points; so, since the genus of these curves is π_{h−1} − 1, one has

π_{h−1} = (π_{h−1} − 1) + ρ + i − 1

whence (since ρ > 0)

ρ = 1, i = 1.

So the residual curves of a L with respect to |C^{(h−1)}| cut out on L one point which is necessarily fixed and is also a common point of the L of our pencil; this yields that these curves are reducible in L of the same pencil.

Now, using the system |C^{(h−1)}|, we transform the surface F into a double surface of order m = r_{h−1} − 1 in a space of dimension m + 1 = r_{h−1}. This surface, on which the curves L are represented by lines, has to be a cone with a certain vertex O and must have a branch curve d of order

2 + 2π_{h−1} = 2m + 2.

Since this curve has to intersect the generators of the surface in at least 3 variable points (in order that the genus of the L be ρ > 0), we must have m = 2; in fact a hyperplane through O cuts out m generators on our cone, and therefore one has

3m ≤ 2m + 2.

In this way we find the conclusion that the system |C^{(h−1)}| leads to the representation of our surface F on a double quadric cone with a sextic branch curve, which, as we have already recalled, is rational (Noether).

Hypothesis II We still have to examine the second hypothesis: every irreducible linear system on the surface F has the genus π and the degree n which satisfy the inequality

n ≤ 2π − 2.

Let us make this hypothesis, and we will derive from it a contradiction. By the assumption, we may suppose that the surface F is transformed into a surface of a suitable space in such a way that it contains no exceptional curves, and, if we like it, that it has no singularities. Its hyperplane sections form a linear system |C|, whose degree we denote by n and whose genus by π; as we have said, one has

n ≤ 2π − 2.

Then the bigenus P of F may be computed by increasing by one the dimension of the bicanonical system, which can be obtained by subtracting a C from the second adjoint system |C′′|; as we have seen, one finds that on a regular surface of geometric genus p = p_{a} = p_{g} = 0, and linear genus p^{(1)}, the bigenus is

P ≥ p^{(1)}.

Accordingly, if P = 0, then one has

p^{(1)} ≤ 0.

Consider then the sequence of successive adjoint systems to |C|, |C′|, |C′′|, …; the number of intersections (2π − 2 − n) which C has with the virtual canonical curves, increases by p^{(1)} − 1 in passing to the adjoint system, that is, it actually decreases by one at least. Therefore at least one of the following facts must be true:

(1) either the difference n − (2π − 2) which is obtained from the characters of a system |C| becomes positive for some of the adjoint systems, and this directly contradicts our hypothesis;

(2) or the above sequence of adjoint systems is finite.

In this case, as it happened in Hypothesis I, one either obtains a final linear system, which is composed with the rational curves of a linear pencil, or a pencil of elliptic curves which is the adjoint system of the hyperelliptic curves of genus 2 of a linear ∞^{2} system of degree at least 4. Whence, with the same argument as we have developed in Hypothesis I, one deduces the rationality of F. But we do not even need to repeat that argument; it is sufficient to point out that, in contrast with our Hypothesis II, we have achieved the construction of a linear system of genus π and degree n > 2π − 2 on F, thus finding a contradiction. It is convenient to remark at this point that, if one has on F a system |L| however reducible of genus π and degree n > 2π − 2, which does not contain any exceptional component, then by adding a multiple of this system to a regular system, one can always obtain an irreducible linear system whose characters satisfy the same inequality. In conclusion we may state the theorem: the necessary and sufficient conditions for a surface to be rational are expressed by the vanishing of the numerical genus and of the bigenus:

p_{a} = P = 0;

where the condition P = 0 a priori yields also p_{g} = 0. So in other words one can say that the rational surfaces are characterized as “regular surfaces with bigenus zero”.

This theorem is due to G. Castelnuovo who proved it essentially by the same method that we have used here: we have only introduced simplifications of some details.

Classification of surfaces of linear genus p^{(1)} = 1

7.1 Surfaces of genus zero and bigenus one, with bicanonical curve of order zero

We have seen (Chapter VI, Section ??) that among the regular surfaces of genus p = 0 and bigenus P = 1 those having a bicanonical curve of order zero are characterized, with respect to the birational transformations, by the value of the trigenus, which is

P_{3} = 0,

or by the value of the sextic genus, which is

P_{6} = 1

(as opposed to P_{6} > 1).

Now we want to investigate in detail these surfaces F, by reducing them to a definite projective type, that is, is the double plane with a branch curve D_{8}, formed by a sextic with two tacnodes M and N and with a double point at the intersection of their tacnodal tangents p and q, plus these two lines p and q. We will then see that such a double plane can be transformed into a surface F_{6} of the 6th order, passing doubly through the edges of a tetrahedron; therefore this surface can also be taken as the representatives of the surfaces of our family (p = 0, P = 1, P_{3} = 0).

To start with let us recall the main properties of our surfaces F, referring to a model without exceptional curves:

(1) every curve C of genus π on F has the degree

Surfaces of genus zero and bigenus one 7.1 229

n = 2π − 2; (2) the dimension of the complete linear system |C| (virtually free of base

points) is

r ≥ π − 1;

(3) for every irreducible linear system |C|, of genus π > 1, one has exactly

r = π − 1

namely the system is regular;

(4) the linear system |C| has an adjoint system |C′|, distinct from |C| but with the same characters; the relationship between the two systems is reciprocal, since also |C| is the adjoint of |C′|: the two systems are two different halves of the same double system

|L| = |2C| = |2C′|;

(5) the surface F may contain elliptic curves, namely irreducible curves of virtual genus π = 1, thus of degree n = 0. These may either be isolated curves, namely forming a regular system of dimension r = 0 or they may belong to a pencil (a superabundant system, of dimension r = 1). If |C| is a pencil of elliptic curves on F, then the adjoint system |C′| consists of the sum of two isolated elliptic curves C/2 and (C/2)′, that are adjoint to one another, and whose doubles belong to the pencil. In fact since the adjoint curve C′ has no variable intersections with the C of the pencil, and it is distinct from a C, it has to consist of parts of curves C, and on the other hand |2C′| must be equivalent to a pair of curves C.

Conversely, if the pencil |C| contains a half curve C/2, then also its adjoint curve (C/2)′ will be one half of a curve of the pencil, and the adjoint curves to C/2 + (C/2)′ will be the curves 2C/2 = C. Then, by property (4), a pencil |C| of elliptic curves contains only one pair of double elliptic curves, which are two halves of the pencil, and are adjoint to one another;

(6) a curve θ, which is not a component of the elliptic curves of a pencil |C|, necessarily cuts out on the C an even number of points. In fact this number is the double of the number of intersections of θ with the curves C/2;

(7) an irreducible curve θ of (virtual) genus zero, thus of degree −2, on the surface F, has no adjoint curves.

In fact, if an adjoint curve θ′ existed, then it would meet θ at −2 points, and therefore it would coincide with θ ; but θ = θ′ yields p = 1, contrary to the hypothesis.

In other words the existence of θ′ implies that θ has genus > 0;

(8) given a pencil of elliptic curves |C| on the surface F, there is no irreducible curve θ of genus 0 which is a bisecant of the pencil.

In fact the system C + θ, which would be a pencil of curves of virtual genus ρ = 2, should have an adjoint system |L| of dimension 1, containing the composite curve θ + C/2 + (C/2)′; but we can see that, by contrast, this curve cannot vary in a pencil. To show this, we remark that C/2 has to meet the curve L at only one point, since

\frac{C}{2} L = \frac{C}{2} θ + \frac{C}{2} \frac{C}{2} + \frac{C}{2} (\frac{C}{2} )′

and

\frac{C}{2} \frac{C}{2} = 0;

therefore the L meet C/2, and similarly (C/2)′, at one fixed point; therefore – if there are ∞^{1} of them which are not reduced to θ itself plus the curve C′ = (C/2) + (C/2)′ – then these curves have no intersections with the C, and thus they have to be composed with the C and with a curve θ′ adjoint to θ: but the existence of θ′ would imply that the genus of θ is > 0, contradicting the assumption.

We will take into account assertions (1–8). If |C| is an irreducible linear system, of dimension π−1, let us consider the

double system |L| which has degree 4n = 8π −8, genus 4π −3, and dimension 4π − 4. Inside the system |L| there are two ∞^{2π−2} subsystems, that we call Σ and Σ′, the first made up of pairs of curves C, and the second of pairs of curves C′. Each one has index

i = \frac{1}{2} ( \begin{matrix}2π − 2 \\ π − 1 \end{matrix} ) .

If we denote by S_{4π−4} the space whose points represent the elements (curves) of the system |L|, then we have inside this space two varieties of dimension 2π − 2 and order i, which represent Σ and Σ′, which have either i^{2} or infinitely many points in common. So one can find curves L which can be decomposed simultaneously as two curves C and as two curves C′; these L are necessarily reducible and in general allow us to construct inside a system |C| of genus π > 1, a system which is partially contained in it and has genus strictly less than π.

More precisely we can find a curve composed of two C:

C = C_{1} + C_{2} and C = C_{3} + C_{4},

and decomposed in such a way that we get a curve C′ by adding together one summands of the first plus one summand of the second, say

C_{1} + C_{3} = C′, C_{2} + C_{4} = C′,

so that, since

C_{1} + C_{3} = (C_{1} + C_{2})′,

we deduce that

C_{3} = C′_{2}

and similarly

C_{4} = C′_{1}.

This yields that the curves C_{1}, C_{2} and similarly C_{3} and C_{4} have genus ≥ 1.

We distinguish two cases:

(a) one of the two curves, C_{1} say, has genus π_{1} > 1.

(b) both C_{1} and C_{2} has genus 1.

In case (a) we can say that C_{1} belongs to a complete linear system |C_{1}| of dimension π_{1} − 1 ≥ 1. Since this system is partially contained in |C|, it has dimension π_{1} − 1 < π − 1, thus π_{1} < π.

Furthermore, the variable parts of the curves C_{1} must be irreducible and again of genus ρ > 1 (with ρ < π). Indeed, assume these parts to be composed with s curves K of a linear pencil (with s > 1). If the genus of the K were greater than 1, then the complete system s-multiple of |K| would be of dimension greater than s, and therefore irreducible.

However, suppose that the genus of the K equals 1. Under this assumption, the groups of sK only make up a part of the curves C_{1}:

|C_{1}| = θ + |sK|.

In order that the genus of C_{1} be π_{1} > 1 it is necessary that θ, or one of its components θ_{1}, meet the K at some point (a curve which does not intersect the K is a part of some K and its genus is therefore ≤ 1). Since the number of these intersections is even, thus greater than or equal to two, by adding θ_{1} to |sK| one certainly succeeds in enlarging the system |sK| if θ_{1} has genus 1, and even if θ_{1} has genus 0; in fact under this assumption the number of intersections with the K has to be greater than 2 (see assertion (8)) and so greater than or equal to 4.

In conclusion, the system of the C_{1}, or rather a complete system contained in it, is an irreducible linear system of genus greater than one.

Therefore, if case (b) never occurs, the iterated application of our reduction process leads to a linear system, partially contained in |C|, of genus two.

Now consider case (b). Therefore let us suppose that among the curves of a system |C| of genus π there is a curve which is reducible into two curves of genus 1,

C = C_{1} + C_{2}.

Since C_{1} and C_{2} are curves of genus 1 and degree 0, their doubles 2C_{1} and 2C_{2} give rise to two pencils of elliptic curves, whose variable parts we shall denote by K_{1} and K_{2}; as before, it is easy to see that neither of these two pencils can have fixed parts which are not components of the curves of the pencil itself. So the curves of the two pencils |K_{1}| and |K_{2}| must intersect at a positive number of points, in order that the degree of |C_{1} + C_{2}| be positive. Now the two halves of K_{1} and K_{2}, say K_{1}/2 and K_{2}/2, are two isolated elliptic curves, with a certain number s of intersections, such that

1 ≤ s ≤ π − 1.

In conclusion we may say that, if our reduction process does not lead to the construction of a pencil of genus 2, then one may in any case arrive at two isolated elliptic curves, which we call again C_{1} and C_{2}, with a certain number s ≤ π − 1 of common points (s > 0). Actually, we may not say that C_{1} and C_{2} have no common parts; it might happen, for instance, that C_{1} breaks up into two curves of genus zero with two common points, C_{1} = θ + θ_{1} and similarly C_{2} = θ + θ_{2}; then C_{1} and C_{2} have the common component θ, but, in the arguments which are based on the use of the virtual characters, the curves θ + θ_{1} and θ + θ_{2} behave “as if” they were irreducible with

s = (θ + θ_{1}, θ + θ_{2})

common points.

The reduction process that we have indicated, starts with an irreducible linear system on the surface F and leads to the construction on F: either of an ∞^{1} linear system |C| of genus 2, or of a pair C_{1} and C_{2} of elliptic curves with s (≥ 1) common points.

In the first case the curves C have to be considered irreducible, since in any case their variable parts, which form a pencil, are irreducible of genus 2: indeed if these were curves K of genus 1, |C| would have a fixed part θ, meeting the K at two points only, which is impossible.

Therefore one can find some curves in |C| which are reducible into two isolated elliptic components C_{1} and C_{2}, with one common point.

In the second case, when one has two isolated elliptic curves C_{1} and C_{2} on F with s > 1 points in common, then one can find a pair of elliptic curves on F intersecting at s′ < s (with s′ ≥ 1) points, and thus also a pair of elliptic curves with one common point. The feasibility of this further reduction can be proved as follows.

If s > 1 then the linear system |C| = |C_{1} + C_{2}| has genus π = s + 1 > 2. Therefore among the pairs of curves 2C, one has in general

i = \frac{1}{4} ( \begin{matrix}2π − 2 \\ π − 1 \end{matrix} )^{2} ≥ 9.

reducible curves. If these contain some components of genus greater than or equal to two, then with the procedure treated above, one arrives at a pair of elliptic curves with a number of intersection s′ < s. If, by contrast, all the aforementioned reducible curves are formed by elliptic components, then we make the following argument.

Precisely, we have to assume that, beside the pair of reducible curves which forms

C_{1} + C_{2} + C′_{1} + C′_{2},

we can find another pair of curves which are reducible into elliptic components

C_{3} + C_{4} + C′_{3} + C_{4},

where C_{3} and C_{4} are distinct from C_{1}, C_{2}, C′_{1} and C′_{2}: the existence of C_{3} + C_{4} can be deduced from the assumption that the two varieties Σ and Σ′ representing the pairs of curves C and C′ have some other common point that is distinct from the two points corresponding to the pairs

C_{1} + C_{2}, C′_{1} + C′_{2}

and

C_{1} + C′_{2}, C_{2} + C′_{1};

and we will see in a while that indeed the pairs of isolated elliptic curves C_{1} + C_{2} and C′_{1} + C′_{2}, cannot correspond to infinitely near intersections of Σ and Σ′.

Then, since C_{3} + C_{4} is a reducible curve of the system |C| different from C_{1} + C_{2} and from C_{1} + C′_{2}, C_{3} and C_{4} must cut out some point on each one of the two curves C_{1} and C_{2}, since otherwise they should contain C_{1} itself, or one of the two curves C_{2} or C′_{2}, as a part; so, for instance, C_{3} has to meet each of the curves C_{1} and C_{2} at fewer than s points; therefore we have obtained, as we wanted, a pair C_{1} and C_{3} of isolated elliptic curves with s′ < s common points.

We are left with the problem of showing that our system |C| cannot contain two pairs of infinitely near isolated elliptic curves C_{1} + C_{2} and C_{3} + C_{4}, where, for instance, C_{3} is infinitely near to C_{1} and C_{4} to C_{2}. In fact the curves 2C_{1} and 2C_{3} would define a pencil of elliptic curves having C_{1} as a fixed component, so that C_{1} itself would vary in a pencil of elliptic curves, whereas we supposed it to be isolated.

From the above discussion we draw the conclusion that our surface F possesses a pair C_{1} and C_{2} of isolated elliptic curves having one common point, so that their adjoint curves C′_{1} and C′_{2} are incident to C_{2} and C_{1}. Therefore the reducible curve C_{1} + C_{2} + C′_{2} turns out to be of genus 3, and defines a net of curves |K| of the same genus: the K cut out on each one of the curves C_{2} and C′_{2} only one point, which must be fixed; therefore the K have two simple base points which coincide precisely with the common points of C′_{1} and C_{2}, and C_{1} and C′_{2}. This happens because C′_{1} belongs to ∞^{1} curves K, since its residual system with respect to |K| is the pencil of elliptic curves |2C_{2}|: this assertion follows from the remark that the curves C′_{1} + 2C_{2} are the second adjoint curves of the curves K = C_{1} + C_{2} + C′_{2}, and are therefore equivalent, since they belong to the same system |K|.

Thus the K are hyperelliptic curves, intersecting two by two at two variable points; moreover these K cut out on C_{1} the pairs of a g_{2}^{1}, since there is only one residual curve of C_{1} with respect to |K|, namely C_{2} + C′_{2}. Let us add that the base points of the K are, on any K, two double points of the g_{2}^{1}.

By means of the net |K| of hyperelliptic curves of genus 3, one may now represent the surface F on a double plane, which has a branch curve D_{8} of order 8.

This curve D_{8} contains two branch lines, p and q, which correspond to the base points of |K|; and the point O = pq is the image of the elliptic curve C′_{1}, thus it will be a quadruple point for D_{8}, that is, a double point for its component D_{6} which is left once the lines p and q have been removed. The lines of the pencil with centre at O will be, in the double plane, the images of the elliptic curves of the pencil |2C_{2}| = |2C′_{2}|; the points corresponding to C_{2} and C′_{2} are two tacnodes of D_{6} with p and q as tacnodal tangents, namely two points of type [3, 3] of the branch curve D_{8}. These singularities of D_{8} actually characterize D_{8} as the branch curve of a double plane of genus p_{a} = p_{g} = 0 and P = 1, with (pure) bicanonical curve of order zero, as one may see from the results of Chapter V, Section ??.

We may therefore conclude by saying

the regular surfaces of genus zero and bigenus 1, with bicanonical curve of order zero – which are characterized by the value P_{3} = 0 of the trigenus, or P_{6} = 1 of the sextic genus – form a unique family, whose typical element is the double plane with branch curve

D_{8} = D_{6} + p + q

formed by a sextic curve with two tacnodes and a double point at the intersection of their tacnodal tangents, plus these two tangents.

This family of surfaces depends on 10 moduli. In fact the construction of the D_{6} with 5 assigned double points, and therefore of the D_{8}, depends on

\frac{6 · 9}{2} − 5 · 3 = 12

parameters, so, adding the 6 constants which depend on the choice of the singular points of the D_{6}, one has ∞^{18} curves D_{8}. From the 18 parameters on which they depend, we have to subtract 8, corresponding to the ∞^{8} homographies, which, of course, define the same surface F.

Let us consider now the sextic surface F_{6} of the ordinary space, which passes doubly through the edges of a tetrahedron T. Since F_{6} is a regular surface with p = 0, P = 1, P_{3} = 0, it must be representable on a double plane with a branch curve D_{8} having the stated singularities. One actually obtains such a representation of F_{6} on a double plane by means of the skew projection from a pair of opposite edges a and b of the tetrahedron T, since the lines which are incident to a and b intersect F_{6} at two simple points. But this construction leads in general to a double plane with a branch curve D_{10}, which can be reduced to our D_{8} only with a further transformation of the plane. It is therefore useful to specialize the above construction, by projecting on a plane passing through an edge c of the tetrahedron, incident to a and b. Then the lines of the double plane appear to be the images of the curves K cut out on F_{6} by the quadrics through a, b and c. These curves K are hyperelliptic of genus 3, and among them one finds a curve which is reducible as the sum of the neighbourhoods of the 3 double lines of the tetrahedron T: c, d and e, where d and e are the edges of T which are the further intersections of F_{6} with the planes ca and cb. Let us remark that the components of this reducible curve K, namely the neighbourhoods of the 3 double lines c, d and e, are elliptic curves connected one to another in the same way as the C_{1}, C_{2}, C′_{2}, the curves we used before in the construction of the net of the K which led us to the double plane with the branch curve D_{8}.

So we can verify in a very simple way the conclusion which we a priori know to be true for the F_{6} by the property of being a regular surface of genus p = 0 and bigenus P = 1, with bicanonical curve of order zero.

We notice that there are ∞^{13} sextics F_{6} passing doubly through the edges of a given tetrahedron (we wrote the equation of these in Chapter III, Section ??), and that there are ∞^{3} homographies with the vertices of T as fixed points; whence it follows that the F_{6} depend on 10 parameters or projective invariants, exactly as the double planes with the branch curve D_{8}, to which, as we know, they can be reduced. The fact that these projective invariants give an equal number of moduli for the surfaces F_{6}, with respect to the birational transformations, can be deduced in this way: the plane section system is complete, and since the surface is regular, it cannot be contained in a continuous series of analogous systems, although indeed one may have on F a discontinuous series of such systems^{(1)}.

It follows from this that the F_{6}, as well as the corresponding double plane with branch curve D_{8}, can be assumed as the general typical element of the regular surfaces of genus p = 0 and bigenus P = 1, with a bicanonical curve of order zero (P_{3} = 0).

The problem of constructing the transformed F_{6} starting from the double plane with a branch curve D_{8} can be solved by determining a cubic K_{3} of this plane passing through the 5 double points of D_{6}, touching D_{6} at 4 points and such that it is the image of a reducible elliptic curve on the surface F represented by our double plane. The fact that among the cubics K_{3} adjoint to D_{6} that are also four-tangent to it there are actually the images of some reducible curves can be proved by using the criterion of Comessatti that we have already mentioned many times. One needs to prove that there is a K_{3} that is four-tangent to D_{6}, and whose contact points lie on another cubic curve that is also adjoint to D_{6}. In order to see this, one has to start with any cubic K_{3} and choose two points P and Q having the same tangential point O; then one chooses a group of 4 points A_{1}A_{2}A_{3}A_{4}, which is a section of K_{3} with another cubic curve K′_{3} passing through O and tangent to our K_{3} at P and Q; then one verifies that there are ∞^{5} curves D_{6} of the 6th order passing doubly through O and having at P and Q two tacnodes with tacnodal tangent lines p = P O and q = QO respectively, and tangent to K_{3} at A_{1}, A_{2}, A_{3}, A_{4}: as K_{3} varies as well as the points we choose on it for the above construction, D_{6} varies in the complete system of the sextic curves with two tacnodes, one double point at the intersection of their tacnodal tangent lines p = P O and q = QO, and tangent to K_{3} at A_{1}, A_{2}, A_{3}, A_{4}. One deduces therefore that, given any such D_{6}, there exist adjoint cubics K_{3} which are four-tangent to it and are the images of two distinct elliptic curves on the surface F.

(1)The actual existence of such a series stems from the property of F_{6} of possessing a discontinuous infinite system of birational transformations into itself. These can be defined in relation to the ∞^{1} elliptic quartics K cut out by the quadrics passing through 4 edges abde of T forming a skew quadrilateral. On each one of these curves one defines a birational transformation, by adding to a point P the virtual group of order zero which is the difference of the pairs G_{d}−G_{e} cut out by two incident edges d and e: we notice here that these pairs G_{d} and G_{e} neither can be equivalent, nor they can have equivalent multiples, since this would imply that d and e themselves should be equivalent, thus contradicting the fact that d and e have a common point. Therefore the transformation defined on each K of our pencil, which carries the point P to the point P′ = P + G_{d} − G_{e} cannot be a cyclic one, and thus it gives rise to an infinite series of powers, which are again birational transformations of the surface into itself.

Now let us consider the curves of genus 2 on F whose images are the lines through one of the two tacnodes of D_{6}, for instance P, which we may assume to be distinct from O; by adding to these curves of genus two a bisecant elliptic curve corresponding to our curve K_{3}, one defines an ∞^{3} linear system of genus 4 and degree 6 which leads to the representation of F onto a sextic F_{6} passing doubly through the edges of a tetrahedron. The images of the plane sections of this F_{6} on the double plane are ∞^{3} curves K_{6} of the 6th order, with the double points O, P, Q, and P′, Q′ infinitely near to P and Q, in common with D_{6}; furthermore they have a variable double point and they are eight-tangent to D_{6}.

The meaning of the construction we worked out on the double plane may also be explained by saying that: on the surface F, which contains, as we know, a simple quadrilateral of isolated elliptic curves, C_{1} + C_{2} + C′_{1} + C′_{2}, there are two other isolated elliptic curves C_{3} and C′_{3} (corresponding to the aforementioned K_{3}), which are incident with the edges of the quadrilateral. Three of the two-by-two incident elliptic curves, say C_{1}, C_{2} and C_{3}, form a curve of genus 4 which belongs to an ∞^{3} linear system; this leads to the transformation of F into a F_{6} where the above-mentioned curves have as their images three double lines, edges of a triangle; their adjoint curves C′_{1}, C′_{2}, C′_{3} give also rise to three double lines, the edges of a tetrahedron containing the triangle and passing through the opposite vertex to the triangle in the tetrahedron.

In conclusion we state the theorem^{(2)}:

The regular surfaces of genus p = 0 and P = 1, with bicanonical curve of order zero – which are characterized either by the value P_{3} = 0 of the trigenus or P_{6} = 1 of the sextigenus – can be transformed into sextic surfaces F_{6} passing doubly through the edges of a tetrahedron.

Remark It is useful to point out some remarkable particular cases of the sextic F_{6}.

As a first example, it may happen that the four planes of the tetrahedron, whose edges are the double lines of F_{6}, all pass through one point. The tetrahedron degenerates therefore to an angoloid, whose vertex becomes a quadruple point for F_{6}. This case has been pointed out by Castelnuovo.

(2)F. Enriques, Sopra le superficie algebriche di bigenere uno, Mem. Soc. It. delle Science, detta dei XL, 1906.

Surfaces with all the genera equal to one 7.2 238

Projecting down from the quadruple point onto a plane, F_{6} is represented as a double plane with a branch curve D_{10}, which, via a quadratic transformation may be reduced to the D_{8} of the type we defined before.

Another possible degeneration of the tetrahedron is obtained by letting two opposite edges become two infinitely near skew lines, and also letting the pairs of faces through these lines come together. One obtains in this way a F_{6} with a triple line, an infinitely near double line, skew with it, and two other skew double lines which are incident to the triple line. This F_{6} corresponds to a double plane with a branch curve D_{8} = p + q + D_{6}, where one of the tacnodes of D_{6} is infinitely near to the double point O.

7.2 Surfaces with all the genera equal to one

We have seen that the regular surfaces with genus p = 1 and canonical curve of order zero are characterized, among those with linear genus p^{(1)} = 1, by the value P = 1 of the bigenus, instead of P > 1. Also all the other plurigenera of the surface turn out to be P_{i} = 1.

If one takes a surface F without exceptional curves as a model, then every system |C| of genus π without base points has degree n = 2π − 2 and is selfadjoint; therefore, for an irreducible system |C| the characteristic series is the canonical series.

On the surface F (with p = P = 1), a connected curve, reducible or irreducible, of virtual genus π ≥ 0 belongs to a complete linear system of dimension r = π (see Chapter IV, Section ??).

By contrast a disconnected curve belongs, in general, to a linear system of dimension greater than its genus π; in particular: the groups of s > 1 elliptic curves of a pencil on F, belong to a (reducible) linear system of genus 1 and dimension s. The linear systems which contain a curve θ of genus zero as a fixed part are also superabundant.

Moreover let us notice that: if we add to a linear system |C| on F without fixed parts a fixed curve θ which intersects the variable components of the C in more than one point, then the linear system becomes larger. The only case in which this enlargement does not occur is when one adds to the groups of s elliptic curves of a pencil a curve of genus zero which is unisecant to these curves.

In order to classify the surfaces with genus one (p = P = 1), we consider on a given F the irreducible linear systems of minimal genus π > 1 which belong to it. To start with we may suppose the genus of such a minimal system |C| to be π = 2, and then π = 3, π = 4, …

The hypothesis π = 2 yields that the degree of the system is n = 2 and its dimension r = 2, so that F can be represented on a double plane with a sextic branch curve D_{6}. And in fact the double plane with branch curve D_{6} has genus

p = p_{a} = p_{g} = 1,

and bigenus P = 1, since on such a surface there is only one canonical curve of order zero.

For π = 3 we are led to the general surface F_{4} of order 4, which actually has the genus p = 1, and has one canonical curve of order zero.

For π = 4 we will be led to consider a surface F_{6} of S_{4}, obtained as a complete intersection of a quadric Q_{2} and a cubic variety V_{3}. In fact the ∞^{14} quadrics of S_{4} cut out on F_{6} the double of the hyperplane sections system, which is of genus

4 + 4 + 6 − 1 = 13,

and so also of dimension 13; thus there is a quadric Q_{2} which contains F_{6}. One similarly proves that there are ∞^{5} irreducible cubic varieties V_{3} passing through the surface F_{6}, which is therefore defined as the complete intersection of a Q_{2} and a V_{3}. On the other hand, it is clear that the complete intersection surface defined in this way has canonical curves as hyperplane sections, and therefore the linear system of these hyperplane sections is an ∞^{4} self-adjoint system of curves of genus 4, whence

p = p_{a} = p_{g} = 1, P = 1.

For π = 5 we are led to a surface F_{8} of order 8 in S_{5}, which is easily seen to be the base surface of a net of quadrics.

It is easy to continue the sequence of the surfaces of genus 1 that we have just constructed for the first values of π, by considering, for instance, the intersections of quadrics with rational threefolds whose curve sections are elliptic. But, without dwelling upon these examples, let us answer the general question which arises here:

The regular surfaces with genus p = 1 and bigenus P = 1 (with canonical curve of order zero) may be classified by means of the minimum value of the genus π (> 1) of the irreducible curves which belong to them; they are therefore divided into infinitely many families whose typical element is a surface F_{n} of order n = 2π − 2, with canonical hyperplane sections, belonging to the space S_{π}.

There are families of surfaces F_{n} corresponding to all the values of π = 2, 3, 4, 5, …, and each family depends on 19 projective invariants, which give an equal number of moduli for the surfaces with respect to the birational transformations. Surfaces corresponding to different values of n and π are in general distinct one from the other with respect to the birational transformations.

The actual existence of surfaces F_{n} for all the integral values of π, or rather for all the even values of n = 2π −2, may be proved by observing that there exist at least ∞^{18} particular surfaces F_{n} which can be obtained as a projection of a F_{n+2} with a double point. In order to prove this, it suffices to notice that, among the ∞^{19} F_{n}, there are ∞^{18} special F_{n} containing a conic. For the time being, we justify this claim with a simple count of parameters: the existence of a four-tangent hyperplane to F_{n} imposes a priori x − π + 4 conditions, if the existence of such a hyperplane yields, as a consequence, the existence of ∞^{x} similar hyperplanes. But if the section of a 4-tangent hyperplane to F_{n} breaks up into a conic and a residual curve of order n − 2 (which must have 4 common points with the conic) the existence of the aforementioned hyperplane leads to the existence of an ∞^{π−3} system of similar hyperplanes, precisely the hyperplanes through the plane of the conic, and then one has x = π − 3. Therefore, since x − π + 4 = 1, there will be in general no four-tangent hyperplane to F_{n} cutting out on the surface a reducible curve of which the conic is a part; thus the surfaces for which such planes do exist, that is, the surfaces containing a conic, must satisfy a condition.

Now we start with an F_{n} containing a conic C and consider the linear system on it which is the sum of C with the hyperplane sections; this is a system of genus π + 1 and dimension π + 1, which has C as a fundamental curve; thus this system yields the existence of an F_{n+2} of S_{π+1} with canonical hyperplane sections, and with a double point O, which corresponds to C; such an F_{n+2} is projected from the double point O onto the F_{n} we started with, or rather onto a surface which is projectively equivalent to it^{(3)}.

But the family of surfaces F_{n+2} with a double point, which depends only on 18 absolute invariants, will be contained in a family of F_{n+2} depending on 19 invariants, and therefore in general without double points.

Indeed the existence of this larger family of F_{n}, that is, the count of their moduli, may be justified (as in Chapter V, Section ??) by representing the surface onto an n-tuple plane, and by computing the dimension of the system of curves (with a given number of nodes and cusps) defined by the branch curve C. Since we deal here with an F_{n} having a double point, the curve C has a double point which has to be considered as virtually nonexistent, and one may a priori guess that it should disappear for the more general curve of the system indicated above.

(3)As we see, the above argument is suggested by the attempt to construct surfaces F_{n} with n = 2π − 2 inductively (passing from π to π + 1) by means of Chisini’s constructive process for multiple planes (see Chapter V, Section ??). But the delicate question encountered in this process (in relation with the choice of a suitable sextitangent conic to the branch curve), may be solved here by using the Riemann-Roch theorem for the surfaces with all the genera equal to one.

However our assertions gives rise to certain critical doubts of a delicate nature. We need to prove that:

(1) the F_{n} which (according to Chapter V, Section ??) have at least 19 moduli, actually have 19 moduli and no more;

(2) the F_{n} (or the F_{n+2}) which contain a conic actually depend on 18 moduli and no more, that is, that the condition of containing a conic is not identically satisfied by the more general F_{n} of its family.

These assertions can be justified by counting the number of projective invariants which belong to a rational surface with canonical hyperplane sections of genus π, having a triple point and lying in a space S_{π}. These surfaces can be represented on the plane by an ∞^{π} linear system of genus π and degree 2π − 2, having a fundamental cubic curve on which there must be 12 base points of the system forming a G_{12}^{(4)}; therefore there are exactly 12 absolute invariants, in correspondence with the 11 independent points of the G_{12} and with the absolute invariant of the cubic. Since, by imposing a triple point for a surface F_{n}, one has 7 (and no more) conditions, we deduce that the surfaces F_{n} of S_{π} cannot form a family depending on more than 19 moduli. The same argument shows that the general F_{n+2}, which depends on 19 invariants, cannot have a double point, since, by imposing that this double point become a triple point, one imposes 6 and no more conditions on the surface. Whence it follows that the F_{n} containing a conic, which also arise as a projection of an F_{n+2} with a double point, depend on 18, and not on 19, moduli.

The question remains of justifying the claim that the surfaces F_{n} (where n = 2π − 2) corresponding to different values of π (> 1), are in general birationally distinct. This may be worked out by induction, with a simple count of parameters like the one we used before to show that an F_{n} in general contains no conic. Indeed with a similar computation one proves that on a surface F_{n} of S_{π} (corresponding to the minimum value π of the genus of the curves which belong to it) in general there are no linear systems of curves other than the linear system of the hyperplane sections and its multiples.

(4)The theorem we refer to is due to P. Du Val, On rational surfaces whose prime sections are canonical curves. (Proc. London Math. Soc., 1932). It is also virtually stated in the treatment of Enriques–Conforto, Le superficie razionali, Libro I, §41, page 184 and §46, page 220. In this book (see also §46, page 223) it is proved that a linear system |C| either may be transformed in such a way that it has no fundamental curves which are parts of the adjoint system, or it can be transformed into a linear system formed by curves of a certain order n having two base points of multiplicities r and s, with n = r + s, or also into a system of curves of a certain order n with a (n − r)-tuple base point and other r-tuple base points which are infinitely near to it in different directions. In the last two cases the system |C|, which is supposed to be of genus π, has degree n > 2π − 2; therefore all the systems |C| with canonical characteristic series fall in the statement of page 184.

Now we wonder whether it is possible to find, among the ∞^{π} hyperplanes of S_{π}, an m-tangent hyperplane to F_{n} such that its section with F_{n} is reducible into two curves of genus π_{1} and π_{2} with m common points: π = π_{1}+π_{2}+m−1. If such a hyperplane exists, then there are ∞^{x} such hyperplanes, with x = π_{1}+π_{2}; therefore the number of parameters on which the hyperplanes we look for depend is π − m − x = −1: this means that the surfaces F_{n} possessing this kind of reducible curves are subject to one condition. In a similar way we see that the section of an F_{n} with a hypersurface of any given order in general cannot be reducible unless it breaks up into two curves which in turn are complete intersections with two hypersurfaces; in the latter case the count of parameters gives again one condition which the surfaces F_{n} should satisfy, but it is obvious that such a condition is as a consequence of the fact that F_{n} contains the hyperplane section system, and therefore it is identically satisfied.

Now it is important to notice that, as we have already said before (in the discussion of the F_{n} which contain a conic), there is an objection to the argument based on our count of parameters, which can already be moved to Noether’s proof of the theorem to the effect that the general surface of order 4 of the ordinary space only contains complete intersection curves.

However the result can be rigorously proved in a simple way, on the ground of the principle of continuity, by reducing an F_{n} to the limit case of a rational surface with a triple point. We shall confine ourselves to briefly showing how this proof can be developed in the elementary case of Noether’s theorem for surfaces of order 4: the extension of the argument to the F_{n} of genus one presents no difficulty.

So let a quartic surface F_{4} vary continuously, until it degenerates to a general monoid Φ_{4} with a triple point O: the monoid, which contains 12 lines a_{1}a_{2}a_{3} … a_{12} through O, can be represented on a plane, by projecting from O, by means of a linear system of curves C_{4} of order 4 passing through the 12 base points A_{1}A_{2} … A_{12} cut out on the plane by the lines a_{i}: these points lie on a fundamental cubic curve C_{3} of genus one.

Our argument consists in showing that: when the surface F_{4} tends to the general monoid Φ_{4}, an irreducible, complete linear system of curves C on F_{4} becomes in the limit a linear system of curves \overline{C}, which are complete intersections of Φ_{4} with surfaces of a certain order d, not containing the triple point O. So, if π is the genus of the \overline{C}, and therefore also of the C, the dimension of |C| is π and its degree 2π − 2, hence the curves C on F_{4} must have 2π − 2 common points with the intersections with the surfaces F_{d} of order d, which are also of genus π and degree 2π − 2; since a C cannot contain a linear series g_{2π−2} of dimension π, we deduce that the system |C| on F_{4} must coincide with the linear system of complete intersections with the F_{d}.

In order to develop the proof that we have just indicated, we remark that, when the surface F_{4} degenerates to Φ_{4}, a system |C| of genus π and degree 2π − 2 belonging to F_{4} will a priori tend to an ∞^{π} system of connected limit curves which will be formed by the intersections of Φ_{4} with surfaces passing a certain number of times through O, plus the lines a_{i}, counted h_{i} times respectively (with h_{i} ≥ 0), plus the neighbourhood of O counted s (≥ 0) times. Therefore the images of such curves on the plane on which Φ_{4} is represented will, a priori, be curves of a given order n, belonging to a complete linear system

|C_{n}| = C_{m}(A_{1}^{r_{1}}A_{2}^{r_{2}} … A_{12}^{r_{12}}) + sC_{3} + ∑ h_{i}A_{i},

where the curves C_{m} (of order m ≤ n) belong to an irreducible ∞^{π} linear system. Since the C_{m} have genus π and degree 2π − 2, we have

n^{2} − ∑(r_{i} + s − h_{i})^{2} = 2π − 2
n(n − 3) − ∑(r_{i} + s − h_{i})(r_{i} + s − h_{i} − 1) = 2π − 2,

and therefore

3n = ∑(r_{i} + s − h_{i}).

This means that for the linear system |C_{n}| the cubic C_{3} is a virtual fundamental curve, that is, it has zero points of intersection with the curves C_{n}.

First of all we need to discuss the assumptions:

(1) C_{3} is also actually a fundamental curve for the complete system |C_{m}|, namely

|C_{n}| = |C_{m}|, s = 0, h_{i} = 0.

Under this assumption we easily prove that |C_{n}| is a multiple of the system C_{4}(A_{1}A_{2} … A_{12}) and therefore the corresponding system on Φ_{4} is formed by complete intersections of this monoid with the surfaces of order d = n/4.

In fact, denoting by G the group of sections of the elliptic cubic C_{3} with a line, we have the following equivalences:

∑ r_{i}A_{i} ≡ nG
∑ A_{i} ≡ 4G;

multiplying the second relation by r_{1} and subtracting the first one from it, we find:

∑_{i=2,...,12}(r_{i} − r_{1}) ≡ (n − 4r_{1})G;

since the group of the 11 points A_{2}A_{3} … A_{12} can be arbitrarily chosen, the latter relation cannot hold unless we have n = 4r_{1} and

r_{1} = r_{2} = ⋯ = r_{12} = r;

thus the curves C_{n} have order 4r and belong to the system |rC_{4}|. Q.E.D.

If, by contrast, assumption (1) is not satisfied, then we have two possibilities which we have to exclude.

(2) First, that the C_{n} do not contain the curve C_{3}, but only the neighbourhood of some point A_{i}:

s = 0, h_{i} > 0.

Now, if C_{3} is a fundamental curve for |C_{m}|, this irreducible linear system is a multiple, of a certain order r = r_{1} = r_{2} = ⋯ = r_{12}, of C, and – if it does not become larger when one adds to it the neighbourhoods of some of the points A_{i} – then there is only one point, for instance A_{1}, to be added and its multiplicity for C_{m} has to be 1, thus r = 1. But it is obvious that a curve C which passes through the 12 points A, plus one of these, cannot be a limit of a curve lying on F_{4}, when this surface degenerates to the monoid Φ_{4}.

If, by contrast, the curve C_{3} is supposed not to be a fundamental curve for |C_{m}|, then this system is regular (and not superabundant); therefore it always gets larger when the neighbourhood of some point A_{i} is added to it. And this contradicts the assumption that |C_{n}| is complete.

(3) We finally need to discuss the hypothesis that the cubic C_{3} appears a certain number s > 0 of times as a fixed part of the C_{n}. In this case the system |C_{m}|, for which C_{3} is not a fundamental curve, is regular, and therefore it gets larger when we add to it the neighbourhood of a point A_{i} which is not a base point for it: therefore in the sum ∑ h_{i}A_{i} only a certain number t of the points A_{i} may appear, and each h_{i} ≤ s times. If we denote by l = C_{m}C_{3} the number of variable intersections of a C_{m} with the cubic C_{3}, we have now

l + ts − 3s ≥ 0.

So we have to distinguish between two a priori possible cases: either l + t ≥ 4, in which case the enlargement of the linear system C_{m} + C_{3} + ∑ A_{i} follows; or l + t < 4, which yields s ≤ 1; in the latter case the dimension of |C_{m}| would be 1 less than the genus of |C_{n}|: and therefore the system |C_{n}| could not be the limit of a linear system belonging to F_{4}, whose dimension equals the genus.

Since the assumptions (2) and (3) are thus ruled out, we are left only with the assumption (1), whence, as we said, we deduce that the only curves belonging to a general F are the complete intersections of F with other surfaces of order d = 1, 2,… Remark As we have said, the above argument can be extended to the case of surfaces of genus 1. However one does not prove in this way that all the curves belonging to a general F with 19 moduli are the complete intersections of F with a hypersurface; it can only be proved that all curves on F, included the hyperplane sections, in case these are not the irreducible curves of minimal genus belonging to F, actually are the multiple systems of one and the same system of curves of genus ρ and degree ν; thus one has

n = sν = 2π − 2 = s(2ρ − 2).

And conversely, by going back from the rational surface with a triple point to a surface of genus one with canonical hyperplane sections, one sees (by induction) that when

n = sν and 2π − 2 = s(2ρ − 2),

there are at least two families of surfaces F_{n} of genus 1: an s-multiple surface and a simple surface, each depending on 19 moduli, and not birationally equivalent.

7.3 A historical note

The first examples of surfaces F_{n} of genus 1, having the same number of moduli as the surfaces of order 4, and not birationally equivalent to these, were pointed out by Castelnuovo^{(5)} (1893). In 1896, Enriques^{(6)} characterized, with respect to the birational transformations, the family of these regular surfaces of genus 1 with canonical curve of order zero, by means of the value of the bigenus P = 1. Later on Enriques^{(7)} and Severi^{(8)}, independently succeeded in proving the existence of families of surfaces F_{n} of S_{π}, of even order n = 2π − 2, for all the values of π. Both took as a starting point the count of moduli made by Enriques (1908, see Chapter V, Section ??). Since the surfaces F_{n} depend on (at least) 19 moduli, the existence of a family of F_{n} with that number of moduli follows by Enriques from the fact that one has some particular hyperelliptic F_{n} with 3 moduli, having 16 double points: hence one also argues that the number of moduli of the F_{n} must be 19 and not larger; the proof of this requires to show that the hyperelliptic F_{n} are characterized by the fact that they possess 16 double points, as it is well known for n = 4 (Kummer surfaces).

(5)See F. Enriques, Ricerche ecc., III, 6.

(6)F. Enriques, Sui piani doppi di genere 1. Mem. della Soc. It. delle Scienze (detta dei XL), 1896.

(7)F. Enriques, Le superficie di genere uno, Rendic. Acc. di Bologna, 1908.

(8)F. Severi, Le superficie algebriche con curva canonica d’ordine zero, Atti Ist. Veneto, Vol. LXV, 2, 1909

Severi gives an algebro-geometric proof of the existence of the F_{n} (with n = 2π − 2) for any value of π, by constructing those particular F_{n} with 4 double points which can be obtained by transforming quartic surfaces F_{4} having 4 curves of (effective and virtual) genus zero, without common points. Assuming that in order to contain these curves the surfaces F_{4} have to satisfy 4 independent conditions (and this implies Noether’s theorem that the curves belonging to a general F_{4} are complete intersections, and even a bit more), he deduces that the general F_{n} cannot have more than 19 moduli, hence they depend on exactly 19 moduli.

It is worth noticing here that some F_{n} can be constructed by starting either from the quartics or from other particular surfaces with the genera equal to one, in many different ways, but all these methods require some attention, as pointed out in the book Lezioni by Enriques–Campedelli (page 311). For example one obtains some F_{n} with n = 2π − 2, where π is of the form π = 2 + 3s, by starting from a double plane with a sextic branch curve which is the envelope of an ∞^{1} series of cubics: if we add to the genus two curves C, whose images are the lines of the plane, s elliptic curves K, whose images are these cubic curves, we find the system C +sK, with genus exactly π = 2 + 3s^{(9)}.

Since we start with particular surfaces having exactly 18 moduli and to the corresponding surfaces are all F_{n} containing a plane cubic, which are therefore subject to one (and only one) condition, we deduce that the F_{n} depend on just 19 moduli, and no more.

The questions we are discussing have been more recently clarified by A. Franchetta^{(10)}, in the way we have treated in the text. The author was looking for a connection between the proof of the existence of the infinite series of F_{n} with the proof of Chisini’s existence theorem. In this way he could give to this proof the simplest form by showing how to pass from particular F_{n} containing a conic to the F_{n+2}. Franchetta also proves that the F_{n} (for any given value of n) cannot depend on more than 19 moduli; in order to do this he makes use of the rational F_{n} with a triple point as we saw above.

The fact that the F_{n} corresponding to different values of n are in general not birationally equivalent, depends, as we have said, on the theorem that on a general F_{n} there are no curves other than complete intersections with hypersurfaces. This statement may be easily guessed if one thinks to Noether’s theorem concerning the quartics F_{4} and the count of parameters on which it is based. But this does not give a rigorous proof. Enriques and Severi thought that such a proof might be given (for n ≥ 4) by continuously deforming a general F_{n} to a hyperelliptic one, with general moduli, and by using the property, which can be proved with transcendental methods^{(11)}, that on these surfaces there are no curves other than the complete intersections with hypersurfaces and the contact curves with the tangent hypersurfaces. And Severi has worked this out in an elegant way (there is only a small gap in his argument, which can be easily fixed). The algebro-geometric proof which is given here by degenerating to the limit case of rational F_{n} with a triple point, is due to A. Franchetta, who gave it in the note referred to.

(9)If π is of the form π = 3+3s, one can makes a similar construction starting with an F_{4} which contains a line; and if π = 4 + 3s, one may start from a particular F_{4} which contains a plane cubic.

(10)A. Franchetta, Sulle superficie regolari di genere uno con curva canonica d’ordine zero.

Ist. Lombardo, 1942.

Now we give some reference about the various proofs of Noether’s theorem on the surfaces of order 4 and on the general surfaces of higher order.

M. Noether, Zur Grundlegung der algebraischen Raumcurven, Abhandlungen der Akademie der Wissenschaften, Berlin, 1883.

K. Rohn, Die Raumcurven auf der Flaechen vierter Ordnung, Berichten der koenigl. Sachs. Gesellshaft der Wissenschaften, Leipzig, 1897.

F. Severi, Le superficie algebriche con curva canonica d’ordine zero, Atti Ist. Veneto, LXV, 2, 1909.

G. Fano, Sulle varietà algebriche che sono intersezioni complete di piu’ forme. Rendic. Acc. Torino, Vol. XLIV, 1909.

S. Lefschetz, On certain numerical Invariants of algebraic Varieties, Trans. Amer. Math. Soc., Vol. 22, 1921); Concerning Noether’s theorem on curves traced on nonsingular surfaces, Bull. Amer. Math. Soc., Vol. 29, 1923.

A. Franchetta, Sulle superficie regolari di genere uno con curva canonica d’ordine zero, Rendic. Ist. Lombardo, 18 giugno 1942.

B. D’Orgeval, Sur les surfaces algébriques dont tous le genres sont 1, Gauthier-Villars, Paris, 1945.

Surfaces of linear genus p^{(1)} = 1 with P > 1 7.4 248

If one looks for a complete classification of all surfaces with the genera equal to 1, there is now a new question, that is, whether the surfaces F_{n} with n = 2π − 2, with canonical hyperplane sections in an S_{π}, form in this space a unique family, or more distinct families, each depending on 19 absolute invariants or moduli. About this question we may notice that, in accordance with the final remark of the previous section, already in the space S_{5} one has 2 types of F_{8}, that is, the simple F_{8} intersection of three quadrics, and the double Veronese surface F_{4}, with branch curve of order 12, which is the image of a double plane with a sextic branch curve. In an analogous way one finds in the space S_{9} two distinct types of F_{16}, one of which may be obtained from a surface of order 4 of S_{3}. Hence one deduces the existence in S_{6} of two types of particular F_{10} with a double point, which are projected from this double point respectively onto the F_{8} and the aforementioned double F_{4}. But P. Du Val^{(12)} has shown that these two F_{10} belong to one and the same family of general F_{10}, without double points; and this agrees with the general result stated by B. D’Orgeval, in his volume already quoted above, who in fact proved that the surfaces with canonical hyperplane sections in S_{π} form a unique family.

(11)Enriques and Severi, Mémoire sur les surfaces hyperelliptiques. Acta Math., Vol. XXXII, no. 27.

7.4 Surfaces of linear genus p^{(1)} = 1 with P > 1

Let F be a surface of (absolute) linear genus p^{(1)} = 1 and geometric genus p > 1. The ∞^{p−1} canonical curves of F form a linear system of elliptic curves, of degree p^{(1)} − 1 = 0, which therefore are composed with the elliptic curves of a (base point free) pencil. The same can be said about the bicanonical or i-canonical curves of F, which form a linear system of genus

\frac{i(i + 1)}{2}(p^{(1)} − 1) + 1 = 1,

and degree

i^{2}(p^{(1)} − 1) = 0.

Thus: the canonical, as well as the pluricanonical, curves of a surface of linear genus p^{(1)} = 1 are in general reducible, and they are composed with the the elliptic curves of a pencil.

Therefore the existence of a pencil of elliptic curves, which is linear for regular surfaces, follows as a consequence of the fact that p^{(1)} = 1 whenever the surface has the genus p > 1 or one of its plurigenera P_{i} is

P_{i} > 1.

But this inequality always holds for the regular, irrational, surfaces which have some canonical or bicanonical curve of order greater than zero, since under this assumption we have seen (Chapter VI, Section ??) that p > 1 yields

P ≥ 2.

and p = 0 implies:

P ≥ 1, P_{6} ≥ 2.

Since, on the other hand, we have already found out that surfaces with p = 0 and P = 1 and a bicanonical curve of order zero (P_{6} = 1) always possess linear pencils of elliptic curves, we can assert that:

(12)P. Du Val, Superficie di genere uno che non sono base per un sistema di quadriche and Osservazioni sulle superficie di genere uno che non sono base per un sistema di quadriche, Rendic. Acc. Lincei, febbraio-marzo, 1932.

The regular (or rational) surfaces with (absolute) linear genus p^{(1)} = 1 possess a linear pencil of elliptic curves, with the exception of those with all the genera equal to 1 (surfaces F_{n} with n = 2π −2, with canonical hyperplane sections in S_{π}, characterized by p = P = 1).

In order to classify the surfaces F of linear genus p^{(1)} = 1, with P > 1, containing a linear pencil of elliptic curves C, it is convenient, to start with, to notice that there are two invariant characters belonging to them, which are capable of taking arbitrary integral values, namely:

(1) the genus p (= p_{a}) which is determined by the number S of curves C of the pencil with a double point, since the Zeuthen–Segre invariant is I = S − 4 = 12p + 8 (= 12p_{a} − p^{(1)} + 9).

(2) and the determinant d, defined as the minimal number of points which can be rationally determined on a C, in terms of the parameter on which C itself depends.

In order to understand the meaning of the latter character, we need to recall that in general on an elliptic curve C of order n, in the plane or in any space S_{r}, one cannot rationally determine (that is, as a function of the coefficients of its equation) any linear series, other than the multiples of the g_{n} cut out by the lines of the plane or by the hyperplanes of S_{r}. Thus in general it is not possible to lower the order n of a variable elliptic curve C by rationally operating on the parameters depending on which the curve itself varies^{(13)}.

Therefore an ∞^{1} series of elliptic curves C, of order n, given in the space, forms a surface F with a pencil of C for which in general there will be no transformation that lowers the order of the C.

Now we can say that: the surfaces F, with a pencil of elliptic curves C, of determinant d > 1, can be transformed in such a way that the C become curves of order d, and so – under the assumption that the surface is regular – in surfaces F_{m} of a certain order m of the ordinary space with a (m−d)-tuple line (for d = 2 in double planes); and there is no transformation of these that lowers the order of the C.

To justify the assertion, we notice that – in view of the definition – F has to contain a curve L which is d-secant for the curves C and no other curve cutting out a smaller number of points on the C. Now if we add to L a C, then the degree of L increases by 2d and the genus by d, hence the dimension of |L + C|, given by the theorem of Riemann–Roch, increases by 2d > d > 1 with respect to the dimension of |L|; so the curves L + C, L + 2C, …, L + sC, will be contained in larger irreducible linear systems of d-secant curves to the C. For d = 2 one obtains a double plane with a certain branch curve D_{2n}, on which the C are represented by the lines through a point of multiplicity 2n − 4 of D_{2n}.

(13)See F. Enriques, Sulle superficie algebriche con un fascio di curve ellittiche Rendic.

Acc. Lincei, 1912. In this note the argument of no. 3 contains a mistake due to a wrong computation. See also Enriques–Chisini, Lezioni, Vol. III, page 345.

In order to better understand our surfaces of linear genus p^{(1)} = 1 (regular, irrational, surfaces with P > p ≥ 1), it is convenient to study in particular those with determinant d = 1.

A surface F of this family contains a linear pencil of elliptic curves C and a curve θ which is unisecant the C. By means of an ∞^{3} linear system contained in |3θ + sC|, F can be transformed into a surface F_{m}, of a certain order m, on which the C are cubic curves cut out by the planes through a line a of multiplicity m − 3. These C, which vary in the pencil of the above mentioned plane sections, have a rationally determined flex, which can be assumed to be a fixed, (m−2)-tuple point A of F_{m} on a; and we can also assume that a is the corresponding inflexional tangent line of the C. Therefore the surface F_{m} can be represented, via projection from A, on a double plane with branch curve D_{2n}, of a certain even order 2n, with a point O (the trace of a) of multiplicity 2n − 3. The curve D_{2n} can have other singular points that lower the genus of the double plane: 4-tuple points or [3, 3] points. But the presence of these singularities enables us in general to lower the order of D_{2n} using quadratic transformations of the plane. If one wants the D_{2n} to be of minimal order, then one has to assume that it belongs to one of the following types:

I. D_{2n} with a point O of multiplicity 2n − 3, without other singularities affecting the genus; and

II. D_{2n} with a point O of multiplicity 2n − 3 and a 4-tuple point B, consisting of the line OB and a D_{2n−1} with multiplicity 2n − 4 at O and multiplicity 3 at B.

The double planes of Type I have genus

p = n − 2,

since they have ∞^{p−1} canonical curves represented by the groups of p−1 lines through O. The i-canonical curves are given by the groups of i(p − 1) lines through O, and therefore

P_{i} = i(p − 1) + 1.

For the double planes of Type II, there are ∞^{n−4} canonical curves, represented by the groups of p − 1 = n − 4 lines through O, plus the exceptional branch line to be added to them^{(14)}; therefore the genus is

p = n − 3.

Similarly the i-genus is

P_{i} = i(p − 1) + 1,

Thus we conclude that:

The surfaces F of absolute linear genus p^{(1)} = 1 and of determinant d = 1, with P ≥ 1, can be represented as double planes of Type I, with branch curve D_{2n} having a point O of multiplicity 2n−3, or of Type II, namely with branch curve D_{2n} having a point O of multiplicity 2n−3 and moreover with a 4-tuple point B: these double planes give rise to many distinct families according to the values of the genus, which for Type I is:

p = n − 2

and for Type II

p = n − 3,

and for them one always has

P_{i} = i(p − 1) + 1.

Every family of our F_{1} contains infinitely many classes of birationally distinct surfaces, each characterized by a certain number of parameters or moduli, which is easy to compute. The D_{2n} of Type I, for which the (2n − 3)- tuple point O has been fixed, depends on

\frac{2n(2n + 3)}{2} − \frac{(2n − 3)(2n − 2)}{2} = 8n − 3

parameters, and – since there are ∞^{6} homographies which leave the point O fixed – it has

8n − 9

absolute invariants. So, for every value p = n − 2 of the genus, the F^{1} of Type I depend on

8p + 7

moduli.

(14)On the line OB there are three singular points of D_{2n}, namely B, O and the point infinitely near to O; therefore the degree of the line in the plane is −2, and the corresponding curve on the surface consists of two curves of degree −1 (and genus 0).

Moreover the D_{2n} with multiplicity 2n − 3 at O and multiplicity 4 at B depend on

8n − 13

parameters and possess

8n − 13 − 6 = 8n − 19

invariants with respect to the ∞^{6} quadratic transformations which preserve the pencil of lines O and B. Therefore, since p = n − 3, one has

8p + 5

moduli.

At first sight, these results seem to contradict what we proved in Chapter V, Section ??, that a class of regular surfaces with genera p and p^{(1)} depends on at least

9p − 2p^{(1)} + 12

moduli; since p^{(1)} = 1, one has

9p − 2p^{(1)} + 12 = 9p + 10

which is greater than 8p + 7 and 8p + 5.

The explanation of the paradox is the following: the surfaces F^{1} of linear genus p^{(1)} = 1 and determinant 1, with P > 1, of a given genus p, are contained in families of surfaces F^{2}, of determinant 2, having the same genus p.

In fact the double plane of Type I is a particular case of the double plane I’ with a branch curve D_{2n} having a point O of multiplicity 2n − 4, which is also of genus

p = n − 2

and depends on

10n − 12

invariants, namely on

10p + 8

moduli; and for

n ≥ 4, p > 1,

this number is always

≥ 9p + 10 (= 9p − p^{(1)} + 12).

Similarly the double plane of Type II is a particular case of the double plane of Type II′ with a branch curve D_{2n} having a point O of multiplicity 2n − 4 and a 4-tuple point B, which is also of genus

p = n − 3

and depends on

10n − 22

invariants with respect to the ∞^{6} quadratic transformations, namely on

10p + 8

moduli.

Now, after having discussed the case of surfaces F^{1} of linear genus p^{(1)} = 1 and determinant d = 1 (with P > 1), we turn to the analysis of the surfaces F^{2} of determinant d = 2 (with a linear pencil of elliptic curves C). Here we meet something unexpected: the classification of such surfaces depends not only on the genus p, but also on another character, which is the number s of the curves of the pencil |C| consisting of double elliptic curves; on the other hand this character may be related to the bigenus P by means of the formula

P = 2(p − 1) + s − 1.

In order to classify the surfaces F^{2}, we start with the remark that a surface with a linear pencil of elliptic curves C, on which there are curves bisecant to the C, contains an irreducible linear system of the type (θ + hC); therefore it can be represented (on a double rational scroll and) on a double plane with a branch curve D_{2n}, of some order 2n, having a point O of multiplicity 2n − 4. The curve D_{2n} may in general have other elementary singularities which lower the genus p: 4-tuple points and [3, 3] points. But if we require that the order of the D_{2n} cannot be lowered with quadratic transformations of the plane, we will have the following irreducible types:

I′. D_{2n} with a point of multiplicity 2n − 4 and without other singularities

affecting the genus:

p = n − 2;

II′. D_{2n} with a point of multiplicity 2n − 4 and a 4-tuple point

p = n − 3;

III. D_{2n} with a point O of multiplicity 2n − 4 and s (≥ 1) [3, 3] points: A_{1}A_{2} … A_{s}. The curve D_{2n} contains s lines through O and a remaining part D′_{2n−s} having s tacnodes with the above lines as tacnodal tangent lines:

p = n − 2 − s.

The surfaces F of linear genus p^{(1)} = 1 and determinant d = 2 (for which P > p ≥ 1) give rise, for each value of the genus p, not only to the double planes of Type I′ and II′ for which

p = n − 2, n − 3,

and

P = 2(p − 1),

but also to infinitely many families of double planes of Type III, which depend on another integer s = 1, 2, 3,…:

p = n − 2 − s,

which can be distinguished by giving, together with the genus p, the bigenus

P = 2(p − 1) + s + 1.

In fact, on a double plane of Type III, the canonical curves are represented by groups of p − 1 = n − 3 variable lines through O (double lines which are the images of irreducible elliptic curves C), plus the s lines OA_{1}, …, OA_{s} (the images of fixed curves C/2), or better, the neighbourhoods of the [3, 3] points A_{1} … A_{s} (see Chapter III, Section ??, ?? and Chapter V, Section ??); and the bicanonical curves are given by the groups of 2(p − 1) + s variable lines through O. We add that for a double plane of Type III the successive plurigenera are expressed by

P_{2i} = 2i(p − 1) + si + 1,

and

P_{2i+1} = (2i + 1)(p − 1) + si + 1.

We can also compute the number of moduli on which a class of double planes of Type III depends. Indeed the construction of a curve D′_{2n−s} with a given (2n − s − 4)-tuple point O and given tacnodes A_{1} … A_{s}, with tacnodal lines OA_{1} … OA_{s}, depends on 5(2n − s) − 6 − 6s parameters, while the group of the s + 1 points O, A_{1}, …, A_{s} depends on 2(s − 3) projective invariants. Therefore our curves D′_{2n−s}, and thus also the D_{2n}, depend on 5(2n − s) − 6 − 6s + 2(s − 3) = 10n − 9s − 12 invariants or moduli. And since

p = n − 2 − s,

we have

M = 10p + s + 8,

or, by introducing the bigenus:

M = 8p + P + 9.

For p = 0, P = 1, with a bicanonical curve of order zero, we find again the 10 moduli of the typical double plane with branch curve D_{8} that we have already met.

The results we obtained for the classification of the surfaces with linear genus p^{(1)} = 1 and determinant 2, suggest by induction the statement of the following general theorem for the surfaces F_{d} of any determinant d:

In the classification of the regular surfaces of linear genus p^{(1)} = 1 (with P > 1) and determinant d > 1 (with a linear pencil of elliptic curves C which can be supposed to be of order d) one has to take into account not only the genus p, but also other integral characters corresponding to the divisors r of d (with 1 < r ≤ d); each of these characters, s_{r}, denotes the number of elliptic curves C/r, which, counted r times, belong to the pencil |C|.

The various families of surfaces F^{d} which correspond to these characters can also be distinguished by means of the values of the plurigenera: in correspondence with each one of the s_{r} curves C/r, the ith plurigenus P_{i} exceeds i(p − 1) + 1 by [i(r − 1)/r], the maximum integer contained in i(r − 1)/2, and therefore the rth genus P_{r} exceeds r(p − 1) + 1 by r − 1, and, in general, P_{ir} exceeds ir(p − 1) + 1 by i(r − 1), etc.

In order to justify the assertion, notice that, if in the pencil of the C on a F^{d} there is an elliptic curve C/r which – counted with multiplicity r – forms a C, then this has no influence on the number of the C having a double point in the computation of the Zeuthen–Segre invariant; therefore it also has no influence on the arithmetic genus p_{a} of the surface F^{d} and on its geometric genus p_{g} = p_{a}; the curve C/r, counted r − 1 times, has to be added as a fixed component to the variable parts of the canonical curves K of F^{d} (which are composed with the C). In fact, let us construct on F^{d} a net of curves |L + C|, containing a pencil of reducible curves L + |C|; it is easy to see that the Jacobian of such a net contains r − 1 times each C/r as a fixed component, whence it follows that (r − 1)C/r appears as a fixed component in the canonical curves.

The classification problem for the surfaces of linear genus p^{(1)} = 1 may be considered as substantially solved by the previous theorem. But it is convenient to add an interesting relation which links our surfaces F^{d}, of determinant d > 1, to the surfaces F^{1} of determinant 1, and also suggests a new way of studying the F^{d}. Let us consider, on an F^{d}, the pencil of elliptic curves C which we may assume to be of order d. For a given C there is in general a variety which represents all the g_{d}^{d−1} which belong to it; and if one considers these g_{d}^{d−1} as elements or “points” of a curve, one has a curve c which is birationally identical to C, but with one point rationally defined on it, in correspondence with the series cut out on C by the planes or the hyperplanes of the space containing it. So we may associate to the surface F^{d}, described by the curves of the pencil |C|, a surface F^{1}, described by the curves c, which are birationally identical to their homologues C. This assertion seems, at first glance, to contradict the fact that F^{1} and F^{d} are birationally distinct surfaces; but one has to notice that we are not given one single transformation between a curve C and its homologue c, but a group of d transformations, that cannot be rationally distinguished one from the other in terms of the parameter on which C depends; therefore one has an algebraic correspondence of type [d, d] between F^{1} and F^{d}. Indeed, if one takes a point P on c, then this determines a series g_{d}^{d−1} on C, and inside this series one can rationally determine a group G_{d} of d points. Now if one associates to a point P a point P′ among the d points of the group G_{d} (where G_{d} = P′ + G_{d−1}), then one defines a one-to-one correspondence between C and c, where to a point A of C corresponds a point A′ of c such that the group A′ + G_{d−1} belongs to the series g_{d}^{d−1}, the homologue of A.

We can therefore say that:

To any surface F^{d} of linear genus p^{(1)} = 1 and determinant d > 1 (P > 1), containing a linear pencil of elliptic curves C, one can associate a surface F^{1} of determinant 1, which is in [d, d] correspondence with F^{d} and contains a pencil of curves which are birationally equivalent to the C.

In particular the curves C of F^{d} with a double point correspond to curves c with a double point on F^{1}, thus the two surfaces F^{1} and F^{d} have the same Zeuthen–Segre invariant and the same genus p_{a}.

Note In this section we have treated, clarified, and in some points amended and completed, the results about the classification of surfaces of linear genus p^{(1)} = 1 obtained by Enriques, especially in the notes of the years 1898, 1914: Sui piani doppi di genere lineare p^{(1)} = 1, Rendic. Acc. Lincei, 1^{o} sem. 1898, Sulla classificazione delle superficie algebriche e particolarmente sulle superficie di genere lineare p^{(1)} = 1, Rendic. Acc. Lincei, febbr. 1914.

The things we have said can be easily extended to the irregular surfaces (p_{g} > p_{a}), as soon as one established for them the characteristic property (concerning continuous, nonlinear systems of curves (Chapter IX, Section ?? and ??)).

Precisely, in the note Intorno alle superficie algebriche di genere lineare p^{(1)} = 1^{(15)}, Enriques proved that: the irregular surfaces of linear genus p^{(1)} = 1 and arithmetic genus p_{a} ≥ 0 (p_{g} > p_{a}) contain an irrational pencil, of genus p_{g} − p_{a}, of elliptic curves. These surfaces may be classified in the same way as the regular ones, by adding for them the character p_{g} − p_{a}.

(15)Rendic. Acc. di Bologna, dic. 1906.

Beside these we are left only with the irregular surfaces with linear genus p^{(1)} = 1 which have the arithmetic genus p_{a} = −1; they will be completely classified as the “elliptic and hyperelliptic surfaces, with an ∞^{1} and ∞^{2} system of birational transformations into themselves” (Chapter XI).

Regular canonical and pluricanonical surfaces

8.1 Introduction

Surfaces with genus p_{g} > 3 have a (pure) canonical system |K| of dimension p_{g} − 1 ≥ 3, via which this they are, as a rule, transformed into canonical surfaces of (p_{g} − 1)-dimensional space S_{p_{g}−1}, whose hyperplane sections are the curves K. In this chapter we shall examine the construction of some canonical surfaces, considering the case of regular surfaces: p_{g} = p_{a} = p.

Canonical surfaces of S_{p−1}, with (absolute) linear genus p^{(1)} (> 1), have in general order

p^{(2)} = p^{(1)} − 1

and the invariant properties of the transformed surfaces of the whole class will be reflected in their projective properties.

However some exceptions occur, concerning the construction of canonical surfaces: firstly the surfaces with linear genus p^{(1)} = 1, on which the canonical curves are reducible into the elliptic curves of a pencil. This is the essential exception which takes us back to the family of surfaces characterized by the property of possessing a pencil of elliptic curves, that we studied in the preceding chapter: for these, not only the canonical surface is missing, but also the pluricanonical surfaces that we shall discuss later, because all the pluricanonical curves are also composed with the elliptic curves of the pencil.

Other exceptions, or cases where the general treatment at least requires some modifications, are:

(1) the surfaces for which the genus p is ≤ 3;

Canonical surfaces of genus p = 4 and linear genus p^{(1)} = 6, 5 8.2 259

(2) those on which the canonical system, although irreducible and at least ∞^{3} (p > 3), belongs to an involution I_{n}, of order n > 1, so that the canonical surface reduces to a multiple (n-tuple) surface; (3) the surfaces, with linear genus p^{(1)} > 1, for which the (pure) canon-

ical curves K are always reducible; as we saw in Chapter IV, Section ??, there are indeed examples in which the curves K contain (nonexceptional) fixed parts of genus zero and > 0; it remains doubtful whether there do exist surfaces on which the variable parts of the curves K are composed with the irreducible curves K of a pencil.

If on a surface F the pure canonical system |K| has a fixed component, exceptional or not, if the genus p is > 3, and if the ∞^{p−1} canonical curves have irreducible variable components, then the canonical system always leads to a canonical surface, but its order will be equal to the effective degree of the canonical system, \overline{p}^{(2)} < p^{(2)} = p^{(1)} − 1.

If instead the variable parts of the canonical curves are reducible, there will no longer be a canonical surface. However, in this case too, as in the previous cases, for p^{(1)} > 1, we may obtain a projective model of the class by appealing to the bicanonical or tricanonical surfaces, and so on, whose hyperplane sections are bicanonical, respectively tricanonical curves. But we will say more on this later.

In the following sections we shall first refer to regular surfaces of genus p > 3 with irreducible (pure) canonical curves (without base points), forming a simple linear system, at least ∞^{3}, of genus p^{(1)} and degree p^{(1)} − 1. The first question we are confronted with is the following: do there exist in the same space S_{p−1}, that is, for the same value of the genus p, canonical surfaces of different order, thus corresponding to different values of p^{(1)}? That is: are the two characters p and p^{(1)} independent of each other? The answer, given by Noether already since 1870, is affirmative.

Indeed the independence of p and p^{(1)} is immediately proved by giving effective examples. It suffices to observe that a triple point and a tacnode, imposed to the surfaces F of a given order and with given singularities, both make p decrease by one, but they make p^{(1)} decrease by 3, respectively by 2.

8.2 Canonical surfaces of genus p = 4 and linear genus p^{(1)} = 6, 5

In this section, and in the ones immediately following it, we intend to classify the regular surfaces with superficial genus p = 4 and linear genus p^{(1)} > 1, by constructing the corresponding canonical surfaces, belonging to the ordinary space S_{3}. We shall assume to deal as a rule with surfaces whose pure canonical curves K form a simple irreducible system (free from base points on a model without exceptional curves), except that we may dedicate some remark to the cases where reducibility occurs; thus we shall firstly look for the simple canonical surfaces, those satisfying the assumption that the system |K| does not belong to an involution I_{n} of order n > 1.

For p = 4 and p^{(1)} > 1, if a canonical surface F is supposed to be simple, then its order must satisfy

p^{(2)} = p^{(1)} − 1 ≥ 5.

The simple type corresponding to the smallest value of p^{(1)} is thus: the general surface F_{5} of the 5th order, a canonical surface with

p = 4, p^{(1)} = 6.

It is clear that a surface with p = 4 and p^{(1)} = 6, whose canonical curves are irreducible, may be birationally reduced to the projective model of a F_{5}. As examples we mention the surfaces indicated by Noether:

(a) The surface F_{7} of the 7th order, with a triple conic and a triple point outside of it. With a quadratic transformation of the first kind the above surface F_{7} is carried to a particular F_{5}, over which a point O and a conic are given; the point O corresponds to the exceptional line in which the plane of the triple conic cuts F_{7}.

(b) The surface F_{7} with a double sextic C_{6} of genus 3. In the usual plane representation of a cubic surface F_{3} passing through C_{6}, the curve C_{6} appears as having as image a curve C′_{6}(1^{3}2^{2}3^{2}4^{2}5^{2}6) whose behaviour with respect to the fundamental points 1, 2, 3, 4, 5, 6 is indicated by the preceding symbol; hence the variable residual intersections of two F_{3} through C_{6} are twisted cubics, represented by C_{3}(2, 3, 4, 5, 6^{2}), and forming a homaloidal net; we deduce from this that the above F_{3} form the system yielding a Cremona transformation of the space, which reduces our F_{7} to a F_{5}.

(c) The surface F_{7} with a double quintic C_{5} of genus 0. Also in this case the F_{3} through C_{5} form a homaloidal system, which has as base locus the C_{5} itself completed by a line meeting it at 4 points, that is an exceptional curve of the surface F_{7}; and with this transforming system of surfaces F_{3} the surface F_{7} is reduced by the Cremona transformation to a surface F_{5}.

We want now to show with the utmost rigour and generality that:

every surface F with genera p = 4 and p^{(1)} = 6, for which the variable part of the canonical curves is irreducible, can be transformed into a canonical quintic F_{5}.

We denote by |k| the variable part of the canonical system, which by assumption is not composed with the curves of a pencil. Taking the projective image of this system we obtain a surface F_{n} whose order n (≤ 5) equals the effective degree of |k|.

If n = 5 the surface F_{n} is a quintic and it turns out a posteriori that

|k| = |K|.

If n < 5, we can have neither a simple canonical surface (it would have genus p ≤ 1) nor a multiple plane (this assumption would imply p = 3), thus only a double quadric F_{2} remains possible – at least a priori: we would indeed encounter this case if there would be an exceptional fixed component of the pure canonical curves, that is, a base point of |K| on a model of F free from exceptional curves. Since the curves k have genus π ≤ p^{(1)} (= 6), the branch curve on the F_{2} will be a C_{2π+2} of order 2π + 2. We shall firstly prove that we certainly do not have

π < 5,

and thus, a priori, only the following cases are possible:

π = 6, 5.

Indeed we cannot have π < 5, that is, π ≤ 4, since then the canonical curves K should intersect the curves k of genus π in residual groups of the characteristic series g_{n}^{2} = g_{4}^{2}, thus in no more than 2 points, while at the same time we must have kK > kk = 4. But, if π = 5, we have on the double F_{2} a branch curve of order 12; and, if π = 6, a curve of order 14. The first case leads to a double plane with a branch curve C_{12} having two 6-tuple points: the canonical curves of this double plane are the conics through the two singular points (completed by the exceptional line joining them) and hence its absolute linear genus is p^{(1)} = 5 instead of p^{(1)} = 6. The second case leads to a double plane with a branch curve C_{14} endowed with an 8-tuple point O and a 6-tuple point O_{1}, or else with two infinitely near 7-tuple points OO_{1}; thus its canonical curves are the ∞^{3} cubics through O^{2} and O_{1} (completed by the exceptional line OO_{1}), hence its genus is p = 5 instead of p = 4.

We have said that p^{(1)} = 6 is the minimal value that the linear genus p^{(1)} can take for a surface of genus p = 4, admitting a simple canonical surface as a projective model. But this is no longer true if the canonical system |K| belongs to an involution I_{2} of order 2, therefore admitting a base point.

Surfaces with p = 4 and p^{(1)} = 7 8.3 262

We have then as the double quadric canonical surface with a branch curve of order 12, which leads, as we have said above, to a double plane with a branch curve C_{12} of order 12 endowed with two 6-tuple points. This double plane with p = 4 and p^{(1)} = 5, corresponds to the minimal value of p^{(1)} for p = 4 (this agrees with the general inequality p^{(1)} ≥ 2p − 3, that we will establish later on). This results from the previous analysis since a lower value of p^{(1)} should lead to a simple canonical surface of order 3 or 2, which would have p = 0, or to a triple plane which would have genus p = 3.

And the same analysis also excludes that a hypothetical reducibility of the pure canonical system (that is, the presence of fixed components, exceptional or not) may lead to other types of surfaces with p = 4 and p^{(1)} = 5.

We thus state that: Every surface with p = 4 and minimal linear genus p^{(1)} = 5 can be transformed in a double plane with a branch curve C_{12} endowed with two 6-tuple points.

As an example let us consider the surface F_{6} of the 6th order with two skew double lines a and b: the adjoint quadrics cut out on such a F_{6} the canonical curves K, which belong to the involution cut by the lines incident to a and b.

8.3 Surfaces with p = 4 and p^{(1)} = 7

We proceed to the construction of the canonical model of surfaces with genera p = 4 and p^{(1)} = 7. We shall have, under assumptions that appear to be more general, surfaces F_{6} of order 6 of ordinary space, whose adjoint quadrics have to split into a variable plane and a fixed plane, intersecting F_{6} along a double cubic.

Thus for p = 4 and p^{(1)} = 7 we have a canonical surface F_{6} of order 6 with a double plane cubic.

Every surface whose genera takes the values p = 4 and p^{(1)} = 7, and with a simple canonical system, can be birationally transformed into such an F_{6}. Thus, for example, this happens for the surfaces of 6th order F_{6} having 6 triple points, whose canonical curves (intersections with the quadrics through the 6 points) have precisely genus 7 and degree 6.

Note But if we want to go deeper into the study of the family of surfaces for which p = 4 and p^{(1)} = 7, it is convenient to examine those surfaces for which the canonical system |K| (which we assume to be irreducible except for some possible fixed components) is not simple, whence the canonical surface reduces either to a double cubic F_{3} or to a double or triple quadric F_{2}. The main aim of this investigation is to answer the question whether all the surfaces for which p = 4 and p^{(1)} = 7 belong to the same family represented by the family of canonical F_{6}, or else there are several distinct families of surfaces with the stated characters.

As far as the double cubics F_{3} is concerned, it is easy to understand that they occur as a particular case in the family of the F_{6}.

Firstly the F_{3} cannot be ruled, since it would not be normal, whence we would have p = 5; thus it has elliptic plane sections and possesses a branch curve C_{12} of order

2p^{(1)} − 2 = 12.

This C_{12} must be the complete intersection of the F_{3} with a surface of 4th order, otherwise we would find on F_{3} a pencil of conics cutting C_{12} at 6 points, and these conics would be images of curves of genus 2, cutting the canonical curves K at 2 points, therefore the K themselves would be hyperelliptic and the plane sections of F_{3} rational, which has just been excluded.

Hence the double F_{3} has a representation on the double plane with a branch curve C_{12} of order 12 endowed with six 4-tuple points, which – in fact – has the genera p = 4 and p^{(1)} = 7, and on which the canonical curves have as images the cubics through the 6 points counted twice.

We show now that the double plane mentioned above occurs as a particular case in the family of the surfaces F_{6} with a double plane cubic; considering a pencil of surfaces determined by the F_{6} and by an adjoint F_{3} counted twice:

F_{3}^{2} + λF_{6} = 0,

for λ = 0 we obtain in this pencil the double cubic surface F_{3}^{2} with branch curve cut out by F_{6}=0.

To prove it, it is enough to see that a plane section f_{6} of F_{6}, belonging to a pencil f_{3}^{2} +λf_{6} = 0, reduces by continuity (for λ = 0) to the double cubic f_{3} with branch points at the 18 simple base points of the aforementioned pencil. Now if we assume the f_{6} to be general (that is, without double points), it gives rise to an envelope of class 30 which, at the limit for λ = 0, must reduce to the cubic envelope f_{3} = 0 counted twice (whose class is 2 · 6 = 12), plus the 18 pencils having as centres the branch points of the given double cubic; and it is evident^{(1)} that the contact points of the tangents issued from any point P to the f_{6} end up, for λ = 0, at the 18 intersections of f_{6} and F_{3}. If moreover the f_{6} has 3 collinear double points, which are taken as base points of the pencil f_{3}^{2} + λf_{6} = 0, the 3 pencils with those centres counted twice have to be taken away from the above 18 points, hence there will remain a G_{12} as branch locus of the double f_{3}.

While the canonical surfaces given by double cubics with a C_{12} as branch locus occur as a particular case in the more general type of the F_{6} with a double plane curve of the 3rd order, on the contrary those canonical surfaces which are given by double quadrics form a distinct family.

Without dwelling upon a more detailed investigation, let us say that there is a type of surfaces F, with p = 4 and p^{(1)} = 7, on which the canonical curves are hyperelliptic with two base points (chosen among the double points of the g_{2}^{1}), which lead to canonical double quadrics representable over the double plane with a branch curve C_{10} of order 10, endowed with two [3, 3] points.

(1)Cf. Enriques–Chisini, Lezioni, Vol. III.

And indeed these double planes do form a second family of surfaces with p = 4 and p^{(1)} = 7, which do not occur among the simple canonical F_{6}, since they depend on 39 moduli (39 = 65 − 18 − 8), whereas we have seen that the F_{6} depend on 38 moduli.

More difficulties are encountered in the investigation of those canonical surfaces, with p = 4 and p^{(1)} = 7, which are given by triple quadrics. The surfaces of this family are characterized by the property that the pure canonical curves K do not have base points and contain a g_{3}^{1}, therefore |K| turns out to belong to a rational involution I_{3} of order 3. The triple canonical quadric Q_{2}^{3}, studied by G. Pompilj^{(2)}, is a cone and has a branch curve C_{18} of order 18.

One can see that the triple canonical quadrics Q_{2}^{3} do not occur as particular cases in the family of the simple F_{6} with a double plane cubic, but they constitute instead a third family of surfaces with p = 4 and p^{(1)} = 7, which is distinct from the previous ones.

To prove this, it is enough to observe that a section of F_{6}, that is, a plane curve C_{6} with 3 collinear double points, cannot degenerate by continuity to a section of Q_{2}^{3}, that is, a triple conic C_{2}^{3} having 18 branch points. For this, we first observe that the ∞^{27} system of plane sextics cannot contain the ∞^{29} system of the triple conics of the same genus with 24 branch points. One cannot nevertheless exclude that among the ∞^{27} C_{6} there might occur particular C_{2}^{3} with 24 critical points; but if one wants to find inside this family a C_{2}^{3} with 18 distinct branch points, that is, a degenerate C_{6} with 3 collinear double points, it is necessary that 6 among the aforementioned 24 critical points come pairwise together to 3 collinear apparent critical points, and this is impossible for an irreducible C_{2}^{3}.

8.4 Surfaces with p = 4 and p^{(1)} = 8

Let us proceed to the next case: p = 4 and p^{(1)} = 8. The (simple) canonical surface F_{7} corresponding to these values of the numerical characters must possess in general a double curve C_{7}, which, counted twice, forms the complete intersection of the F_{7} with the adjoint quadric Φ_{2}. The curve C_{7}, by a formula we have already seen^{(3)}, has a triple point that is double for the adjoint Φ_{2}, which is therefore a cone. This agrees with the fact that the C_{7} is the contact curve of the Φ_{2} with a surface of order 7, hence it belongs to the system of halves of the linear system |C_{14}| given by the intersections of Φ_{2} with the surfaces of 7th order in space. But from the plane representation of the quadric it appears that the above system |C_{14}| cannot be divided by two in general; in order that this happens, it is necessary for the quadric to be a cone, whose representative system in the plane is given by the conics through two infinitely near points O, O_{1}: then the images of the C_{14} are curves of the same order C′_{14}(O^{7}O_{1}^{7}) and the system of halves of the system |C′_{14}| is a priori a system C′_{7}(O^{4}O_{1}^{3}) or a more particular system contained in it. But the C′_{7}(O^{4}O_{1}^{3}) which do not have multiplicity greater than 4 at O, are images of curves C_{7} of Φ_{2} which are contact curves of Φ_{2} with some F_{7}. And they cannot be double for an irreducible F_{7} since the generators of the cone Φ_{2} would have eight intersections with such a F_{7}. Hence the double curve of one of our surfaces F_{7} has to belong to the linear system whose image in the plane representing Φ_{2} is C_{7}(O^{5}O_{1}^{2}); therefore the double curve passes three times through the vertex of the cone and has genus ρ = 4.

(2)G. Pompilj, Sulle superficie algebriche le cui curve canoniche posseggono una g_{3}^{1}, Rendic. Ist. Lombardo, 1940.

(3)Cf. Chapter V, Section ??

We have shown that, if it exists, the canonical F_{7} (with p = 4, p^{(1)} = 8) has a double curve C_{7} of genus ρ = 4, with a triple point V, belonging to the system of halves of the system cut on Φ_{2} by the surfaces of the 7th order passing triply through V (or behaving like those that have a triple point there). The question now is to see whether such surfaces F_{7} do actually exist. Their existence can be deduced inductively from the computation of their number of moduli; if we assume the more expressive formula given in Chapter V, Section ?? to be true:

M ≥ 10p − 2p^{(1)} + 12 = 36,

we are led to suppose that every C_{7} (defined as above over a quadric cone with vertex V that is triple for the curve) can be a double curve for ∞^{26} canonical surfaces F_{7}: in fact the C_{7} over the quadric cone are ∞^{17} and thus, in view of the ∞^{7} projective transformations of the cone, they give rise to ∞^{10} projectively distinct curves; on the other hand, the surfaces F_{7} having one and the same double curve C_{7} are ∞^{26}, since splitting off the cone we have ∞^{25} residual F_{5}, passing through the curve C_{7}.

The above conclusion is based only on an inductive reasoning. Indeed the formula for the moduli we used depends on an assumption which has not been rigorously proved; moreover this formula can be applied to compute the dimension of a continuous family of surfaces with given p and p^{(1)}, when we assume the existence of some surface in the family; thus at least we have to verify the actual existence of a particular surface F_{7} (with genera p = 4, p^{(1)} = 8) endowed with a double C_{7}: one can succeed for example in constructing such a F_{7} starting from a sextic F_{6} with 5 triple points and a tacnode, for which in fact p = 4, p^{(1)} = 8. But we can prove more directly, by the method indicated by A. Franchetta^{(4)}, that every C_{7} (belonging to the series we considered above on a quadric cone) is the double curve of some (canonical) surface F_{7}. Thus for these surfaces, the computation of the number of moduli is also confirmed.

The method we shall use consists in the following: we verify that a C_{7} is always a double curve for one (and in fact for ∞^{6}) rational surfaces of the 7th order F′_{7}, possessing moreover an assigned quadruple point; taking a linear combination of this particular F′_{7} with a degenerate F_{7} passing doubly through the C_{7}, we obtain the canonical F_{7} as required.

To realize this program, let us firstly observe that there do exist rational surfaces of the 7th order F′_{7} with a double curve of the 7th order and an isolated quadruple point: one such a surface is represented on the plane by the system of curves of order eleven C_{11}(A^{5}B_{1}^{3} … B_{9}^{3}D_{1} … D_{8}) where the 18 points A, B and D belong to a fundamental curve of order seven K_{7}(A^{3}B_{1}^{2} … B_{9}^{2}D_{1} … D_{8}); the C_{11} are curves of genus 8 and the characteristic series of the ∞^{3} system is autoresidual, since the system |C_{11}| is equivalent to the sum of the fundamental curve K_{7} and of its adjoint system; thus indeed the surface represented in this way is a F′_{7} with plane sections of genus 8, hence it has a double curve C_{7} and moreover possesses a quadruple point corresponding to the curve K_{7}.

Now, the F′_{7} thus constructed depend upon a certain number of absolute projective invariants which it is easy to compute: this number is 19. Moreover one can calculate the dimension of the system of the F′_{7} admitting the same double curve C_{7} and the same quadruple point O not on it; this dimension exceeds by one the dimension of the characteristic system, cut out over one F′_{7} by the other F′_{7} of the same system, hence its value is 6. With this set-up, if we let the quadruple point vary, we have (at most) ∞^{9} surfaces F′_{7} with the same double curve C_{7}, hence – since there are ∞^{19} F′_{7} – we have (at least) ∞^{10} double curves C_{7} for some F′_{7}. But the C_{7} thus obtained are just those curves of the 7th order and of genus 4, lying on a quadric cone, and passing triply through the vertex,which we defined above. Taking a linear combination of a F′_{7} with the ∞^{25} degenerate F_{7} formed by the quadric cone containing C_{7} and by a surface F_{5} through the curve C_{7}, we obtain ∞^{26} canonical surfaces F_{7} which pass doubly through the given C_{7}. With this argument we have proved that: there exist ∞^{36} projectively distinct canonical surfaces of the 7th order F_{7}, whose double curves are the ∞^{10} curves of the 7th order and of genus 4 which can be defined through the bisection of the linear system cut out on a quadric cone by the surfaces of the 7th order passing triply through the vertex.

The curves C_{7} can be characterized as twisted hyperelliptic curves of genus 4, having a triple point: in fact the twisted hyperelliptic curves of the 7th order of genus 4 are ∞^{26} and to possess a triple point imposes one condition to them, as it is a priori apparent from the fact that the g_{7}^{3} admitting a neutral triple over a hyperelliptic curve of genus 4 are defined by the condition of containing the double of the g_{2}^{1} and thus they depend on three parameters.

(4)A. Franchetta, Su alcuni esempi di superficie canoniche, Sem. Mat. di Roma, 1939.

We explained the general construction of the canonical surfaces F_{7} with a double curve C_{7}: a hyperelliptic curve of genus 4 with a triple point. But if instead of looking for the construction of a F_{7} with a double curve C_{7}, we ask for a canonical F_{7} possessing some multiple curve with multiplicity greater than 2, soon occurs the very simple example of the F_{7} endowed with a triple line a and with two double conics lying in planes through it. When the triple line and the two double conics are assigned, we have ∞^{33} surfaces F_{7}: in fact, by splitting off the planes of the two conics we let the number of parameters decrease by 2, and we are left with the dimension of the system of the F_{5} passing simply through a line and through two conics in planes through it, that is indeed 31.

Let us notice now that the curve formed by a line and two conics in planes through it has an absolute invariant, that is, the cross-ratio of the 4 points on the line. We therefore have ∞^{34} projectively distinct canonical surfaces F_{7} possessing a triple line and two double conics. We are now faced with the question: do these F_{7} occur as particular surfaces in the family of the canonical F_{7} with a double C_{7}, or do they form a distinct continuous series of surfaces, which depends only upon 34 instead of 36 moduli? Were the formula for the number of moduli established without exceptions in the most expressive form we mentioned above, the question would be solved a priori choosing for the first alternative. However we can verify directly in which way the F_{7} with a triple line and two double conics are specializations of the F_{7} with a double C_{7}.

The question is to show that the curves of ordinary space C_{2} + C′_{2} + 3a, formed by two generic conics, and by the line intersection of their planes, occur as a particular case in the family of our C_{7}. Since these curves C_{2} + C′_{2} + 3a satisfy the three conditions of having a triple point and of being hyperelliptic (that is, of having a g_{2}^{1}), it suffices to show that those reducible curves occur in the family of the C_{7} of genus 4. Now, the C_{7} of genus 4 of ordinary space can be obtained as projections of curves C_{8} of the space S_{4} from a simple point O; and among those C_{8} we can find curves that are reducible into two quartics C_{4}, C′_{4} of the 1st kind passing doubly through O and having there the same tangent plane α. Such a curve projects from O down to a curve composed of the triple line a, trace of the plane α, and of the two conics C_{2}, C′_{2}, projections of C_{4}, C′_{4} respectively. Thus it appears that the curve C_{2} + C′_{2} + 3a can be viewed as a specialization of our hyperelliptic curves C_{7}, endowed with a triple point. One can see directly that the number of conditions required for this degeneration of the curve C_{7} is 9, hence all the reducible curves C_{2} + C′_{2} + 3a occur in the family of our C_{7}; as we have already seen, this implies that the F_{7} with a triple line and two double conics fall within the family of the general canonical surfaces with p = 4 and p^{(1)} = 8, satisfying two particular conditions.

The canonical surfaces F_{7} which we have learnt to construct correspond to the values p = 4 and p^{(1)} = 8 of the genera (p^{(1)} is the absolute linear genus) when the canonical curves K form, on a birational transform F without exceptional curves, a simple irreducible system without base points. If instead we assume that the system |K| has on F a (simple) base point, this system has effective degree p^{(2)} − 1 = p^{(1)} − 2 = 6, hence the corresponding canonical surface becomes a sextic F_{6}, over which the image of the base point of the canonical system is an exceptional curve counted twice: this line belongs simply to the impure canonical system and then appears as a fixed component of the pure canonical system.

It is now easy to actually construct a surface F_{6} of the 6th order possessing an assigned double conic C and, in the plane γ of this, a tacnode O whose tacnodal plane α, different from γ, is also assigned; one obtains all the F_{6} satisfying the indicated conditions by linear combination of two particular F_{6}: the F_{6} formed by a quadric through C counted twice, and by the plane α also counted twice, and a F_{6} formed by the plane γ and by a F_{5} passing simply through C and having a biplanar point at O, where one of the two branches touches the assigned tacnodal plane α.

The F_{6} thus constructed has in fact a fixed component of the canonical curves, that is, the line a along which the plane α touches the F_{6} and which thus appears twice in the intersections of the F_{6} with the ∞^{3} adjoint quadrics, formed by a variable plane and by the fixed plane α. Denoting by ρ = 0 and by −ν the genus and the degree of this line a, the genus and the degree of 2a are respectively given by:

2 · 0 − ν − 1 = −(ν + 1) and − 4ν;

hence one can compute the degree \overline{p}^{(2)} and the (virtual) genus \overline{p}^{(1)} of the impure canonical system on F_{6} (the latter is its relative genus):

\overline{p}^{(2)} = 6 − 4ν + 4 = 10 − 4ν
\overline{p}^{(1)} = 8 − (ν + 1) + 2 − 1 = 8 − ν (with p^{(1)} = 8);

and from the relation

\overline{p}^{(2)} = \overline{p}^{(1)} − 1

one obtains

10 − 4ν = 7 − ν, ν = 1.

Thus it remains proved a posteriori that the line a is actually an exceptional line for F_{6} (−ν = −1), therefore because of its presence the relative linear genus becomes

\overline{p}^{(1)} = 7 = p^{(1)} − 1

and the absolute linear genus is p^{(1)} = 8; moreover this line, after having been subtracted once from the impure canonical curves, still remains as a fixed component of the pure canonical curves, and thus corresponds to a simple base point^{(5)} of |K| over a birational transform F.

Let us now count the number of parameters on which our F_{6} depends. The F_{6} passing through the conic and having a biplanar point O with a branch tangent to the plane α depend on 37 parameters; hence the F_{6} with a double conic, a tacnode and an assigned tacnodal plane are ∞^{38}.

Since the assigned singularities of the F_{6} can be projectively transformed into any other set of similar singularities however given in the space (indeed they do not possess any absolute invariant), and actually those singularities are transformed into themselves by ∞^{3} homologies, we see that: our canonical surfaces F_{6} with p = 4 and p^{(1)} = 8, corresponding to a canonical system with a base point, depend on 35 absolute invariants, that is, on 35 moduli.

Also in this case the number of moduli is less than the number of moduli (36) belonging to the canonical surfaces F_{7}, therefore one may ask whether the given F_{6} are in fact a specialization of the F_{7}, corresponding to the degeneration of F_{7} to a F_{6} plus a plane.

We shall restrict ourselves to a hint at how to see this. If we add to our F_{6} the plane γ we obtain a surface F_{7} possessing a triple conic C and moreover two infinitely near double lines coinciding with a; the same F_{7} can be viewed as being endowed only with the triple conic C and the double line a, viewing as virtually nonexistent the double line infinitely near to a, which we want to consider as a connecting line between F_{6} and the plane γ; thus the reducible F_{7} has the (numerical) genus of F_{6}, which has to be increased by zero (genus of the plane) and by zero (genus of the connecting line). The question now is to verify that the degenerate F_{7} thus defined, with the point singularity that our construction assigns to it at the point O, can be considered as a specialization of a canonical F_{7} with a double curve C_{7} of order 7 and genus 4 (hyperelliptic and endowed with a triple point). In order to see this it is useful to consider the above C_{7} as a projection from a point P of a C_{8} of order 8 and genus 4 of the space S_{4}, and then to let the curve C_{8} degenerate into a curve C_{6} of genus 3 belonging to a quadric cone with vertex P, and having a double point at P, and into two incident lines a and O, the former being a chord of C_{6}, the latter passing through P: the projection of this degenerate C_{8} from P to ordinary space can be viewed as a limit of a variable C_{7} which degenerates into a triple conic C and into a line a lying in the plane of C; this degenerate curve must be completed by adding to it an infinitesimal curve formed by the neighbourhood of the point O trace of the line O, taken inside a plane through a. We understand in this way how also the property of the F_{6} of possessing a tacnode may result as a limit of the property of possessing a double curve C_{7}, a property which defines the family of the F_{7}.

(5)One can verify also in another way that the F_{6} is a surface with absolute linear genus p^{(1)} = 8 whose pure canonical curves admit the exceptional curve as a fixed component.

Indeed it suffices to transform the F_{6} quadratically by means of the system of quadrics passing through the conic C and a point A of it; we thus obtain a surface of the 7th order with a triple conic, and possessing moreover a double point (of Noether type) O, with an infinitely near tacnode; this singular point (image of a fundamental elliptic curve of degree −1) gives rise to a simple base point for the canonical curves cut out by the quadrics, the variable parts of the adjoint cubic surfaces.

Note To go more deeply into the study of the surfaces with p = 4 and p^{(1)} = 8 it is necessary to examine the cases in which the canonical system |K|, though having irreducible variable components, belongs to an involution, giving rise to a multiple canonical surface. We are led to consider canonical surfaces given by double cubics F_{3}, or by double or triple quadrics F_{2}. Without dwelling on this analysis, we shall limit ourselves to indicate some remarkable families that can be obtained in this way.

Firstly, we find a double F_{3} which corresponds to the case where the K (viewed as curves on a birational transform without exceptional curves) belong to a rational involution I_{2} and have a base point. One sees, as in the previous section, that the surface F_{3} has elliptic plane sections (that is, it is not ruled) and its branch curve turns out now to be a curve of order 2p^{(1)} − 2 = 14. If we represent this F_{3} on the plane we obtain a double plane with a branch curve C_{12} of order 12, endowed with five 4-tuple points and with a [3, 3] point. Actually this double plane has as (double) images of the canonical curves the cubics through the six singular points of the C_{12} and thus

p = 4, p^{(1)} = 8.

We can now verify that the family of surfaces with genera p = 4, p^{(1)} = 8, defined by the above double planes is contained in the family of simple canonical F_{6}, corresponding to the case where the canonical system has a simple base point. In fact, let us consider the adjoint cubics to one of the mentioned F_{6}, that is, passing through its double conic C_{2} and through its tacnode O (in the plane of C_{2}); let us take among these surfaces a F_{3} which touches the tacnodal plane of the F_{6} at O and let us consider the pencil

F_{3}^{2} + λF_{6} = 0.

In this pencil we see that, for λ = 0, the F_{6} degenerates with continuity to the F_{3}^{2}, with a branch curve of order 14, which is the section of the F_{6} off C_{2}: the line that, counted twice, forms the further section of F_{6} with the plane of C_{2} turns out to be a part of the branch curve.

A second family of multiple canonical surfaces is the family of double quadrics which can be reduced to the type of double plane with branch curves C_{12} = r + C_{11} including an exceptional line r and a C_{11} of order 11 endowed with two infinitely near 4-tuple points A, A_{1} on r and of two [3, 3] points B and D, the former still belonging to the line r (with the infinitely near triple point B_{1} not on r). The double images of the canonical curves K are here the conics through A and D, completed by the exceptional line AA_{1}D; hence

p = 4 and p^{(1)} = 8,

and on a model of the surface without exceptional curves the canonical curves have 3 base points. It seems that the question whether these double planes occur in the family defined by the canonical surfaces F_{7}, or in the family of the F_{6} (pure canonical system with a base point or an exceptional fixed component), should receive a negative answer, in spite of the fact that the number of moduli upon which these double planes depend is 33, that is, it is smaller than the corresponding number for the mentioned F_{7} (which is 36) and for the F_{6} (which is 35). Indeed, it is difficult to imagine that a surface formed by a double quadric and two planes may possess the singularities which are characteristic of the F_{6}.

We shall not further dwell upon the multiple canonical surfaces with the genera p = 4 and p^{(1)} = 8, leaving aside the problem of the triple canonical quadrics with the above genera; we shall only say that a research by G. Pompilj (that we shall have the opportunity of mentioning later on) excludes the existence of this type.

We shall finally limit ourselves to point out that, for a complete classification of the surfaces with p = 4 and p^{(1)} = 8, it would also be convenient to examine the multiple canonical surfaces which would arise in the hypothesis of reducible canonical curves with a fixed part, and whose variable components have genus < p^{(1)}.

8.5 Surfaces with p = 4 and p^{(1)} = 9

The canonical surfaces F_{8} of genus p = 4 and (absolute) linear genus p^{(1)} = 9 and with a simple irreducible, base point free, canonical system, can be constructed with a method which is similar to the one we used before for p^{(1)} = 7, 8. The F_{8} must have, according to our formulas

t = 4

triple points, and an adjoint surface Φ_{3} having 4 double points there. One has to consider the F_{9} passing triply through the 4 double points of Φ_{3}, and then take the system of halves of the linear system of curves cut out by them on the Φ_{3}.

We appeal to the usual representation of the Φ_{3} on a plane, where the images of the plane sections are the curves C_{3} passing through three pairs of infinitely near points, P P_{1}, QQ_{1} and RR_{1}, lying on a conic; then the half system we are looking for is given by the curves of the 9th order,

C_{9}(P^{4}P_{1}Q^{4}Q_{1}R^{4}R_{1}).

These are ∞^{21} curves C_{12} of the 12th order and of genus 10 on the Φ_{3}, and a priori it is possible for them to be double curves of canonical surfaces F_{8}. We observe now that a C_{12} which is a double curve of an irreducible F_{8} will also be double for ∞^{13} surfaces F_{8}, since splitting off the adjoint Φ_{3} there remain ∞^{12} surfaces of the 5th order F_{5} passing through C_{12}. Since the cubic surfaces with 4 double points are ∞^{12}, projectively equivalent between them, our C_{12} belong to a family which depends on 21 absolute invariants. If we admit that every C_{12} is a double curve for some (hence for ∞^{13}) irreducible F_{8}, we will have ∞^{34} projectively distinct F_{8}, that is, the number of moduli of our canonical surfaces should be exactly given by the formula

M = 10p − 2p^{(1)} + 12 = 40 − 18 + 12 = 34.

The arguments treated in Chapter V, Section ??, aiming to prove the above expression of M at least as a lower bound for the number of moduli for the surfaces of genus p > 3 (with simple canonical system), let us believe that, as a matter of fact, all the C_{12} defined above are actually double curves for irreducible F_{8} (each for ∞^{13} of these). The result thus inductively achieved can be confirmed with Franchetta’s method, hence it looks at least plausible that the computation of the number of moduli can also be confirmed for this class of surfaces, according to the above formula. First of all we verify that there actually exist irreducible canonical surfaces F_{8} having a double C_{12}, and this at least when this C_{12} is a double curve for a rational surface F′_{8} with a further isolated quadruple point: the F_{8} that can be constructed with this method starting with such a rational F′_{8} are obtained as linear combination of the F′_{8} with the surfaces formed by the subadjoint Φ_{3} and by a F_{5} passing through the C_{12}. As to the effective existence of the rational F′_{8}, this is proved by giving to representative system in the plane, which is an ∞^{3} linear system of curves of genus p^{(1)} = 9 with autoresidual characteristic system: it can be defined for example by the curves of eleventh order C_{11}(A_{1}^{3} … A_{12}^{3}B_{1} … B_{5}) where the 17 points A and B belong to a fundamental curve of the 7th order K_{7}(A_{1}^{2} … A_{12}^{2}B_{1} … B_{5}).

After having thus proved that there do exist irreducible canonical surfaces F_{8}, we are furthermore lead to admit that the number of moduli on which they depend is exactly

M = 10p − 2p^{(1)} + 12 = 34,

and thus that all the C_{12} obtained by the bisection problem we explained above are actually double curves of some canonical surface F_{8}; this statement can be justified if one admits at least that imposing a quadruple point to our surfaces F_{8}, yields for them 17 conditions and not less; since we recover in this way the number 34 − 17 = 17 of the absolute invariants of the rational F′_{8}, as can be computed by starting from the plane representation.

Note Also for p = 4 and p^{(1)} = 9 one encounters multiple canonical surfaces, which can give rise to several cases. We shall limit ourselves to mention two cases. Firstly, the double F_{4} with plane sections of genus 2 (that is, having a double line) with a branch curve of order 12, that admits a representation as a double plane with a branch curves C_{14} of order 14, with a 6-tuple point O and with eight 4-tuple points A_{1}, A_{2}, …, A_{8}. Here the (double) images of the canonical curves are the quartics through O^{4}A_{1}A_{2} … A_{8}, whence one has indeed

p = 4 and p^{(1)} = 9.

It is easy to verify that the stated double planes, or rather the corresponding F_{4}^{2} occur, as a particular case, inside the family of surfaces with p = 4 and p^{(1)} = 9 defined by the simple F_{8}. This follows, quite surely, from the computation of the moduli, since the projective invariants of the branch curve C_{14} of the given double plane are in the number of

119 − 19 − 64 − 8 = 28,

and this number is less than the number of moduli of the family to which the F_{4}^{2} belong, which is certainly

M ≥ 9p − 2p^{(1)} + 12 = 30.

A second remarkable case – for p = 4 and p^{(1)} = 9 – in which we encounter a multiple canonical surface is the case of the double cubic F_{3}, with a branch curve of order 16, which admits a representation as a double plane with branch curves C_{12} with four 4-tuple points, A_{1}A_{2}A_{3}A_{4}, and two [3, 3] points: B_{1} and B_{2}. Here the images of the canonical curves K are given by the cubics through A_{1}A_{2}A_{3}A_{4}B_{1}B_{2}; and we have two base points of |K| corresponding to B_{1} and B_{2}. Our double planes depend upon 32 invariants or moduli; we shall not dwell upon the investigation whether they do or do not lie inside the family defined by the simple F_{8}, which has 10p − 2p^{(1)} + 1 = 34 moduli.

Surfaces with p = 4 and p^{(1)} > 9 8.6 274

8.6 Surfaces with p = 4 and p^{(1)} > 9

The difficulties which already appear in the construction of the canonical surfaces F_{8}, with p = 4 and p^{(1)} = 9, tend to increase when we proceed to higher values of p^{(1)}. The first question which arises in the method we used is finding the half of a linear system over a surface Φ_{n−5} (with n = p^{(1)} − 1), a problem which cannot be so simply solved when one deals with irrational surfaces. And this case already occurs for the F_{9} with p^{(1)} = 10; since we encounter here an adjoint surface Φ_{4} of order 4, with

t = 10

double points, and the matter is to bisect the linear system cut on the Φ_{4} by the surfaces F_{9} of order 9 which pass triply through these 10 double points.

The problem which presents itself for p^{(1)} = 10, 11,… suggests the following remarks:

(1) One cannot in general say that the bisection of the system cut on a Φ_{n−5} with t double points by the F_{n} passing triply through them is possible, and one cannot say either that it gives a uniquely determined system.

(2) In any case one should then decide whether the hypothetical curves C of the half system of the aforementioned system are all, or a part of them, actually double curves for irreducible surfaces F_{n}.

(3) From (1) and (2) it follows that the (simple) canonical surfaces of ordinary space (p = 4, p^{(1)} = n+1), instead of forming a unique continuous system, may give rise, for values of n > 8, to several distinct families (perhaps possessing the same number of moduli M = 10p−2p^{(1)} +12 = 50 − 2n). There is no surprise in this for those who have already seen, for the lowest values of n, that there do exist several distinct families of surfaces with the genera p and p^{(1)} = n + 1, corresponding to the multiple canonical surfaces.

However, if the present formula M = 48 − 2n is right also for higher values of n, we would find for n the upper bound n = 24, hence p^{(1)} = 25.

But we cannot say either that these bounds are actually achieved, nor can we claim that they cannot be exceeded. Franchetta’s method, according to which the construction of the F_{n} depends upon the construction of rational surfaces F′_{n} with an isolated quadruple point, yields the effective existence of the canonical surfaces F_{n} only up to the order n = 10.

Canonical surfaces with p = 5 and p^{(1)} = 9 8.7 275

(4) Although the results of this analysis conclude in a rather negative manner, leaving the most vital problems unsolved, still the analysis itself succeeds in confirming Noether’s intuition, according to which the surfaces cannot in general possess other independent numerical characters, in addition to p_{a} and p^{(1)}; in fact, the simple canonical surfaces F_{n} (considered here for the value p = 4) do not seem to give rise to other numerical characters which do not depend upon p_{a} and p^{(1)}, since all the characters of the double curve (defined by taking the half of a linear system on the adjoint Φ_{n−5}) turn out to be perfectly defined. Nevertheless it is convenient to put a question mark concerning the multiple canonical surfaces F_{n}, whether these (also for higher values of p^{(1)}) may give rise to families distinct form the simple ones: here a new character would come into play, that is, the order of multiplicity m of F_{n}, or even the number s of the base points of the pure canonical system, being

nm + s = p^{(1)} − 1.

8.7 Canonical surfaces with p = 5 and p^{(1)} = 9

The method we sketched above for the construction of canonical surfaces of genus p = 4 in ordinary space S_{3}, may also be applied for the construction of surfaces of genus p > 4: without having to go out of ordinary space S_{3}, we shall succeed in constructing there some nonnormal canonical surfaces, projections, from external points, of normal canonical surfaces of a hyperspace.

We have the simplest example for p = 5 and p^{(1)} = 9. We deal here with a canonical surface F′_{8}, of order p^{(1)} − 1 = 8, which should have a double curve C_{12} of the 12th order, belonging to the adjoint cubic surface Φ_{3}. When p = 4, the Φ_{3} has to possess 4 double points and the curve C_{12} originates form the bisection of the linear system of the C_{24} cut on Φ_{3} by the surfaces of the 8th order passing triply through these 4 points. Instead, for p = 5 we find t = 0; whence, we have a completely general cubic surface Φ_{3}, and we have to bisect on it the system of the C_{24} which are complete intersections with the surfaces of order 8. This leads to the curves C_{12} which are complete intersection of the Φ_{3} with surfaces of the 4th order F_{4}; and each of the C_{12} is effectively double for ∞^{4} surfaces F′_{8}. Each of these F′_{8}, possessing ∞^{4} adjoint surfaces Φ_{4} (through C_{12}), appears as a projection from an external point of a canonical surface F_{8} which is normal in S_{4}. The normal canonical F_{8} of S_{4} is readily seen to be a complete intersection of a three dimensional quadric V_{3}^{2} and of a variety V_{3}^{4} of the 4th order. In fact, the system of surfaces Φ_{4} adjoint to F′_{8} in the space S_{3} represents a monoid V_{3} of the 4th order, which projects simply over the space S_{3} from its triple point O (corresponding to the Φ_{3} through C_{12}). Hence the surfaces F_{8} of the 8th order passing doubly through C_{12} appear as images of the complete intersections of the mentioned monoid with the quadrics V_{3}^{2}. Let us add that there are ∞^{15} varieties of the 4th order V_{3}^{4} passing through a F_{8} of S_{4}, among which there is one possessing a triple point at an assigned point of 4-space.

We conclude that: the surfaces of genus p = 5 and absolute linear genus p^{(1)} = 9 (with simple irreducible, base point free, canonical system) admit in general a representation as canonical surfaces F_{8} of S_{4}, complete intersections of a quadric and of a variety of the 4th order.

Let us add that these canonical surfaces F_{8} of S_{4} possess 44 absolute invariants, and thus we find again that the number of moduli upon which the surfaces with p = 5 and p^{(1)} = 9 depend is given by the formula

M = 10p − 2p^{(1)} + 12 = 44.

Every other surface with the characters p = 5, p^{(1)} = 9, different from the canonical surface (and with irreducible and simple pure canonical system) can be easily transformed into the canonical surface F_{8}. For example, let us consider the surface F_{6} of the 6th order with a double conic C in the ordinary space S_{3}. This F_{6} is a projection in S_{3} of a surface F_{8} with a double point O, which is a complete intersection of a quadric V_{3}^{2} (represented on S_{3} by the surfaces of the 2nd order through C) and of a variety V_{3}^{4} of the 4th order; in fact the F_{6}, taken together with the plane of C counted twice, gives a particular F′_{8} passing 4 times through the conic C.

8.8 Surfaces with p = 5 and p^{(1)} = 10

Proceeding now to the surfaces F with p = 5 and p^{(1)} = 10 (with irreducible and simple canonical system) we shall firstly give an example where it is easy to construct the associated canonical surface F_{9}, of order 9 (= p^{(2)} = p^{(1)} − 1) in the space S_{4}. Let us consider indeed in S_{3} the surface F = F_{6} of the 6th order with 5 triple points A_{1}A_{2} … A_{5}, for which we have in fact p = 5 and p^{(1)} = 10. The quadrics of S_{3} passing simply through the above points, are the surfaces Φ_{2} (= Φ_{n−4}) adjoint to the F_{6} (= F_{n}), cutting out the canonical curves on F_{6}. Now, the system of such Φ_{2} represents a cubic variety V_{3}^{3} of the space S_{4}, a particular V_{3}^{3} (studied by Segre and Castelnuovo) which has 10 double points corresponding to the 10 edges of the pentahedron A_{1}A_{2} … A_{5}. The cubic varieties of S_{4} (which in general do not pass through any of the double points) cut on this V_{3}^{3} surfaces F_{9} of the 9th order, whose image in S_{3} are the F_{6} passing triply through A_{1}A_{2} … A_{5}. Therefore the F_{6} of S_{3} can be transformed to a canonical surface of S_{4}, a complete intersection of two cubic varieties of this space, one of which is a V_{3}^{3} with 10 double points.

But the F_{6} with 5 triple points which we started from form only a particular family of surfaces with p = 5 and p^{(1)} = 10, and it is clear that any other surface with such characters gives rise in general to a canonical F_{9} of S_{4}, a complete intersection of two cubic varieties: since on such a F_{9} it is easily verified that the ∞^{34} cubic varieties S_{4} cut out a complete linear system (adjoint of the system cut by the quadrics) of dimension 32.

On the other hand all the F_{9} of S_{4}, complete intersections of two cubic varieties, form a continuous system with the same characters p = 5 and p^{(1)} = 10, hence we may say that the general type of simple surfaces, corresponding to the genera p = 5 and p^{(1)} = 10, is the surface F_{9} of S_{4}, complete intersection of two cubic varieties, which depends on

2 · 34 − 2 − 24 = 42

absolute invariants, hence it possesses M moduli

M = 10p − 2p^{(1)} + 12 = 42.

It is interesting to see how the above canonical surface may be constructed via the method explained in the previous sections, starting from one of its projections of the same order, which is a nonnormal canonical surface F′_{9} in the space S_{3}, with a double curve C_{18} of order 18 belonging to a surface Φ_{4} of the 4th order. In order to do this it is useful to remark that the projection F′_{9} of a F_{9}, made from an external point O, must have t = 6 triple points, B_{1}B_{2} … B_{6}, corresponding to the 6 lines through O of the cubic variety passing through F_{9} and O, and^{(6)} therefore they belong to a conic^{(7)} C_{2}.

(6)It is known and obvious that on a V_{3}^{3} of S_{4}, and through one of its points O, there are 6 straight lines, which belong to the surface obtained as the section of the V_{3}^{3} with the S_{3} tangent to it at O: therefore these lines lie on the quadric cone osculating the above section at O. (7)One might a priori believe that the surfaces of the 4th order having 6 double points lying on a conic form a system contained in a larger system of Φ_{4} with 6 double points not belonging to a conic any longer. As a matter of fact the surfaces of the 4th order with 6 double points fall into several ∞^{28} continuous systems, one of which is indeed formed by the Φ_{4} with 6 double points lying on a conic, which cannot be extended to a larger system. One can immediately see that there do exist Φ_{4} with 6 double points not belonging to a conic, since one can assign 5 double points in general position, A_{1}, A_{2}, …, A_{5}, and then impose a double point (not at a prescribed point) to the above system of Φ_{4}, of dimension 34-20=14: we obtain in this way ∞^{28} Φ_{4} with 6 double points not belonging to a plane. But, on the other hand, if we impose to a Φ_{4} to possess a plane tangent along a conic C, the ∞^{4} sextics cut out on Φ_{4} by the quadrics through C (among which one finds ∞^{2} quadrics reducible into the plane of C and another variable plane), meet C at 6 fixed points, which turn out to be double points for Φ_{4}. It is now easy to see that imposing a tangent plane along a conic implies in fact 6 conditions for Φ_{4} (these can be expressed by requiring that Φ_{4} contains a conic which is self residual with respect to the system of plane sections): whence the Φ_{4} with 6 double points lying on a conic form themselves too a continuous system ∞^{28} of surfaces of the 4th order with 6 double points.

We believe that on a surface Φ_{4} with 6 double points not belonging to a conic, it is not

Now, one can see that the double curve C_{18} of the surface F′_{9} has to belong to a half-system of the system cut out on the adjoint Φ_{4} by the surfaces of the 9th order passing triply through the 6 points B_{1}, B_{2}, …, B_{6}, which are double for the aforementioned Φ_{4}. The bisection of this system |C_{36}| is easily obtained by observing that the C_{18} of the half-system one looks for can be defined as curves belonging to the system sum of the system |C_{16}|, cut out on Φ_{4} by the general surfaces of the 4th order passing through B_{1}, B_{2}, …, B_{6}, and of the conic C_{2} along which the plane of C_{2} touches Φ_{4}.

Φ_{4} is a regular surface of genus 1, with a canonical curve of order zero, therefore every linear system on it has dimension equal to the genus π, if it is free from base points (at simple points of the surface). Therefore the system |C_{16}|, of degree 64 − 2 · 6 = 52, has genus equal to its dimension, that is, 26 + 1 = 27. Summing to |C_{16}| the conic C_{2} of genus 0, which cuts the C_{16} (away from B_{1}, B_{2}, …, B_{6}) at 2 points, we obtain C_{18} of genus

π = 27 + 0 + 2 − 1 = 28.

Hence also the dimension of the system |C_{18}| is 28, and exceeds by 1 the dimension of the system |C_{16}| to which we added the fixed conic C_{2}: this means that the system |C_{18}| is irreducible. Thus we have obtained on the Φ_{4} with 6 double points B_{1}, B_{2}, …, B_{6} belonging to a conic, ∞^{28} curves C_{18}, in general irreducible, having triple points at B_{1}, B_{2}, …, B_{6}. These are a priori “possible” double curves for nonnormal canonical surfaces F′_{8}. Let us assume that, among these C_{18}, there are ∞^{x} (with x ≤ 28) curves through which actually pass some irreducible surfaces F′_{9} with multiplicity two. Then each of these C_{18} belongs, as a double curve, to ∞^{5} irreducible F′_{9}, since there exist ∞^{4} F′_{9} which are reducible into the Φ_{4} and a F_{5} through it. We therefore have ∞^{x+5} F′_{9} with a double curve C_{18} belonging to a given Φ_{4} (endowed with 6 double points lying on a conic). But there are ∞^{14} Φ_{4} with the same 6 double points and, by varying these, we obtain ∞^{28} Φ_{4}: whence we have ∞^{28+x+5} F′_{9} in the space S_{3}. Among these there are ∞^{15} which are projectively equivalent to a given one, and ∞^{19} which are birationally equivalent, and corresponding to projections of F_{9} from ∞^{4} different centres in S_{4}; therefore the number of birationally distinct F′_{9} and F_{9}, that is, the number of moduli of the above mentioned canonical surfaces is:

M = 28 + x + 5 − 19 = x + 14;

since we know on the other hand that

M = 42,

possible in general to bisect the system cut by the surfaces of the 9th order passing triply through the 6 points: the problem should be examined by representing the surface Φ_{4} on a double plane, via a projection from one of its double points.

Surfaces on a variety of dimension 3 or more 8.9 279

we infer

x = 28.

This means that

all the curves C_{18} obtained with the above described bisection are double curves of some nonnormal canonical surfaces F′_{9}, which do actually exist and are in general irreducible.

We end proposing to the researcher some open problems.

QUESTIONS Do there exist, among the surfaces with simple canonical system and with p = 5 and p^{(1)} = 10, some surfaces for which this system possesses a base point; these surfaces would therefore lead to canonical surfaces F_{8} of S_{4}, or to surfaces F′_{8} of S_{3}, with a double curve C_{11} of order 11. And whether these surfaces occur as particular cases inside the family of the F_{9}, or of the F′_{9}, which we have just described. Whether, on the other hand do occur as particular cases inside the family of the F_{9}, or of the F′_{9}, some surfaces whose canonical system (possessing a base point) is composed with a rational involution I_{2}. These would yield F_{8} which reduce to double F_{4}, representable on a double plane with a branch curve C_{12}, of order 12, with four 4-tuple points and with a [3, 3] point: notice that these double planes have exactly the characters p = 5 and p^{(1)} = 10, and they depend only on 41 moduli.

Note Passing to the case p = 5 and p^{(1)} = 11, let us point out now the double plane with a branch curve C_{12} having three 4-tuple points and two [3, 3] points, which has the genera p = 5 and p^{(1)} = 11, and a (non-simple) canonical system with 2 base points. This double plane, which leads to a double canonical quartic with a branch curve C_{18}, depends upon 40 moduli. If the simple canonical surfaces F_{10} of S_{4} (with p = 5 and p^{(1)} = 11) depend upon M = 10p − 2p^{(1)} + 12 = 40 moduli as well, this means that, as it is likely, we shall have, for p = 5 and p^{(1)} = 11, two distinct families of surfaces: the F_{10} and the aforementioned double F_{4}. It is worthwhile to remark that certainly there do exist simple canonical surfaces F_{10} of the space S_{4}; one can obtain particular F_{10} (with a double line) starting with a F_{6} of ordinary space having 4 triple points and a tacnode.

8.9 Surfaces on a variety of dimension 3 or more

Instead of pursuing further the analysis of the previous sections, we prefer to indicate how one can obtain remarkable examples of canonical surfaces in hyperspace, by appealing to the geometry of the varieties of dimension larger than 2.

When one passes from surfaces to varieties of dimension 3 or more, one encounters three kind of properties:

(1) properties of general order, which appear as natural extensions of the properties of curves and surfaces, which we are thus allowed to assume as well known; some of these are already indicated in M. Noether’s fundamental memoir in Math. Ann., Band 8;

(2) properties which can more or less be foreseen as extensions of those of surfaces, but which may nevertheless require less obvious, and sometimes also difficult, proofs;

(3) completely new and unexpected properties which do not have any analogue in the theory of surfaces^{(8)}

We shall only consider problems of the first type, concerning the general theory of linear system of surfaces belonging to a variety of dimension 3, the adjoint systems, the canonical system, and the invariants which depend on it.

For the sake of simplicity we refer to a 3 dimensional variety V, free from singularities in a suitable hyperspace, and we consider linear systems of surfaces on it which are free from base points and curves. A surface F on V belongs in general to a complete linear system |F |, and one can operate on such systems – as on the linear systems of curves on a surface – with sums and subtractions.

If the dimension of the system |F | is at least three, then every triply infinite system contained in |F | has a Jacobian surface F_{j}, that is, the locus of double points of surfaces in it; the F_{j} associated to the various systems totally contained in |F | are equivalent one to the other, and determine a complete linear system: the Jacobian system of |F |, that we shall denote by |F_{j}|.

As well as in the case of systems of curves on a surface the fundamental theorem expressed by the relation

|(F + F)_{j}| = |F_{j} + 4F | = |4F + F_{j}|,

holds, and the difference

|F_{j} − 4F |

(8)We recall, as an example, the results of Fano and Enriques concerning the rationality questions; irrational involutions in the space; different types of varieties containing a congruence of rational curves, and so on.

defines the canonical system, effective or virtual, of V^{(9)}, while the adjoint system to F is given by

|F′| = |F_{j} − 3F |.

In general a variety V_{n} of order n belonging to the 4-dimensional space S_{4} possesses a double surface Γ, whose points correspond to the neutral pairs of the system of hyperplane sections on a transformed model of V; and, except if one has more complicated singularities, the canonical system is cut by the adjoint varieties Φ_{n−5} of order n−5, which are simply defined by the condition of passing through the double surface Γ. Instead, the adjoint Φ_{n−4} cut out on V_{n} the adjoint surfaces to the hyperplane sections.

If the variety V_{n} of S_{4} possesses higher singularities, that is, either a multiple surface of order i^{2}, or an i-tuple curve (i^{2}) or an isolated i-tuple point (i > 3), then the adjoint varieties Φ to V_{n} are in general defined by the conditions of passing i − 1 times through every i-tuple surface, i − 2 times through an i-tuple curve and i − 3 times through an isolated i-tuple point.

On a 3-dimensional variety V (belong it to a space S_{4}, or not) the surfaces F′ of the adjoint system to a system |F | cut a generic F along canonical curves; conversely, when the system |F | does not have fundamental surfaces, this property characterizes the adjoint surfaces F′.

The characters of the canonical system of a variety give corresponding invariant characters of V itself, already encountered by Noether.

These are:

(1) the dimension of the canonical system, increased by one, that is, the number of linearly independent canonical surfaces, which is the principal geometric genus P (or P_{g}) of V;

(2) the geometric genus p_{g} and the linear genus p^{(1)} of the canonical surfaces which are respectively defined to be the superficial genus p_{g} and linear genus p^{(1)} of the variety V.

The genus x of the curve K intersection of two canonical surfaces, and the degree y of the canonical system (that is, the number of points in common to three canonical surfaces) do not give new characters of V, since they can be expressed as functions of the previous ones by the formulas

2x + y − 1 = p^{(1)} 4y = p^{(1)} − 1 = p^{(2)},

(9)It is hardly necessary to remark that the canonical surfaces, thus defined with respect to the birational transformations on the variety V, should not be confused with the projectively canonical surfaces, which are defined in a space S_{p−1} by the property that their hyperplane sections are canonical curves (of the surface itself).

Examples of canonical surfaces in hyperspace 8.10 282

which are a consequence of the fact that |2K| is the canonical system of the canonical surface. We have therefore only three independent characters, as was already remarked by Noether.

Together with the geometric characters one may define for V the numerical characters: the principal numerical genus P_{a} and the numerical superficial genus p_{a}^{(10)}.

From what we said above it follows that if we have a surface F in S_{4} which is the complete intersection of two varieties V^{m} and V^{n} of respective orders m and n, with no singularities along F, the canonical system is cut out on F by the varieties of order m + n − 5^{(11)}; we have here an extension of the well-known theorem of Noether concerning skew curves^{(12)}.

If F is not the complete intersection of V^{m} and V^{n}, but these meet in a further surface F^{x}, then the canonical system of F is cut out by the varieties of order m + n − 5 passing through F^{x}.

In an analogous way, if in r-dimensional space we have r −2 hypersurfaces (r − 1 dimensional varieties), of respective orders n_{1}, n_{2}, …, n_{r−2}, free from singularities and meeting along a reducible surface F +F^{x}, then the canonical curves are cut out on F by the hypersurfaces of order n_{1}+n_{2}+⋯+n_{r−2}−r−1 passing through F^{x}.

8.10 Examples of canonical surfaces in hyperspace

The previous notions lead to simple examples of projectively canonical surfaces in hyperspaces. Let us start by considering the surface in S_{4} which is the complete intersection of two varieties V^{m} and V^{n} without singular points. The latter will be a canonical surface if

m + n − 5 = 1,

since, by what has been said, the hyperplanes cut on it the canonical curves. We obtain again here the two types already mentioned above: the surface F_{8}, intersection of a quadric and of a variety V_{3}^{4}, for which p = 5 and p^{(1)} = 9; and the surface F_{9}, intersection of two cubic varieties, for which p = 5 and p^{(1)} = 10.

More generally, examples of canonical surfaces of the space S_{r} will be given by the complete intersections of r−2 hypersurfaces, of respective orders n_{1}, n_{2}, …, n_{r−2}, when these orders (≥ 2) are such that one has

(10)Cf. Severi, Ibidem.

(11)Cf. Enriques–Chisini, Ibidem, Libro V, Chapter V, §41 (Vol. III, page 528).

(12)Cf. Severi, Su alcune questioni di postulazione, Rendic. Circ. Mat. di Palermo, Vol. XVII, 1903.

n_{1} + n_{2} + ⋯ + n_{r−2} − r − 1 = 1,

that is,

n_{1} + n_{2} + ⋯ + n_{r−2} = r + 2.

Taking into account that we must have n_{i} ≥ 2, this relation gives us some remarkable canonical surfaces, as the following ones.

In the space S_{5} we have the surface F_{12} with p = 6 and p^{(1)} = 13, complete intersection of two quadrics and of a cubic variety. And in the space S_{6} we have the surface F_{16} with p = 7 and p^{(1)} = 17, complete intersection of four quadrics.

These examples are particular issues of more general families of canonical surfaces, which can be defined as the intersection of a normal 3-dimensional variety of a hyperspace, having elliptic or projectively canonical curve sections, with a hypersurface of order 3, respectively 2.

About the V_{3}^{n} of the space S_{r}, with n = 2r−4, with projectively canonical curve sections^{(13)}, which we assume to be free from singular points, we notice that their hyperplane sections F are surfaces of genus 1 with canonical curve of order zero, whence the adjoint system |F′| to F is formed only by the surface of order zero. Therefore the adjoint system to |2F | is

|(2F)′| = |F |;

hence the surfaces F, that is, the hyperplanes of the space S_{r}, cut the surfaces of |2F |, the intersections with the quadrics, along canonical curves.

Concerning instead the V_{3}^{n} of the space S_{n+1} (n > 3) with elliptic curve sections^{(14)}, which are rational, it suffices to make the following remark: since their sections are normal, the intersections with the quadrics, that is, the surfaces of the system |2F | double of the system |F | of hyperplane sections, are surfaces with all the genera equal to 1 and with a canonical curve of order zero. In fact the surfaces of |2F | have as hyperplane sections curves of order 2n, of genus n+1, belonging to n-dimensional spaces, thus they are canonical curves. Hence it follows that the system |2F | admits as the unique adjoint surface the surface of order zero, and therefore the adjoint system to |3F | is given by

(13)Varieties of such a type are the cones obtained by projecting a surface having all the genera equal to 1 from an external point of the space which contains it. For the first values of the order these cones are contained in larger families of varieties without singular points.

However, it seems possible to extend, with minor modifications, the argument of the text also to the cones.

(14)The 3 dimensional varieties with elliptic curve sections are classified in the note by F. Enriques, On the linear systems of algebraic surfaces whose variable intersections are elliptic curves. Rendic. Acc. Lincei, may-june 1894. Cf. Math. Ann., Band 46.

|(3F)′| = F,

so that the surfaces F, and also the hyperplanes of the space S_{n+1}, cut the intersections of the V_{3}^{n} with a cubic variety along their canonical curves.

Summing up, we have obtained two series of canonical surfaces of genus p in the space S_{p−1}:

(1) the surfaces F_{3p−6}, of order 3p−6 (r = p−1, n = p−1),with p^{(1)} = 3p−5, complete intersections of a cubic hypersurface with a normal rational variety V_{3}^{p−2} having elliptic curve sections; and

(2) the surfaces F_{4p−12}, of order 4p − 12, with p^{(1)} = 4p − 11 (r = p − 1, n = 2p − 6), complete intersections of a quadric with a variety V_{3}^{2p−6} having projectively canonical curve sections.

We shall see later on other classes of canonical surfaces defined on 3dimensional varieties with rational curve sections. We shall not dwell further on the construction of canonical surfaces in hyperspace, limiting ourselves to mention a few recent papers, which are somehow related to this topic, and in which this investigation is pushed further with the construction of other interesting examples:

G. Fano, Superficie algebriche e varietà a tre dimensioni a curve sezioni canoniche, Atti Acc. Lincei, 1936; Su alcune varietà algebriche a tre dimensioni aventi curve sezioni canoniche in Scritti matematici offerti a L. Berzolari, page 329 (1936); Sulle varietà algebriche a tre dimensioni a curve sezioni canoniche, Mem. Acc. d’Italia, 1937.

L. Godeaux, Sur la construction d’une surface canonique, Bull. Soc. Math. Liège, 5, (1936); Sur une surface canonique appartenant à la variété de Segre representant les couples de points de deux plans, Bull. de l’Académie Belgique, 22, (1936); Sur les surfaces algébriques dont les sections hyperplanes sont des courbes canoniques, I, II, III, Bull. Soc. Math. Liège, 6, (1937); Construction d’une surface canonique du septième ordre, Bull. de la Soc. des Sciences de Liège, 1944; Construction d’une surface canonique du huitième ordre, Bull. des Sciences Math., 1944; Construction d’une surface canonique du neuvième ordre, Bull. de l’Acad. de Belgique, 1944. P. Burniat, Note sur quelques surfaces canoniques et sur des surfaces de genre un, Bull. de l’Acad. de Belgique, 5, 22, (1936); Note sur quelques surfaces canoniques, Acad. de Belgique, 3, 22, (1936); Sur des hypersurfaces et variétés canoniques, Acad. de Belgique (5), 23, (1937).

E. A. Maxwell, Regular canonical surfaces of genus three and four, Proc. Cambridge Phil. Soc., 33, (1937).

Minimal value of the linear genus with respect to the superficial genus 8.11 285

8.11 Minimal value of the linear genus with respect to the superficial genus

It appears clearly, already from the construction of canonical surfaces, that, once a fixed value is assigned to the superficial genus p (or p_{g}) of a surface, then the (absolute) linear genus p^{(1)} cannot go below a certain minimal value, at least if the canonical curves are irreducible, an assumption which excludes, for p > 2, the surfaces with p^{(1)} = 1. We intend now to show that, with no exceptions, for the surfaces with p^{(1)} > 1, the minimal value of p^{(1)} with respect to p or p_{g} (> 2) is given by

p^{(1)} = 2p − 3.

In other words, we have

p^{(1)} ≥ 2p − 3,

or, equivalently,

p ≤ (p^{(1)} + 3)/2 :

that is, the maximum of p with respect to p^{(1)} (> 1) is (p^{(1)} + 3)/2.

This inequality is established by Noether, in the case of irreducible (pure) canonical curves, in his memoir in Math. Ann., Band 8 (1875). If we assume the canonical curves K to be irreducible, it is a consequence of Clifford’s theorem, according to which the order of a special series g_{n}^{r} on an arbitrary curve is n ≥ 2r^{(15)}. In fact the degree of the characteristic series of |K| is equal to n = p^{(1)} − 1 and its dimension is r = p − 2, whence

p^{(1)} − 1 ≥ 2p − 4.

But the same inequality holds, however reducible the canonical curves K might be, provided the variable part splits into curves of genus π ≥ 3. Firstly, if their variable components k are irreducible, their degree is n ≤ p^{(1)} − 1 (Chapter IV, Section ??), whereas the dimension of |k| is the same of |K|, that is, p − 1; we have therefore

n ≥ 2p − 4

and a fortiori

p^{(1)} − 1 ≥ 2p − 4 p^{(1)} ≥ 2p − 3.

If instead the variable curves k are assumed to be composed with p−1 curves of a pencil of genus π ≥ 3, the genus of |k| (≤ p^{(1)}) will be

≥ (p − 1)(π − 1) + 1;

(15)Cf. Enriques–Chisini, Ibidem, Libro V, Chapter I, §11 (Vol. III, page 87).

we infer

p^{(1)} ≥ 2p − 1.

Now, if a surface is such that its genus p (> 2) attains the maximum p = (p^{(1)} + 3)/2(p^{(1)} > 2), then its canonical curves are irreducible and hyperelliptic.

In fact, if the curves K are irreducible, then the characteristic series g_{n}^{r} on a curve K (n = p^{(1)} − 1, r = p − 2) has to have n = 2r, and this implies indeed that K is hyperelliptic^{(16)}. If instead the curves K were reducible, the argument could be repeated for their variable components, which are certainly irreducible themselves (otherwise p^{(1)} ≥ 2p − 1): then the K turn out to be hyperelliptic and lead to a double canonical surface related to the double plane we shall describe in what follows, and therefore it results a posteriori that there are no fixed components for |K|.

It is also remarkable that, for our surfaces with maximum p, it turns out to be excluded the case that the pure canonical curves K contain exceptional fixed components, that is, that |K| has some base point (on a model free from exceptional curves): indeed, if there are r base points, it follows that

p^{(1)} ≥ 2p − 3 + r.

The canonical surface corresponding to the maximal value of p (p^{(1)} = 2p−3) is a normal surface with rational hyperplane section curves, of order p − 2 in S_{p−2}, and therefore it can be represented on the plane with a linear system of curves belonging to one of the following types^{(17)}:

(I) net of the lines,

(II) ∞^{5} system of the conics,

(III) system of the curves of order n with a base point O of multiplicity n−1, and possibly also another simple base point distinct from O, or several simple base points, infinitely near to O.

The double planes whose canonical systems have such systems of rational curves as images are respectively:

(I) the double plane with a branch curve C_{8} of the 8th order: p = 3, p^{(1)} = 3;

(II) the double plane with a branch curve C_{10} of the 10th order: p = 7, p^{(1)} = 9; (III) the double plane with a branch curve C_{2n+6}, of order 2n + 6, having a

(16)Cf. Enriques–Chisini, Ibidem, Libro V, Chapter I, §12, Vol. III, page 97.

(17)Cf. Enriques–Chisini, Libro V, Chapter II, §22, Vol. III, page 194.

2n-tuple point O: p = 2n + 1, p^{(1)} = 4n − 2; and

III′ particular cases of the type III), where the branch curve (whose order we may assume cannot be lowered by quadratic transformations) possesses, beyond O, a 6-tuple or a 4-tuple point, yielding p = 2n−2, p^{(1)} = 4n−8, respectively p = 2n, p^{(1)} = 4n − 4.

Indeed, as we have already said, one has to exclude those types of double planes for which the characteristic series of the pure canonical system |K| has some fix point (base point of |K|): e.g., a [5,5] point or a [3, 3] point.

We found so far that:the surfaces of superficial genus p > 2 with p^{(1)} > 1, for which the absolute linear genus p^{(1)} attains the minimum, that is, p^{(1)} = 2p − 3, can be represented on a double plane belonging to one of the types I), II), III) and III′)^{(18)}.

We may replace Noether’s inequality

p ≤ (p^{(1)} + 3)/2

with a stronger one, valid for surfaces with a simple canonical system (p > 3), which has been pointed out by Castelnuovo^{(19)}. When the canonical system is irreducible and simple, the superficial genus (p or p_{g} > 2) cannot go beyond the value (p^{(1)} + 6)/3; we have therefore

p ≤ (p^{(1)} + 6)/3, p^{(1)} ≥ 3p − 6.

In order to prove this inequality, it is convenient to start from a lemma of Bertini concerning the autoresidual g_{n}^{r} which belong to a curve of genus π. Let us consider, on a curve which contains such a g_{n}^{r}, two groups of points G and G′ of the series, having in common a group Γ of r − 1 generic points; if the series is simple (that is, not composed with an involution), G and G′ do not have points in common other than the r − 1 points of Γ, and the group Γ cannot belong to a linear series g_{r−1}^{s} of dimension s > 0. With these assumptions, the theorem of Riemann–Roch says that the group G of the g_{n}^{r} imposes n − r conditions to the groups of the canonical series in order to contain it^{(20)}. Moreover, the group G’ imposes n − r − (r − 1) = n − 2r + 1 conditions (a priori not independent) to the groups of the canonical series which already contain r − 1 points of it, and therefore to the groups which already satisfy the condition of containing G. Hence, the number of conditions in order that a canonical group contains G + G′ − Γ is less than or equal to

(18)Cf. Enriques, Sopra le superficie algebriche di cui le curve canoniche sono iperellittiche Rendic. Acc. Lincei, s. V, Vol. V, page 191 (1896).

(19)G. Castelnuovo, Osservazioni intorno alla geometria sopra una superficie. Nota II.

(Rendic. Ist. Lombardo, s. II, Vol. XXIV, 1891). Memorie scelte, no. XVIII.

(20)Cf. e.g., Enriques–Chisini, Ibidem, Libro V, Chapter I, Vol. III, page 85.

n − r + (n − 2r + 1) = 2n − 3r + 1.

Now, the canonical groups satisfying these conditions cannot be more than one, that is, they coincide with G + G′, whence the points of Γ are double; since otherwise Γ (which is formed by r − 1 generic points of the curve, where r < π) should belong to a series g_{r−1}^{s}, with s > 0. We infer that

2n − 3r + 1 ≥ π − 1,

π −1 being the number of conditions which determine the group G+G’ inside the canonical series.

But, since the g_{n}^{r} is autoresidual,

n = π − 1,

hence

π − 3r ≥ 0.

Let us apply this formula to the simple characteristic series of the canonical system on a surface of genera p and p^{(1)}. We have

n = π − 1 = p^{(1)} − 1; r = p − 2,

and therefore

p^{(1)} − 3p + 6 ≥ 0, Q.E.D.

Castelnuovo, after having written down this inequality, which he in fact derives from Bertini’s formula, gives actual examples of surfaces with irreducible simple canonical system, for which the genus p attains the maximum with respect to p^{(1)}, and he indeed succeeds in determining the whole family of surfaces for which the maximum is attained. The first examples are:

(1) the surface of the 5th order in S_{3}, for which p = 4 and p^{(1)} = 6;

(2) the canonical surface F_{8} of the four-dimensional space, intersection of a quadric with a variety of the 4th order, for which, as we just saw, p = 5 and p^{(1)} = 9;

(3) the surface obtained in S_{6} on the three dimensional cone of the 4th order (projecting from a point O a Veronese surface) by intersecting it with a hypersurface of the 4th order passing through a two dimensional quadric cone, with vertex O, and lying on the cone:

p = 7, p^{(1)} = 15.

Upper bound for the linear genus p^{(1)} in terms of the superficial genus 8.12 289

We leave it to the reader to verify that the F_{14} thus constructed is in fact a canonical surface.

For p > 7 the simple canonical surfaces of order p^{(1)} − 1, whose linear genus has the minimum value p^{(1)} = 3p − 6, can be obtained as intersections of a variety V_{3}^{p−3}, described by a simply infinite system of planes in S_{p−1}, and of order p − 3, with a hypersurface V_{p−2}^{4}, of the 4th order passing through p − 5 planes of the V_{3}^{p−3}.

We shall limit ourselves to sketch the proof of the fact that a surface F of order 3p − 7, thus obtained, is in fact a canonical surface of S_{p−1}. To this purpose it is convenient to recall how the variety V_{3}^{p−3} with rational hyperplane sections^{(21)} is represented on the space via projection from p − 4 of its points: the representative system is formed by ∞^{p−1} surfaces of order p − 3 passing p − 4 times through a straight line r and simply through p − 4 further lines t incident with it (these are the traces of the planes through the centres of projection).

One can assume that the surface F contains the aforementioned p − 4 centres of projection, and therefore that its image on the S_{3} chosen for the representation, is a surface F_{2p−3} containing r as a (p − 4)-tuple line and the p−4 lines t incident to r as triple lines. It is therefore evident that the adjoint surfaces Φ_{2p−7} to the F_{2p−3} split into the p − 4 planes rt and into a residual surface Φ_{p−3}, of order p − 3, passing simply through the p − 4 lines t. It is thus proved that the hyperplane sections of the normal surface F of S_{p−7} are the canonical curves, Q.E.D.

8.12 Upper bound for the linear genus p^{(1)} in terms of the superficial genus

The very construction of canonical surfaces, as well as the count of moduli, suggests that the linear genus p^{(1)} cannot exceed a certain maximum value with respect to p (p_{g} or p_{a}). Although there has been yet no success in determining the maximum effectively achieved by p^{(1)}, we should say that a first inequality in this direction has been written down by A. Rosenblatt^{(22)}:

p^{(1)} ≤ 16p_{a} + 27.

(21)The three dimensional varieties with rational curve sections are: simply infinite rational series of planes, or cones of the 4th order projecting Veronese surfaces from a point, which are represented on ordinary space by quadrics tangent at one point. Cf. F. Enriques, Sui sistemi lineari di superficie algebriche ad intersezioni variabili iperellittiche. Acc. Lincei, 1893. Cf. Math. Ann., Band 46.

(22)A. Rosenblatt, Sur quelques inégalités etc., Comptes Rendus Acad. des Sciences, Paris, t. 154 (1912).

Canonical systems belonging to an involution 8.13 290

More recently, F.Severi^{(23)}, relying on a transcendental result by HODGE, has obtained the inequality:

p^{(1)} ≤ 2p_{g} + 8p_{a} + 11 − ρ,

where ρ denotes the number of the base: that is, the number of independent continuous systems of curves traced on the model of the surface free from exceptional curves.

The question is still open from the geometric viewpoint, even in the case of regular surfaces: taking up again the count of moduli which (in the case p_{g} = p_{a} = p > 3 and for simple canonical surfaces) has led us to the formula

M = 10p − 2p^{(1)} + 12 + ω,

one has to investigate the upper bound, and possibly the maximum effectively reached, for the superabundance ω. The question indeed consists in determining the minimum number of conditions imposed by the group of cuspidal points of a surface of order n to the intersections with the adjoint surfaces of the same order.

8.13 Canonical systems belonging to an involution

The constructions of the preceding sections clearly show that, even if the canonical system of a surface of genus p > 3 is assumed to be irreducible, yet it can belong to an involution I_{n} of order n ≥ 2, so that we have a corresponding multiple canonical surface. It is convenient now to say something about this possibility.

The simplest case, which we met in the previous examples, corresponds to the case where the surface contains a rational involution I_{2} of the 2nd order, that is, where the surface can be represented on a double plane with any branch curve C_{2n} of even order equal to 2n: in such a case – as it was already noticed by Noether – the canonical curves (having as their images double curves of order n − 3 which form a half-system of the second adjoint system to C_{2n}), always belong to the involution I_{2}; we have thus as canonical surface a double rational surface (which may also possibly further reduce to a multiple surface).

It is particularly worth of note the fact that the possess of such an involution immediately follows from the hypothesis that the linear genus p^{(1)} is particularly small with respect to p: p^{(1)} < 3p − 6, since we established that, if the canonical system is simple and irreducible, then

(23)Rendic. Circ. Mat. di Palermo, t. LVI, page 79 (1932).

p^{(1)} ≥ 3p − 6.

But the converse assertion does not hold, since for instance there do exist double planes with p^{(1)} > 3p − 6; for n large enough such is the double plane with a branch curve C_{2ν} free from singularities, for which

p = n(n − 3)/2 + 1, p^{(1)} = 2(n − 3)^{2} + 1

and the ratio p^{(1)}/p tends to 4 for n = ∞.

The double canonical surfaces always correspond to surfaces possessing an involution I_{2} of the 2nd order, but this involution is not necessarily rational. A few examples of surfaces, whose canonical surfaces belong to a irrational involution I_{2}, have already been described by other authors, but now we do not remember the reference. In our circle of ideas, we may construct an interesting example of double canonical surface, which cannot be referred to a double plane, starting from the canonical surface F_{10}, with p = 4 and p^{(1)} = 11, whose double curve possesses t = 20 triple points. This surface has an adjoint surface Φ_{5} of the 5th order, having 20 conical double points. In the pencil

λF_{10} + Φ_{5}^{2} = 0

one will see the surface F_{10} tend by continuity to the Φ_{5}^{2}, on which there is no proper branch curve, but t = 20 conical points giving rise to as many infinitesimal branch curves. Thus the quintic Φ_{5}, with genera p = 4 and p^{(1)} = 6, forms a double canonical surface with genera p = 4 and p^{(1)} = 11. We shall be content to have mentioned this case, which we point out to the interested reader as a topic of study.

The first examples of surfaces whose canonical system belongs to an involution I_{n} of order n > 2 were pointed out by B. Segre^{(24)}. The simplest is given by the surface F_{6} of the 6th order possessing two oscnodes A and B: p = 4, p^{(1)} = 9. The ∞^{2} adjoint quadrics to F_{6} touch the oscnodal planes at A and B, whence they pairwise meet along conics tangent to these two planes, and lying in planes through A and B: the above conics form a congruence of the 1st order of the space and cut the sextic F_{6} along an involution I_{4}, with which the canonical system turns out to be composed. The corresponding canonical surface is a quadruple quadric.

We are led to a quadric, and more precisely to a 4-tuple canonical quadric cone, with the same characters p = 4 and p^{(1)} = 9, by the double plane with a branch curve C_{18}, of order 18, with eight 6-tuple points. In fact in this case the canonical curves have as their double images the sextics of genus two passing doubly through the eight points, which belong, as it is well known, to an involution I_{2} (of the 3rd type according to Bertini).

(24)B. Segre, Sulle superficie algebriche aventi il sistema canonico composto con un’involuzione, Rendic. Acc. Lincei, s. VI, Vol. XVI, page 316 (1932).

We leave it to the reader to compare the two examples and to find out whether they correspond to distinct types of surfaces and whether they do occur or not as members of a larger family of surfaces.

The second example of B. Segre is the sextic F_{6} with an oscnode A and a triple point B having an infinitely near double line: p = 4, p^{(1)} = 7. It is clear that also here the adjoint quadrics pairwise cut each other along conics of a congruence of the 1st order, and these conics meet the sextic at triples of points of an involution I_{3}, with which the canonical system is composed. The sextic F_{6} leads therefore to a triple canonical quadric, which has, as we have already said,

p = 4 and p^{(1)} = 7.

This case shown by B. Segre is a concrete issue of a remark by Castelnuovo, who noticed, since 1892, that if the canonical curves of a surface contain a series g_{3}^{1}, then the characteristic series of the system has to be composed with it, hence the canonical system has belong to a rational involution I_{3} of the 3rd order.

The problem of determining all the surfaces (with p > 2) whose canonical curves (irreducible and without base points) contain a g_{3}^{1}, which are therefore transformed into triple canonical surfaces with rational sections, was recently solved by G. Pompilj^{(25)}. There are three types of canonical surfaces corresponding to these conditions:

(1) a triple plane with branch curve C_{12} of order 12 (already mentioned in the Lezioni by Enriques–Campedelli)^{(26)}, which has

p = 3 and p^{(1)} = 4;

(25)G. Pompilj, Sulle superficie algebriche le cui curve canoniche posseggono una g_{3}^{1}, Rendic. Ist. Lombardo, 27 marzo 1941.

(26)It is pointed out there that such a triple plane with p = 3 and p^{(1)} = 4 (which the count of moduli shows to be a particular case of a larger family) arises from the surface F_{5} of the 5th order endowed with a tacnode, by projection with centre at this singular point.

One can similarly start from a canonical surface F_{6} of S_{3}, acquiring a triple point.

These considerations can easily be extended; thus from the canonical surfaces F_{6}, F_{7}, F_{8}, … of S_{3}, acquiring a triple point, one obtains the construction of particular multiple canonical planes, quadruple, quintuple, in general (p^{(1)} − 1)-tuple planes, of genus p = 3, with p^{(1)} = 5, 6, …

Letting instead the above canonical surfaces acquire two triple points, one obtains surfaces with p = 2 and linear genus p^{(1)} = 2, 3, …, where the pencil of canonical curves possesses p^{(1)} − 1 base points.

(2) a triple quadric cone with branch curve C_{18} of order 18, for which

p = 4 and p^{(1)} = 7,

in this type is included (as a particular case?) the example of B. Segre;

(3) a triple rational normal cubic cone in S_{4}, with branch curve C_{24} of order

24 for which

p = 5 and p^{(1)} = 10.

These three types are represented analytically by the following equations:

(1) the triple plane

z^{3} + 3p_{4}z + q_{6} = 0,

where p_{4}(x, y) = 0 and q_{6}(x, y) = 0 define general curves of respective orders four and six: the branch curve is

4p_{4}^{3} + q_{6}^{2} = 0;

(2) the triple plane

z^{3} + 3p_{6}z + q_{9} = 0,

p_{6} and q_{9} being curves of orders six and nine with p_{6} passing through two infinitely near triple points O^{3} and O_{1}^{3}, q_{9} through O^{5} and O_{1}^{4}; the branch curve is

4p_{6}^{3} + q_{9}^{2} = 0;

(3) the triple plane

z^{3} + 3p_{8}z + q_{12} = 0;

p_{8} and q_{12} being curves of orders 8 and 12, passing through a point O and two points infinitely near to O, O_{1} and O_{2}, as indicated by the symbols p_{8}(O^{5}O_{1}^{2}O_{2}^{2}) and q_{12}(O^{8}O_{1}^{4}O_{2}^{4}); the branch curve is

4p_{8}^{3} + q_{12}^{2} = 0.

We shall not dwell on Pompilj’s analysis, based on the theory of triple planes established by himself^{(27)}, which can also be carried on in a different way^{(28)}. Rather we would like to indicate how to justify Castelnuovo’s remark to the effect that if the canonical curves of a surface contain a g_{3}^{1}, then the characteristic series is necessarily composed with it.

(27)G. Pompilj, Sulla rappresentazione algebrica dei piani tripli. Sem. Mat. Università di Roma, s. 4, Vol. III. Osservazioni sui piani tripli, Rendic. Ist. Lombardo, 27 febbraio 1941.

(28)Cf. A. Franchetta, Sulle superficie le cui curve canoniche posseggono una g_{3}^{1}. Boll.

U.M.I., 1942.

Surfaces with linear genus p^{(1)} = 2: first case p = 2 8.14 294

In order to prove the assertion it suffices to consider a curve of genus π > 3, containing a g_{3}^{1} and an autoresidual series of dimension r ≥ 1, inside which we may choose a g_{n}^{1}, with n = π − 1: we shall derive a contradiction from the assumption that the g_{n}^{1} is not composed with the g_{3}^{1} (away from the fixed points). Indeed, if the g_{n}^{1} is not composed with the g_{3}^{1} we can transform the given curve into a C_{m} of order m = n + 3, on which both the g_{n}^{1} and the g_{3}^{1} are cut out by systems of lines, through points A and B respectively: A is a n-tuple point, B a triple point. Now, a pair of lines through B should belong to an adjoint curve to C_{m} of order m − 3, therefore there should exist a curve of order m − 5 passing n − 1 = m − 4 times through A: which is obviously absurd.

8.14 Surfaces with linear genus p^{(1)} = 2: first case p = 2

When either for a surface F the canonical system has dimension smaller than 3, that is, the genus p is < 4, or this (pure) system is reducible, but the linear genus is p^{(1)} > 1, or it belongs to an involution: then in all these cases where we no longer have a simple canonical surface of order p^{(1)} − 1, it is useful choose as a projective model of the class of F the bicanonical surface of the space S_{P −1} (with P = p+p^{(1)}), whose hyperplane sections are the bicanonical curves, or choose as well the tricanonical surface, and so on. We show the advantage we can draw from this consideration, determining in such a way the regular surfaces of the lowest linear genus: p^{(1)} = 2, 3.

Let us assume that the (absolute) linear genus p^{(1)} of F is equal to 2, and that its superficial genus is p_{a} = p_{g} = p. According with the relation giving the maximum value for p with respect to p^{(1)} we have:

p ≤ 2;

indeed, the canonical curves cannot form a net of irreducible curves, whose degree would be p^{(2)} = p^{(1)} − 1 = 1, neither can they be reducible containing as their parts the curves of a pencil, since they should have genus ¡ 2, and thus be elliptic, what implies that p^{(1)} = 1. We shall therefore have to examine three cases:

p = 2, p = 1, p = 0.

And, corresponding to the three values of p above, the bigenus will respectively be

P = p + p^{(1)} = 4, 3, 2,

and the trigenus

P_{3} = p + 3p^{(1)} − 2 = 6, 5, 4.

Surfaces with p^{(1)} = 2 and p = 1 (P = 3, P_{3} = 5) 8.15 295

First case: p^{(1)} = 2, p = 2, P=4. We have here a (linear) pencil of canonical curves K, of genus 2, and a linear system ∞^{3} of bicanonical curves, of genus 3p^{(1)} − 2 = 4, cutting the K along the g_{2}^{1} and thus belonging to a rational involution I_{2}. We shall therefore have, as a projective model of the class of the given surfaces F, a bicanonical surface, of order 4(p^{(1)} − 1) = 4; but this is reduced to a double quadric with a branch curve C_{10} of order 10, intersecting the plane conic sections at 10 points; since the generators of the quadric have to possess an even number of branch points, such a quadric will be a cone, on which one adds, as a branch point, the vertex V to the C_{10}. By the way, this point corresponds to the base point of the pencil |K| on F, which (since it is autoresidual) must be, for each curve K, a double point of the g_{2}^{1}.

Summing up:

The surfaces with general p = 2, p^{(1)} = 2, admit as a model the bicanonical surface given by the double quadric cone with a branch curve of order 10, complete intersection with a surface of the 5th order not passing through the vertex, and to which the vertex itself is added as a branch point. The above cone can also be represented on a double plane with a branch curve of order 10 endowed with two infinitely near 5-tuple points.

Remark One could object to the above analysis that it assumes the canonical curves K to be irreducible. The curves K could a priori be composed with curves k variable in a (linear) pencil and with other fixed curves θ: K = nk + ∑ θ. But, in any case, the curves k could not have genus π = 0 or π = 1, since then it would imply that the surface is either rational (p = 0) or it has p^{(1)} = 1. Hence, the curves k, whose genus cannot exceed p^{(1)} (cf. Chapter IV, Section ??), will have genus equal to 2.

Now, the ∞^{3} system |k′| adjoint to |k| – and which is contained in |2K| – will have degree equal to 4 and will belong to a rational involution I_{2}. It will lead to a model of our surface given by a double quadric cone with a branch curve C_{10} of order 10: this is the general model for the bicanonical surfaces (p = 1, p^{(1)} = 2) defined above, and it turns out a posteriori that there are neither fixed parts of the canonical pencil |K|, nor of the bicanonical system.

8.15 Surfaces with p^{(1)} = 2 and p = 1 (P = 3, P_{3} = 5)

Here there is a canonical curve K of genus p^{(1)} = 2 and ∞^{2} bicanonical curves of genus 3p^{(1)} − 2 = 4, meeting pairwise at 4(p^{(1)} − 1) = 4 points.

If we assume K to be irreducible, then also the bicanonical curves of |2K| are irreducible, and they cannot be hyperelliptic, otherwise the characteristic series of the net, formed by the pairs of a g_{2}^{1}, would not be complete, as required by the regularity of the given surface F. With this in mind, we have an ∞^{4} tricanonical linear system |3K| on F, also formed by irreducible curves, and which is necessarily simple, since it cuts on any bicanonical curve its canonical series, and since p > 0. Thus it leads to a tricanonical surface F_{9} of order 9(p^{(1)} − 1) = 9 of the space S_{4}, with hyperplane sections of genus 6p^{(1)} − 5 = 7.

The direct construction of the F_{9} in the space S_{4} seems to present some difficulty. It is instead easy to characterize a projection F_{6} of it to the ordinary space. Let us observe to this purpose that F_{9} has a triple line, the image of the canonical curve, on which the tricanonical curves cut out a g_{3}^{1}. From a point of this triple line the F_{9} is projected to a surface F_{6} of the 6th order of S_{3}, which has to contain a priori three exceptional lines. The F_{6} has a singular double point O, the trace of the triple line of F_{9}, such that the neighbourhood of O corresponds to the genus two curve K. Now, the plane sections of F_{6} have genus seven, that is, they are sextics with three double points, whereas the plane sections through O have genus four, since the singularity of the double point O corresponds to a fundamental curve of genus two which lowers by 3 the genus of the plane sections through it; therefore these do not have double points off O. We infer that the F_{6} possesses 3 double lines through O (which cannot coalesce in a triple line, which would yield a pencil of elliptic plane sections, thus p^{(1)} = 1); and these lines a, b and c lie^{(29)} in the tangent plane to F_{6} at O (this plane, counted with multiplicity two, forms the osculating cone): the singularity on any plane section of F_{6} through O consists of two branches having a contact of the 5th order (a sixfold contact).

A surface F_{6} with the above singularities has an adjoint quadric given by the plane of the three double lines counted twice; in order that there are three exceptional lines, the above mentioned double lines have to be cuspidal lines.

The F_{6} that we have constructed has geometric genus p_{g} = 1, since it has an adjoint quadric formed by the plane abc counted twice; it has also the numerical genus p_{a} = 1, that is, it is regular; in fact for this surface, whose plane sections have genus 7, there do exist ∞^{7} (here 7 = p_{a} + π − 1) adjoint surfaces of the 3rd order, defined by the conditions of passing through the lines a, b, c, and of having a contact of the 4th order with the branches of F_{6} in O; and, finally, it has absolute linear genus p^{(1)} = 2 because its bigenus equals P = p + p^{(1)} = 3, since there are ∞^{2} bicanonical curves cut out on it by the planes through O, to which one has to add the plane abc counted three times: the calculation of p^{(1)} also follows from the order of F_{6}, with tricanonical plane sections, taking into account the 3 exceptional lines:

(29)Otherwise O would be a triple point.

6 = 9(p^{(1)} − 1) − 3 = 9 − 3^{(30)}.

A F_{6} with the singularities indicated above can be constructed by starting with a cubic surface F_{3} containing three lines a, b, c lying in a plane and passing through a point O. There do exist ∞^{3} cubic surfaces F_{3}’ touching F_{3} at O in such a way to give a sixfold contact on any plane section: these surfaces F_{3}’ touch as a consequence the surface F_{3} along the three lines a, b, c, as it clearly appears from the plane representation of F_{3}, and cut F_{3} further along a plane cubic. They are indeed characterized by this last property. Now, a reducible surface F_{3} + F′_{3} yields a particular sextic possessing a, b, c, as tacnodal lines, and it presents the singularity we indicated above at O (two branches with a sixfold contact on any plane section through O). Let us take a linear combination of F_{3} + F′_{3} with the degenerate sextic given by the plane abc counted three times, and by a cubic cone with vertex O; in this way one obtains a F_{6} with the required singularity in O, and for which the lines a, b, c turn out to be cuspidal lines.

One verifies immediately that the F_{6} thus constructed possesses an ∞^{7} system of adjoint cubic surfaces Φ_{3}, defined by the conditions of passing through the three lines a, b, c, and of touching simply the F_{6} at O, in such a way that the section of Φ_{3} with a plane through O contains five of the six consecutive double points of the section of F_{6}. Whence it follows that the surface F_{6}, whose plane sections have genus 7, is regular, since the above mentioned Φ_{3} cut out, on a plane section of F_{6}, the complete linear system of its adjoint curves.

We summarize the result thus obtained by stating that:The regular surfaces with the genera p = 1 and p^{(1)} = 2 can be transformed into sextics F_{6}, endowed with three cuspidal lines lying in one plane and passing through a singular double point O, such that on any plane section through it there are two branches with a contact of the 5th order.

Remark In the previous analysis we assumed the canonical curve K to be irreducible, hence the net of the bicanonical curves |2K| to have no fixed part. Let us briefly sketch how we can get rid of this restrictive hypothesis, and let us therefore examine whether it is possible, for p = 1, p^{(1)} = 2, to have a bicanonical system

|2K| = |k_{2}| + ∑ θ

having some fixed component θ, and whose variable parts are irreducible curves k_{2} (cf. Chapter IV, Section ??). We denote by π and n the genus and the degree of a curve k_{2}, and respectively by ρ and ν the genus and the degree of some ρ. Then the additive character expressing the virtual number of intersections of the bicanonical curves with the canonical ones is given by

(30)The classification of the surfaces with p^{(1)} = 2, p = 1, 2 is given in the Note by F. Enriques, Le superficie algebriche di genere lineare p^{(1)} = 2. Rendic. Acc. Lincei, February 1897.

2p^{(1)} − 2 = 2 = 2π − 2 − n + ∑(2ρ − 2 − ν).

And, assuming the surface to be free from exceptional curves (thus, for ρ = 0, ν ≤ −2), we have for all the components θ:

2ρ − 2 − ν ≥ 0,

hence

2π − 2 − n ≤ 2.

Now, if some θ does actually exist, in order that K, and therefore also the curves of |2K|, are connected with order of connectivity ≥ 2, the number of intersections of some k_{2} with the ∑ θ (residual with respect to the bicanonical system) has to be

k_{2} · ∑ θ = 4π − 4 − 3n ≥ 2.

Whence it follows that

4π − 4 − 2n = (4π − 4 − 3n) + n ≤ 4, for n ≤ 2,

and therefore n = 2 (since n > 1),

2π − 2 ≤ n + 2 = 4;

that is, the net |k_{2}| consists of curves of genus π ≤ 3, and of degree n = 2. But this conclusion is absurd since (in the most favourable condition: π = 3) it yields as a consequence the possibility of representing the given surface F over a double plane with branch curve C_{8} of order 8, which – since p = 1 – has to possess two elementary singularities, quadruple points or [3, 3] points, whence one has

p^{(1)} = 1.

So the bicanonical system of a F with genus p = 1 and with p^{(1)} = 2 has no fixed parts, and the classification of the above surfaces turns out to be perfect.

8.16 Surfaces with p^{(1)} = 2 and p = 0 (P = 2, P_{3} = 4)

We are led to a tricanonical surface of ordinary space S_{3}: a surface F_{9} of order 9, with plane sections of genus 7, and therefore endowed with a double curve C_{21} of order 21, which does not belong to any surface Φ_{5}, but is nevertheless a double curve for ∞^{4} biadjoint surfaces Φ_{10} of order 10, which cut on it the skew sextics of genus 4 of a pencil, that is, the bicanonical curves.

It is not easy to directly construct F_{9} or to precisely specify its singularities, but we can at least prove its existence, by starting from particular surfaces of the family of which it gives the projective model. To this purpose, we can start from Campedelli’s double plane, having a branch curve C_{10} of order ten with a 4-tuple point A and five [3, 3] points, and for which

p = 0 and p^{(1)} = 2,

In fact there is a pencil of bicanonical curves of genus 3p^{(1)}−2 = 4, represented by the (double) quartics passing doubly through A and simply through the five pairs of infinitely near triple points of C_{10}, which form the singularity of type [3, 3]. The double plane thus defined represents a surface F on which the bicanonical curves are not hyperelliptic and therefore the tricanonical curves form a simple system (see the remark we shall make later on). Using this system F can be transformed into the above-mentioned tricanonical F_{9}, or, to be more precise, to a particular case of it.

Indeed our double planes depend on 4 moduli, since there are 4 projective invariants of the branch curve C_{10}:

4 = 65 − 8 − 5 · 9 − 8;

instead, the number of moduli of the family of surfaces F_{9}, with p = 0 and p^{(1)} = 2, is certainly

M ≥ 9p − 2p^{(1)} + 12 = 8.

Remark Let us add that the tricanonical surface corresponding to a regular surface F with p = 0 and p^{(1)} = 2, is certainly simple, at least if the bicanonical curves of F are irreducible.

Let us sketch briefly how one can justify this assertion.

First of all, it is convenient to establish that that the irreducibility of the bicanonical curves K_{2} implies that the tricanonical curves of the adjoint system |K′_{2}| = |K_{3}| are irreducible and without base points. Since the curves K′_{2} cut on a K_{2} the complete canonical series, if the tricanonical curves are reducible then they have to contain an irreducible fixed part θ, which is fundamental for the pencil |K_{2}|. Then the tricanonical system, once deprived of θ, will be the adjoint to the curve K_{2} −θ, whence (K_{2} −|θ|)′ would have the same dimension as |K′_{2}|, that is, dimension 3: now our aim is to see that this is impossible since K_{2} − θ is a connected curve of genus < 4 (cf. Chapter IV, Section ??). To this purpose we shall realize the K_{2} as canonical sextics in ordinary space, which are projected to the plane over sextics with two triple points, and then we shall study their possible degenerations: indeed in this way one sees that subtracting an irreducible θ from |K_{2}| leads necessarily to a connected curve of genus ≤ 2.

A similar analysis excludes that the irreducible system |K_{3}| has some base point on the given surface F: one starts proving that there is no base point which is simple for the curve K_{2} passing through it, and so on,… After having settled this point, since |K_{3}| is an irreducible, base point free, linear system of degree 9, if it belongs to an involution I_{n} of order n, then n must be a divisor of 9, thus n = 3. If this is the case we are lead to examine a triple rational cubic surface F_{3} with elliptic sections, with branch curve C_{12} of order 12. This surface should be the tricanonical image of F with genus p_{a} = p_{g} = 0 and with p^{(1)} = 2. But it is easy to see that such a triple cubic F_{3}, with genera p = 0 and p^{(1)} = 2, cannot exist. In fact we claim that our F_{3}, which counted three times should be the tricanonical image of a surface F with p = 0 and p^{(1)} = 2, should possess at least a (singular) double point. Indeed, if we assume the F_{3} to be free from double points, then we have two homaloidal nets of twisted cubics |C_{3}| and |L_{3}| on it, complementary with respect to the system of sextics cut out by the quadrics; and we see right away that these must represent two nets of curves, |C| respectively |L|, of degree 3 and genus 4. If it were not so, since the C_{3} + L_{3} cut the branch curve C_{12} of the triple F_{3} at 24 points, then one of the two nets, say |C|, would have genus π < 4; but, if π < 3, then all the plurigenera of F turn out to be zero; if instead π = 3, then the bicanonical curves of F, which-by assumption-form a pencil, would cut on a curve C the pairs of a g_{2}^{1}, and this is incompatible with the fact that the C contain a g_{3}^{1} without fixed points.

With this in mind, since the curves C and L have genus 4, we see that the characteristic g_{3}^{1} is autoresidual (with respect to the canonical series) on them. If in fact it were not so, for instance for C, then one would prove that the genus of the virtual canonical curves K, which have three points in common with the C, should be p^{(1)} = 1 instead of p^{(1)} = 2. Let us indeed assume as a model of F a simple F_{n}, of a sufficiently high order n, on which the C are cut out by the planes through a (n − 3)-tuple point O: the curves of the adjoint system |C′| = |C + K| belonging to F are projected from O to plane curves of the 6th order, and, in particular, the C′ passing through a group of the characteristic g_{3}^{1} (which are ∞^{1}) go to sextics with a triple point: this means that the genus of the C is

7 = 4 + p^{(1)} + 3 − 1,

whence

p^{(1)} = 1.

Therefore we have proved that each of the two nets of curves |C| and |L| on F has an autoresidual characteristic series. As a consequence the bicanonical curves K_{2} (of |2K|) cut C in groups of points of the series double of the mentioned g_{3}^{1}, thus they are partly contained in the double system |2C|, and similarly in |2L|.

Since the complete system |2C| has dimension 6 and the minimum double system |2C| has dimension 5, there is one curve of the pencil |K_{2}| belonging to the minimum double |2C|, and this is a characterized by the property of belonging to the involution I_{3} whose triples correspond to the points of F_{3}, thus it should belong as well to the minimum double system |2L|.

Let us now represent the cubic F_{3} pointwise on a plane α: the images of the plane sections of F_{3} are the cubics through six points A_{1}A_{2} … A_{6}, not belonging to a conic; the curves of the minimum double system |2C| may be assumed to be represented by the conics of α; and those of the minimum double system |2L| by C_{10} of order 10 passing 4 times through the points A_{i}. Hence there should be, in the plane α, a conic passing through some point A_{i}, which simultaneously is part of some C_{10} and meets the C_{10} at 2 variable points: but such a conic does not exist. We have thus seen that the assumption of a F_{3} (a triple image of a tricanonical F with p = 0 and p^{(1)} = 2) with no double points is absurd.

Let us suppose on the other hand that there does exist a F_{3} with a double point O (however singular), whose plane sections are the images of the tricanonical curves K_{3} of F.

The plane sections C_{3} of F_{3} through O (as we said before for the possible twisted cubics belonging to this surface) will be images of curves C of genus 4, and the bicanonical curves K_{2} of F have as their images on F_{3} some simple sextics, meeting the mentioned C_{3} at six variable points, and therefore not passing through O. As a consequence the quadricanonical system, adjoint to |K_{3}|, that is,

|2K_{2}| = |K′_{3}| = |4K|,

will in general be formed by curves not passing through O; and O corresponds to a curve θ on F which is fundamental for the mentioned system |K_{3}|. Whence we deduce that, subtracting the curve θ from such a system, we get the adjoint to |C|:

|C′| = |K′_{3} − θ|, (|K_{3} − θ| = |C|),

thus the genus of the C must equal to the genus of the K_{3} minus one:

7 − 1 = 6;

this consequence is absurd since the genus of the C equals, as we said,

π = 4!

We have thus carried through the discussion: the triple bicanonical F_{3} of genus p = 0 and p^{(1)} = 2 do not exist. Q.E.D.

8.17 Surfaces with linear genus p^{(1)} = 3: first case p = 3

(31) Let us examine the (regular) surfaces with linear genus p^{(1)} = 3. Their superficial genus

p ≤ (p^{(1)} + 3)/2

cannot exceed the value 3 (cf. Section ??). Therefore we will have a priori 4 cases:

p = 3, 2, 1, 0.

First case: p^{(1)} = 3, p = 3. The canonical system |K|, which we assume to be irreducible, is a net of degree p^{(2)} = p^{(1)} − 1 = 2, which leads to a representation of the surface on the double plane with branch curve C_{8} of order 8.

The hypothesis about the reducibility of |K| in a fixed component ∑ θ and in a variable part |k| does not lead to any new case. Let us denote by π and n the characters of the k, and by ρ and ν those of a θ. Calculating the additive character which indicates the number of intersections of the canonical curves with these curves, we have:

2π − 2 − n + ∑(2ρ − 2 − ν) = 2.

Now, the number of the intersections of a (irreducible) k with the rest of the canonical curves, ∑ θ, is

2π − 2 − 2n ≥ 0,

and the degree of the system |k| is n ≥ 2, hence

2π − 2 − n ≥ 2.

On the other hand, if we suppose to have eliminated any exceptional curve from the surface, we have for each θ

2ρ − 2 − ν ≥ 0;

as a consequence

2π − 2 − n = 2, 2ρ − 2 − ν = 0, n = 2 andπ = 3.

Henceforth the k form a net using which the given surface can be transformed into a double plane with a branch curve C_{8} and, a posteriori, the θ turn out not to exist.

(31)The classification of the surfaces with linear genus p^{(1)} = 3, for p = 1, 2, 3, which is here more deeply investigated, and extended to the case p = 0, is given by F. Enriques in the note Sulle superficie algebriche di genere lineare p^{(1)} = 3, Rendic. Acc. Lincei, may 1897. In that note only the simple type for p^{(1)} = 3 and p = 2, of Section ?? is missing.

Surfaces with p^{(1)} = 3 and p = 2 (P = 5) 8.18 303

8.18 Surfaces with p^{(1)} = 3 and p = 2 (P = 5)

We assume that the canonical pencil |K| is irreducible, disregarding the possible exceptional fixed components corresponding to its base points. Then also the ∞^{4} bicanonical curves, cutting on each K the complete canonical series g_{4}^{2}, form an irreducible, base point free, system, of genus 3p^{(1)} − 2 = 7 and degree 4(p^{(1)} − 1) = 8. And this system has to be simple, if the canonical curves of the pencil |K| are not hyperelliptic, as it is convenient to assume in a first moment. We are thus lead to a simple bicanonical F_{8} of S_{4}, with hyperplane sections of genus 3p^{(1)} − 2 = 7, on which the K are plane quartics having two points in common and lying therefore in planes through a line a, which does not lie on F_{8}. The hyperplanes through a cut the F_{8} along pairs of quartics K; whence it follows that the planes of the K form a quadric cone Q of the 2nd kind, having a as its vertex. Let us now project F_{8} into the space S_{3} from a generic point P of Q: we have a surface F′_{8} in S_{3} with two infinitely near quadruple lines o and o′, issuing from a point O. This means that the F_{8} is the complete intersection of the quadric cone Q with a hypersurface of the 4th order. But (as we have already seen) the intersection of a quadric and a quartic in S_{4} is in general a canonical surface with genera p = 5 and p^{(1)} = 9: in order to have a bicanonical F_{8} (p = 2, p^{(1)} = 3) one has to impose to the above intersection some convenient singularities which lower its genus, and precisely – since the genus 9 of the hyperplane sections has drop by 2 – it is convenient to impose to F_{8} a double curve C_{2} of the 2nd order. Let us consider now to the F_{8}’, which is the projection of F_{8} into S_{3}, from a point of the quadric cone which contains it. We see that the C′_{2}, projection of C_{2}, cannot consist of two lines, incident or not, which are not in a plane through O, hence it must be a conic lying in a plane through o. We may conclude that the type of the simple bicanonical surface, for p = 2 and p^{(1)} = 3, is a F of order 8 of the space S_{4} belonging to a quadric cone Q and having as its projection from a point of Q into the ordinary space a F′_{8} with two infinitely near incident quadruple lines o and o′, and with a double conic lying in another plane through o.

The F′_{8} may be constructed by combining linearly: a quartic, counted twice, having two double lines o and o′ and passing through the conic C′_{2}, and a degenerate F′_{8} formed by the plane ω = oo′ counted four times, by the plane γ of the conic C′_{2} counted twice, and moreover by a generic quadric. And one verifies right away that the surface F′_{8} with the resulting singularities is just our nonnormal bicanonical surface. In fact, the adjoint surfaces to it will reduce:

1) by the presence of the two 4-tuple lines o and o′: to tangent cones along o (to which one has to add the plane ω = oo′ counted twice); and

2) by the presence of the double conic C′_{2} (whose plane γ splits from the above cones) to planes through o cutting on F′_{8} the quartics K; hence it follows that p = 2, p^{(1)} = 3. There is moreover a biadjoint Φ_{7}, Φ_{7} = ω^{5}γ^{2}, cutting F′_{8} only along the singular curves, so that the plane sections of F′_{8} turn out to be bicanonical curves.

Every surface with p = 2, p^{(1)} = 3, with (irreducible) nonhyperelliptic canonical curves, can be transformed to the simple bicanonical F_{8} of S_{4}.

A particular example is given by the double plane with branch curve C_{12} of order 12 with eight 4-tuple points A_{1}A_{2} … A_{8}. Here the canonical curves are represented by cubics C_{3} through A_{1}A_{2} … A_{8}, which-taken as double curves, with branch points in the intersections with C_{12}-are, in general, nonhyperelliptic curves of genus three. Indeed there are ∞^{4} bicanonical curves whose images on the plane are curves of order twelve passing four times through A_{1}A_{2} … A_{8}, and being 8-tangent to the branch curve C_{12}; and these curves represent a complete ∞^{4} linear system which is simple. Inside the above complete system there are ∞^{3} bicanonical curves whose images are the sextics C_{6} passing doubly through the 4-tuple points of C_{12}. They form an incomplete linear system belonging to an involution of the 4th order.

In the previous example we see a particular bicanonical F_{8} which-from a certain external point-is projected to a quadruple cone of the second order. Another remarkable example is given by a F_{8} which can be projected from an external point O into a double quartic F_{4} of genus 1 with plane branch curve of the 4th order C_{4}, and which has moreover plane tangent along a conic, on which are to be found six conical branch points.

Indeed if the F_{8} is transformed into itself by an involution I_{2}, induced by a harmonic homology with centre O, its projection from O will be a F_{4}, on which the images of the canonical curves K are the plane sections of genus 3 of a pencil; therefore it must be free from double curve: since moreover the canonical curves of the pencil |K| on F_{8} are pairwise conjugated in the I_{2}, in this pencil there must be two invariant K, one of which projects to the branch curve C_{4} of F_{4}2, the other (a hyperelliptic curve) projects to the conic C section with a tangent plane α, and on this curve are to be found (beyond the two intersections with C_{4}) six branch points, which are the conical points of F_{4} on α. Conversely, assume a quartic F_{4} is given with a tangent plane α along a conic C and let us construct the surface F = F_{42} by extracting the ν αβ on F_{4}, in such a way as to possess a branch quartic C_{4}, β = 0, to which the apparent critical conic C has to be added. It is easy to see that the ∞^{1} sections with the planes passing through the common line of the two planes α and β form double quartics, images of reducible curves as the section with the plane β, and therefore – if they are taken simply – they represent ∞^{1} canonical curves of F, so that for the given F one has p_{g} = 2. One can also verify that p_{a} = 2, either by showing a priori the regularity of the surface, or by calculating p_{a} with the formulas established for the [1, 2] correspondence between F_{4} and F: here we must take into account the fact that the six double points of F_{4} on the conic C are six infinitesimal branch curves (to be added to the quartic section with the plane β) corresponding to exceptional curves or simple points of the surface F.

But another completely different type, of surfaces not included among the simple bicanonical F_{8} (with p = 2, p^{(1)} = 3) is given by the bicanonical surface given by a double Segre surface F_{4}^{(32)} (with two double points), or by the double plane with branch curve C_{10} of order 10, with two 4-tuple points and two [3, 3] points^{(33)}.

Here the canonical curves are represented by ∞^{1} conics in a pencil (each one to be taken as double with 6+2=8 branch points) and the bicanonical curves by ∞^{4} quartics with two double base points and two contacts:

p = 2, p^{(1)} = 3.

One shows easily that every surface F with p = 2 and p^{(1)} = 3, yielding a double bicanonical image in S_{4}, has hyperelliptic canonical curves and can be represented either on the F_{4}^{2} of S_{4} or on the aforementioned double plane.

Indeed if the bicanonical system |2K| of F is not simple, this means that its curves passing through a point A have to pass as a consequence through (at least) another point A′; but this lie off the curve K through A, and if it lies on K (which we assumed to be irreducible) then the canonical series g_{4}^{2} cut out by |2K| on K is composed and K is hyperelliptic.

Now, if the K are hyperelliptic, then the bicanonical transform of F is a double surface F_{4}, and in order that the F_{4} be normal in S_{4}, it has to have elliptic hyperplane sections, that is,it is a Segre surface, the complete intersection of two quadrics in S_{4}. On the F_{4}^{2}, whose double hyperplane sections have genus 3p^{(1)} − 2 = 7, we have a branch curve C_{12} of order 12. Furthermore our F_{4} in S_{4} is not the more general intersection of two quadrics, but has two double points, which lie among the base points of the pencil of canonical conics (they are the images of two double points of the g_{2}^{1} on the K). The one-to-one representation of the F_{4} on the plane leads to a representing system of the hyperplane sections formed by curves of the 4th order with two double points and two simple base points where the curves have a contact: these are the images of the bicanonical curves of a double plane with p = 2 and p^{(1)} = 3, whose branch curve C_{10} has two 4-tuple points and two [3, 3] points.

We have claimed that the double Segre F_{4} form a second type of surfaces with p = 2 and p^{(1)} = 3, not occurring as a particular case inside the family which has as typical element the simple bicanonical surface F_{8}. In order to prove this one can compute the number of moduli: the simple bicanonical F_{8}, as well as the F_{4}^{2} depend on 24 = 9p − 2p^{(1)} + 12 moduli, and therefore they form two distinct families. But there is also a direct argument for this. In fact if we try to reduce by continuity the F′_{8}, projection of the F_{8} in S_{3}, to the F_{4}’, projection of the Segre F_{4}, counted twice, we do not arrive at a quartic having an irreducible double conic (as it usually occurs for our second type), but instead to a quartic with two infinitely near double lines: therefore we actually have two distinct families of surfaces with p = 2 and p^{(1)} = 3, having in common a particular subfamily.

(32)Cf. F. Enriques and F. Conforto, Le superficie razionali, page 144.

(33)Or, if one likes it better, a C_{1}0 formed by a line r and by a C_{9} with three 3-tuple points on r and two infinitely near triple points away from r.

Remark Also in this case we can complete the analysis, by examining the assumption that the canonical system |K| is reducible.

First of all, we claim that the curves k, variable components of the K, are in any case irreducible, otherwise, being composed with the curves of a linear pencil (by the regularity of the surface), one would have at least ∞^{2} K and p > 2.

Then we can assume that the K possess, beyond the k, some fixed and non-exceptional components θ

|K| = ∑ θ + |k|.

If we denote by π and n the characters of |k| and by ρ and ν those of a θ, we have

2π − 2 − n + ∑(2ρ − 2 − ν) = p^{(1)} − 1 = 2,

while π ≥ 2.

Moreover, if we refer to a surface without exceptional curves, we have for each θ

2ρ − 2 − ν ≥ 0.

Since moreover a k has 2π − 2 − 2n intersections with the residual curve ∑ θ, then:

2π − 2 − 2n ≥ 0.

The above inequalities allow only three possibilities for the values of π and n:

1) π = 2, n = 0;

2) π = 2, n = 1;

3) π = 2, n = 2.

Assumption 1) can immediately ruled out. The adjoint system |k′| to |k| would be an ∞^{2} system of curves of degree

0 + 2 · 2 + 2 = 6,

formed by hyperelliptic curves (bisecting the k) and would therefore lead to a double ruled cubic, which is not normal, hence |k′| (which has dimension p + 2 − 1 = 3) should be contained in a complete system of dimension > 3: this is impossible.

Also assumption 2) (π = 2, n = 1) can easily be disposed of.

In fact the adjoint system |k′|, which is a system of hyperelliptic curves of genus 4 and degree 7, with a simple base point, would again lead to a double nonnormal ruled cubic, whence its dimension would be > 3, what is impossible.

Hence only assumption 3) (π = 3, n = 2) remains. But under in this case the system |2k| leads us just to the double surface F_{4} which we found using |K′| = |2K| under the assumption that the K were irreducible. We therefore fall again in the case of the same double plane as before and, a posteriori, we find out that the K are irreducible: |K| = |k|.

8.19 Surfaces with p^{(1)} = 3 and p = 1 (P = 4)

Assuming the canonical curve K to be irreducible, the ∞^{2} bicanonical system |2K| is also irreducible and base point free on K. If it is simple it leads to transform the given surface into a F_{8} of order 8, in the space S_{3}, with plane sections of genus 3p^{(1)} − 2 = 7, and thus endowed with a double curve C_{14} of order 14. This C_{14} belongs to a biadjoint surface Φ_{7}, of order 2. 8-9=7, which together with a plane forms a Φ_{8} cutting on F_{8} a bicanonical curve; actually, the C_{14} is the complete intersection of the F_{8} and Φ_{7} (7 · 8/4 = 14). On the other hand, there must exist an adjoint quartic Φ_{4}, which cuts F_{8}, off C_{14}, along the canonical K of order 4 (4 · 8 − 2 · 14 = 4). With this set-up, it is easy to characterize the curve C_{14}: it is, in general, the double curve of the surface Φ_{7} with elliptic sections, a surface possessing an adjoint quartic and therefore rational.

Therefore the simple bicanonical surface F_{8} with genera p = 1 and p^{(1)} = 3 is defined, in general, by the property of possessing of a double curve C_{14}, of order 14, which is the double curve of a rational surface Φ_{7} of the 7th order with elliptic sections^{(34)}. In order to construct the surface F_{8}, it suffices to combine linearly the adjoint quartic Φ_{4} to Φ_{7}, counting it twice, with the Φ_{7} itself, taken together with a generic plane.

One can evaluate the number of moduli on which F_{8} depends: we find out that the C_{14} (or the Φ_{7}) has 12 projective invariants and is a double curve for ∞^{4} F_{8}, thus there are ∞^{16} projectively distinct F_{8}: the number of moduli of the F_{8} equals:

M = 16 = 10p − 2p^{(1)} + 12.

Now, within the family of the simple bicanonical F_{8} with p = 1 and p^{(1)} = 3 we see to occur, as particular cases, also the multiple bicanonical surfaces, which-at least under the assumption of an irreducible canonical curve-are only double quartics F_{4}^{2}, and give rise to two types.

If on a surface F with p = 1 and p^{(1)} = 3 we have an irreducible canonical curve K, and the bicanonical system |2K|, cutting on K the complete g_{4}^{2}, is not simple, then it necessarily belongs to an involution I_{2} of the 2nd order for which the K is either: 1) a double curve, or: 2) a hyperelliptic curve transformed into itself. Since in neither case can |2K| have base points, we infer that “the only multiple bicanonical surfaces for p = 1 and p^{(1)} = 3 are double quartics F_{4}^{2}”.

Let us study, one after the other, the two cases to which respectively lead the above-mentioned assumptions 1) and 2) concerning the property of |2K| of belonging to an involution I_{2}.

First of all: if we assume hypothesis 1), then the bicanonical transform of the given F will be a double quartic surface F_{4} with a plane quartic branch curve K. And in order that that K has the genus p^{(1)} = 3, and also that the F_{4}^{2} has genus p = 1, one needs that the plane sections of F_{4} have no double points, and that the F_{4} itself has genus 1. One can construct a double F_{4} of genus 1, representing a surface F with p = 1 and p^{(1)} = 3, by considering the pencil formed by the bicanonical F_{8} (with the above genera) and by its adjoint F_{4} counted twice; in this pencil

λF_{8} + F_{4}^{2} = 0

we see the generic surface (that is, a F_{8}) to tend by continuity (for λ = 0) to the double F_{4}. The F_{4} has as double points the triple points of F_{8}, and this add up as branch points (infinitesimal branch curves) to the proper branch curve which is the quartic K, simple intersection curve with F_{8}; the remaining part which is the C_{14}, the contact curve of F_{4} and of a Φ_{7}, is an apparent critical curve.

Now, the number of triple points of the C_{14}, double curve of F_{8} (and also of Φ_{7}), can be counted by a well-known formula, yielding

t = 10.

This number can also be derived otherwise from the correspondence formula relating the numerical genera of the two surfaces of genus 1, F and F_{4}, in a [1,2] correspondence.

From the previous discussion it turns out to exist, as a particular case of the bicanonical F_{8} with p = 1 and p^{(1)} = 3, a double F_{4} of genus 1 with a plane quartic section and 10 conical points as branch locus. One could nevertheless think that the double F_{4} constructed above represent only a subfamily of a more general type of double bicanonical surfaces with p = 1 and p^{(1)} = 3, not contained inside the family of the F_{8}. In order to remove this doubt, we have to go more deeply into the study of our F_{4}^{2}. They can be defined by extracting on F_{4} the square root of a polynomial of the 8th order, which can be reduced to αΦ_{7}, where α = 0 is the plane of the branch curve K, and Φ_{7} the surface of the 7th order having the C_{14} as double curve (an apparent critical curve). One can replace this polynomial αΦ_{7} with another polynomial αF_{7}, by letting the surface vary freely inside the systems of the F_{7} of order seven passing with multiplicity three through the ten triple points of F_{4} and touching it along a curve of the complete system |C_{14}|. Actually, we can enlarge this system by adding to it twice the neighbourhoods of the ten conical points, taking thus instead of the old surface F_{7} a new F_{7} passing simply through those ten points and touching F_{4} along the apparent critical curve of the 14th order (which we still denote by C_{14}). And we are also allowed to add or subtract to such a F_{7} a surface of even order 2n, touching F_{4} along a curve C_{4n}: thus substituting to the C_{14} an apparent critical curve C_{14} ± C_{4n}. More precisely, we want to show that we can subtract from the system |C_{14}|, cut out on F_{4} by the tangent F_{7} passing through its ten conical points, two plane quartic sections (each counted twice), so to define the F_{4}^{2} by the extraction of a ναF_{3}, where F_{3} = 0 represents a cubic surface tangent to F_{4} along a sextic C_{6} of genus three; the last curve can then be assumed as an apparent critical curve of the F_{4}^{2}.

We observe to this purpose that the linear system of the C_{28} cut out on the F_{4} by the F_{7} through its ten conical points has degree

4 · 7^{2} − 2.10,

hence the half system of the C_{14} cut by the tangent F_{7} has degree

7^{2} − 5 = 44,

hence its genus and dimension are (44+2)/2=23. Now, the system |C_{14}| cuts on a plane quartic C_{4} a series g_{14}^{11}, therefore by subtracting the C_{4} from |C_{14}| we obtain a system |C_{10}| of dimension and genus 23-12=11. And since this |C_{10}| cuts on another plane quartic section a g_{10}^{7}, by subtracting also this curve we shall have an ∞^{3} linear system of sextics C_{6} of genus three, which are contact curves of cubics passing through the ten conical points of F_{4}.

We conclude: the bicanonical surface with p = 1 and p^{(1)} = 3, represented on a double surface F_{4} of the 4th order and of genus one, is related to a F_{4} which is touched along a sextic C_{6} of genus three by a surface F_{3} of the 3rd order and thus it has ten conical points on this C_{6}; the F_{4}^{2} is obtained by extracting on the F_{4} the ν αF_{3}, where α denotes the plane of the branch quartic.

It remains to be verified that the above F_{4}^{2} occur as a particular case inside the family of the simple bicanonical F_{8} s (p=1 and p^{(1)} = 3).

It suffices for this to compute the number of invariants or moduli on which our F_{4}^{2} depend. As a first observation, we note that “the existence of a genus three sextic C_{6}, which is autoresidual with respect to the system cut out by the cubic surfaces, yields ten conditions for the F_{4}, and also the existence of ten double (and in general double conical) points on the C_{6}”. In fact, the existence of a genus three sextic C_{6} on the F_{4} yields just one condition for F_{4}; once this condition is satisfied, we shall have ∞^{6} cubics F_{3} cutting on F_{4} two sextics C_{6}, of the same genus three and residual to each other; and two such sextics – forming together a curve C_{12} of genus nineteen – have fourteen points in common; but in order that the residual C_{6} are equivalent it is needed that among the fourteen above points there are four variable points and ten base points, which necessarily have to occur at ten double points of F_{4}; now, the existence of a base point, a fixed point for the series g_{14}^{11} cut out by the C_{6} of a system on the residual C_{6}, yields one condition, but the existence of ten base points whose residual series is the canonical g_{4}^{2} yields only nine conditions. Therefore all together the conditions which the F_{4} has to satisfy in order to possess an autoresidual C_{6} of genus three are in the number of ten, and yield as a consequence ten double points of F_{4} on C_{6}.

With this set-up, the F_{4} possessing an autoresidual C_{6} depend upon 34 − 10 = 24 parameters; we have to add to these the three parameters on which depends the choice of a plane section K, and subtract the fifteen parameters of a space homography: it turns out that our F_{4}^{2} depend on

M = 24 + 3 − 15 = 12

projective invariants or moduli. And since

M < 9p − 2p^{(1)} + 12 = 15,

we deduce that the family of the above F_{4}^{2} is contained in a larger family of simple bicanonical F_{8}, and precisely in the family that we have defined above.

There is another type of bicanonical F_{4}^{2}, corresponding to a surface F with the genera p = 1 and p^{(1)} = 3, and with (irreducible) hyperelliptic canonical curve. This is obtained by taking a double quartic F_{4} with genus two sections (that is, with a double line) and with a branch curve C_{12} of order twelve; and also this F_{4}^{2} occurs as a particular case inside the family of the simple bicanonical F_{8}.

By representing the F_{4} on a plane, we obtain a double plane with branch curve C_{10} of order ten endowed with a 4-tuple O and four [3, 3] points: A_{1}A_{2}A_{3}A_{4}. Indeed on this double plane there is a canonical curve K of genus p^{(1)} = 3, whose image is the conic through OA_{1}A_{2}A_{3}A_{4}, and there is a system of bicanonical curves, represented doubly by ∞^{3} quartics C_{4} with O as a double point and having a contact at A_{1}A_{2}A_{3}A_{4}. These curves are the representative system of the plane sections of the F_{4}.

We see that this double plane occurs as a particular case inside the family of the simple bicanonical F_{8} (with p = 1 and p^{(1)} = 3) because its branch curve C_{10} depends upon

65 − 8 − 36 − 8 = 13

projective invariants or moduli, and one has

13 < 9p − 2p^{(1)} + 12 = 15.

We can summarize the results of the previous discussion by stating that: all the surfaces with p = 1 and p^{(1)} = 3 occur, at least if the canonical curve is irreducible,within the more general family of the simple bicanonical F_{8} of ordinary space. It looks likely that the possible reducibility of the canonical curve does not give rise to any new case. But we will not dwell on a subtler analysis of the matter.

8.20 Surfaces with p = 0 and p^{(1)} = 3

Here the bicanonical curves, forming a net of degree 4(p^{(1)} − 1) = 8, lead in general to a multiple plane of order 8. Therefore it is more convenient to look for a model of the given surfaces by considering the tricanonical surface, which is a F_{18} of order 18 in the space of dimension 3p^{(1)} − 2 = 7.

The high values of the numerical characters of this surface make it difficult to construct it directly inside S_{7}, or to define precisely a projection F′_{18} inside S_{3}: whose plane sections have genus 6p^{(1)} − 2 = 16, and which therefore possesses, in general, a double curve C_{120} of order 120. But we can at least show the existence of the F_{18}, since we know a particular case of the family, that is, a Campedelli double plane, defined by a branch curve C_{10} of order ten with six [3, 3] points not lying on a conic.

It is easy to see that the above double plane leads to a simple tricanonical surface F_{18}. But this is not the more general F_{18}, since the above-mentioned branch curve C_{10} depends on three projective invariants or moduli, and we have

3 < 9p − 2p^{(1)} + 12 = 6.

Simple and multiple pluricanonical surfaces 8.21 312

Let us add that, at least if the net of the bicanonical curves of F (a regular surface) is irreducible,the corresponding tricanonical surface (p = 0 and p^{(1)} = 3)turns out certainly to be simple. In fact, a multiple tricanonical surface (of order 18/2=9) could only arise under the hypothesis that the bicanonical curves K_{2} of F are hyperelliptic; since then the (complete) canonical series cut out by the tricanonical curves on a K_{2} would be composed. In this case the characteristic series of the bicanonical net on a generic curve K_{2} would be a g_{8}^{2} composed with the g_{2}^{1} belonging to the K_{2}, and therefore it would be contained in a complete g_{8}^{4}; but (if the surface has to be regular) this yields P=6 instead of P=3.

8.21 Simple and multiple pluricanonical surfaces

In the previous sections we have met several types of surfaces which give rise to double bicanonical surfaces, in particular the following general types:

(I) For p = 3 and p^{(1)} = 3, the double plane with branch curve C_{8} of order 8;

(II) For p = 2 and p^{(1)} = 2, the double plane with branch curve C_{10} of order 10, endowed with two infinitely near quintuple points;

(III) For p = 2 and p^{(1)} = 3, the double plane with branch curve C_{12} of order 12, endowed with two quadruple points and two [3, 3] points.

For Types I and II, not only the bicanonical, but also the tricanonical model turns out to be a double surface, since the bicanonical curves (of which the tricanonical curves are the adjoint) are hyperelliptic curves.

We intend now to show that these types I) and II) are the only types of surfaces, with p > 0 and p^{(1)} > 1, whose tricanonical models are multiple surfaces.

We notice that, since the bicanonical curves or their variable parts are irreducible (cf. Chapter IV, Section ??), then the adjoints and thus the tricanonical curves, or their variable parts, certainly form a simple system if the bicanonical curves are not hyperelliptic (thus belonging to a rational involution I_{2}); if therefore a regular tricanonical surface (for p > 0 and p^{(1)} > 1) is not simple, it must be representable on a double plane in such a way that the (hyperelliptic) bicanonical curves have as their images the rational curves of a linear system at least ∞^{2}, to which possibly some fixed component has to be added.

It is convenient to examine separately the three cases in which the images of the given bicanonical curves reduce to:

1) the ∞^{2} lines of the plane;

2) the ∞^{5} conics of the plane;

3) finally, the curves C_{n} of a certain order n (> 1) forming a linear system of dimension 2n−s with a (n−1)-tuple base point O and without further base points (s = 0), or with another simple base point O_{1} (s = 1), or with s (> 1) simple base points O_{1}O_{2} … O_{s}, infinitely near to O (in distinct directions).

Hypothesis 1) – yielding P = p + p^{(1)} = 3, p = 1, and p^{(1)} = 2 – is impossible as it appears form the examination of the surfaces with such characters, carried out in Section ??.

Hypothesis 2) leads to the double plane with a branch curve C_{8}, that is, to type I) with p = 3, p^{(1)} = 3, which we encountered in Section ??.

Hypothesis 3) occurs for n = 2, when we assume as branch curve of the double plane a C_{8} of order eight with two infinitely near triple points O and O_{1}: the images of the bicanonical curves are the conics through O and O_{1}.

But the double plane thus defined occurs inside the more general type where we assume as branch curve a C_{10} of order ten with two infinitely near 5-tuple points, which is type II) for p = 2 and p^{(1)} = 2: if moreover the C_{10} has a double point, we can reduce it with a quadratic transform to the case of a C_{8} with a [3, 3] point.

In order to discuss in general the possibilities to which hypothesis 3) gives rise, let us suppose that the bicanonical curves of our double plane contain a fixed part of order d, to be added to the C_{n}; in order that the addition of this part does not enlarge the system |C_{n}|, it has to consist of fundamental lines for C_{n}, counted respectively h_{1}h_{2} … h_{s} times, that is, of lines OO_{i}, thus the fixed part should pass with multiplicity d through O and with multiplicity h_{i} through O_{i}. So the branch curve of the double plane will be a C_{2m} of order 2m which has the curves C_{n} + ∑ h_{i}OO_{i}, of order n + d = n + ∑ h_{i}, as second adjoints; and we will have

2m = n + d + 6.

Besides, it will suffice to think that the C_{2m} must have multiplicity n−1+ d + 2 = n + d + 1 = 2m − 5 at O. Now, on a double plane with branch curve C_{2m} endowed with a (2m − 5)-tuple point O, the images of the bicanonical curves are in general curves of order 2m − 6 = n + d, passing through O with multiplicity

2m − 8 = n + d − 2;

in order that this multiplicity grows by 1, becoming n+d−1 (as it happens for the C_{n} + ∑ h_{i} · OO_{i}), it is necessary to have near to O another (2m − 5)-tuple point of C_{2m}; but one has

2m − 5 + 2m − 5 ≤ 2m,

whence

m = 5, 2m = 10

or (in particular)

m = 4, 2m = 8;

thus we find back type II).

We end stating the theorem:

The tricanonical image of a regular surface with genera p > 0 and p^{(1)} > 1 is always a simple surface, of order ≤ 9(p^{(1)} − 1), except for the two types I) and II) of surfaces which can be referred to a double plane, corresponding to the values p = 3 and p^{(1)} = 3 and p = 2 and p^{(1)} = 2.

For these two types I) and II) both the bicanonical and the tricanonical surface turn out to be double: in case I) the bicanonical surface is a double Veronese surface (F_{4} of S_{5}) and in case II) we have a double quadric cone of S_{3}; the tricanonical surface is, in the former case, a double surface F_{9} of order 9 with elliptic sections in S_{9},in the latter it is a double surface F_{6} of order 6 with elliptic sections in S_{6}.

Since the tricanonical curves of our surfaces are not hyperelliptic in any case, it follows (for p > 0) that their adjoints do not belong to an involution, that is: on a regular surface with p > 0 and p^{(1)} > 1, the quadricanonical curves (or their variable components), hence also the i-canonical curves for i > 4, always form a simple system.

It is likely that this statement, about the simplicity of the pluricanonical surfaces (p^{(1)} > 1) for i > 3, can be extended to the case p = 0; but one needs for it a more subtle analysis, upon which we shall not dwell (cf. the discussion of Section ?? for the case p^{(1)} = 2).

Irregular surfaces and continuous systems of inequivalent curves

9.1 Introduction

We recall some remarks made in Chapter IV, Section ??.

To start, with we point out that, when speaking of curves and of linear or continuous systems of curves C on a surface F, we usually mean that we are dealing with irreducible curves without multiple points^{(1)}.

We have seen that on a regular surface a complete linear system |C| of curves has complete characteristic series. And a surface containing a continuous system of curves C, formed by ∞^{d} distinct linear systems, that is, a system {C} containing ∞^{d} inequivalent curves, with d > 0, is irregular, with irregularity

p_{g} − p_{a} ≥ d.

This statement can be immediately justified for the ∞^{d+r} continuous systems {C} of irreducible curves, formed by ∞^{d} linear systems having a certain (generic) dimension r: in fact the characteristic series cut out on a generic C by the infinitely near curves of {C} has dimension

r + d − 1,

thus the characteristic series of the linear system |C| turns out to have deficiency (at least) equal to d; it therefore suffices to invoke the theorem of Castelnuovo (Chapter IV, Section ??) which says that

d ≤ p_{g} − p_{a}.

(1)If one wants to consider also curves C with multiple points, these points should be assumed to be assigned base points for the linear or continuous systems one is referring to.

Introduction 9.1 316

Let us add that this inequality holds also for the continuous systems {C} which are formed by reducible curves; in fact first of all one may leave out of consideration the fixed components which can possibly be parts of the C; next, if the variable C are reducible, then by subtracting them from a large enough linear system |L| one may obtain a continuous system {L − C} which – away from fixed components – consists of irreducible curves and contains ∞^{d} inequivalent curves – just as {C} does.

The main purpose of this Chapter is to treat the arguments leading to the converse of the previous theorem, in the sense that: on an irregular surface, of genera p_{a} and p_{g} > p_{a}, thus of irregularity q = p_{g} − p_{a}, there always exist continuous systems of curves consisting of ∞^{q} distinct linear systems.

Thus the presence of continuous systems of inequivalent curves is the characteristic property of irregular surfaces. And this statement, or rather the theorem which makes this assertion more precise by involving the value of p_{g} − p_{a}, may be referred to as the fundamental theorem in the theory of irregular surfaces.

But before approaching the proof of the fundamental theorem, and in consideration of the difficulties of rather subtle kind which appear in it, it is convenient to consider the most general assumptions; therefore we shall examine what can be said about the surfaces containing ∞^{q} continuous systems of inequivalent curves, where we only know that, as we have seen before,

0 ≤ q ≤ p_{g} − p_{a}.

We will thus distinguish between two notions of irregularity, which a priori might not always coincide: the numerical irregularity

p_{g} − p_{a}

and the geometric irregularity q, which is the largest possible dimension of a continuous series of inequivalent curves lying on the surface:

q ≤ p_{g} − p_{a}.

In any case the number q denotes an invariant character of the surface itself: all the known examples of irregular surfaces (surfaces with an irrational pencil of curves, surfaces of the pairs of points of a curve, etc.) give q > 0, or rather q = p_{g} − p_{a}.

The arithmetic condition for an ∞^{q} continuous series 9.2 317

9.2 The arithmetic condition for a curve on a surface with geometric irregularity q to belong to an ∞^{q} continuous series of inequivalent curves

Let q > 0 be the geometric irregularity of a surface F: this means that F contains an ∞^{q}, and not larger, continuous series {D} of inequivalent curves.

We ask the following question: can one give an arithmetic criterion in order that a curve C on F belong to an ∞^{q} series of inequivalent curves as above, that is, to a series formed by ∞^{q} distinct complete linear systems?

The answer is given by computing the virtual dimension of the system determined by the curve C (of genus π, degree n, and index of speciality i):

r = p_{a} + n − π + 1 − i.

Indeed, let |C| be a linear system of curves, either irreducible or reducible, of degree n, of genus π, and of index of speciality i, which is arithmetically effective, that is, such that its virtual dimension is

p_{a} + n − π + 1 − i ≥ 0.

We associate with |C| another regular (irreducible, etc.) large enough linear system |D| so that |C + D| is also regular. We may assume that the system |D| is contained in a continuous system {D} formed by ∞^{q} inequivalent linear systems, one of which – to contrast with |D| – we will denote by |\overline{D}|.

Now the system |C| can be constructed by subtracting a curve d from |C + D|, and its dimension turns out to be ≥ p_{a} + n − π + 1 − i, since |C + D| cuts out on the curve d a (complete or not) series which has precisely the index of speciality i (see Chapter IV, Section ??).

Let us consider the series cut out by the same system |C +D| on a \overline{D}. It is not possible that, for a generic choice of \overline{D} in \{D\}, the aforementioned series has an index of speciality greater than i, since – by varying \overline{D} continuously – one may reduce it to d and, in this limiting process, the index of speciality of the series cannot decrease. Therefore the dimension of the linear system |\overline{C}| = |C + D − \overline{D}| turns out to be (according to the well known computation which gives the theorem of Riemann–Roch) ≥ p_{a} + n − π + 1 − i: so, as \overline{D} varies, one obtains ∞^{q} distinct complete linear systems, which form an irreducible series {C}.

Let us state the theorem that we have just proved:

On a surface of geometric irregularity q > 0, a linear system of degree n, genus π and index of speciality i, which is arithmetically effective, that is, such that

p_{a} + n − π + 1 − i ≥ 0,

The Picard variety corresponding to an irregular surface 9.3 318

always belongs to an ∞^{q} series of inequivalent linear systems.

With this theorem we have a very simple understanding of the notion of geometric irregularity (thus of the difference p_{g}−p_{a}, if we take into account the fundamental theorem which gives the characteristic property of the irregular surfaces).

9.3 The Picard variety corresponding to an irregular surface

In the previous section we have seen how, on a surface of geometric irregularity q (> 0), the ∞^{q} linear systems forming a certain continuous system {|C|} or {C} are related to those of another analogous continuous system {D}: there is a one-to-one correspondence between the elements (linear systems) of {C} and those of {D}, in which two homologous linear systems are residual one of the other with respect to a |C + D|. This remark (due to Castelnuovo) can be made more precise by saying that: The elements (linear systems) of the ∞^{q} continuous series belonging to a surface of geometric irregularity q (¿0), can be regarded as the points of one and the same Abelian variety V, which Castelnuovo wanted to dedicate to the French geometer, naming it the Picard variety corresponding to the surface.

Indeed the operation +D − \overline{D} by means of which one passes from a linear system |C| to another linear system |\overline{C}| contained in \{C\}, can be interpreted as a pointwise transformation of the variety V into itself; it is obvious that all the analogous transformations are permutable and form an ∞^{q} simply transitive group without exceptions of the aforementioned variety V = V_{q} (of dimension q): precisely for this reason V is characterized as a nondegenerate Abelian variety.

The meaning of the theorem we have obtained is explained by the following considerations.

Let us consider the question of determining all the curves of a given order n belonging to an algebraic surface F.

In order to belong to F, these curves have to satisfy some algebraic conditions, thus they fall a priori into a certain number of families, hence forming, in general, some continuous systems.

These systems are linear systems when F is regular. If, by contrast, the geometric irregularity of F is q (= p_{g} > p_{a}) then these systems – which may be seen as varieties whose elements are distinct linear systems – are not arbitrary algebraic varieties, but they are Abelian varieties V_{q} or, at least, their multiples form Abelian varieties.

In order to understand better the implications of this fact we will just recall a few facts about the Abelian varieties (see Section ??).

The characteristic property of the irregular surfaces 9.4 319

9.4 The characteristic property of irregular algebraic surfaces: the fundamental theorem for p_{g} = 0

The proof of the fundamental theorem of the irregular surfaces, which we have stated in Section ??, can be given in a very simple way for surfaces with p_{g} = 0, whereas it requires subtler considerations for p_{g} > 0. It is therefore convenient to distinguish between these two cases.

Let us first of all assume p_{g} = 0, thus p_{a} = −p < 0.

Let us consider, on our surface F, a complete linear system |C|, which is regular, nonspecial, of genus π and degree n, and therefore of dimension

r = −p + n − π + 1.

Let us associate to it another linear system |D|, also regular, of genus ρ and degree ν, and therefore of dimension

r′ = −p + ν − ρ + 1,

and such that the sum |C + D| is also a regular system (see Chapter IV, Section ??).

Denoting by m the number of intersections of C and D, we write down the genus and the degree of |C + D|=—E—:

Π = π + ρ + m − 1 N = n + ν + 2m,

and then its dimension:

R = −p + N − Π + 1 = −p + (n − π) + (ν − ρ) + m + 2,

that is,

R = r + r′ + p + m.

The fundamental theorem for p_{g} > 0 9.5 320

Let us recall that, in view of the principle of degeneration or of reducibility^{(2)}, an irreducible curve, with a certain number m of double points, cannot be continuously deformed to a reducible curve whose components, if they have no multiple points, have less than m + 1 common points (that is, the limit curve must have a further double point, by means of which the two components of the reducible curve are connected). From this principle one deduces that, by imposing that the curves E possess m double points, one cannot obtain a continuous system \overline{E} of dimension r + r′ + p > r + r′ formed by irreducible curves, strictly containing the system of the curves of |E| = |C + D| which are reducible in a curve of |C| and in a curve of |D|; on the contrary one defines in this way a system such that one at least of its parts (of the same dimension) contains a continuous system of curves which are reducible into a curve of the same order as C and a curve of the same order as D; therefore these two components describe two continuous nonlinear complementary systems, say {C} and {D}.

(2)See Enriques–Chisini, Libro V, Chapter III, 35 (Vol. III, page 405).

To any curve C of the larger continuous system {C} there corresponds, as residual with respect to |E|, a linear system of curves |D|, and conversely; so it is easy to compute the dimension of the continuous system {C}, which has to be (at least) r + p, and similarly the dimension of {D}, which has to be r′ + p. Furthermore we can say that {C} consists of (at least) ∞^{p} inequivalent linear systems, of dimension equal to r, since the dimension of the complete linear systems forming {C}, which a priori is ≥ r, cannot be > r; in fact these linear systems can be continuously deformed to the system |C| we started with, which has exactly dimension r.

But if the geometric genus of F is p_{g} = 0, we know that F cannot contain a continuous system of curves formed by ∞^{d} distinct linear systems, where d > p_{g} − p_{a} = p; thus we conclude that, on the irregular surfaces with geometric genus p_{g} = 0, there are continuous systems of ∞^{p} inequivalent curves, and no system formed by a larger series of such curves.

We summarize the results obtained by stating the Restricted Fundamental Theorem:

A surface with negative arithmetic genus p_{a} = −p certainly contains continuous systems formed by ∞^{p} inequivalent curves, and, if the geometric genus is p_{g} = 0, it does not possess any larger system of inequivalent curves;

in other words, the geometric irregularity is

q ≥ −p

and for p_{g} = 0 it is equal to the numerical irregularity

p_{g} − p_{a} = p.

9.5 The fundamental theorem for p_{g} > 0

Now we try to extend the fundamental theorem to the surfaces of geometric genus p_{g} > 0 and of any arithmetic genus p_{a} < p_{g}. In order to do this, we take into consideration again, on the surface F (which we suppose to be of genera p_{a} and p_{g}) two regular, nonspecial, linear systems |C| and |D| with characters n, π and ν, ρ such that the sum |E| = |C + D| is also regular and cuts out on a d a complete and nonspecial series g_{m} + ν.

If we denote by N and Π the characters of |E|, then we see (as before in the case p_{g} = 0) that the linear system |E| has the dimension

R = p_{a} + N − Π + 1 = p_{a} + (n − π) + (ν − ρ) + m + 2 where (m = CD),

that is,

R = r + r′ + m − p_{a},

where r and r′ are the dimensions of |C| and |D|. Now, if the linear system |C| has to be contained in a continuous system {C} of dimension r+q = r+p_{g}−p_{a} (formed by ∞^{q} distinct linear systems), then it is necessary that imposing to the E to have m double points on F, so that these curves break up into C of {C} and d of the residual system {D}, yields only m − p_{g} independent conditions, and not m conditions (or m − p_{a}, as the existence of |C| and |D| seems to imply). If this happens, then the continuous system {C}, of dimension p_{g} + n − π + 1, has, on a given C = C_{0}, a complete characteristic series, and therefore it cannot be contained in a larger continuous system.

In order to show that

imposing to the E to have m double points on F, so that these curves break up into curves of {C} and {D}, yields at most m−p_{g} conditions,

and therefore that the geometric irregularity q of F and the numerical irregularity (q = p_{g} −p_{a}) are equal, we are led to examine the characteristic series of the continuous system {C}; it is apparent that we will accomplish the proof if we see that such a series is complete.

In order to prove that, it suffices to consider the curves E of |E| = |C + D| which pass through the group G of the m points common of the two components C_{0} and D_{0} of a reducible E, E_{0} = C_{0} + D_{0}; these curves E cut out on C_{0} a complete characteristic series, and one of them, together with E_{0}, determines a linear pencil of curves which have a contact at the points of G: whence it follows that also the curve of the pencil which is infinitely near to E_{0} has m double points near to those of G^{(3)}. Therefore, if the degeneration principle can be applied also to these infinitely near curves, then this curve has to be reducible into two curves C_{1} and D_{1}, the first of which – near to C_{0} – cuts out on C a group of the characteristic series (that can be arbitrarily given).

Have we thus obtained the proof (for any p_{g}) of the fundamental theorem?

(3)See e.g., Enriques–Chisini, Lezioni, Libro II, §5, Vol. I, page 183 and Libro III, Chapter II, §20, Vol. II, page 167.

No, we must not be misled by the appearance of the proof: the theorem we deal with concerns an integral property, and we have only established a differential property that is a consequence of it, and which does not necessarily imply it. Indeed, we have at most proved that

the characteristic series of C_{0} is completely cut out by curves C_{1}, which are infinitely near to it,

but we have not proved that the C_{1} actually belong, as curves which tend to a limit, to a continuous system of proper curves {C}.

Here there is a critical point of the proof, whose historical significance we will point out later (Section ??): the condition that imposing m − p_{g} double points on the E yields as a consequence p_{g} more double points has not been verified “globally” but only “locally”, for the curves infinitely near to E_{0}.

The “local” differential property follows, as we have said, from the integral, or “global”, property that we want to establish, but the latter is not in turn a consequence of the former.

In order to explain the matter better, let us consider a variety V_{r} of dimension r, which is defined in a space S_{R} (with R > r) as the intersection of a certain number of hypersurfaces (or of some differentially distinct linear branches of one and the same algebraic hypersurface): in order to compute the dimension r, we are led to consider the number s of linearly independent points in the first order neighbourhood of a generic point P of V_{r}; but it is clear that the varieties as the intersection of which V_{r} is defined may have a certain contact at each of its points; in that case the number of the points infinitely near to P, which we are discussing, will be s > r.

For instance, if r = 1 then V_{1} may be the contact curve of two or more surfaces of the space S_{3}, a curve to be counted twice; then the number of its independent points which are infinitely near to a point P, is no more r = 1, but s = 2.

This explanation also clarifies which is the gap that has to be filled in our proof of the general fundamental theorem for p_{g} > 0, in passing from the local property we have just proved, to the global one we have mentioned above.

Let us consider the V_{r} of S_{R}, defined as the intersection of m branches of hypersurfaces, which correspond to the curves of |E| = |C + D| having m double points (and forming the larger continuous system consisting of the curves E which are reducible into a curve C of the same order as C_{0} and in a d of the same order as D_{0}): we need that these m branches do not touch each other all along V_{r} or at one of its generic points P. This means that we cannot have a linear branch of a curve with centre at P having a contact of finite order s (≥ 1) with the aforementioned branches but not with V_{r}; in other words: if such a linear branch has h < s + 1 (> 0) proximate points in common with V_{r}, then it must also have the proximate (h + 1)-tuple point in common with this variety.

Returning now from the V_{r} to our surface F, we need to show that for any infinitely near curve C_{1} in the first order neighbourhood of C0, there is a sequence C_{1}, C_{2}, … of infinitely near curves in the successive neighbourhoods of order 1, 2, 3,…; and this series has to be regular, (that is, it has to be similar to a sequence of points on a linear branch). So C_{1} is a part of a reducible curve E_{1} = C_{1} + D_{1} near to C_{0} + D_{0}; C_{2} will be a part of a successive curve C_{2} + D_{2}, and so on.

In order to do this and thus accomplish the general proof of the fundamental theorem, an idea arises in a very natural way^{(4)}: since two infinitely near inequivalent curves, C_{0} and C_{1}, are given on F, the operation C_{1} − C_{0} can be successively applied to a curve, for instance to C_{1}, defining in this way the sequence of successive curves C_{2}, C3, …, or rather the sequence of the linear systems which they determine. Here a new idea comes into the picture, namely one assumes that “one may operate on the surface F by summing and subtracting curves, even when dealing with the infinitely near ones”. We want to justify this idea.

We start from a somewhat analogous question which concerns the curves. If, on a curve L of genus p > 0, we consider two inequivalent groups of p points, G and G′, then the operation +G′ − G can be successively applied starting with G and it leads to a series, that is (in general) infinite, of groups

G, G′, G′′, G′′′, …,

which correspond to the homologous of G in the powers of a transformation T of the first kind of the Jacobian variety V associated to L. Therefore the sequence of the successive groups G′′, G′′′, …, is still defined^{(5)} when G′ becomes infinitely near to G: in this case the transformation T becomes an infinitesimal transformation that, according to S. Lie, generates the continuous group of V, and defines an analytic series t, which is an orbit of the Jacobian group of L.

In order to pass from this case to the consideration of the infinitely near curves on a surface, it is first of all convenient to justify the degeneration principle for a curve infinitely near to C_{0} + D_{0}, having (as in the above mentioned construction) m = C_{0}D_{0} double points near to the common points of C_{0} and D_{0}; thus we need to show that such a curve is a C_{1} + D1, formed by two parts which are respectively near to C_{0} and D_{0}.

(4)The part which follows was not completed by the Author and therefore the arguments which are developed here contain many gaps; we thought it was better to present it anyway, because it contains ideas which perhaps, if suitably completed, may lead to a satisfactory conclusion of the theory. (Editors’ note).

(5)Precisely in the sense of the differential conditions that define the infinitely near points.

See Enriques–Chisini, Lezioni, Libro IV, Vol. II.

In order to do this, we may assume that C_{0} and D_{0} are normal, nonspecial curves belonging to spaces of very high dimension compared with their genus: we may indeed always reduce to this case by means of a transformation of the given surface F. Then it is easy to verify that, in the space of F, the irreducible curves E of order N and genus π, describing the family defined by the reducible C + D, form a variety of a certain dimension R; inside this, one has a variety of dimension R − m consisting of the (reducible) curves E = \overline{E} with m double points, containing the family of the curves C + D, so that the independent reducibility conditions for the E (not on the surface but) in the space of F are just m, and not fewer.

As for the proof, let us project the reducible curve C_{0} +D_{0} with m double points (the intersections of C_{0} and D_{0}) from a suitable linear space on a multiple (reducible) N-tuple line, with 2N + 2Π − 2 critical points.

This N-tuple line may be thought of as of an N-sheeted Riemann surface, which – for a certain choice of the loops, taken in a given order, and for a suitable ordering of the sheets – can be defined by a system of transpositions of the sheets, similar to those which we find in the statement of the theorem of Lüroth–Clebsch^{(6)}; thus (if n, π and ν, ρ are the characters of C_{0} and D_{0} respectively) we have:

(1) first of all 2n + 2π − 2 loops corresponding to an even number of similar transpositions (12)(12) ⋯ ((n − 1)n), ((n − 1)n);

(2) then m loops (going to the m apparent critical points, images of the common points of C_{0} and D_{0}) each of which corresponds to the product of two transpositions

(n(n + 1)) (n(n + 1)) = 1;

(3) and finally 2ν + 2ρ − 2 loops corresponding to an even number of transpositions like

((n+1)(n+2)), ((n+1)(n+2)), …, ((n+ν−1)(n+ν)), ((n+ν−1)(n+ν)).

Now, by the existence theorem^{(7)}, we may let the multiple line defined in this way vary continuously, in such a way to separate the two branch points that come together at one of the apparent critical points. It is therefore evident that the reducible curve \overline{E} = C_{0} + D_{0} in the space of F is susceptible of varying in a certain ∞^{R} variety V of irreducible curves E, and inside V there is an ∞^{R−m} variety \overline{V} of reducible curves with m double points; in this way the infinitely near curves to C_{0} +D_{0} with m double points precisely form the linear neighbourhood of this curve-element on \overline{V}.

(6)See Enriques–Chisini, Lezioni, Libro V, Chapter I, §3, Vol. III, page 26; Chapter III, §33, Vol. III, pages 359–376.

(7)See Enriques–Chisini, Lezioni, Libro V, Chapter III, §33, ibidem.

Therefore we have finally proved the fundamental lemma:

every curve E with m double points, infinitely near to C_{0} + D_{0} (in the space of F), belongs to an ∞^{1} (regular) series of reducible curves containing C_{0} + D_{0}, and therefore it is actually reducible in two improper curves C_{1} and D_{1} of the same order as C_{0} and D_{0} respectively, near to these curves.

The fundamental lemma that we have just established means that

the infinitely near curve to C_{0}, which is a part of a curve near to C_{0} + D_{0} with m double points, constructed on F as we have just explained, depends only on C_{0} and not on the choice of a D_{0}, which can be associated to it in infinitely many different ways, in a regular system |D|.

On this ground, we want to recognize that after the mentioned curve C_{1}, we have again a curve C_{2}, in the second order neighbourhood of C_{0}, and then a curve C_{3} and so on.

Let us assume that here we can speak about infinitely near curves, defined by differentiable conditions in the same way as the points on a curve, as about proper curves; we intend to justify the use we make of this terminology later on.

Let us construct on the surface F, which contains the above considered linear systems |C| and |D|, a normally large linear system |L| of a certain genus P; so |C| and |D| cut out on a generic L, and even on every (irreducible) L with a double point, complete series of order smaller than P^{(8)}. Let us consider a general pencil of curves L in |L|, with a certain group Γ of base points: first of all we claim that

the infinitely near inequivalent curves C_{0} and C_{1} cut out inequivalent groups on the L of the pencil.

This claim is a consequence of the fact that |C_{0}| cuts out on each L a complete series: so – since M = CL > C_{0}^{2} – the equivalent infinitely near curves to C_{0} are as many as the equivalent infinitely near groups to C_{0}L on L; therefore if C_{1} would cut out on a curve L a group equivalent to C_{0}L, then C_{1} should also be equivalent to C_{0}.

(8)See Chapter IV, Section ??.

Now we can slightly modify and make more precise our assumptions, by assuming that C_{0} belongs, inside {C}, to a regular system of dimension r = p_{a} + n − π + 1, so that the curve itself is determined by r simple base points which form a subgroup A of Γ (and similarly that |D_{0}| too, if needed, is of dimension r′ = p_{a} + ν − ρ + 1, so that the curve d can be determined by imposing r′ simple base points forming again a subgroup of Γ). Choose then in the base group Γ of the pencil of the L, away from A and B, P − M more points (which may also be a single point counted P − M times), forming a certain group S. By adding S to the group of M points cut out by C_{0} on L, one has a nonspecial group G of P points on L, and by adding S to the group cut out by C_{1}, one similarly has a group G_{1} of P points, which is infinitely near to G and inequivalent to it. The correspondence between G and G_{1} (an infinitesimal transformation of the group of the Jacobi variety of L) gives rise on the L of the pencil to an analytic series t of groups of P points, containing a group G_{2} infinitely near to G in the second order neighbourhood of G, that is, a point that follows G_{1}. As L varies in the pencil, the group G_{2}, which is algebraically and rationally determined^{(9)}, describes on the surface F a curve C_{2} infinitely near to C_{0}, in its second order neighbourhood, and successive to C_{1}; to this curve one has to add the infinitesimal curves, not connected with it, corresponding to the points of the group A.

In order to justify the claim to the effect that, as L varies in the pencil, G_{2} traces out a curve C_{2} near to C_{1} and of the same order, we need to prove that the locus of G_{2}, which is defined up to irreducible curves L of the pencil, does not contain any of the points of its base group Γ away from A, and therefore that on any L of our pencil (which was chosen generically inside |L|) the group G_{2} (near to G in its second order neighbourhood) cannot be special.

To see this we notice that:

(1) on a curve L of genus P, the group G_{2} cannot be special unless the groups G and G_{1} are also special; inasmuch as a group which is infinitely near to a nonspecial one is certainly nonspecial;

(2) since the group of M points LC_{0} (away from A) does not belong to a linear series of dimension > 0, by adding to it a generic group of P − M points (the group S) one obtains a nonspecial group^{(10)}: in order to have a special group of points P one has to impose one condition on S and in order that the group G = S + LC be special on L one has to impose at least one further condition on S; thus the choice of a group S of P − M points of L that, when added to LC_{0} and to LC_{1}, gives rise to two special groups G and G_{1} yields at least two conditions which S has to satisfy.

(9)The differential expression that defines G_{1}, from which the one defining G_{2} can be derived, has been computed by Chisini. See Enriques–Chisini, Lezioni, Libro VI, Chapter II, §28, Vol. IV, page 143.

(10)This is a simple consequence of the theorem of Riemann–Roch. See for instance Enriques–Chisini, Lezioni, Libro V, Chapter I, Vol. III.

With this in mind we go back to the construction outlined above of a pencil of curves L in |L|. To start with we take a particular (irreducible) curve \overline{L} of |L| (of genus P) and r of its points, that will be imposed on |C| as base points. Then we take a group S of P − M points, that will be added to those of the group LC_{0}, (from which A has been subtracted). Finally, on the same \overline{L}, we take a group Γ of the characteristic series g_{N}^{R−r} of \overline{L} containing A. In order that the group of P points G = S + LC_{0} be special on \overline{L} and have, on the same curve, an infinitely near group G_{1} = S + LC_{1} which is also special, the choice of the group S is subjected, as we have said, to two conditions at least; on the other hand, in order that the analogous conditions be satisfied on a curve L of the pencil which has Γ as base group, the choice of S, and therefore the choice of Γ, must satisfy at least one condition on L, since we have here one more parameter; so we would have a pencil of curves L that is not generically chosen inside |L|.

In this way the following assertion has been proved: the group G_{2} which we have constructed as above on the curve L of a generic pencil of |L| describes a curve near to C_{1}, of the same order (and not passing through any of the base points of Γ). And it is worth noticing that our argument works even if in a pencil of curves L there is, in general, some (irreducible) curves of effective genus P − 1 with a double point X; inasmuch as one may regard these curves as having virtual genus P (considering the double point X as a virtually nonexistent double point). This means that a group S of (P −1)+1 points on such an L has to be considered as nonspecial, unless it belongs to an ∞^{1} linear series having a neutral pair X_{1}X_{2} in correspondence with X. So, also in this case, the speciality of G and G_{1} yields at least two conditions to be satisfied by the choice of the group S.

In the above discussion we talked about the curves C_{1}, C_{2}, …, that are infinitely near to C_{0}, as if they were proper curves. To justify the use of this terminology, it is convenient to show that the above mentioned curves may be actually defined through a limiting process of proper curves. This happens first of all for the curve C_{1} which is infinitely near to C_{0} in the first order neighbourhood: in fact we have seen that C_{1} is a part of the reducible curve near to C_{0} inside a limit pencil for λ = 0 of C_{0} + D_{0} + λE (|E| = |C + D|) whose base points are the m points of the group C_{0}D_{0}. It is in fact true that the curve of this pencil is irreducible, for λ ≠ 0, and tends to C_{0} + D_{0}; but, according to the fundamental lemma proved above, as λ approaches zero, this curve acquires m base points and breaks up into C_{1}+D1: this means that the ∞^{1} analytic series t of groups of points determined by G and G_{1} on the L of the aforementioned pencil does not depend on the choice of D_{0} or of the system |D| which we associated with |C|.

Now we can construct a proper curve whose variation defines C_{2}, by considering on a generic curve L of the pencil:

(1) the group G cut out by C_{0} with the addition of the points of A;

(2) the group G′ formed (by A and) by the points of C_{0} + D_{0} + λE near to those of G;

(3) the group of P points G′′, which corresponds to G′ in the transformation (GG′) of the Jacobi variety of L.

The locus described by G′′, as L varies in the pencil, defines the curve C_{2} which is successive to C and C_{1}.

To tell the truth, as long as λ ≠ 0, this locus is an irreducible curve tending to a curve larger than C_{0}, that is, to C_{0} + D_{0}; in fact a point of C_{0} + D_{0} + λE, near to C_{0}, may be interchanged, as L varies in the pencil, with a point near to D_{0}, so that the curve which is defined by the variable G′′ is generated by the sum of G′′ and of a \overline{G}′′ analogously defined with respect to D_{0} and near to this D_{0}; and the sum G′′ + \overline{G}′′ describes a curve E2, that cuts out on the curves L groups of the linear series cut out by |E| = |C + D|, or (a priori) a multiple of this series, whose limit – for λ = 0 – is C_{0} + D_{0}^{(11)}.

But the groups G, G′, G′′ belong on L to an ∞^{1} analytic series of groups of P points that – as λ → 0 – that is, when they become infinitely near, only depends on C_{0} and C_{1} and not on D_{0} or on |D|, and therefore it actually defines, in the second order neighbourhood of C_{0}, the curve C_{2}, that is, the irreducible part of E_{2} for λ → 0 whose existence we wanted to prove.

We have thus constructed on F a curve C_{2} near to C_{1} and successive to it, in the second order neighbourhood of C_{0}, which is a part of a reducible E_{2} = C_{2} + D_{2} of the linear system |E| = |C + D|. Similarly, starting with C_{2}, one constructs a successive curve C_{3}, near to C_{0} in its third order neighbourhood, etc. So we accomplish the proof of our theorem, that is, we prove that near to the reducible E_{0} = C_{0} + D_{0} there are, in all the successive neighbourhoods of any finite order, reducible curves of the system |E| = |C + D|, which form a regular series (like the linear branch of a curve); therefore the fundamental theorem is proved in general.

(11)The justification of this assertion is the following: let us have on a curve L of genus P two groups G and \overline{G} of P points, and two more G′ and \overline{G}′ such that G + \overline{G} = G′ + \overline{G}′ and let us construct the groups G′′ = G′ + |G′ − G| and \overline{G}′′ = \overline{G}′ + |\overline{G}′ − \overline{G}|, then G′′ + \overline{G}′′ = G′ + \overline{G}′ = G + \overline{G}.

The history of the theory of continuous systems 9.6 329

On a surface of geometric genus p_{g} ≥ 0 and of arithmetic genus p_{a} < p_{g}, any linear regular nonspecial system |C| of irreducible curves is contained in a continuous system {C} which cuts out on a C a complete characteristic series and is therefore formed by ∞^{q} (q = p_{g} − p_{a}) distinct linear systems. So: the geometric irregularity of a surface is always equal to the numerical irregularity:

q = p_{g} − p_{a}.

And (Section ??) on a surface of irregularity q any curve C, with the characters n, π and i, which is arithmetically effective, that is, such that

p_{a} + n − π + 1 − i ≥ 0,

belongs to an ∞^{q} continuous series of inequivalent curves.

9.6 The history of the theory of continuous systems

The theory of continuous systems of curves lying on the irregular surfaces has a fundamental significance, not only in the algebro-geometric study of the surfaces, but also for the (transcendental) theory of the simple integrals (of total differentials) which belong to them, thus giving a way of relating these two different aspects of the science of the surfaces. Therefore we will take the general point of view consisting in understanding the genesis of this theory. In particular we want to go more deeply into the general theory of these continuous systems by putting them into a historical perspective, so that one may also see the meaning of the errors through which the theory itself had to go: this seems to us even more relevant since we believe that the development of the theory has not yet attained its final stage.

The memoir of Castelnuovo–Enriques of 1900 Sopra alcune questioni fondamentali nella teoria delle superficie algebriche^{(12)} on the one hand fruitfully closes a series of efforts, but on the other it raises new problems, such as the determination of the surfaces with all plurigenera equal to zero (thus of the surfaces with genus p_{g} = 0 and negative arithmetic genus). In order to investigate these problems and in general to make a progress in the knowledge of the surfaces, also in connection with the transcendental theory of the French school, the study of the continuous systems of inequivalent curves arose in a natural way.

The general consideration of such systems goes back to a Note by G. Humbert of 1893^{(13)}: the author proves that the possess of such a system of curves yields the existence of simple integrals of the first kind (Picard integrals of total differentials) attached to the surface. C. Segre called the attention of his colleagues to this Note and after a while this subject became a field of research of the Italian school of geometry. Already in 1896 Enriques gave an indirect contribution^{(14)} proving that “a rational series of curves on a surface is always contained in a linear system”.

(12)Ann. di Mat., t. VI, s. II, page 165.

(13)Sur une propriété d’une classe de surfaces algébriques, Comptes Rendus, 1893).

In 1899 Enriques starts the geometric study of the surfaces containing continuous systems {C} of inequivalent curves^{(15)}, proving that they are irregular. The author shows that the adjoint system of a linear system |C| contained in {C} cannot cut out on a C the complete canonical series. When he communicated, before the publication, this result to Castelnuovo, the latter replied by giving another proof of the theorem, in which one shows that the characteristic series of |C| is necessarily deficient. And in the Note published by Enriques, he gives also an account of the proof of his colleague, after having treated his own proof.

After this (in 1901) Enriques himself – having in mind the objective of understanding the connection between the geometrically defined irregular surfaces (p_{g} − p_{a} > 0) and those possessing simple integrals of the first kind – proved that “the surfaces which possess q simple integrals of the first kind with 2q periods, contain continuous systems of inequivalent curves^{(16)}”.

On the other hand Castelnuovo succeeded in proving that “a surface of genus p_{g} = 0 and p_{a} < 0, containing a continuous system of inequivalent curves, consequently possesses also an irrational pencil of curves” (Section ??). This was a property whose meaning depended on further possible developments of the theory, and Castelnuovo communicated it to Enriques, who gave it publicity only later (1904), when he could deduce from it that “any surface with p_{g} = 0 and p_{a} < 0 always possesses such an irrational pencil”.

Meanwhile, in 1902, F. Severi, who had recently graduated at the University of Torino, following advice of his teacher C. Segre came as an assistant to the chair of geometry at the University of Bologna, which at that time was held by Enriques, in order to get in closer relationship to those who studied the theory of surfaces. Enriques soon attracted the attention of the young geometer to the subject of the continuous systems; he especially suggested to him the study of a particular family of surfaces containing such systems, that is, the surfaces that represent the pairs of points of a curve; Severi in fact devoted a paper, published in 1903^{(17)}, to the study of these surfaces. The study of the surfaces representing the pairs of points of a curve gave the appropriate examples which shed light on some general questions: in particular Severi encountered there the problem of the basis, which later on has been the object of some of his most important studies. Meanwhile it gave him the chance of introducing the more general concept of characteristic series.

(14)Un’osservazione relativa alla rappresentazione parametrica delle curve algebriche. Rendic. Circ. Mat. di Palermo, 1895.

(15)Una proprietà delle serie continue di curve appartenenti ad una superficie algebrica regolare. Rendic. Circ. Mat. di Palermo, 1899.

(16)Sur les surfaces algébriques admettant des intégrales de différentielles totales de première espèce, Ann. de Toulouse, (2), 3, page 77.

The idea was contained in germ in the above-mentioned proof that Castelnuovo communicated to Enriques, about the deficiency of the characteristic series of a |C| contained in a {C} formed by inequivalent curves; here in fact one sees that the deficient series of |C| can be enlarged with the limit of the characteristic series of an infinitely near linear system. More precisely Castelnuovo pointed out in an explicit way this concept by referring to the (hyperelliptic) surfaces which represent the pairs of points of a curve L of genus 2; on these surfaces one has a continuous system of curves C of genus two, which is of degree two and contains the images of the series of pairs of points of L with a fixed point: now a curve C of such a system is isolated, so that the linear system which should cut out on it the characteristic series is missing; but this characteristic series, which (since p_{g} = 1 and the canonical curve is of order zero) is the canonical g_{2}^{1}, is cut out on the aforementioned C by the infinitely near curves in {C}.

Castelnuovo communicated this idea to Severi, who was induced to develop it in general; first of all he pointed out that the characteristic series of a C may be defined, independently on the existence of a larger linear system containing C, as the difference

(C + D)C − DC,

where |D| is any auxiliary linear system (such that |C + D| cuts out on C a complete series). In fact this difference series does not depend on the choice of the auxiliary system |D|^{(18)}. As we see, here we can find the geometric or functional interpretation of the procedure which was already used by Enriques to define the virtual degree of the curve C, just by considering |C| as a difference of two linear systems of curves.

Once the characteristic series of a curve C has been thus defined, independently of the existence of a larger linear system |C|, there is no difficulty in understanding that “if C belongs to a continuous system {C}, then the curves of this system which are infinitely near to C cut out on it groups of the characteristic series”. Therefore Enriques’ proof of the theorems that we have recalled above was reduced to its simplest form.

(17)Sulle superficie che rappresentano le coppie di punti di una curva algebrica. Atti Acc.

Torino, Gennaio 1903. It is worth saying that, independently from Severi, also M. De Franchis meanwhile published a paper Sulla varietà ∞^{2} delle coppie di punti di due curve o di una curva algebrica. Rendic. Circ. Mat. di Palermo, 1903.

(18)Severi, Osservazioni sui sistemi continui di curve appartenenti ad una superficie algebrica. Atti Acc. Torino, Febbraio, 1904.

After the result was proved to the effect that the possess of a continuous system of inequivalent curves yields the irregularity of a surface (p_{a} < p_{g}), the problem naturally arose of proving the converse of the theorem, that is, of establishing that the possess of such continuous systems is the characteristic property of the irregular surfaces. But more than that, this problem took a more precise form. In fact the geometers, in force of a number of examples, had departed from the hypothesis implicitely assumed by Noether of the completeness of the characteristic series of complete linear systems. Thus they naturally wondered whether the characteristic series of a larger continuous system were in fact complete. But the point was to find a way of discussing the question; a criterion in fact was needed in order to count in some way the conditions for the reducibility of a curve contained in a linear system. Enriques (1904) found such a criterion in the degeneration or reducibility criterion: if an irreducible curve C, varying with continuity, becomes reducible in two components, these must have at least a common point of connection, which is not the limit of a variable double point of C.

From the topological point of view this is completely evident: in fact by continuity the Riemann surface of C, which is connected, has to stay connected in any passage to the limit which leads to a reducible C^{(19)}.

By means of the reducibility principle Enriques believed he could prove that “the complete continuous systems have complete characteristic series”^{(20)} so that a regular nonspecial linear system (with characters n and π), on a surface of irregularity q = p_{g}−p_{a}, belongs to a continuous system of dimension p_{g} + n − π + 1, thus composed of ∞^{q} inequivalent linear systems. In order to do this, he used a representation of the surface on a multiple plane which – for the proof itself – is not at all essential; in fact the argument may be repeated in a substantially identical way on the surface itself, as was noticed by Severi^{(21)}.

Now it is worth pointing out that Enriques’ argument contained a gap, as we explained in Section ??: the author was not aware of the critical point which one meets here and he passed, without realizing it, from a differential property to an integral one. The difficulty, that is, the fact that the reducibility principle is no longer sufficient for p_{g} > 0, has been discussed in Section ??, where we pointed out the real meaning of this principle:

(19)Only recently a remark by Noether was pointed out in which the aforementioned principle was anticipated. In the memoir Ueber die reductiblen algebraischen Curven (Acta Math., 1896), he develops the theory of the series of groups of points (cut out by linear systems of curves or surfaces) on a reducible, plane or twisted, curve, and points out that for the twisted curves (!) the theory differs from the case of the irreducible curves, unless the components have at least a common point. The reducibility principle, rather than from a topological point of view, may be also established in an algebro-geometric way.

See for instance Enriques–Chisini, Libro V, Chapter III, §36, Vol. III, page 405.

(20)Sulla proprietà caratteristica delle superficie algebriche irregolari. Rendic. Acc.

Bologna, dicembre 1904.

(21)Intorno alla costruzione dei sistemi completi non lineari, Rendic. Circ. Mat. di Palermo,

in order that a curve of a linear system |C + D| becomes reducible into two curves of the same order as C and d with m common points, one has to impose (not more than) m conditions;

by contrast, for p_{g} > 0 one needs to see that these conditions are, in general, only m − p_{g}. To show that this holds for the curves infinitely near to a reducible curve C + D of our linear system just means that one proves a differential property, which one is not allowed to take in the global sense. The argument only works for p_{g} = 0 and (p_{a} < 0), in which case indeed there is no need to consider the characteristic series.

But the mistake or the gap in Enriques’ argument was not noticed, for fifteen years, by the geometers, and in fact we have to wait until 1921 to find the criticism expressed by Severi; and at that time the fundamental theorem, concerning the existence of ∞^{q} series of inequivalent curves, had been proved, in a different way, by means of transcendental techniques, by Poincaré (1910)^{(22)}. In 1904 Enriques’ argument seemed to be confirmed also by the treatment which Severi himself gave of it, which preserves – as we said above – its essential structure^{(23)}.

Therefore the theorem, accepted in this way by the geometers, became the starting point for important developments, which essentially proceeded in two directions:

(1) Enriques first of all attacks the classification of surfaces with p_{g} = 0, and then the more general problems of the classification of surfaces, which form the subject of the next two chapters X and XI. Luckily enough these developments are mainly based, or can be based, on the restricted fundamental theorem.

This is a priori clear for Chapter X concerning the surfaces of genus p_{g} = 0, but it holds also for the results about the surfaces of arithmetic genus p_{a} < 0, which we treat in Chapter XI, where the proof should anyhow be slightly modified in some points. Exceptions are given by the surfaces with p_{a} = 0 and p_{g} = 1: these surfaces cannot have a canonical curve of order zero and always possess a pencil of elliptic curves.

(22)H. Poincaré, Sur les courbes tracées sur les surfaces algébriques, Ann. Ecole Norm.

Sup., page 3 (27).

(23)See the quoted memoir: Intorno alla costruzione dei sistemi completi non lineari che appartengono ad una superficie irregolare. Rendic. Circ. Mat. di Palermo, 1905.

(2) On the other hand, Castelnuovo and Severi turned to draw the consequences of the fundamental theorem in view of the transcendental theory: the question here was to clarify from a quantitative point of view the connection between the irregular surfaces and those possessing simple integrals of the first kind, upon which the mentioned theorem shed some light. Concerning this we have already mentioned the first attempt we made in order to prove the converse of Humbert’s remark, that is, the theorem that the surfaces possessing q simple integrals of the first kind with 2q periods contain continuous systems of inequivalent curves and are therefore irregular. This theorem would have acquired a larger significance if Picard’s transcendental theory enabled us to know a priori that the number of the aforementioned integrals of the first kind attached to any surface is always equal to one half of the number of the periods.

In the absence of this result, already before Enriques’ note Sulla proprietà caratteristica, etc., Severi investigated in a different way the question about the surfaces possessing simple integrals of the first kind, proving that “the possess of integrals of the second kind (and therefore of the first kind) yields that the characteristic series of a complete linear system is deficient, and so also that the surface is irregular”^{(24)}. Although this proof establishes a fundamental fact and contains an interesting analysis concerning the singularities on the curve of poles of the integral, it does not seem possible to complete it in order to find the number of the integrals of the first kind in terms of the irregularity, without adding to it some essential argument^{(25)}.

On the other hand the consideration of the ∞^{q} continuous systems of inequivalent curves enabled Castelnuovo and Severi^{(26)}, independently of one another, to refine Humbert’s remark from a quantitative point of view, proving that “the irregular surfaces, with irregularity q = p_{g} − p_{a}, possess exactly q simple integrals of the first kind, with 2q periods”. The proof of this essential theorem, for which the consideration of the Fuchs’ linear equation used by Picard is not sufficient, is also achieved by Poincaré in a different way in the quoted memoir; there he rediscovers, as we have already said, also the existence of the ∞^{q} continuous systems of inequivalent curves.

(24)Sulle superficie algebriche che posseggono integrali di Picard della seconda specie, Rendic. Acc. Lincei, 1904. See Math. Ann., 1905.

(25)The analysis of this question leads Severi to a result (Severi, Sulla differenza fra i numeri degli integrali di Picard della prima e della seconda specie appartenenti ad una superficie irregolare, Atti Acc. Torino, 1905). The same results were obtained by Picard by his methods (Comptes rendus, 1905): see the Traité by Picard and Simart, II, page 417.

(26)G. Castelnuovo, Sugli integrali semplici appartenenti ad una superficie irregolare, Rendic. Acc. Lincei, Maggio-Giugno, 1905, F. Severi, Il teorema di Abel sulle superficie algebriche, Ann. di Mat., agosto, 1905.

For the proof of the theorem on the integrals that we have mentioned above, Castelnuovo introduces the important concept of the variety he called the Picard variety attached to an irregular surface (see Section ??). This notion enables one to construct in general the continuous system defined by a curve C, starting from any other continuous system {D} belonging to the surface. Enriques remarked that, in force of the theorem of Riemann–Roch, this construction more precisely yields that “a linear system of curves with characters n, π and i, which is arithmetically effective, that is, such that p_{a} + n − π + 1 − i ≥ 0, always belongs to an ∞^{q} series of inequivalent linear systems” (see Section ??). This remark was communicated to Severi by the author, and since the former had succeeded just at that time in remarkably simplifying the proof of the theorem of Riemann–Roch, Enriques left it to him to publish it^{(27)}.

We come now to the criticism which revealed the gap in the proof of Enriques’ theorem about the continuous systems. As we have said, this was made by Severi in 1921. It may be found in a series of Notes “Sulla teoria degli integrali semplici di prima specie appartenenti ad una superficie algebrica”^{(28)}, in which one finds the treatment, with some simplification at some points, of Poincaré’s proof of 1910, including the existence of the continuous systems and of the integrals of the first kind^{(29)}. The author remarks that (thanks to Poincaré) at least the essential part of the theorem (the fundamental theorem as stated here in Section ??) holds. Later on there will be other attempts to extend this theorem by proving that the characteristic series of any complete continuous system is always complete; will will discuss this later (Section ??).

Attempts to prove and extend the fundamental theorem 9.7 336

Enriques tried, in a few Notes in 1936-1937, Curve infinitamente vicine sopra una superficie algebrica^{(30)} and La proprietà caratteristica delle superficie algebriche irregolari e le curve infinitamente vicine^{(31)}, to fill the gap which had been pointed out in the proof of the fundamental theorem. He tried to prove that, given a curve C, (if the characteristic series of C has deficiency q = p_{g} − p_{a}) then not only there are ∞^{q} inequivalent curves infinitely near to it in the first order neighbourhood, but also other successive curves in the second order neighbourhood etc. The fundamental idea of this proof (which already appears in the Lezioni by Enriques and Campedelli), relies on the fact that “starting from two infinitely near points on an Abelian variety one defines, as Lie does, an infinitesimal transformation which generates an ∞^{1} group.

(27)Severi, Sul teorema di Riemann–Roch e sulle serie continue di curve appartenenti ad una superficie algebrica, Rendic. Acc. Torino, 1905.

(28)Rendic. Acc. Lincei, 1921.

(29)Poincaré’s proof, with further simplifications by Lefschetz, has been treated by O. Zariski, Algebraic Surfaces (Berlin, Springer, 1935). See VII, 4, 5 (page 128, 133).

(30)Rendic. Acc. Lincei, aprile 1936 and Rendic. Sem. Mat. R. Università di Roma, 1936.

(31)Rendic. Acc. Lincei, ottobre 1937.

Thus, in particular, in the Jacobi variety formed by all groups of p points of a curve of genus p, one can define, starting with two infinitely near groups, an ∞^{1} analytic series of elements (groups of points). This construction can be extended to the continuous systems of inequivalent curves on a surface, if one assumes the existence of a system whose elements form the Picard (Abelian) variety. Enriques’ considerations tend to justify this construction, passing from infinitely near curves in the first order neighbourhood to curves in the second order neighbourhood and so on; and this independently of the assumption of the existence of the Picard variety, but rather with the purpose of justifying its construction. The starting point is therefore that, near to a reducible curve C + D, and in its first order neighbourhood, there are curves \overline{C} + \overline{D} which are reducible into two components, one of the same order as C, the other one of the same order as D, having the same number m of common points. Here, as the author points out, first of all one has to justify an implicit application of the infinitesimal reducibility principle; but this is not a difficulty, inasmuch as one may see that, by getting out of the surface, one may construct, in the space, a series of reducible curves which contains C + D and \overline{C} + \overline{D}.

Once the effectiveness of the reduction process for the infinitely near curves, and thus the effective existence of a inequivalent \overline{C} infinitely near to C has been justified in this way, Enriques’ construction gives a series of operations on C + D and \overline{C} + \overline{D} which turns out to depend on C and \overline{C} only, and just leads to the extension of the series of curves defined by \overline{C} in the successive neighbourhoods of C.

On occasion of a Note in 1937, this proof was disputed by B. Segre.

In any case Enriques (replying to Segre’s objections) was willing to prefer another proof, proposed by Segre himself, simply based on the continuity principle, whose foundation anyhow was still to be discussed: we will expressly discuss this interesting attempt and its developments for the contribution of other geometers in the next Section ??.

Only after the failure of the aforementioned attempt we have taken up again, as it was natural to do, the criticized proof, which – with some explanation and clarification – we have treated in Section ?? above.

9.7 On various attempts to prove and extend the fundamental theorem

The proof of the fundamental theorem, which expresses the characteristic property of the irregular surfaces, was obtained in general (p_{g} ≥ 0) by means of some infinitesimal considerations which take the place of a differential computation. Considerations of this type, even if one remains in the field of the algebraic analysis, are rather delicate and raise doubts in those mathematicians who are not sufficiently familiar with them, or at least they give the impression that, in contrast with the need of logic rigour, some intuitive element has been introduced.

This impression was even better understood in view of some explanation a little bit too fast and not so well justified in the quoted notes: whose treatment^{(32)} we have taken up here again trying to explain and clarify the most delicate points. For these reasons it appeared, and it may still appear, desirable to give other proofs of the same theorem, which could be accepted by everybody, without doubts and hesitations.

We have already said that such a proof, which makes use of the integrals belonging to the surface, was given by Poincaré (1910), and it can be seen today in the simplified treatment by Lefschetz, reported on by Zariski. But, in the field of the algebro-geometric theory, the attempt which, as we have said, was undertaken by B. Segre^{(33)} deserves a special quotation.

The leading idea of this author may be expressed as follows. One considers the linear system |E| = |C + D| which contains the reducible curve E_{0} = C_{0} + D_{0}, having m double points forming the group C_{0}D_{0}; and one imposes on the curves E, near to E_{0}, m − p_{g} double points (near to m − p_{g} points of this group); one wants to prove that the curves \overline{E} thus defined are necessarily reducible into a curve of the same order as C_{0} and a curve of the same order as D_{0}: and in order to do so it is convenient to deduce a contradiction from the hypothesis that the aforementioned curves \overline{E} are irreducible. Let us therefore construct on a curve \overline{E} with m − p_{g} assigned double points the complete characteristic series g of the continuous system \{\overline{E}\}, and let us pass to the limit for \overline{E} → E_{0}; the series g has dimension

2p_{g} + (n − π) + (ν − ρ) + 1,

but, in the limit, it apparently goes to the series g_{c} + g_{d} formed by the groups of the (completed) characteristic series of C_{0} and D_{0} with the addition of the p_{g} fixed points of the group C_{0}D_{0} (the new double points acquired by the reducible E_{0}); but now the dimension of the limit series g_{c} + g_{d} is

2p_{g} + (n − π) + (ν − ρ)

one less than the dimension of g!

The author claimed that this reduction of the dimension of a series is absurd, being in contrast with the continuity principle. But Enriques, who had made an analogous attempt (in the simplest case p_{g} = 1) on his own, was stopped by an embarassing paradox, and he treated the difficulty he had met^{(34)}. Let us consider, in the plane, the system of curves C_{8} of order 8 with 15 double points; among these C_{8} there are the reducible curves formed by two double quartics C_{4} + \overline{C}_{4}, which have 15 common points A_{1}, …, A_{15} limit of the double points of C_{8} and another sixteenth point O. Now on a variable C_{8} the canonical series g = g_{10}^{5} is cut out by the quintics C_{5} through the 15 double points; in the limit C_{8} → C_{4} + \overline{C}_{4} the aforementioned canonical series is nothing but the series which is composed by the groups of the two canonical series g_{4}^{2} on C_{4} and \overline{C}_{4}, cut out by the curves C_{5} through A_{1}, …, A_{15}, which consequently have the common fixed point O; so it seems that in the limit the above series g goes to a series of lower dimension. The doubt arises whether, as it seems to follow from the continuity principle, the dimension is preserved!

(32)Especially following the way shown in the Notes of 1936.

(33)Un teorema fondamentale della geometria sulle superficie algebriche e il principio di spezzamento, Ann. di Mat., 1938.

Thanks to Segre, who explained the paradox, this difficulty has been clarified. The principle of conservation of the dimension of a linear series actually follows from considerations of continuity and can be explained in an explicit and irrefutable way; but when one looks for the limit of the series g, as C_{8} → C_{4} + \overline{C}_{4}, one has to consider not only what it becomes of g as an adjoint curve C_{5} of C_{8} tends to a generic C_{5} through A_{1}, …, A_{15}, but also the limit of g in correspondence with a variation in which C_{8} and C_{5} both tend to reducible curves containing C_{4} (or \overline{C}_{4}) as a part, and where the section of C_{8} with C_{5} becomes undetermined. In other words: the paradoxical example brings to light the necessity of considering, when the series g goes to the limit, the whole limit!

B. Segre well explains the importance of this necessity by means of the example of a varying hyperbola which tends, in the limit, to the line pair of its two asymptotes: the g_{2}^{1} cut out on the hyperbola by the lines through the centre, reduces to the fixed point which is the centre, but also to the two symmetries with respect to the centre itself which one has on the asymptotes. In a similar way, if one takes into account the fact that the curves C_{8} passing through A_{1}, …, A_{15} on C_{4} do not pass through O as a consequence, then the limit of the g = g_{10}^{5} in Enriques’ example is given by two series of dimension 5 which are given by associating groups of C_{4} and of \overline{C}_{4}. Just to describe one of these series, we may say that it can be constructed by associating with the canonical groups of \overline{C}_{4} the groups of a g_{6}^{3} on C_{4}, that no longer has O as a fixed point, but is defined by setting g_{6}^{3} = g_{4}^{2} + 2O.

After having explained in this way the embarassing paradox, B. Segre saw a gap in his own proof of the fundamental theorem, and thought about the problem of filling it. He meant to do this by looking for a criterion for the determination of the limit of the intersection of two curves which, as they vary on a surface, come to acquire a common part. He gave an account of this infinitesimal criterion in a Note, Sur un théorème fondamental de géométrie sur les surfaces algébriques^{(35)}, which was accompanied by another Note by Enriques^{(36)} who was glad to welcome the success of the Colleague.

(34)Enriques, Sulla proprietà caratteristica delle superficie irregolari, Rendic. Acc. Lincei, Giugno, 1983.

Meanwhile B. Segre left Italy and, as far as we know, he has not published so far any more detailed paper on this subject.

Successively Severi made a critical revision^{(37)} of the whole theory of the continuous systems of curves on the algebraic surfaces. Unluckily this treatment is affected by some mistakes, which were partly pointed out by Enriques in a Note, Sui sistemi continui di curve appartenenti ad una superficie algebrica, and which we need to consider here.

In order to complete the proof following Segre’s argument, as we explained above, Severi takes up again from Segre – in a slightly modified version – a lemma of differential geometry, to the effect that, in passing to the limit \overline{E} → E_{0} = C_{0} + D_{0}, the (complete) characteristic series g of the continuous system \{\overline{E}\} on \overline{E} necessarily has the p_{g} points P of the group C_{0}D_{0} which are not in the limit of the m − p_{g} double points of \overline{E} as fixed, and this no matter how the passage to the limit is made, even if the curves which cut out the g on \overline{E} tend to curves having C_{0} as a part, so that lim g is undetermined. The lemma of Segre–Severi which would enable us to draw this conclusion expresses a differential geometry property, that is, that the aforesaid points P should be the limits of points belonging to groups of the series g. This property should in turn be a consequence of the fact, which was pointed out by Enriques in the original proof of the fundamental theorem, that “the infinitely near curves to C_{0} cut out on it the complete characteristic series”; in this way the gap in the proof of that theorem would have been filled.

Now Enriques, who had already found some difficulties in reconstructing Segre’s infinitesimal criterion on the basis of the indications which the Author had given to him, was not able to understand this point and therefore declared the lemma to be “obscure and unconvincing”^{(38)}. But he believed it might be possible to accomplish the proof by realizing the series \overline{g} on the variable \overline{E} as cut out by curves which stay irreducible in the limiting process which one considers.

In this he was wrong and Severi was right in pointing out his mistake^{(39)}.

(35)Comptes rendus, 9 Janvier 1939.

(36)Sur la propriété caractéristique des surfaces algébriques irrégulières. Comptes rendus, 3 Janvier 1939.

(37)La teoria generale dei sistemi continui di curve sopra una superficie algebrica. Mem.

Acc. d’Italia, Vol. XIX, 1941.

(38)Comm. Math. Helvetici: presented for publication on the 20th of March, 1942.

(39)Intorno ai sistemi continui di curve sopra una superficie algebrica. Comm. Math. Helvetici (30 Novembre 1942). Enriques was not allowed by the direction of the Commentarii to reply to the latter article, and therefore he could neither admit his mistake, nor point

But the understanding of this mistake throws a shadow of doubt on the whole proof proposed by Segre and Severi. The point is not only to correct the wrong lemma in Severi’s Memoir, as the Author himself^{(40)} did after an example pointed out by BOMPIANI, but rather to ask whether a differential property as it is expressed in the aforesaid lemma actually suffices to exclude that the limit of g can give rise on C_{0} to a fixed point free series. If one makes a careful analysis of the question then one will be easily convinced that this doubt cannot be removed, and, more than that, that the lemma itself appears here to be completely inadequate, that is, that “this lemma of Segre–Severi does not answer the questions for which it has been introduced”. We do not insist on this, since we believe that Severi himself has recognized the inadequacy of his argument by now. In any case it is worth mentioning here an idea, which – if not today – maybe tomorrow could have a fruitful application.

We have mentioned the fact that Severi (and B. Segre before him) had tried to extend the fundamental theorem on the irregular surfaces, by proving that on any surface “the characteristic series of an irreducible continuous system without multiple points (however special or superabundant) is always complete”.

Anyhow at the present time the Author recognized, with examples, that certainly there are exceptions to more general result he had in mind^{(41)}.

But we may restrict, as it is necessary, the choice of the continuous systems of irreducible curves {C} in such a way that the property holds just by hypothesis: so let {C} be an ∞^{d+1} continuous system having on C a complete characteristic series of dimension d; it is worth explaining the various cases which may occur.

The system {C} certainly contains an infinite series of inequivalent linear systems of curves, provided that d + 1 is greater than the dimension of the complete linear system |C|, that is, that this system of dimension

r = p_{a} + n − π + 1 − i + ω (with ω ≥ 0)

has a deficient characteristic series, with deficiency

q − ε (q = p_{g} − p_{a}).

Now we will distinguish between the elementary exceptional cases which, together with all their combinations, give rise to all the possible cases. It may a priori happen:

out the mistakes which, in his opinion, Severi had made.

(40)Note presented to the R. Acc. d’Italia, 16th of January, 1942.

(41)And there are also exceptions to the uniqueness of the continuous system to which a given irreducible curve belongs. The Author believed he could prove this with the argument at the end of page 378 and at the beginning of page 379 of the memoir La teoria generale ecc., of 1941.

(1) that the system |C| is regular and special, with index of speciality i:

0 < i < p_{g}.

In the above notation:

ω = ε = 0;

(2) that |C| is nonspecial and superabundant, with superabundance ω > 0, whereas its characteristic series has the maximal deficiency q = p_{g} − p_{a}: i = 0, ε = 0.

(3) that |C| is nonspecial and superabundant with superabundance ω = ε, and has the characteristic series with deficiency q − ε: i = 0, ε > 0.

In case (1) (if it occurs) the dimension of |C| is

r = p_{a} + n − π + 1 − i,

and that of {C} is r + q, therefore the continuous system is formed by ∞^{q} linear systems with (in general) the same index of speciality.

In case 2), since the dimension of |C| is r = p_{a} + n − π + 1 + ω, the dimension of {C} is again r + q, so {C} is formed by ∞^{q} linear systems with the same superabundance.

But in case 3), since the dimension of |C| is r = p_{a} + n − π + 1 + ε, the dimension of {C} turns out to be r+q−ε, and we cannot say a priori whether, for instance, {C} is formed by ∞^{q} regular linear systems generically having dimension r − ε, among which there is the superabundant |C| of dimension r, or by an ∞^{q−ε} series of linear systems all with the same superabundance ω = ε.

We have just mentioned only the extremal cases among all those which in the hypothesis 3) appear to be a priori possible. The effective possibility that these different assumptions occur, and in particular the first one (which by a very hidden mistake was at a first moment excluded) was recognized by Enriques in the quoted critical Note^{(42)}, where the example which we briefly mention here is carefully studied.

Let us consider the scroll F_{5} of the 5th order with a triple and a double line: F_{5} has geometric genus p_{g} = 0, arithmetic genus p_{a} = −p = −2 and irregularity q = 2.

On F_{5} the plane section system |C| is superabundant of superabundance ε = 1, since it has degree n = 5, genus π = 2, and dimension r = p_{a} + n − π + 1 + ε = 3, while the deficiency of the characteristic series is q − ε = 1.

The linear system |C| is contained in a continuous system {C} of dimension 4, and one can see that it is not formed by ∞^{1} linear ∞^{3} system, but by ∞^{2} linear regular ∞^{2} systems, among which there is the above mentioned |C| of dimension 3. In order to verify this assertion, one may add a pair of generators a, b to the system |C| and, by means of the linear system thus obtained (which is ∞^{6}), one may construct a surface F_{9} of order 9, a normal scroll in the space S_{6}, of which the F_{5} appears to be the projection from a pair of intersecting generators a′, b′. F_{9} has a double point at the intersection point of a′ and b′, but it has no double curve; therefore there are at most finitely many pairs of intersecting generators on it and thus the ∞^{2} linear systems forming {C}, that is, the systems cut out on F_{9} by the hyperplanes through a pair of generators, are, in general, of dimension 2. Q.E.D.

(42)Comm. Math. Helvetici, 20th of March, 1942.

On the other hand one may show the existence on F_{9} of a continuous system {L} formed by ∞^{1} superabundant linear systems, all of dimension 3 (= 2 + 1): this is the continuous system cut out by the hyperplanes passing through the double point O and through a generator of the scroll. And it should be pointed out that this example has to be considered as contained in case 3), rather than 2), since the characteristic series of {L} is a nonspecial g_{5}^{3}.

Remark The ∞^{1} linear systems |L| forming {L} are-on the scroll F_{5}- residual of the generators with respect to the ∞^{5} linear system

|C + a + b − A − B|,

that has two base points, A and B, on a and b respectively, forming a neutral pair for the system

|C + a + b|.

Now, if we replace a and b with two other generators \overline{a} and \overline{b}, then the pair AB will no longer be neutral for |C + \overline{a} + \overline{b}|, thus by subtracting a generator from these linear systems one is led to ∞^{2} linear systems |\overline{L}| = |C + \overline{a} + \overline{b} − A − B|.

The two systems, that is, the system {L} formed by ∞^{1} linear systems and the system \{\overline{L}\} formed by ∞^{2} linear systems, are distinct, namely \{L\} + \{\overline{L}\} is reducible, as a system of curves, in two ∞^{4} varieties; by contrast |\overline{L}|, which contains the ∞^{1} variety of the systems |L|, has to be regarded as an irreducible ∞^{2} variety of linear systems. This circumstance is expressed by saying that the system {L} is exorbitant. The phenomenon of the exorbitance (either of a linear system or of a continuous system of linear systems) has been pointed out by Rosenblatt and Albanese, and later by Severi^{(43)}.

Other examples of exorbitant systems are given by the system of the plane sections of a completely general cone of order n ≥ 3.

(43)Nuovi contributi ecc., Lincei, 1916.

The paracanonical system 9.8 343

For n = 3 the ∞^{3} linear system of the plane sections is contained in the ∞^{1} series of (∞^{2}) linear systems formed by the triples of generators of the cone, with the neighbourhood of the vertex added.

For n = 4 the ∞^{3} linear system of the plane sections of the cone is contained in the ∞^{3} series of (∞^{1}) linear systems formed by the 4-tuples of generators of the cone, with the neighbourhood of the vertex added.

For n ≥ 5 the ∞^{3} linear system of the plane sections of the cone is contained in the ∞^{n} series of (∞^{0}) linear systems formed by the groups of n generators of the cone, with the neighbourhood of the vertex added.

9.8 The paracanonical system

According to the fundamental theorem (Section ??) and to the numerical condition of Section ??, the canonical system |K| of a surface F, of arithmetic genus p_{a} ≥ 0 and of geometric genus p_{g} > p_{a}, is always contained in an ∞^{q} (q = p_{g} − p_{a}) continuous system of distinct linear systems, of dimension

r ≥ p_{a} + p^{(1)} − 1 − p^{(1)} + 1 = p_{a}

It was believed that one could say more precisely what the dimension of this paracanonical system is, by determining the deficiency of the characteristic series of the canonical system |K|, at least under the hypothesis that the pure canonical curves are irreducible, and with no multiple points, of genus p^{(1)} > 1^{(44)}.

This deficiency should be q = p_{g} − p_{a} and therefore the paracanonical system (supposed to have a complete characteristic series on a given K) would have dimension 2p_{g} − p_{a} − 1. But this argument is wrong^{(45)} as one may see by revisiting the Author’s argument and bringing into light a rather delicate objection by which it is invalidated.

Let us indeed consider a regular, nonspecial linear system |C|, of degree n and genus π on the surface F, whose adjoint system |C′| = |C + K| has dimension

p_{a} + π − 1

and cuts out on C a deficient canonical series, of dimension

π − 1 − q.

(44)Enriques, Des courbes paracanoniques appartenant á une surface algébrique irrégulière.

Soc. Royale des Sciences de Liège (19th of October, 1939).

(45)It seems that essentially the same mistake is contained in the proof of the same theorem which has been given by Severi in the quoted memoir on La teoria generale dei sistemi continui ecc., of 1941.

We denote by G = CK the group of 2π −2−n common points of a C and a K, and we want to compute the number of conditions which G imposes on the curves of |C′| that have to contain it. In order to do this, we assume that the C′ which contain G cut out on C the complete characteristic series of |C|, that is, the characteristic series g of {C}, that is cut out on the aforesaid curve C by the (in general inequivalent) curves \overline{C} infinitely near to it. We would like to justify this assumption by saying that, among the residual curves of a \overline{C} with respect to |C′|, there is at least a curve \overline{K} near to K, so that C + K and \overline{C} + \overline{K} give rise to a pencil of curves C′ through G, passing through an arbitrarily given group of g.

But this is in fact the mistake! If the paracanonical linear systems have dimension lower than the dimension of the canonical system |K|, that is, p_{g} − 1, then a paracanonical system |\overline{K}| near to K does not contain a curve \overline{K} near to any given K: the limit of a linear system |\overline{K}| is a smaller linear system contained in the canonical one.

So we have explained the mistake which invalidates the above argument, to the effect that the deficiency of the characteristic series of |K| is q = p_{g}−p_{a}. Now we can actually see with examples that the theorem itself is far from being true. Rather it seems likely that “on a surface of irregularity q = p_{g} − p_{a} > 1, (p_{a} ≥ 0) with irreducible canonical curves, without multiple points, the ∞^{q} paracanonical linear systems are, in general, regular and therefore the continuous system {K} is ∞^{p_{g}}: The characterstic series of |K| (at least if this system is not exorbitant) should have deficiency 1”.

Let us see that this is verified by the surface which represents the pairs of points of a quartic curve L of genus q = 3.

This surface F has the geometric genus p_{g} = 3 and the arithmetic genus p_{a} = 0^{(46)}: the canonical curves of F are given by the series of pairs of points of L taken from the 4-tuples of the special g_{4}^{1}, and therefore there are ∞^{2} of them; by contrast, the paracanonical curves are given by the series of pairs belonging to 4-tuples of nonspecial g_{4}^{1}, and therefore they form an ∞^{3} continuous system, consisting of ∞^{3} inequivalent curves; inside this system one finds the superabundant system |K|.

We can easily verify that the paracanonical systems |\overline{K}| have, in general, dimension p_{a} = 0 and not > 0; otherwise the characteristic series g_{6}^{x} of {K} on a K (of genus p^{(1)} = 7) would have dimension

x ≥ 3,

whereas a nonhyperelliptic curve of genus bigger than 4, like K, cannot contain a g_{6}^{3}.

(46)See F. Severi, Sulle superficie che rappresentano le coppie di punti di una curva algebrica. Atti Acc. Torino, 25th of January, 1903.

It seems that what we have proved for the surface representing the pairs of points of a quartic curve L of genus q = 3, can be extended to the surface F which represents the pairs of points of a curve L of genus q = 4, 5, …

For q = 4, L can be realized as a sextic curve in S_{3}, and the canonical curves K of F are given by the scrolls which are the intersections of the congruence of the chords of L with the line complexes of S_{3}, and therefore one has

p_{g} = 6;

by contrast the paracanonical curves \overline{K} of F will be obtained in correspondence with the series of all the pairs of points of L belonging to its nonspecial g_{6}^{1}. If, as we believe, it happens that the g_{6}^{1} of a nonspecial g_{6}^{2} correspond to the curves \overline{K} of a complete linear system |\overline{K}|, then the paracanonical system \{\overline{K}\} of F turns out to have dimension

p_{g} = p_{g} − p_{a} + p_{a} = 4 + 2 = 6

and contains ∞^{4} regular linear systems of dimension

p_{a} = 2.

But the ∞^{5} canonical system is not contained in this total paracanonical system \{\overline{K}\}, but it has to be exorbitant, since the scroll of the chords of L which belong to a special line complex cannot be the limit of a scroll whose generators join the pairs of a (nonspecial) g_{6}^{1} of the curve L itself.

Also for q = 5, when the curve L may be realized in S_{4} as a curve of order 8 (the base locus of a net of quadrics) it seems that the surface F, which represents the pairs of points of L, possesses a paracanonical system {K} formed by ∞^{q} = ∞^{5} regular linear systems of dimension

p_{a} = 5 (p_{g} = 10);

one has to verify that the g_{8}^{1} belonging to the nonspecial g_{8}^{3} form a (nonlinear) system contained in an ∞^{5} complete linear system.

The examples that we have mentioned, the analysis of which has still to be completed, suggest various problems about the irregular surfaces, and in particular “whether the paracanonical system {K} of a surface with p_{a} ≥ 0 and p_{g} > p_{a}, has dimension p_{g} and consists, in general, of ∞^{p_{g}−p_{a}} regular linear ∞^{p_{a}} systems”.

But at least the case in which the linear genus is p^{(1)} = 1, and therefore the canonical curves, for p_{g} > 2, are reducible, and are composed with the elliptic curves of a pencil, gives rise to an exception.

A digression on Abelian varieties 9.9 346

In this case one deals with surfaces of irregularity q = p_{g}−p_{a}, with a pencil of elliptic curves k of genus q; in fact one can prove that that the continuous system {C}, to which a curve C belongs, cuts out on the k (the components of the canonical curves K) a series of equivalent groups: if that were not the case, then one could find, together with the pencil |k|, another elliptic pencil of curves and one could deduce from this that the k have a constant modulus and p_{a} = −1 (see Chapter X).

Thus, if p_{a} > 0, then the canonical curves of F are composed of p_{g} − 1 > q − 1 curves k, forming a nonspecial series within the pencil: therefore the paracanonical systems are equally of dimension p_{g} − 1, and they give rise to a superabundant continuous system of dimension

2p_{g} − p_{a} − 1.

By contrast, if p_{a} = 0, then the canonical system |K| is the ∞_{g}^{p} − 1 linear system formed by the 2q − 2 = 2p_{g} − 2 curves of the canonical groups in the pencil |k|, and the paracanonical systems have dimension p_{g} − 2, giving rise to a continuous system {K} of dimension

2p_{g} − p_{a} − 2.

Remark It is convenient to add here that for p^{(1)} = 1 and p_{a} = 0, the case of irregular surfaces (of genus p_{g} = 1) with a pure canonical curve of order zero cannot occur. In fact for a surface F of genus p_{g} = 1 with a canonical curve of order zero, one cannot have p_{a} = 0^{(47)}. Indeed a linear system |C| = |C′| of genus π and degree n = 2π − 2 on F would give rise to an ∞^{1} continuous series of linear systems of dimension π−1, and one of these, different from |C| and not containing |C|, should cut out on a C a nonspecial series of degree 2π − 2 and dimension π − 1.

9.9 A digression on Abelian varieties

An Abelian variety V_{q} (of rank 1) of dimension q is defined as an (algebraic) variety which admits a parametric representation by means of (Abelian, that is,) 2q-periodic functions of q independent variables, a representation which becomes one-to-one in the prismatoid of the periods: if these functions are, as we assume, nondegenerate, then V_{q} has an ∞^{q} simply transitive group of permutable transformations without exceptions: this means that given any two (simple) points of V_{q}, there is a birational transformation which sends the former point to the latter. The simplest example of Abelian variety is the variety of the groups of p (= q) points G_{p} of a curve C of genus p, that is, the Jacobi variety corresponding to the curve^{(48)}.

(47)See Enriques, Intorno alle superficie algebriche di genere lineare p^{(1)} = 1, Rendic. Acc.

Bologna, 9th of December, 1906 (n. 1).

This V_{p} admits a mixed simply transitive group of birational transformations into itself which is generated by the ∞^{p} involutory transformations of the second kind defined by the g_{2p}^{p} which belong to the curve; furthermore it has the ∞^{p} permutable transformations of the first kind, forming a continuous group Γ_{p}, which are defined as the product of an even number of transformations of the second kind. The Jacobi variety V_{p} contains an exceptional variety V_{p−1} corresponding to the series of the special groups Gp on the curve; such a V_{p−1} is transformed into a V_{p−2} of dimension p − 2 by the transformations of first or of the second kind, for instance by the involution between the groups G_{p} defined on C by a g_{2p}^{p}^{(49)}.

When among the p independent integrals of the first kind of the curve of genus p there are q < p of them (linearly independent) reducible with 2q periods, inside the V_{p} of Jacobi corresponding to the curve one finds an ∞^{p−q} series of Abelian varieties V_{q} which form an imprimitive system for the Γ_{p}; each of them has in fact an ∞^{q} simply transitive algebraic group of birational transformations into itself; moreover V_{p} also has, by the theorem of Picard–Poincaré, an analogous ∞^{q} series of Abelian varieties V_{p−q}^{(50)}.

Abelian varieties V_{q} of this type, imprimitively contained in a V_{p} of Jacobi, are the Picard varieties attached to irregular algebraic surfaces^{(51)}: in fact, on an irregular surface F, with irregularity q, the inequivalent curves of an ∞^{q} continuous system cut out on a curve of genus p (> q) a series formed by inequivalent groups of points or by ∞^{q} linear series, which just form an Abelian variety V_{q} in the V_{p} of Jacobi corresponding to the curve.

The previous remark has a general meaning since one can prove^{(52)}, with transcendental techniques, that any Abelian variety (of rank 1) is a Picard variety attached to some irregular algebraic surface; so we may state immediately the theorem of Torelli: There are always Jacobian varieties imprimitively containing a given Abelian variety.

If one does not like to use the result of Castelnuovo–Enriques, which was obtained in a transcendental way, then one can make the following considerations which lead to a theorem less precise than Torelli’s theorem, but sufficient for the further development of our treatment.

(48)See, for instance, Enriques–Chisini, Lezioni, Libro VI, Chapter III, §36, Vol. IV, page 223.

(49)See Enriques–Chisini, Lezioni, Libro VI, Chapter II, §27 (vol. IV, page 143).

(50)See, for instance, Enriques–Chisini, Lezioni, Libro VI. Geometric interpretation by Castelnuovo, Chapter III, 4, 38, (Vol. IV, page 234).

(51)See R. Torelli, Sulle serie algebriche semplicemente infinite di gruppi di punti appartenenti a una curva algebrica, Rendic. Circ. Mat. di Palermo, t. XXXVII, 1914 (§ II, n.

11).

(52)See G. Castelnuovo and F. Enriques, Sur les intégrales simples…, Ann. Ecole Norm.

Sup., s. 3, t. XXIII, 1906.

We take a more general geometric point of view and we define a q dimensional Abelian variety V_{q} to be a V_{q} having an ∞^{q} simply transitive group Γ of birational transformations without exceptions into itself, thus such that there is one nondegenerate transformation of Γ in which any two (simple) points of V_{q} correspond to each other.

The property of the group Γ, of being simply transitive without exceptions on the V_{q}, yields as a consequence that: there is no linear system of dimension¿0 of varieties V_{q−1} on V_{q} which is transitively transformed into itself by ∞^{r} (r > 0) transformations of Γ. In other words: a variety V_{q−1} in V_{q}, which is not transformed into itself by a continuous subgroup of Γ, cannot be transformed by Γ in the varieties of a linear system |V_{q−1}|.

Let us consider, on the Abelian variety V_{q}, a linear system of dimension r > 0 of (irreducible) varieties |V_{q−1}|, which is transitively transformed into itself by an ∞^{r} (algebraic) subgroup of Γ. We may represent |V_{q−1}| by means of a linear space S_{r}, in which a transitive continuous group of linear transformations is given. Now we can find, inside this group, a continuous subgroup Γ′, of dimension h ≥ 1, which fixes a given line a of S_{r} and permutes its points in ∞^{1} ways. Therefore we find in the group Γ of V_{q} a subgroup Γ_{h} of dimension h, which permutes in ∞^{1} ways the varieties \overline{V}_{q−1} of a linear pencil inside |V_{q−1}| and carries a generic point of a \overline{V}_{q−1} to the points of a \overline{V}_{h} formed by ∞^{1} varieties \overline{V}_{h−1} belonging to the \overline{V}_{q−1} of the pencil. But in this pencil there is (at least) one \overline{V}_{q−1} which is invariant for Γ_{h} (corresponding to a fixed point on the fixed line of S_{r}) and therefore the \overline{V}_{h} has to contain a \overline{V}_{h−1} which is invariant for Γ_{h}: by associating two points of this \overline{V}_{h−1} we have not one, but ∞^{1} transformations of Γ_{h}, thus contradicting the assumption that Γ is simply transitive, without exceptions, on V_{q}.

Thus there is no linear system |V_{q−1}| on V_{q} which is invariant for a continuous subgroup of transformations of Γ, but there may be |V_{q−1}| which are invariant for a finite group, which is a subgroup of Γ itself.

Now we may choose a (q − 1)-dimensional variety V_{q−1} in V_{q} which is not transformed into itself by any (non-identical) transformation of the group Γ. Indeed, since V_{q} can be projectively realized in a suitable space S_{r}, one of its generic hyperplane sections certainly is not mapped to itself by any transformation of Γ.

In this setting, the ∞^{q} series of the transformed varieties of the above V_{q−1}, which we denote by {V_{q−1}}, forms an algebraic system which is identical with the group Γ and therefore also with the variety V_{q}. In general we may find in the above series a finite number s of V_{q−1} equivalent to the one we started from. So if we cut the V_{q−1} of the above series with a generic algebraic curve traced on V_{q}, then we obtain a series consisting of ∞^{q} inequivalent groups of points, which can be represented as an Abelian variety W_{q} (contained in the Picard variety of the given V_{q}), that, for s = 1, turns out to be identical with the system V_{q−1} and with the V_{q} itself. If, by contrast, s > 1, then the mentioned system, and the V_{q}, are represented on the multiple W_{q} with multiplicity s, since s equivalent V_{q−1} correspond to a single point of W_{q}. If we consider the latter case, s > 1, then we have a (1, s) correspondence between W_{q} and V_{q}, by means of which we may define an involution I′ of order s on V_{q}; this involution is invariant for the transformations of Γ, and therefore it has no double points, whose locus would give rise to exceptions for the simple transitivity of Γ.

In conclusion: an Abelian variety V_{q} may always be represented without branch points on another, simple or multiple, Abelian variety W_{q} which is imprimitively contained in the Jacobi variety corresponding to a curve of genus p (≥ q).

And conversely: any irreducible variety V_{q}, which can be represented without branch points on a, simple or multiple, Abelian variety W_{q} imprimitively contained in a variety of Jacobi, is itself an Abelian variety.

In order to prove this theorem, let us consider the correspondence of type (s, s), in which a given group (A, B, …) of the involution I′ corresponds to another group of conjugate points (A′, B′, …); this correspondence is defined by a transformation of the Abelian group of W_{q} which sends its homologous points, \overline{A} and \overline{A}′, one to another. The problem is to prove that this correspondence of type (s, s) is the sum of s birational correspondences, and so, for instance, that the point A′ depends on A not only algebraically, but also in a one-to-one way, thus rationally. In order to do this, we let the point A vary continuously on V_{q} describing a closed cycle γ, and we see that A′ moves also continuously, equally describing a closed cycle, and therefore it cannot be interchanged with, say, B′, but it certainly has to come back to A′.

If the cycle described by A is a zero cycle, then our assertion is obvious, since it cannot enclose any branch point of the (s, s) correspondence and the involution I_{s} has no double points. But the same can be proved for a nonzero cycle on the Riemannian space which corresponds to the variety V_{q}. And the proof stems from a continuity argument, by taking into account that, if we let A′ vary in such a way that at the end it coincides with A, then the correspondence (AA′) between the cycle γ and the path γ′ unambiguously described by the point A′ homologous of A, is susceptible of varying continuously until it becomes the identity; thus if the path γ′ is assumed to be open, it cannot become closed in the limit, since the conjugate points of the involution Is are always distinct.

Continuation: the genus of the Abelian varieties 9.10 350

So it turns out that the variety V_{q} possesses a simply transitive group of birational transformations into itself, since there is just one of its transformations carrying A to another point A′. In particular there will be a birational transformation which carries A to a point B conjugate to it in the involution I_{s}, so the involution will be generated by a finite group G_{s} of s transformations.

Now the birational transformations (AA′) which we have found on V_{q} (and that correspond to the permutable transformations of the s-tuple variety W_{q}) cannot be said, a priori, to be permutable. Since they preserve the involution generated by the group G_{s}, whose groups correspond to the points of W_{q}, we may only say that, by transforming one of them, π, by means of another one, T, one gets a transformation

T ΠT^{−1} = Π′

which differs from Π by a cyclic transformation Ω of G_{s}:

Π′ = ΠΩ.

But Ω has to remain unchanged as T varies in the continuous group which contains the identity, so

Ω = 1.

We thus deduce that V_{q} possesses an ∞^{q} simply transitive group of permutable transformations into itself, that is, it is an Abelian variety.

Note The transcendental theory of the Abelian varieties V_{q} is contained in the classical papers by Jacobi, Riemann and then by Poincaré, Picard; a deeper study of the (hyperelliptic) surfaces – the case q = 2 – has been made by G. Humbert and then also by the Italian school.

9.10 Continuation: the genus of the Abelian varieties

We want to complete what we have said about the Abelian varieties, by computing their (principal, geometric) genus.

We start from the Jacobi varieties, proving that: The Jacobi variety, V_{p} = J_{p}, corresponding to a curve of genus p, has genus one.

We shall prove that, on the J_{q} of the q-tuples of points of a curve C of genus p (q ≤ p), the variety K of the q-tuples which belong to the groups of a g_{2p−2}^{q−1} contained in the canonical series of C, is a canonical variety. Let us first of all take q = 2 (p ≥ 2). On the surface J_{2} of the (unordered) pairs of points of the curve C, one has ∞^{1} irreducible curves M, which are birationally equal to C, each corresponding to the series of the pairs with a fixed point; thus the curves M form, on the surface, a system of index two.

Let us take a g_{2p−2}^{1} contained in the canonical series of C and let K be the image curve of the pairs which belong to this ∞^{1} series; let us determine the intersections of K with a curve M. In order to do this, let us consider the curve M corresponding to the series of the pairs of points with a (generic) fixed point A of C and let

A, A_{1}, A_{2}, …, A_{2p−3}

be the group of the g_{2p−2}^{1} which contains A. The points corresponding to the pairs

(AA), (AA_{1}), …, (AA_{2p−3})

belong to M, and form a canonical group on it; on the other hand, all these points, except the first, also belong to K. Now the group of these common points of K and M is residual to the characteristic series of M with respect to the canonical series. Let us in fact consider, together with M, the curve M′ which represents the pairs of points of C having another common point B: M and M′ intersect at the point homologous to the pair (AB), and when B indefinitely approaches A this tends to the point (AA); thus (AA) is a group of the characteristic series on the curve M. Now, since K cuts out on every curve of the system M a residual group of the characteristic series, K itself will be a canonical curve of J_{q} by the third criterion of equivalence (see Chapter III, Section ??). Q.E.D.

After having thus proved our theorem for q = 2, we shall prove it for q = 3 and so on by induction. In fact, if we assume that the theorem is true for the variety of the groups of q − 1 points of C (q ≤ p), we can prove that the theorem is also true for the J_{q} of the q-tuples.

To see this, let us consider on J_{q} the ∞^{1} series of irreducible and birationally equivalent varieties M_{q−1}, each representing the q-tuples with a fixed point: this series has index q. Let K_{q−1} be the variety of the q-tuples belonging to the groups of a g_{2p−2}^{q−1} contained in the canonical series of C, and let us determine the intersection of K_{q−1} with M_{q−1}. Let g_{2p−3}^{q−2} be the residual series of the point A with respect to the series that we are considering. If we add A to the groups of q − 1 points belonging to the groups of g_{2p−3}^{q−2}, then we obtain a variety K_{q−2} which is contained in both the M_{q−1} corresponding to the point A and the K_{q−1} which is the image of the g_{2p−2}^{q−1}. Now if we add to K_{q−2}, on the M_{q−1}, the variety M_{q−2} of the (q − 1)-tuples having A as a fixed point (that is, the variety of the groups of q points in which A appears twice) we obtain a (impure) canonical variety of the M_{q−1}; this follows from the theorem which we assume to be true by induction, since this variety corresponds to the series of the groups of q − 1 points belonging to the groups of the series

g_{2p−3}^{q−2} + A,

a series having the fixed point A and contained in the canonical series of C.

So K_{q−1} cuts out on M_{q−1} a residual variety of M_{q−2} with respect to a canonical variety; therefore, by the third criterion of equivalence (which can be easily extended from the case of the curves to the case of varieties of any dimension) it suffices to prove that M_{q−2} is a variety of the characteristic system of M_{q−1}. Indeed, the M_{q−1} corresponding to a fixed point A and the M′_{q−1} corresponding to a fixed point B have the variety \overline{M}_{q−2} of the q-tuples with the two fixed points A and B as their common variety, and, when B tends to A, M′_{q−1} becomes infinitely near to M_{q−1} and \overline{M}_{q−2} just tends to M_{q−2}.

So the theorem that we have stated is proved, and we may set p = q; we have: on the Jacobi variety, which represents the groups of p points of a curve of genus p, the variety K_{p−1} of the special p-tuples is a (a priori impure) canonical variety. But this is just an exceptional variety, which can be transformed in a variety of dimension p − 2 on the variety, birationally equivalent to J_{p}, which represents the groups of inequivalent p points on the curve C of genus p. Indeed one obtains this transformation by using the involutory transformation which interchanges on C the groups of p points which are residual with respect to a g_{2p}^{p}.

Therefore as a consequence we have that the Jacobi variety J_{p} has genus one.

The theorem we have established for the Jacobi varieties J_{p} = V_{p} can now be easily extended to the more general Abelian varieties; we briefly sketch its proof. First of all we refer to a simple V_{q} imprimitively contained in a V_{p} (q < p); this V_{q} belongs to a series of varieties of the same dimension, a series of index one; thus the (impure) canonical variety of V_{p} cuts out one canonical variety on each one of these V_{q}. Secondly the theorem also extends to the case of a V_{q} which is represented, without a branch locus, on a multiple variety W_{q} contained in a V_{p} (see Section ??). In fact the (1, s) transformation of W_{q} to V_{q} maps the canonical variety of W_{q} to a canonical variety of V_{q}^{(53)}.

In conclusion we can state: The Abelian varieties V_{q} (q ≥ 1) always have genus one, and their pure canonical variety is of order zero.

Note The computation of the genus of the Jacobi varieties V_{p}, and in general of the Abelian varieties, is immediate from a transcendental point of view, since one constructs at once a p-tuple integral of the first kind attached to V_{p}. The geometric proof of the theorem that we have treated above, was given by Severi in a note of 1911^{(54)}.

(53)Theorem of Painlevé–Castelnuovo, proved for q = 1 in Enriques–Chisini, Lezioni. Libro V, Chapter I, §9 (Vol. III, page 72). For q = 2 see Chapter V, Section ?? of this treatise. The theorem may be extended without difficulties for all values of q.

(54)Sulle superficie e varietà algebriche irregolari di genere geometrico nullo, Rendic. Acc.

Lincei, 1911.

Irrational pencil on the surfaces with geometric genus zero 9.11 353

Another geometric proof of the same theorem was given by B. Segre^{(55)}; it consists in showing that the images on the corresponding Jacobi variety of the linear series g_{2p}^{p−1} contained in a g_{2p}^{p} on a curve of genus p are the varieties V_{p−1} of a linear system whose characteristic system plus the intersection of the image of the variety of the special groups, forms the canonical system on a V_{p−1}. We limit ourselves to explain this fact in the simplest case p = 2, but we also point out that the extension of the argument for p > 0 is indeed less easy than one could wish.

Let us consider a curve C of genus p = 2, and a linear series g_{4}^{2} on it: the g_{4}^{1} contained in this series give rise, on the Jacobi surface V_{2}, to the curves of a net, without fundamental curves, which have a variable intersection A with the exceptional curve V_{1}, image of the canonical g_{2}^{1} of C, and a base point B = V_{0} at the point which is its transform by means of the involution defined by the g_{4}^{2}. The canonical curves K of V_{2} are, a priori, defined on the surface by the property of cutting out on a generic L residual groups of the characteristic series of |L|, to which has to be added the intersection A of L with the exceptional curve V_{1}, or the base point B (that is, the transform of V_{1}).

Therefore one has to prove that the series A + B + g_{6}^{2} on a L is (totally) contained in the canonical series g_{8}^{4}. And the assertion essentially derives from the geometric interpretation of the Zeuthen’s formula which gives the relation between the two canonical series of the curves C and L in a (2, 3) correspondence^{(56)}.

Thus the net of the curves L on V_{2}, or rather on a transformed surface without exceptional curves, turns out to be of genus π = 5, of effective degree 6 and virtual degree n = 8, (n = 2π − 2), it has two base points, and canonical characteristic series. The possess of such a net |L|, without proper fundamental curves, yields in fact that V_{2} is a surface of genus p_{g} = 1 with a canonical curve of order zero. Q.E.D.

9.11 Irrational pencil on the surfaces with geometric genus zero

From the fact that an Abelian variety V_{q} of dimension q ≥ 2 has geometric genus equal to one, one deduces that “any non-exceptional surface belonging to an Abelian variety V_{q}, has the geometric genus p_{g} > 0”.

(55)Determinazione di certi gruppi covarianti di due o piu’ serie lineari, Rendic. Circ. Mat.

di Palermo, 1932.

(56)This passage turns out to be very easy (for p = 2) if one introduces the auxiliary curve M which represents the pairs of points of L complementary with respect to a g_{4}^{1} contained in the g_{4}^{2}.

For q = 2 this is evident, since F coincides with V_{2}. To prove the theorem in general one needs to remark that, for q > 2, “F is contained in a continuous system of analogous surfaces on V_{q}, of dimension at least q − 2”.

In fact, if one applies to F the ∞^{q} transformations of the group Γ of V_{q}, one has, in general ∞^{q} surfaces analogous to F; the case in which F admits a continuous series of transformations into itself gives an exception; but in any case (by the simple transitivity of the group Γ) the transformations of F will be at most ∞^{2}, and therefore one has at least ∞^{q−2} surfaces which are transformed of F.

With this in mind, the above-stated property that F is of positive geometric genus is a consequence of the general property that: on a variety V_{q} of dimension q, which has geometric genus P > 0, any surface which belongs to a continuous system of dimension at least q − 2, has the genus p_{g} > 0.

Let us prove this proposition referring first of all to the case q = 3, then to the case q = 4, etc.

If we assume q = 3, then the canonical surface which by hypothesis exists on V_{q} (even if it is just a single surface of order zero) intersects F along a curve which belongs to the residual system of the characteristic system of F with respect to the canonical system^{(57)}; this suffices to prove the existence of the latter, that is, that the genus of F is p_{g} > 0.

If q = 4, then the surfaces F, which are at least ∞^{2}, can be arranged in at least ∞^{1} simply infinite continuous systems, which therefore give rise to an at least ∞^{1} series of varieties V_{3}; now, since the genus of V_{4} is bigger than zero, a canonical variety of V_{4} cuts out on each one of the aforementioned V_{3} some special surfaces, which are residual of the characteristic system of V_{3} with respect to the canonical system, thus these V_{3} have genus P¿0. Whence it follows that the ∞^{1} surfaces describing them are also of genus p_{g} > 0.

With an analogous argument one may pass from q to q + 1 and prove the same assertion for a variety V_{q} of any dimension.

The theorem that we have proved, to the effect that the surfaces belonging to an Abelian variety have genus p_{g} > 0, enables us to make more precise the statement of the characteristic property of the irregular surfaces for surfaces with genus p_{g} = 0, by proving that: The irregular surfaces of genus p_{g} = 0 possess an irrational pencil of genus −p_{a} = p of curves. And conversely, any irregular surface of genus p_{g} = 0 possessing an irrational pencil of genus p (> 0) is an irregular surface of arithmetic genus −p.

Let F be an irregular surface – for the time being we do not make any assumption about the values of its characters – and let us consider a simply infinite continuous system of curves {C} on it, of index i > 2, which we may assume to be simple, that is, such that the i curves C passing through a generic point of the surface do not have other common points as a consequence.

(57)See Chapter VIII, Section ??.

Let us denote by

C_{i} = ∑ C

the sum of the i curves of {C} which go through a point of the surface F. Three cases may occur:

(a) the ∞^{2} curves C_{i} are all equivalent curves on the surface F;

(b) the aforementioned C_{i} are arranged in ∞^{1} simply infinite systems of equivalent curves;

(c) the C_{i} are pairwise inequivalent, or rather each of them is equivalent to a finite number, s ≥ 1, of others.

Now we shall prove that in case a) all the curves C are equivalent, that is, the series {C} is contained in a linear system.

In order to do this it suffices to prove that the ∞^{1} curves C cut out a series of equivalent groups on any curve M traced on F, and use, for example, the first criterion of equivalence. Indeed the property we claim for the series cut out on the curve M is a consequence of a well known equivalence criterion by Severi–Castelnuovo which concerns the ∞^{1} series of groups of points belonging to a curve^{(58)}.

Secondly we will prove that in case b) an irrational pencil of curves can be constructed on the surface F.

In fact let a point A move on F, so that the curves C_{i} going through it are equivalent to each other; since the condition that we have imposed is algebraic, the point A describes an algebraic curve K; and it is clear that the curves K defined in this way on F cannot intersect, thus they have to form a pencil (which has no base points, simple for the surface). We assume, for simplicity, that the K are irreducible (since otherwise we may consider, instead of the K, their irreducible components), and we shall prove now that the pencil (K) is irrational.

Indeed the same consideration made before shows that the curves C of {C} must cut out ∞^{1} equivalent groups of points on any K. So, by the second equivalence criterion^{(59)}, the C are equivalent up to curves of the pencil, that is, – denoting by C and C_{0} any two curves of {C} – one has

C + ∑ K = C_{0} + ∑′ K

(58)See III, 13. See also Enriques–Chisini, Lezioni. Libro V, Chapter IV, §42 (Vol. III, page 483)

(59)See III, 13.

where the summations ∑ and ∑′ contain the same number of summands. Consequently, if the curves C are not all equivalent one to another, then the pencil (K) cannot be linear, since otherwise by adding and subtracting the same number of curves K one could pass from a C to an equivalent C_{0}.

An objection can be moved to this argument, based on the second equivalence criterion, if the pencil (K) contains some reducible curves, like

K = K_{1} + K_{2} and \overline{K} = \overline{K}_{1} + \overline{K}_{2};

in fact then we could have, for instance,

C_{0} = C + K_{1} − \overline{K}_{1} + · · · .

But the doubt raised here can be eliminated if one considers that the curves C_{0} and C belong to one and the same continuous system {C} and, as C_{0} varies in {C}, the components of the reducible K, whose number is finite, stay fixed; whence, by continuously moving C_{0} to C, it necessarily follows that

C_{0} → C, K_{1} = \overline{K}1.

Finally, in case c) one immediately proves that the genus of the surface must be p_{g} ≥ 1. In fact if one represents the curves of {C} with the points of a curve L, then the ∞^{2} curves C_{i} define on the same L a series g_{i}, which can be considered as a doubly infinite, simple or multiple, variety, contained in the Jacobi variety corresponding to the curve L, which represents the inequivalent linear series of L.

The above discussion makes clear that for an irregular surface F of genus p_{g} = 0 any given series {C} on it, of index i, formed by inequivalent curves, gives rise to case b), so that one finds an irrational pencil of curves K on the surface.

Then we can see that the genus of (K) is equal to the arithmetic genus with the sign changed: p = −p_{a}. Because, if it were p < −p_{a}, in force of the second equivalence criterion, one could construct a series of inequivalent curves on F, that would cut out on any K inequivalent groups of points, and so (since case b) has to occur), another irrational pencil of curves; but the existence of two irrational pencils, (K) and (M), on F would imply that this surface must have the geometric genus p_{g} > 0: this is evident if the two pencils are formed by unisecant curves^{(60)}; whereas the case in which the curves of the two pencils cut each other at a certain number s > 1 of points leads to the representation of F on a multiple (s-tuple) surface, with two pencils of unisecant curves, whence it follows, a fortiori^{(61)}, that its genus is p_{g} > 0.

(60)In fact a canonical curve of F is obtained by adding to the curves of (M) which pass through the points of a canonical group of a K, the curves of (K) which pass through the points of a canonical group of a M.

(61)See V, 7.

A note on the surfaces of irregularity one 9.12 357

In conclusion the characteristic property of the irregular surfaces of genus p_{g} = 0, which consists, according to the above statement, in possessing an irrational pencil of genus p = −p_{a}, has been proved.

Note The theorem on the existence of an irrational pencil on the irregular surfaces of genus p_{g} = 0, has been stated by Enriques as a consequence of the characteristic property of the irregular surfaces, and precisely in his quoted note in Rendic. Acc. Bologna in 1904. Indeed, already in 1900 Castelnuovo remarked and communicated to the colleague that “from the existence of a continuous system of inequivalent curves on a surface of genus p_{g} = 0 the existence of an irrational pencil can be deduced”. This property, recalled by Enriques in his Note, mentioning its author, was justified with a transcendental argument as follows.

Let {C} be an ∞^{1} continuous system of inequivalent curves, on the surface F of genus p_{g} = 0. Then, as G. Humbert for the first time remarked^{(62)}, one can construct some simple integrals (of total differentials) of the first kind on F, by summing the values taken by the Abelian integrals of the first kind attached to the series {C} in correspondence with the i (> 1) curves C passing through a given point A of F. And these simple integrals I_{1}, I_{2}, … will necessarily be one a function of the other, otherwise – as was indicated by Noether^{(63)} – one could deduce from them the existence of a double integral of the first kind, whence p_{g} > 0.

After this Castelnuovo remarks that the curves I = const. are composed with the algebraic curves of an irrational pencil which are defined by moving A on the surface in such a way that the i curves C going through a point describe equivalent groups of curves. This argument, especially in the form that it acquires with the considerations of Castelnuovo’s Note^{(64)}, was translated by Severi^{(65)} into the geometric one we treated above.

9.12 A note on the surfaces of irregularity one

In view of possible applications it is convenient to remark that the arguments which we developed in the previous section also lead to the characterization of the surfaces of irregularity q = p_{g} − p_{a} = 1; for these the following theorem holds: The surfaces F of irregularity q = 1 possess an elliptic pencil of curves (of genus π ≥ 0). Any continuous system of curves on them is obtained from a linear system by adding to it the difference of two curves of the aforesaid pencil.

(62)Comptes rendus, 1893; J. de Math., s. 4, t. X, 1894.

(63)Ueber die totalen algebraischen Differentialausdrücke, Math. Ann., Band 29, 1887. See Picard et Simart, ibidem, Vol. I, Chapter V, §15.

(64)Sugli integrali semplici appartenenti ad una superficie irregolare, Rendic. Acc. Lincei, 21st of May, 4th and 18th of June, 1905. See Memorie scelte, page 473 and following pages (see especially n. 10, page 490).

(65)Sulle superficie e varietà algebriche irregolari di genere geometrico nullo. Rendic. Acc.

Lincei, s. V, Vol. XX, 1911.

Let us remark that the Picard variety of F is an elliptic curve. Let us take now on F an ∞^{1} system of inequivalent curves, {C}, of index ν > 1: There will be ν curves C passing through any point P of F, and, as P varies, one obtains ∞^{2} groups of ν curves; therefore, starting with a particular one P = \overline{P} of these points, there are ∞^{1} points P such that the group of ν curves C passing through them is equivalent to the group of the curves passing through \overline{P}. The locus of these points P is a curve K through \overline{P}, and all the analogous K form an elliptic pencil of curves (K) on F. Once the first part of the theorem has been thus proved, in order to prove the second part it suffices to point out that the curves C of the continuous system {C} cut out on any K equivalent groups, and accordingly (by the second equivalence criterion) two inequivalent C only differ by the difference of two K.

Surfaces with geometric genus zero

10.1 Introduction

The characteristic property of the irregular surfaces of genus p_{g} = 0, which we have established in Section ?? and ?? of the previous chapter, enables us, as we shall see, to characterize – in a complete way – the surfaces with genus p_{g} = 0, and – in particular – to assign the necessary and sufficient conditions defining the family of ruled surfaces by the vanishing of the plurigenera.

The development of our deductions will proceed in the following order. First of all we shall treat the case of surfaces which contain systems of

curves of genus π and degree n > 2π − 2, showing that these can be transformed into ruled surfaces, either rational or irrational; the rational case corresponds (as we know) to the surface being regular.

After this, we shall treat a lemma concerning the reducible curves in an irrational pencil; this will enable us to give to the second equivalence criterion a more precise and simpler form, which we will need in what follows.

Whence we are meanwhile going to deduce that the surfaces of genus p_{g} = 0 and p_{a} < −1 are certainly birational to ruled surfaces of genus p = −p_{a}. Then we will be left with the study of the surfaces with the genera p_{g} = 0 and p_{a} = −1, which give rise to:

(1) surfaces birational to elliptic ruled surfaces, and

(2) surfaces which contain an elliptic pencil of curves of genus π ≥ 1.

We shall deal with the two cases which occur, that is, π > 1 and π = 1, separately, and we shall prove that, in both cases, the surface contains another linear pencil of elliptic curves, which are birationally identical.

Surfaces with curves of genus π and degree n > 2π − 2 10.2 360

Finally we shall go deeper into the analysis of the family of (elliptic) surfaces defined by the above properties, we shall give their construction, compute their plurigenera and see that they admit a simply infinite, elliptic group of birational transformations into themselves, etc. We will finish the chapter with the characterization of the surfaces with all plurigenera equal to zero and of those with the plurigenera equal to zero and one.

10.2 Surfaces containing a system of curves of genus π and degree n > 2π − 2

Let F be a surface which contains an irreducible linear system of curves |C|, at least ∞^{2}, for which n > 2π − 2. The curves C′, C′′, etc., the successive adjoint curves of |C| cut out on C a decreasing number of points:

2π − 2, 4π − 4 − n, 6π − 6 − 2n, …

and therefore there are a finite number of them; in other words, the adjunction process starting with the linear system |C| has to terminate after a finite number of steps, giving rise to a last adjoint system |C^{i}|, of dimension ≥ 0, which has no further adjoint curves of order > 0.

We already know of one case in which the adjunction terminates in this way: that is, the case of the regular surfaces of genus p_{a} = p_{g} = 0, with bigenus P = 0, in which the surface F is known to be rational^{(1)}. After this case, it is convenient then to consider the hypothesis that the surface on which our system lies is irregular, of genus p_{g} = 0 and p_{a} = −p ≤ 1, and therefore contains an irrational pencil of genus p of curves K. If the K have genus ρ = 0, then F can certainly be transformed into a ruled surface of genus p^{(2)}. Then we need to examine the hypothesis that the genus of the K is ρ ≥ 1 and derive a contradiction from this. It will be useful to treat the two cases separately:

(1) p > 1

(2) p = 1.

The first case: p > 1.

Let us denote by s the number of intersections of the C with the K, and let us notice that the curves C +pK −p\overline{K} give rise to an ∞^{p} series of distinct linear systems cutting out equivalent series on the K; whence it follows that

s > 1.

(1)Castelnuovo’s theorem, see Chapter V, Section ??.

Now, since the (virtual) canonical curves of F intersect the curves K, of genus ρ, in 2ρ − 2 points, the curves C′ intersect the K at s + 2ρ − 2 points, the curves C′′ at s+2(2ρ−2) points, and so on. Finally the curves of the last adjoint system |C^{i}| cut out on the K a number of points which a fortiori is m ≥ 2. But this conclusion immediately yields a contradiction, provided that the C^{i} are irreducible: in fact, since they possess an irrational involution γ_{m}^{1} of genus p, which is cut out on them by the K, they must have the (effective and virtual) genus greater than or equal to m(p − 1) + 1 > p, and therefore they must possess an adjoint system of dimension > 0.

The conclusion (that contradicts the hypothesis we started with) can be extended to the case in which C^{i} is a sum of irreducible curves, since in any case there is a component of C^{i} which has to cut out at least two points on K: this follows from the remark that the C^{i} belong to a continuous series of inequivalent curves, which can be obtained by adding to them the difference of two curves of the pencil (K): K − \overline{K}.

The second case: p = 1.

We proceed as in the previous case: starting with the system |C| with characters π and n > 2π − 2, we obtain a last adjoint system |C^{i}|, consisting of curves meeting the K at m ≥ 2 points; but it seems here that the adjoint system of C^{i} can actually be empty if the genus of C^{i} (effective and virtual) is equal to 1, as well as the genus of the pencil of the K. But our hypothesis implies that the curves C^{i}, if irreducible (or their components), belong to an ∞^{1} elliptic series of inequivalent elliptic curves; and since the curves of this series cannot form another irrational pencil (distinct from |K|), we must admit that the curves C^{i} + K − \overline{K} give rise to a continuous system \{C^{i}\} of curves of degree ν ≥ 1.

Whence we can deduce that the surface F possesses ∞^{1} rational curves, and therefore it has to be birational to an elliptic ruled surface, thus contradicting our assumptions. In fact, if ν = 1, then the ∞^{1} elliptic series of unisecant curves C^{i} contains ∞^{1} pairs of equivalent curves, and (by Lüroth’s theorem)^{(2)} their points of intersections describe ∞^{1} rational curves on F. If one has instead ν > 1, then the (irreducible) curves C^{i} with ν −1 fixed points give rise to an ∞^{1} series of unisecant elliptic curves, so that the previous argument may again be applied.

So we have proved the fundamental theorem:

The surfaces containing an irreducible linear system at least ∞^{2}, of genus π and degree n ≥ 2π − 2, are birationally equivalent to ruled surfaces, either rational or irrational.

(2)See Enriques–Chisini, Lezioni, Libro II, Chapter I, §3 (Vol. I, page 169) and Libro V, Chapter I, §7 (Vol. III, page 48)

And from this theorem, as we announced in Chapter IV, Section ??, one deduces that:

The surfaces belonging to the family of the ruled surfaces are the only surfaces containing infinitely many exceptional curves; all the surfaces outside of this family can be transformed into surfaces without exceptional curves.

As for the fundamental theorem, it is convenient to add here that it holds in the most general assumptions, regardless of the dimension of the system |C| whose characters verify the inequality n > 2π − 2, provided that the curves C do not contain exceptional curves as components.

In order to prove this, let us remark that the character a(C) = n−(2π−2) is an additive character of the linear systems to which it refers, and therefore it cannot be positive for the sum of two systems without being positive for some of the summands. Thus it suffices to prove that:

Any surface F on which one has an irreducible, non-exceptional curve C of genus π and degree n > 2π − 2 certainly belongs to the family of the ruled surfaces.

In fact the s-multiple of the curve C, with s sufficiently large, has genus and degree

π_{s} = sπ + \frac{s(s − 1)}{2} n − s + 1, n_{s} = s^{2}n;

and therefore it has dimension

r ≥ p_{a} + n_{s} − π_{s} + 1 > 0,

and it forms an irreducible system (for n > 0) for which

n_{s} > 2π_{s} − 2.

And we also explicitly point out that what we have said still holds for π = 0, since, having excluded that C be exceptional and so that n = −1, we have

n ≥ 0;

and indeed, in the case n = 0, the only one which has to be examined, the multiple |sC|, for very large s, turns out to be always composed with the rational curves of a pencil, whence it follows that F can be transformed into a ruled surface.

Note The theorem that the surfaces on which the adjunction terminates, and therefore those containing linear systems with the characters n and π for which n > 2π − 2, belong to the family of the ruled surfaces is the fundamental result of the Memoir by Castelnuovo–Enriques, Sopra alcune questioni fondamentali nella teoria delle superficie algebriche, Ann. di Mat., 1901.

The original proof of this theorem did not use the property of the irregular surfaces of genus p_{g} = 0 of containing an irrational pencil of curves (this property was discovered, as we have said, in 1904). Instead of this, it exploited an inequality coming from the comparison of the Noether’s expression of the Zeuthen–Segre invariant and of the theorem of Riemann–Roch. We give a brief sketch of this argument.

If (on the surface F that has to be of genus p_{g} = 0) the adjunction process starting with a linear system |C| terminates, so that one reaches a last adjoint system |C^{i}|, then the genus of the latter system must be π ≤ −p_{a} = p.

Let us assume, for simplicity, that |C^{i}| is irreducible, ∞^{1} at least; if n denotes its degree, we have that the Zeuthen–Segre invariant of the surface is

δ − n − 4π = −12p − p^{(1)} + 9, (p = −p_{a})^{(3)}

whereas the theorem of Riemann–Roch gives its dimension

r ≥ n − π + 1 − p.

Whence we deduce that

5π + r ≥ 11p + p^{(1)} − 8 + δ ≥ 11p + p^{(1)} − 8.

Then a suitable discussion, in which one has to distinguish between the assumptions p^{(1)} ≤ 1 and p^{(1)} > 1, leads us to find on the surface: either

(1) a linear system of curves of genus π and dimension

r ≥ 3π − 5,

whence it follows that the surface is rational or birational to a ruled surface of genus π^{(4)}; or

(2) a linear system of curves that are reducible into rational curves, whence it equally follows that the surface is birational to a ruled one.

This sketch just gives an idea of the way which was followed: a very detailed analysis of various troublesome circumstances and a special treatment of the cases in which p < 3 are then needed.

A lemma on the reducible curves in a pencil 10.3 364

The considerably simpler proof of our fundamental theorem, based on the irrational pencil belonging to an irregular surface of genus p_{g} = 0, has been described in the Lezioni by Enriques–Campedelli^{(5)} published in Rendic. Sem. Mat. R. Università di Roma (1934). This proof has been developed here in a simpler form, avoiding to invoke, as it was done there, the theorem by Castelnuovo and Enriques on the double planes with all plurigenera equal to zero. Instead of this, we have the elementary consideration we made above, that an ∞^{1} series of pairs of equivalent elements on an ∞^{1} elliptic system is a g_{2}^{1}, thus it is certainly rational.

(3)Here p^{(1)} denotes the relative linear genus of the surface.

(4)This is a consequence of the argument which Enriques used in the Note Sulla massima dimensione dei sistemi lineari di curve di dato genere appartenenti ad una superficie algebrica, Atti Acc. Torino, 1894.

We finally say that the extension of the fundamental theorem to the case of a single, non-exceptional curve, of genus π (≥ 0) and degree n > 2π − 2, was brought into light by L. Campedelli, Intorno alle superficie algebriche su cui esistono curve di genere π e di grado n > 2π − 2, Rendic. Acc. Lincei, 19332) and the same Author pointed out how an explicit use of this property leads to a simplification of our theory in many points.

Note A special importance also has the proposition that we have used in order to prove the fundamental theorem of this section. The surfaces on which the adjunction terminates belong to the family of the ruled surfaces.

Therefore, recalling the results we gave at the end of Chapter VI, Section ??, we may state that: The surfaces which do not belong to the family of the ruled surfaces, that are therefore birational to a model without exceptional curves, have the absolute linear genus

p^{(1)} ≥ 1.

In Chapter VI, Section ??, we only saw that for p^{(1)} ≤ 0 all the plurigenera must be P_{i} = 0, since the adjunction terminates; here we see more, that is, that the surface must be birational to a ruled surface (what contradicts the hypothesis that the exceptional curves can be eliminated).

10.3 A lemma on the reducible curves in a pencil

In view of the developments which follow, we need to analyze what is the valency of a however reducible curve, belonging to a rational or irrational pencil, in the count of the curves with a double point, whose number appears in the expression of the Zeuthen–Segre invariant (Chapter V, Sections 2–3). For simplicity, we shall consider surfaces without exceptional curves, thus not belonging to the family of the ruled surfaces, and we will prove that: For the computation of the Zeuthen–Segre invariant of a surface possessing a pencil |C|, rational or irrational, of irreducible curves of genus π > 0, without base points, any reducible curve of the pencil has the valency ≥ 0.

(5)See Chapter III.

Accordingly, the number δ or Δ of the curves of the pencil with a double point has always to be considered ≥ 0. If δ = 0 or Δ = 0, then the pencil cannot contain any reducible curve, except, when the curves of the pencil are elliptic (π = 1), for some multiple elliptic curves (of degree zero).

To justify these assertions, it is convenient to recall Godeaux’s formula that, as we have said (Chapter V, Section ??), expresses the valency of a particular curve \overline{C} of the pencil, however reducible into multiple components. Let us denote by \overline{δ} the valency of a particular C = \overline{C} that we assume to be decomposable into multiple components of genus ρ_{1}, ρ_{2},…:

\overline{C} = t_{1}θ_{1} + t_{2}θ_{2} + ⋯

meeting each other at

s_{hk} = θ_{h}θ_{k}

points. One has:

\overline{δ} ≥ ∑(t_{i} − 1)(2ρ_{i} − 2) + ∑ s_{hk}(t_{h} + t_{k} − 1),

where strict inequality > occurs in the presence of possible complications, for instance if some of the components θ_{i} possesses a double point, or if some of the points of connection of two components θ_{h} and θ_{k} happen to coincide with a common point of r > 2 components^{(6)}.

Let us remark that, since the surface has no exceptional curves, one has, for every i,

s_{ii} = θ_{i}θ_{i} ≤ 2ρ_{i} − 2;

and therefore

\overline{δ} ≥ ∑ s_{ii}(t_{i} − 1) + ∑ s_{hk}(t_{h} + t_{k} − 1);

(6)The inequality \overline{δ} ≥ 0 was proved by Castelnuovo–Enriques (Ann. di Mat., 1901) in the simplest case in which \overline{C} breaks up into a multiple irreducible curve θ_{1} of genus ρ_{1} and into another simple component θ_{2} which intersects it at s_{12} = θ_{1}θ_{2} (> 0) points. The general formula which is written above was given, by extending the argument used in that case, by L. Godeaux, Sur le calcul de l’invariant de Zeuthen–Segre. (Mem. de la Soc. des Sciences de Hainart, 1920) and then also by L. Campedelli, Sul computo dell’invariante di Zeuthen–Segre; Ancora sul computo ecc., Lincei, 1934. The latter author has proved that \overline{δ} ≥ 0 and more precisely that one can have \overline{δ} = 0 only when the pencil (C) is of genus π = 1 and the C is a multiple elliptic curve of genus ρ = 1:

\overline{δ} = (t + 1)(2ρ − 2) = 0.

This is in fact the property stated in our lemma, that also appears in the Lezioni by Enriques–Campedelli (Sem. Mat. di Roma, 1934) and of which A. Franchetta has recently given a much simpler new proof.

Surfaces with genus p_{g} = 0 and p_{a} < −1 10.4 366

moreover, since one has s_{hk} ≥ 0 if h ≠ k,

∑ s_{hk}(t_{h} + t_{k} − 1) = ∑ s_{hk}(t_{h} − 1) + ∑ s_{hk}(t_{k} − 1) + ∑ s_{hk} ≥
≥ ∑ s_{hk}(t_{h} − 1) + ∑ s_{hk}(t_{k} − 1).

Therefore one has

\overline{δ} ≥ ∑ s_{ii}(t_{i} − 1) + ∑ s_{hk}(t_{h} − 1) + ∑ s_{hk}(t_{k} − 1).

Let us introduce the curve

C^{∗} = θ_{1} + θ_{2} + ⋯

and denote by ρ and ν its virtual characters. The expression on the right hand side of the above inequality equals the number of the virtual intersections C^{∗}(\overline{C} −C^{∗}); thus the theorem will be proved if we show that C^{∗}(\overline{C} −C^{∗}) ≥ 0, that is, if we deduce a contradiction from the hypothesis

C^{∗}(\overline{C} − C^{∗}) = −i (i > 0).

Noting that C^{∗}\overline{C} = C^{∗}C = 0, this hypothesis implies that

ν = C^{∗}C^{∗} = i > 0;

so the curve rC^{∗} would determine a linear system |rC^{∗}| of dimension d_{r} which, for sufficiently large r would be

d_{r} ≥ p_{a} + r^{2}i − rρ − \frac{r(r − 1)}{2} i − r + 1 = p_{a} + \frac{r^{2}i}{2} − \frac{r}{2}(2ρ − 2 − i) + 1

that is, an infinite of the second order as r goes to infinity.

On the other hand the curve r\overline{C} determines a linear system |r\overline{C}| composed with the curves of the pencil; its dimension is given, for r large enough, by D_{r} = r − p, where p is the genus of the pencil; that is, it is an infinite of the first order as r goes to infinity. But it is clear that the system |rC^{∗}| is partially contained in the system |rC|, thus that d_{r} ≤ D_{r}. The hypothesis we started with gives therefore a contradiction.

Moreover it turns out that we can have \overline{δ} = 0 only if ∑ s_{hk} = 0, and this is possible only if C^{∗} is formed by a single component θ, and if θ θ = 2ρ−2 = 0, that is, if ρ = 1.

10.4 Surfaces with genus p_{g} = 0 and p_{a} < −1

Surfaces F with p_{g} = 0 and p_{a} = −1 10.5 367

Let F be a surface with geometric genus p_{g} = 0 and negative arithmetic genus p_{a} = −p < −1. We know (Chapter IX, Section ??) that it must contain a pencil of genus p of curves K, with genus π ≥ 0. Let us prove, by drawing a contradiction from the hypothesis π > 0, that we must have π = 0 and therefore F must be birational to a ruled surface of genus p.

So let us assume that π > 0 and that F (not birational to a ruled surface) can be reduced to a surface without exceptional curves; its absolute linear genus will be (Section ??, Note)

p^{(1)} ≥ 1.

Let us now write the formula that gives the Zeuthen–Segre invariant of F in relation to the irrational pencil K. We have (Chapter V, Section ??)

Δ − 4 + 4(p − 1)(π − 1) = 12p_{a} − p^{(1)} + 9

and so, for π > 0,

Δ ≤ 13 − 12p − p^{(1)}.

Whence it follows Δ < 0, since p > 1, which contradicts the lemma of Section ??.

Then we conclude that π = 0. Thus

the surfaces with genus p_{g} = 0 and p_{a} = −p < −1, are birational to ruled surfaces of genus p.

10.5 Surfaces F with p_{g} = 0 and p_{a} = −1, possessing an elliptic pencil of curves K of genus π > 1: Lemma I

The analysis of the surfaces with genera p_{g} = 0 and p_{a} = −p = −1 leads to the study of the surfaces F that possess an elliptic pencil of curves K of genus π ≥ 1; and it is convenient to distinguish between the two cases that appear to be possible when F does not belong to the family of the ruled surfaces, namely:

π > 1 and π = 1.

We discuss here the case π > 1.

To start with, we notice that our surface F, that we may assume to be without exceptional curves, has the absolute linear genus p^{(1)} = 1. In fact, by computing the Zeuthen–Segre invariant for the elliptic pencil (K)^{(7)}, we find

I = Δ − 4 + 4(p − 1)(π − 1) = Δ − 4 = −12 − p^{(1)} + 9,

(7)See V, 2 and 3.

so that, since Δ ≥ 0, we deduce p^{(1)} ≤ 1, and then (Section ??, Note)

p^{(1)} = 1.

It is also useful to remark that for our pencil (K) we have as a consequence

Δ = 0,

and therefore (Section ??) the pencil itself (formed by curves of genus greater than one) contains neither a curve K with a double point nor, in any case, a reducible one.

With this in mind, let us proceed by stating a series of lemmas which will finally lead us to a theorem.

LEMMA I.-The surface F has at least an elliptic parabicanonical curve, cutting out on the curves K of the pencil groups of the bicanonical series g_{4π−4}^{3π−4}; and similarly an elliptic para-i-canonical curve for i > 2.

Let us in fact consider the linear system |K′′|, the second adjoint system of a curve K of our pencil. Since p^{(1)} = 1 and the virtual canonical curves cut out groups of 2π − 2 points on the K, the genus of the curves K′ is

π′ = π + 1 + 2π − 2 − 1 = 3π − 2

and so the dimension of |K′′| is

r ≥ −p + π′ − 1 = 3π − 4,

thus at least equal to the dimension of the bicanonical series of K. Now, if the series cut out by |K′′| on K is not complete, then there is at least a pencil of curves K′′ passing through any of its groups, whence we deduce the existence of a curve K′′ which breaks up into K and a bicanonical curve, of genus

3p^{(1)} − 2 = 1 :

this is either an irreducible elliptic curve or a curve which is composed of elliptic curves without common points.

In the case in which |K′′| cuts out on K the complete bicanonical series, let us consider the (second paradjoint) continuous system {K′′}, formed by ∞^{1} linear systems, that we obtain from a particular |K′′| by adding the difference of two curves K_{1} − K_{2} of the pencil K.

Let us remark that all the linear systems forming {K′′} cut out the same bicanonical series on K, and similarly the groups cut out by two inequivalent curves of a paradjoint system (of order 1, 2, 3, …) always turn out to be equivalent.

Lemma II: a sketch of the proof 10.6 369

Whence it follows that we have (at least) ∞^{1} curves of {K′′} passing through a fixed bicanonical group of K, and therefore one has a curve of this system which is reducible into a K and a residual parabicanonical curve: the latter, as the virtual bicanonical curve, has genus

3p^{(1)} − 2 = 1

and therefore it is formed by one or more irreducible elliptic curves without common points.

In a completely analogous way one proves that the paradjoint system {K^{i}} of order i = 3, 4, …, of dimension (2i−1)π −2i, contains (at least) ∞^{1} curves passing through a chosen group of the i-canonical series on a K, and therefore it also contains a curve that is reducible into the K and a para-i-canonical curve, and this is either elliptic or a sum of elliptic curves without common points.

But we cannot say that all these elliptic curves, for instance the parabicanonical and the paratricanonical ones, are formed by distinct irreducible components: in fact all of them might coincide with a multiple paracanonical curve, namely an elliptic curve whose order (the number of its intersections with the K) is 2π − 2.

10.6 Lemma II: a sketch of the proof

On a surface F (of genera p_{g} = 0 and p_{a} = −1, with a pencil of curves of genus π > 1) there exist at least two elliptic curves, not formed by the same components, whose multiples are two parapluricanonical curves, which cut out equivalent groups of points on the K.

We have already constructed a parabicanonical curve (or a parapluricanonical curve of higher order), say C, which is a curve of genus one, formed by simple or multiple irreducible components without common points. Let us try now to construct another curve D, of the same order 4π − 4, which is not parabicanonical, and does not cut out on any K a bicanonical series, but a series whose s-multiple is the 2s-canonical series, cut out by the curve sC.

More precisely, in the case in which the genus π of the K is even, and so we have the divisor s = 2, we aim to construct an elliptic curve d which, although it has the same order 4π − 4 as C, does not cut out on a K a bicanonical group, but a group which belongs to a semiquadricanonical series of this curve. And we will succeed in doing this in the easiest case, in which the following two simplifying assumptions, that we shall discuss in the next sections, are satisfied:

(I) on any curve K a semibicanonical (but not canonical) series \overline{g} can be rationally determined;

(II) a group can be rationally determined inside the aforementioned \overline{g}.

In these assumptions we may first of all rationally determine on a K a semibicanonical series \overline{g} = g_{2π−2}^{π−2} (different from the canonical series g = g_{2π−2}^{π−1}) and then we may also determine a group G_{2π−2} in it; so we may construct a sufficiently ample linear system |L| of curves, which cuts out the corresponding series \overline{g} on any K. Therefore the adjoint linear system of |L|, that is, the system |L′|, cuts out on any K the semiquadricanonical series \overline{g} + g. If we compare now |2L′| and |2K′′| (|K′′| is the second adjoint system of a K) by using the second equivalence criterion (in its more precise form that was given here with the lemma of Section ??), we see that these only differ by a certain number r of curves K_{1}, …, K_{r} of the pencil (K), to be counted twice:

|2L′| = |2K′′ + 2(K_{1} + ⋯ + K_{r})|,

whence we deduce that the system |L′ − (K_{1} + ⋯ + K_{r})| has the same characters (degree, genus and thus virtual dimension 3π − 4) as |K′′|. Therefore it gives rise to a continuous system of dimension 3π − 3 (one more than the dimension of the series cut out on a K), from which we may subtract a K: the residual curve is the semiparaquadricanonical curve that we were aiming to construct: C and d are certainly not equivalent, and they are not even paraequivalent, whereas 2C and 2D are paraequivalent.

In case the assumptions II and I we mentioned above are not verified, the argument can be repeated mutatis mutandis in the following sense:

(1) If the hypothesis II is not verified any more (but I still is), instead of a semibicanonical (noncanonical) series \overline{g}, one may construct on a K a series \overline{g}′ which is an i-multiple of the mentioned \overline{g}, with i an odd integer, for which the the hypothesis is satisfied. So, instead of the system |L|, or rather of its adjoint |L′| which we have mentioned before, one may construct a linear system |M|, cutting out on any K the series \overline{g}′. Then the comparison of |2M| and |2K^{i}| (where K^{i} denotes the ith adjoint system of a K) shows that the former system, from which a certain number of curves K_{1}, …, K_{r} of the pencil (K), counted twice, have to be subtracted, is paraequivalent to the latter. Whence we deduce that the system |M − (K_{1} + ⋯ + K_{r})| has the same characters as |K^{i}|; thus (Section ??) it belongs to a continuous system in which there is a curve reducible into a K and a residual elliptic curve D. d has the same order as C = |K^{i} − \overline{K}|, the two curves are neither equivalent nor paraequivalent, but their doubles are paraequivalent and cut out on the K equivalent groups.

(2) If the hypothesis I does not hold, then we may pass from the surface F to an F^{∗} with the same characters, which is represented on the m-tuple F (m > 1) without branch curve, so that the simplifying hypothesis (which is not verified by F) holds true for F^{∗}. Then we construct a semiparapluricanonical elliptic curve d (of order 4π − 4 or i(2π − 2)) on F^{∗}, whose image on F is still an elliptic curve d (of order m(4π − 4), or im(2π−2)). This curve d is not paraequivalent to the parapluricanonical curve C of the same order (that is, to the parabicanonical curve C = |K′′−\overline{K}| that can be constructed on F according to Section ??); but the doubles 2D and 2C are paraequivalent, and cut out equivalent groups on the K.

In the case in which π is an odd number, we denote by s one of its (odd) prime divisors, and we aim to construct on F an elliptic curve of order 4π −4, cutting out on any K a non-bicanonical group, belonging to a series which is an s-submultiple of the s-bicanonical series. And we will succeed in doing this in the simplest case in which the assumptions I and II, analogous to the ones occurring in the case π even and s = 2, are satisfied.

Indeed in such assumptions a nonspecial series \overline{g} = g_{2π−2}^{π−2}, s-submultiple of the s-canonical series, is given on any K, and a group of points is also rationally given inside it; then we can construct an ample enough linear system |L|, cutting out on any K the aforementioned series \overline{g}. Therefore the adjoint system |L′| (of order 4π − 4) cuts out on K the series \overline{g} + g which is an s-submultiple of the s-bicanonical series. And we can also see that |sL′| differs from |sK′′| (|K′′| is the second adjoint system of a K) by a certain number r of curves K_{1}, …, K_{r} of the pencil (K) to be counted s times:

|sL′| = |sK′′ + s(K_{1} + ⋯ + K_{r})|.

As a consequence of this relation the system |L′ − (K_{1} + ⋯ + K_{r})| has the same characters (degree, genus and dimension 3π − 4) as |K′′| and it is contained in an ∞^{3π−3} continuous system, containing a curve reducible into a K and a D: the latter is a curve of genus 1 and degree 0, the s-multiple of which is paraequivalent to the s-multiple of the parabicanonical curve C.

But, as in the first case (π even and s = 2), it is convenient to examine in which way the argument has to be modified if the aforesaid assumptions I and II are not verified.

The first case: the genus π of the curves K is even 10.7 372

(1) If the hypothesis II does not hold, then, instead of a \overline{g}, of order 2π − 2, different from the canonical series, but an s-submultiple of the scanonical series, we may consider on K a non-pluricanonical series \overline{g}′ = (2π − 2)\overline{g} − (π − 2)g, of order π(2π − 2), rationally depending on \overline{g}; recall that according to the hypothesis I, \overline{g} is supposed to be rationally determined. In this \overline{g}′ one group can be rationally determined. In this way we may construct, instead of |L|, or rather of |L′|, a linear system |M| cutting out on any K the corresponding series \overline{g}′. Then one proves that

|sM| = |sK + s(K_{1} + ⋯ + K_{r})|

and therefore

|M − (K_{1} + ⋯ + K_{r})|

has the same characters as |K^{π}|, and it contains a curve which is reducible into a K and a residual elliptic curve D: d has the same order π(2π − 2) as the para-π-canonical curve C = |Kπ − \overline{K}|, but the two curves are neither equivalent nor paraequivalent; only their multiples, sC and sD, give para-s-canonical curves, cutting out equivalent groups on the K.

(2) Furthermore, if the hypothesis I does not hold, then one succeeds in rationally constructing on a K not one, but a group of m > 1 series \overline{g} (or \overline{g}′) and one therefore obtains a surface F^{∗}, with the same characters, represented on the m-tuple F without branch curve; and on this F^{∗} each of the elliptic curves K^{∗} corresponding to the K contains a determined series \overline{g} (or \overline{g}′). Therefore (as the hypothesis I is verified for F^{∗}) we can construct on F^{∗} an elliptic curve D^{∗} which is a suitable submultiple of a parapluricanonical curve, whose corresponding curve d on F is the elliptic curve we were looking for.

In this way we have explained the general plan of the proof of our lemma II; now we shall develop this proof in all its details, separately discussing the two cases which occur: π even and s = 2, π odd and s one of its prime divisors (different from 2).

10.7 The first case: the genus π of the curves K is even

To start with we recall that the bisection of a linear series g_{2n}^{2n−π} with n > π, on a curve of genus π, always leads to 2^{2π} distinct series^{(8)}.

In particular the double of the canonical series on a K, that is, the bicanonical series g_{4π−_{4}}^{3π−4}, gives rise to 2^{2π} halves; one of these is the canonical series g = g_{2π−2}^{π−2} and the others, which we will now consider, are 2^{2π} − 1 semibicanonical series \overline{g} = g_{2π−2}^{π−2}, that are distinct from g and distinct one from the other. Since the curve K, as it varies in the pencil, stays irreducible and never acquires double points, the aforementioned series \overline{g} never come together; still, they can be permuted by a circulation of the curve K in its own (irrational) pencil, and it is convenient to suppose that in general one can rationally define – not a single series – but a group of m (≥ 1) distinct series \overline{g} = g_{2π−2}^{π−2}.

(8)See for instance Enriques–Chisini, Lezioni, Libro V, Chapter III, §35 (Vol. III, page 394). See also Libro VI, Chapter III, §34 (Vol. IV, page 199).

As we have announced, we shall introduce here two simplifying assumptions, which will be justified later, in relation with the consequences that we shall deduce from them.

Hypothesis I.-One has m = 1, that is, for any K it is possible to determine uniquely, and therefore rationally, one semibicanonical (and not canonical) series.

Hypothesis II.-In a series \overline{g}, which is assumed to be rationally determined, it is possible to determine uniquely, and therefore rationally, one group, which, as K varies on F, describes a curve L cutting out on the K semibicanonical groups of \overline{g}. The latter hypothesis may seem evident, since, as soon as we have the series \overline{g} on K, we are led in general to transform the K into a K_{2π−2} of the space S_{π−2} defined up to projective transformations, on which the hyperplanes cut out the series \overline{g}, thus this series is linearly given, in such a way that a hyperplane of S_{π−2} cuts out one of its groups. But it is not a priori clear that a rationally given series \overline{g} = g^{π−2} on K can be linearly defined in a unique way. In fact, among the infinitely many projectively equivalent curves K_{2π−2} which are the images of the series \overline{g}, the choice of a particular K_{2π−2}, that can be uniquely determined starting with a group of \overline{g}, depends in general on some numerical irrationality that one has to add to the field of rationality of K and \overline{g}, and such irrationalities give-as K varies-some algebraic irrationalities^{(9)}.

In force of the assumptions I and II, let us then construct a curve L on F which cuts out on each K of our elliptic pencil a group of 2π − 2 points of the semibicanonical series \overline{g} (that we supposed to be given). The adjoint curves L′ of L cut out on the K the groups of a semiquadricanonical series \overline{g} + g, that is different from the bicanonical series, and form a linear system |L′|; we may assume that |L′| is as ample as we like, since we may add to it – if needed – a certain number of curves K. Let us now compare the linear system |2L′| with |2K′′| (the double of the second adjoint system to a K). Since the two systems cut out equivalent groups on any K, by the second equivalence criterion, made more precise by the lemma of Section ??, they differ only by curves of the pencil (K). But on this elliptic pencil the difference of two groups of elements (the first of which large enough) turns out to be equivalent to a positive group of elements, thus we may write

(9)In order to explain the difficulty, let us consider a plane quartic K_{4} defined as the curve enveloping an ∞^{1} series of quadritangent conics K_{2} of index 2; in this way the g_{4}^{1} of the 4-tuples of the points of contact is rationally determined on K_{4}; but this is not linearly defined by means of a well determined ∞^{1} linear system of conics with 4 base points. From an abstract point of view the g_{4}^{1} can be regarded as a rational curve which can be represented on a line; but, in order to determine the representation, a point of the rational curve is needed, and this yields the addition of a quadratic irrationality. (Noether’s theorem, see Enriques–Chisini, Lezioni, Libro V, Chapter III, §32 (Vol. III, page 341).

|2L′| = |2K′′ + K_{1} + K_{2} + ⋯ |.

We distinguish now between two cases that a priori appear to be possible, that is, that the number of the K_{1}, K_{2}, … is an even number 2r, or that it is an odd number 2r + 1.

In the first case, if we let the group of the curves K_{1}, K_{2}, … vary in the linear series g_{2r}^{2r−1} that it determines in the elliptic pencil, we can arrange the group to be equal to a group counted twice:

2K_{1} + 2K_{2} + ⋯ + 2K_{r} (K_{1} = K_{r+1}, K_{2} = K_{r+2}, …).

In the latter case we can arrange the group to be equal to another one formed by r + 1 curves, r of which are counted twice:

2K_{1} + 2K_{2} + ⋯ + 2K_{r} + K_{0}.

So we have

|2L′ − 2K′′ − 2(K + K_{1} + ⋯ + K_{r})| = |K_{0}|,

and therefore the characters (π and n = 0) of K_{0} can be expressed in terms of the characters (genus ρ and degree ν) of the virtual curve

L′ − K′′ − (K + K_{1} + ⋯ + K_{r})

whose double is equivalent to K_{0}.

Then we have

n = 4ν = 0 π = 2ρ + ν − 1 = 2ρ − 1,

and consequently π has to be odd. Since we assumed π to be even, the latter case is not possible and we must have

|2L′ − 2(K_{1} + ⋯ + K_{r})| = |2K′′|.

Now the linear system |2L′−2(K_{1}+⋯+K_{r})| has two inequivalent halves:

|L′ − (K_{1} + ⋯ + K_{r})|

and

|K′′|.

Therefore the first system has the same characters, genus and degree, as the latter, that is,:

4π − 4 + π + 1 − 1 = 5π − 4

and

2(4π − 4) = 8π − 8.

Its dimension is therefore

3π − 4,

which is the dimension of the complete linear series that it cuts out on a K. Accordingly, a group of 3π − 4 points of this series belongs to ∞^{1} curves of the continuous system {L′ − (K_{1} + ⋯ + K_{r})} formed by ∞^{1} inequivalent linear systems, and there is at least one curve of this continuous system which breaks up into a K and into a residual curve D: the genus of d turns out to be equal to the genus of C = |K′′ − \overline{K}|, that is, equal to one. And – as required – the elliptic curve D, of the same order as the parabicanonical curve C, is not equivalent (but semiequivalent) to it and therefore it is essentially distinct from it, that is, it certainly contains some other (elliptic) component.

It is now convenient to discuss the assumptions I and II which we assumed in order to develop the previous argument. Let us start from the

Discussion of the hypothesis II

The hypothesis II, that is verified for π = 2^{(10)}, a priori is not true any more for higher values of the even number π (π = 4, 6, …). But we prove that, if we cannot determine in general on K one group G of the \overline{g} or of the semibicanonical series series \overline{g} + g (that is rationally obtained from the given \overline{g}), we may in any case determine a group formed by i = π − 1 groups G (i odd), belonging to a non-pluricanonical series – say \overline{g}′ = i\overline{g} – which is a half of the 2i-canonical series. Therefore we may construct on F a curve M cutting out on any K a group of the corresponding \overline{g}, and by operating on |M| as we did on |L|, or rather on |L′|, that is, by subtracting a certain number of curves K, we will obtain an elliptic, semipara-2i-canonical curve d (essentially distinct from iC), whose double is paraequivalent to the bicanonical curve C counted i times.

Thus our argument based on the hypothesis II can be repeated with slight modifications (by substituting the series \overline{g}′ for \overline{g} or for \overline{g} + g), if we prove what we have stated before: it is possible to determine one group formed by i = π − 1 (an odd number of) groups G of \overline{g} (and therefore to determine \overline{g}′ linearly). In order to do this, we consider in the space S_{π−2} the family of all the projectively equivalent curves K_{2π−2}, which are the images of the semibicanonical series \overline{g} = g_{2π−2}^{π−2} given on K. We try to construct two quadrics Q and Q’, or rather a pencil of quadrics in this space, with the property of being a (projective) covariant of K_{2π−2}: then the self conjugate i-gon, with i = π − 1, common to Q and Q′, is determined as a covariant of the pencil, and its faces cut out π − 1 groups of the \overline{g} on K_{2π−2}.

(10)In this case the series \overline{g} consists of a unique pair, which is therefore given together with the \overline{g}.

The construction of the aforementioned pencil of Q and Q′ can be uniquely, and therefore rationally, performed by starting with the K on which the \overline{g} is given. This is immediately seen if K_{2π−2} does not belong to any quadric of S_{2π−2}: the minimal double of the series \overline{g} is a (complete or not) bicanonical series cut out on K_{2π−2} by the quadrics of its space and in that case it is linearly defined on the corresponding bicanonical curve K_{4π−4} of S_{3π−4} (which is also linearly given) by the hyperplanes of a linear system α that is well determined by the curve K_{4π−4}. So a pencil of quadrics in S_{π−2} can be rationally determined as a function of \overline{g} by giving a pencil of hyperplanes of S_{3π−4}, chosen in α by imposing that they pass through a suitable group of points of S_{3π−4}, that can be fixed independently of the K_{4π−4}; this is the same as giving a series g_{4π−4}^{1} contained in the bicanonical series g_{4π−4}^{3π−4} (and more precisely in the double of the series \overline{g}).

But if the K_{2π−2} belongs to ∞^{r} quadrics of S_{π−2} (r ≥ 1), then the above argument fails, since a group of the minimal double of the series \overline{g}, regarded as a hyperplane section of K_{4π−4}, does not correspond any more to a single quadric of S_{π−2}, but to an ∞^{r+1} linear system of quadrics instead; similarly an ∞^{t} linear system of hyperplanes of S_{3π−4} contained in α corresponds to an ∞^{r+t+1} linear system of quadrics, which cuts out on K_{2π−2} a series g_{4π−4}^{t} of lower dimension. But, if we set

t = 3π − 4 − 2 = 3π − 6,

then we have a linear system |Q| of quadrics in S_{π−2}, whose dimension is two less than the dimension of the total system: there is now a harmonic system of dual quadrics associated^{(11)} with |Q|, which defines a conjugate i-gon (i = π − 1), and therefore a group of i groups of \overline{g}. Q.E.D.

Discussion of the hypothesis I

After having justified the hypothesis II (at least in the sense that, if it is not verified for the series \overline{g}, then it is verified for \overline{g}′), let us see how we can relatively justify also the hypothesis I, in the sense that-if it does not hold for our surface F -it certainly holds for an F^{∗} with the same characters as F which is represented on F counted m (≥ 1) times, without branch curve; so it is possible to construct an elliptic curve D^{∗} on F^{∗}, whose image on F is a semiparapluricanonical curve D.

(11)If in a linear space of any dimension a quadric a_{x}^{2} = 0 and a dual quadric a_{u}^{2} = 0 are given, then (according to Battaglini and Rosanes) there is a simultaneous linear invariant of them, the harmonizer a_{a}^{2}, whose vanishing defines the harmonic relation between a quadric and the associated linear system of dual quadrics. See Enriques–Chisini, Lezioni, Libro III, Chapter I, §9 (Vol. I, page 61).

Let us assume that on each K of F we cannot uniquely determine a single semibicanonical series \overline{g}, but we can determine a group of m (m > 1) such series \overline{g}, that are permuted by a circulation of K in the pencil (K). And, for simplicity, let us assume π > 2 and therefore π ≥ 4, and let us also assume that for each one of these series \overline{g} the hypothesis II holds; thus we can construct a linear image of this series, as a rational function of it, that is, we can define a single curve K_{2π−2} of S_{π−2}, chosen among the infinitely many corresponding projective curves.

The ∞^{1} algebraic variety whose elements are our series \overline{g} on the K (that is, the series associated to the K) can be abstractly regarded as an ∞^{1} series Σ of spaces S_{π−2}, contained in a space of higher dimension S_{R}. Since two of the m series \overline{g} on a K never come together, the series Σ contains an elliptic involution γ_{m}^{1} of spaces without double elements, and it is therefore also elliptic.

Now in each one of the spaces S_{π−2} of Σ we can (by hypothesis) construct a single curve K^{∗} = K_{2π−2}, an image of the homologous \overline{g} on the corresponding K: the locus of the curves K^{∗} (which are birationally equivalent to the K) is a surface F^{∗} that is represented on the m-tuple F (without branch curve) in such a way that any K of F corresponds to a group of m distinct curves K^{∗} on F^{∗}. The curves K^{∗} on F^{∗}, birationally equivalent to the K, form an elliptic pencil (K^{∗}).

It is easy to see that the surface F^{∗}, as well as F, and according to the formula of the Zeuthen–Segre invariant of F^{∗} in relation to the elliptic pencil of the K^{∗} without double points, has the arithmetic genus p_{a} = −1 and the linear genus p^{(1)} = 1.

In this setting we may say of F^{∗} what – in the hypothesis I – we have said of F, since for any curve K^{∗} (of genus π) on F^{∗} a single semibicanonical series \overline{g}, defined by the hyperplanes of S_{R}, is rationally and linearly given. Whence we deduce that F^{∗} contains an elliptic semiparabicanonical curve D^{∗}, whose image on F is again an elliptic, semipara-2m-canonical curve D.

Remark In the above argument we have excluded the hypothesis π = 2, in which in any case the same discussion could be repeated with slight modifications. But it suffices to notice that in this case, if the K are of genus π = 2, on each of them the group of the 6 distinct double points of the g_{2}^{1} is determined, and – as K varies – it describes an elliptic semiparasexticanonical curve.

The second case: the genus π of the curves K is odd 10.8 378

10.8 The second case: the genus π of the curves K is odd

We denote by s an (odd) prime number which divides π, and we consider on any K one of the nonspecial series \overline{g} = g_{2π−2}^{π−2} obtained from the division by s of the s-canonical series. For the time being we introduce two simplifying assumptions analogous to I and II of the previous section.

Hypothesis I: on the variable K of the pencil (K) a series \overline{g} can be uniquely, that is, rationally, determined.

Hypothesis II: if a \overline{g} is rationally given on any K, we can rationally determine a single one of its groups, which is uniquely defined as K varies in (K); in other words the rationally determined \overline{g} can be linearly defined (giving rise to a single K_{2π−2} in S_{π−2}).

In force of these assumptions, we can construct a curve L on F cutting out on any K a group of \overline{g}; and |L|, defined up to summands equal to K, can be assumed to be as ample as we like.

Let us denote by |L′| the linear system, of order 4π − 4, adjoint to |L|, and by |K′′| the second adjoint system to a K, which is of the same order.

Let us compare, using the usual equivalence criterion, the two systems |sL′| and |sK′′|; they must differ by a certain number of curves of the pencil (K) to be added to the less ample system |sK′′|; moreover we may assume that the curve to be added to this smaller system is formed by a certain number of s-tuple curves

sK_{1}, sK_{2}, …, sK_{r}

and maybe also by a t-tuple curve tK_{0} of the pencil, where t < s:

|sL′| = |sK′′ + s(K_{1} + ⋯ + K_{r}) + tK_{0}|.

But we prove that

t = 0,

since the genus of the virtual curve

((t/s)K_{0}) = |L′ − K′′ − (K_{1} + ⋯ + K_{r})|

must be an integer ρ, whereas the genus of tK_{0} (of degree 0) is

tπ − t + 1 = sρ − s + 1

so that

t(π − 1) = s(ρ − 1) :

this relation cannot hold with ρ an integer, since s (a prime divisor of π) is coprime with π − 1 and t.

Therefore we have

|sL′| = |sK′′ + s(K_{1} + ⋯ + K_{r})|

and

|sL′ − s(K_{1} + ⋯ + K_{r})| = |sK′′|.

So

|L′ − (K_{1} + ⋯ + K_{r})| and |K′′|

have the same characters (degree 8π − 8 and genus 5π − 4) and therefore the same virtual dimension 3π − 4, which is the dimension of the series they cut out on a K. Whence we deduce that in the ∞^{3π−3} continuous system {L′ −(K_{1} +⋯+K_{r})} there is a curve reducible into a K and a residual curve D; d (as well as the parabicanonical C = |K′′ −\overline{K}|) is elliptic, of order 4π −4, and sD, as well as sC, is a para-2s-canonical curve, and the two curves cut out on the K equivalent (2s-canonical) groups.

If the assumptions I and II do not hold, then the above argument can still be repeated with some modification.

Assume that the hypothesis II does not hold, that is, it is not possible to uniquely determine a group of the nonspecial series \overline{g} = g_{2π−2}^{π−2}, which an s-submultiple of the s-canonical series and, according to the hypothesis I, it is supposed to be rationally given; then, instead of the nonspecial series \overline{g}, s-submultiple of the s-canonical series, we may consider on K the series

\overline{g}′ = (2π − 2)\overline{g} − (π − 2)g;

we can rationally determine one of its groups by taking the sum of the groups \overline{G}_{2π−2} of \overline{g} having a point of a (canonical) group G_{2π−2} of g as a (π − 2)-tuple point and by subtracting then the group G_{2π−2} counted π − 2 times: it is important to notice that \overline{g}′, whose s-tuple is the sπ-canonical series, is not the π-canonical series, since otherwise from the relations

(2π − 2)\overline{g} − (π − 2)g = πg

and

(2π − 2)\overline{g} = (2π − 2)g

that is,

s\overline{g} = sg,

we would deduce

\overline{g} = g^{(12)}

(12)It suffices to solve the diophantine equation (2π − 2)x − sy = 1; then we deduce that [(2π − 2)x − sy]\overline{g} = [(2π − 2)x − sy]g.

Lemma III 10.9 380

since 2π − 2 and s are coprime numbers.

Now, instead of |L|, or rather of |L′|, we may construct a sufficiently ample linear system |M| cutting out on each K the series \overline{g}′ (constructed as a function of \overline{g}, which in turn is rationally given). We compare |sM| with the linear system |sK^{π}| multiple of |K^{π}|,the πth adjoint system of a K; if we denote by K_{0}, K_{1}, …, K_{r} suitable curves of the pencil (K), then we have

|sM| = |sKπ + s(K_{1} + ⋯ + K_{r}) + tK_{0}|,

where t = 0 and therefore

|sM| = |sK^{π} + s(K_{1} + ⋯ + K_{r})|.

Whence we deduce that the linear system

|M − (K_{1} + ⋯ + K_{r})|

has the same characters as |K^{π}| – degree, genus and virtual dimension – and therefore it belongs to a continuous system whose dimension is one more than the dimension of the series \overline{g}′; therefore there is inside this system a curve reducible into a K and a residual elliptic curve D, of order π(2π − 2), whose s-multiple is a para-sπ-canonical curve.

Let us finally discuss what can be done if the hypothesis I fails to hold for the surface: this means that on a variable curve K we cannot rationally define a nonspecial series \overline{g}, among those whose s-multiple is the canonical series, but we may only define a group of m (> 1) series \overline{g}, permuted by a circulation of K in the pencil (K). As we have already seen in the first case (π even and s = 2), it is convenient to pass from the surface F to an F^{∗} with the same characters, represented without branch curve on the m-tuple F and such that the hypothesis I holds for it. Then on F^{∗} an elliptic curve D^{∗} of a certain order (2π − 2), or more generally π(2π − 2), can be constructed and the corresponding curve d on F, of order mπ(2π − 2) is also elliptic: and D, which is an s-submultiple of a para-smπ-canonical curve, certainly contains some component distinct from the components of a para-mπ-canonical curve C, constructed on F according to Section ??.

10.9 Lemma III

Let F be a surface of genera p_{g} = 0 and p_{a} = −1, possessing an elliptic pencil of curves of genus π > 1, and let E be a curve on F cutting out on the curves K of the pencil groups of points equivalent to those cut out by a para-i-canonical curve C; if E has no common points with C, away from the possible common components, then it is a para-i-canonical elliptic curve, like C.

Conclusion: the surfaces F also possess a linear pencil of elliptic curves 10.10 381

Let us compare E and C with the second equivalence criterion (made more precise with the lemma of Section ??): they are equivalent up to the difference K_{1} − K_{2} of two curves of the pencil (K):

|E| = |C + K_{1} − K_{2}|.

Therefore E has the same genus one and the same degree zero as C and it is, as well as C, a para-i-canonical curve.

10.10 Conclusion: the surfaces F also possess a linear pencil of elliptic curves

The lemmas I, II and III that we established in the above sections lead us now to a theorem which closes the analysis of our surfaces F.

The surfaces F, of genus p_{g} = 0 and p_{a} = −1, with an elliptic pencil of curves K of genus π > 1, also possess a base point free linear pencil of elliptic curves.

Indeed let us construct on F (Lemma II) two elliptic para-i-canonical curves C and D, cutting out on the curves K equivalent groups of n = i(2π−2) points. On each K the above mentioned curves determine a linear series g_{n}^{1}, possessing a certain number

m ≤ 2n + 2π − 2

of groups G_{n} with a double point, away from those which can be absorbed by the groups cut out by C and D. The locus of these groups G_{n} is a curve E that (with respect to the K) has the same order mn as mC and mD, and cuts out on the K groups equivalent to sections with these curves; moreover, it can never happen that, as K varies in the pencil (K), a group G_{n} coincides with the group cut out by C: in fact this would imply the existence of double points for the elliptic involution cut out by the K over an elliptic component of C. Therefore the curve E has zero intersections with mC, and it is consequently a para-mi-canonical curve, as well as mC and mD.

Surfaces with p_{g} = 0 and p_{a} = −1 and an elliptic pencil 10.11 382

With this in mind, let us represent F onto an n-tuple elliptic ruled surface Φ, in such a way that the K correspond to the generators, each of which represents the g_{n}^{1} defined, as before, by the sections of C and D. We may realize Φ as a scroll in a certain space S_{r}, and on this we have two minimal directrices corresponding to C and D, belonging to two complementary spaces γ_{1} and γ_{2}, of dimensions s and r − s − 1; for instance, we may refer Φ to a cone whose vertex corresponds to C and such that d is one of its hyperplane sections. And on Φ we have an elliptic curve E without common points with γ_{1} and γ_{2}. Now the ∞^{1} linear transformations of S_{r} whose spaces of fixed points are γ_{1} and γ_{2} act on the image of E by transforming it in a rational ∞^{1} system of elliptic curves of degree 0, thus in a base point free linear pencil of elliptic curves; accordingly, there is a base point free linear pencil of elliptic curves C on F which corresponds to this pencil of the n-tuple Φ; so we have proved the existence of our pencil. Q.E.D.

It cannot be excluded that the C are reducible: in any case their irreducible variable components form a linear pencil, which we again denote by |C|.

Let us add that the computation of the Zeuthen–Segre invariant for this irreducible linear pencil |C| gives us

δ − 4 = 12p_{a} − p^{(1)} + 9 = −4,

thus δ = 0: this means that the elliptic curves C, varying in |C| and never acquiring a double point, all have the same modulus, that is, they are birationally equivalent.

10.11 Surfaces with genera p_{g} = 0 and p_{a} = −1 with an elliptic pencil of curves of genus π = 1

The theorem which asserts the existence of a linear pencil of elliptic curves on the surfaces with p_{g} = 0 and p_{a} = −1 possessing an elliptic pencil of curves K of genus π > 1, can now be extended to the surfaces with the same characters (p_{g} = 0, p_{a} = −1 and p^{(1)} = 1) possessing an elliptic pencil of curves K of genus π = 1.

It is convenient to start with the remark that, since the Zeuthen–Segre invariant is

Δ − 4 = 12p_{a} − p^{(1)} + 9 = −4,

we must have Δ = 0;

thus the elliptic K, varying in (K) and never acquiring a double point, are birationally equivalent one to the other.

Let us transform F, on which the K are projectively realized as curves of order m, into a surface, that we call F again, of a suitable space, in such a way that the K become elliptic normal K_{m}, of the same order m, each belonging to a space S_{m−1}: each K of the original F is, in general, a projection of the corresponding K_{m}. Now two elliptic normal K_{m} with the same modulus are linearly equivalent in a certain number n of different ways: and precisely one has n = 2m^{2} if the curves have general modulus, whereas n = 4m^{2} or respectively n = 6m^{2}, if they are harmonic or equianharmonic. Thus the set Γ_{n} of n linear transformations of the space referring two K_{m} one to another is rationally defined; by fixing one of the two K_{m} and one of its points P, one defines a curve C passing through P and intersecting the other variable K_{m} at the n points corresponding to P in the n linear transformations of Γ_{n}.

At a first glance one could believe that – since the K_{m} are birationally identical – the aforementioned linear transformation are always distinct and therefore send a point P, fixed as above, to n distinct points; thus the curve C would contain an elliptic involution γ_{n}^{1} without double points and so it would certainly be elliptic and, as P varies, it would give us a linear pencil |C| of elliptic curves, directrices of the elliptic pencil.

These curves can be reducible, in which case we denote again by C their irreducible components, that are also elliptic without common points, and we denote by n again the number of their intersections with the K = K_{m}. So we have, on the surface of irregularity p_{g} − p_{a} = 1, possessing an elliptic pencil of elliptic curves (K), another pencil |C| of elliptic curves, directrices of the K, and this pencil is necessarily rational^{(13)}.

But an objection can be moved to the argument used to prove that the curves C have genus one; this objection can only be overcome after a deeper analysis, in which one has to take into account the fact that we are actually dealing with surfaces of genus p_{a} = −1. In fact one cannot a priori exclude that the linear transformations existing between two variable K_{m} become degenerate, and so – supposing for instance that n = 6 – that one has a pencil of normal sextics K_{6} in S_{5}, inside which there are degenerate sextics consisting of a double plane cubic K_{3}^{2}: in that case there are not 36 any more, but 18 degenerate linear transformations relating a generic K_{6} and the K_{3}, and these induce as many [1,2] correspondences between the two curves.

One also cannot a priori exclude that among the generically projectively equivalent K_{m} of an elliptic pencil there are cuspidal curves or curves reducible in several simple or multiple parts; but this case cannot occur on our surface of genus p_{a} = −1 (and p^{(1)} = 1), where, as we have seen, the number of K_{m} with a double point has to be

Δ = 13 + 12p_{a} − p^{(1)} = 0

Therefore, since p_{a} = −1, we only have to remove the exception that the pencil (K_{m}) may contain some multiple elliptic curves, which have zero valency in the computation of the Zeuthen–Segre invariant Δ (see Section ??).

In order to do this, we notice that, by means of the two pencils (K) and |C| our surface F can be represented on a n-tuple cubic cylinder Φ, where the curves C correspond to the normal plane sections c of Φ and the K correspond to its generators: on Φ we have a branch curve formed by a certain number of plane sections c, corresponding to multiple curves C = sC_{s}, and (possibly) also by a certain number of generators k, corresponding to degenerate K consisting of multiple elliptic curves K = rK_{r}. If p_{a} = −1, and so (for the K) Δ = 0, then the multiple cylinder Φ gives the representation of an F that may be assumed without singularities, on which a C_{s} and a K_{r} meet at n/rs points (simple points for the surface) corresponding to the point A = ck of Φ: indeed if one of these intersection points of C_{s} and K_{r}, say A′, were a double or multiple point of F, then the limit degenerate curve of a K would not be rK_{r} (as we have supposed) but rK_{r} plus the neighbourhood of A′, so that we would have (Section ??) Δ > 0. But now we shall prove that if, by contrast, the point A of the n-tuple cylinder Φ comes from a point of the surface F, then this has to be a singular (double or multiple) point of F, and thus the hypothesis of a degenerate K of the type rK_{r} leads to a contradiction. Let us indeed consider a plane \overline{α}, which is transversal with respect to the generators of Φ and passes through A, and a variable plane α which does not pass through A but tends to \overline{α}. The section of Φ with the plane α is a n-tuple multiple curve ℓ, on which there are two branch points A_{1} and A_{2}, the intersections with c and k. Let us describe on the Riemann surface of the multiple ℓ a closed path γ which goes around A_{1} and A_{2}; it can be considered as the sum of two elementary paths which go around A_{1} and A_{2}. The first path gives rise to a substitution on the sheets equivalent to the product of

(13)See Chapter IX, Section ??.

(s − 1)\frac{n}{s}

transpositions, and the latter to a substitution equivalent to

(r − 1)\frac{n}{r}

transpositions. Therefore the closed path γ, that in the limit – as α → \overline{α} – becomes a path going around A, gives rise to a substitution formed by at least n transpositions: this means that the curve L, homologous to ℓ, cannot have any simple point corresponding to A, since such a curve would give rise to a branch of order n at worst, which would correspond to a substitution product of n − 1 transpositions on the sheets of the Riemann surface of ℓ!

We may therefore conclude that:

The surfaces F of genera p_{g} = 0 and p_{a} = −1, possessing a pencil of elliptic curves K, also possess a linear pencil of elliptic curves C which are directrices of the K.

Note In the above argument we have used the fruitful concept of the singularity of a curve which appears in the Memoir of O. Chisini, Le singolarità di un ramo superlineare di curva piana definite mediante un prodotto di sostituzioni^{(14)}.

It is worth pointing out explicitly that, if we remove the hypothesis p_{a} = −1, then there are indeed surfaces possessing an elliptic pencil of elliptic curves K which contains reducible curves: say surfaces F for which p_{a} ≥ 0 and p_{g} = p_{a} + 1, containing a linear pencil of curves C of genus π > 1 which are directrices of (K). If we call elliptic surfaces the surfaces with p_{g} = 0 and p_{a} = −1 studied in the previous sections (the motivation for this name will be explained later) then these other F, presenting some remarkable analogies with the former, can be called paraelliptic surfaces. The more relevant analogy is that on the paraelliptic surfaces, as well as on the elliptic ones with p_{a} = −1, there are two pencils of birationally equivalent curves: an elliptic pencil of elliptic curves (K) and a linear pencil of curves, which here are not elliptic any more but of genus π > 1. But, while the elliptic surfaces have – as we shall see – an ∞^{1} group of birational transformations into themselves, this property is not shared by the paraelliptic surfaces.

We limit ourselves to pointing out here two examples of paraelliptic surfaces, constructed as multiple coverings of a cubic cylinder f(x, y) = 0:

\begin{matrix}u^{2} = (z − a)(z − b)(z − c)(z − d)ψ(x, y) \\ f(x, y) = 0 (p_{a} = 3, p_{g} = 4); \end{matrix}  (10.1)

ψ represents a quadric cylinder parallel to f which cuts f = 0 in six branch generators, to be added to the 4 branch cubics

\begin{matrix}z = a, z = b, z = c, z = d; \\ u^{3} = (z − a)(z − b)(z − c)(z − d)ψ(x, y) \\ f(x, y) = 0 (p_{a} = 0, p_{g} = 1); \end{matrix}  (10.2)

ψ = 0 denotes here a cubic cylinder having a triple contact with f along two generators, touching it along another generator k and cutting it further along a generator k′: the branch curve of the triple cylinder f consists therefore of the three cubics

z = a, z = b, z = c,

and of the two generators

k and k′.

Finally we point out to the young researchers this interesting subject of study: to determine and classify all the paraelliptic surfaces; we refer to the Notes at page. 123 and 134 of the Enriques–Campedelli’s Lezioni, in Rendic. Sem. Mat. R. Università di Roma (1934), where the reader can find some useful remarks and suggestions.

(14)Atti Ist. Veneto, t. XXX, parte 2a, 1921. This memoir, published after the second volume of the Lezioni by Enriques–Chisini, gives a further important contribution to the theory developed there.

Elliptic surfaces 10.12 386

10.12 Elliptic surfaces

Let us summarize the results we have obtained in the previous sections: Any irregular surface F of genus p_{g} = 0 which does not belong to the family of ruled surfaces (rational or not) has the arithmetic genus p_{a} = −1 and contains an elliptic pencil (K) of curves of genus π ≥ 1, and a linear pencil of elliptic curves |C|; therefore it can be represented on a multiple cubic cylinder with multiplicity n = CK, with a branch curve formed by normal plane sections of Φ.

But conversely an n-tuple cubic cylinder Φ, with the branch curve formed by normal plane sections (which are fixed in such a way to satisfy the existence conditions of an irreducible surface), does not always represent a surface of genus p_{g} = 0, since the images of the plane sections of Φ can be reducible elliptic curves whose components form a nonlinear pencil (C) of genus ρ ≥ 1.

It is apparent here that our surfaces, with p_{g} = 0 and p_{a} = −1, not birational to ruled surfaces, are contained in a more general family of surfaces F, with p_{a} = −1, possessing an elliptic pencil of curves K of genus π ≥ 1 and a pencil of genus ρ ≥ 0 of elliptic curves C; for such F the geometric genus is p_{g} = ρ.

Indeed, since p_{a} = −1, the existence of the two pencils (C) and (K) that characterize these more general surfaces F yields that the irregularity of F is

q ≥ ρ + 1

and therefore

p_{g} ≥ ρ;

but, on the other hand, the canonical curves of F, which are sums of curves C, must cut out on each K groups of the g_{2π−2}^{π−1} composed with the double point free involution of genus ρ^{(15)} cut out by (C); therefore they have to belong to the canonical series g_{2ρ−2}^{ρ−1} in the pencil (C). Then one has

p_{g} ≤ ρ

and therefore

p_{g} = ρ, Q.E.D.

These more general surfaces, characterized-as we have said-by the arithmetic genus p_{a} = −1 and by the existence of two pencils (K) and (C) of irreducible curves on them, that is,

(1) an elliptic pencil of curves K of a certain genus π ≥ 0 (for which, since p_{a} = −1, one has Δ = 0); (2) and a pencil of genus p_{g} of elliptic curves C, such that n = CK ≥ 1, are

(15)Painlevé-Castelnuovo’s theorem. see Enriques–Chisini, Lezioni, Libro V, Chapter I, §9 (Vol. III, page 72).

called elliptic surfaces: and more precisely proper elliptic surfaces those, with π ≥ 1, which are not birational to ruled surfaces, and improper elliptic surfaces those, with π = 0, which are birational to ruled surfaces of genus 1.

The name “elliptic” comes from a parametric representation of the aforementioned surfaces which is related to the ∞^{1} group of their transformations about which we will say later, and that has been pointed out by P. Painlevé^{(16)}.

In relation to this parametric representation the number n = CK has been called determinant of the surface.

The parametric representation we have mentioned above translates into the following geometric property: Any elliptic surface F has an ∞^{1} elliptic group of birational transformations into itself, whose trajectories are the curves of the pencil (C).

Let us prove this theorem, by showing that there exists a birational transformation of F, for which all the curves C are invariant and in which two arbitrary points P_{0} and P′_{0} of one of these curves C_{0} are sent one to the other.

Indeed let us choose another curve C of the same pencil (C) and a point P on it. If the C, which are birationally equivalent, are elliptic curves with general modulus, there are-as it is well known-two transformations of C_{0} in C mapping P_{0} to P^{(17)}; and only one^{(18)} of these (rationally determined) preserves the groups cut out by the curves K, forming on each curve an elliptic involution γ_{n}^{1}. Therefore, since the transformation of the first kind (P_{0}P′_{0}) is given on C_{0}, a single transformation of the first kind of C is also determined together with it and this sends a point P to a point P′, in such a way that (P P′) goes to (P_{0}P′_{0}) as C tends to C_{0}.

The proof can be easily extended to the cases in which the C are harmonic or equianharmonic elliptic curves.

It is convenient to remark that the property we have established for the elliptic surfaces characterizes them:

Any surface F which admits an ∞^{1} elliptic group Γ of birational transformations into itself is a, proper or improper, elliptic surface of arithmetic genus p_{a} = −1 possessing an elliptic pencil of curves K of genus π (≥ 0) and a pencil of genus p_{g} of elliptic curves C.

In fact in the first place F must have a pencil (C), of a certain genus ρ ≥ 0, formed by the trajectories of the group Γ, which are elliptic curves birational to Γ itself. But secondly, we shall show with the following construction the existence of an elliptic pencil (K) on F: let us consider an (irreducible) curve L on F and the ∞^{1} elliptic series {L} formed by its images via the transformations of the group Γ; the curves L of this series are certainly not equivalent, since they have to cut out on a C the inequivalent groups of points corresponding to a given one in a transformation of the first kind; but among the groups of curves L passing through a variable point P on F we will find ∞^{1} equivalent groups: this means that there is an elliptic pencil of curves on F, on which the L of the series {L} cut out equivalent groups (see Chapter IX, Section ??).

(16)P. Painlevé, Leçons sur la théorie analytique des équations différentielles. (Paris, Hermann, 1897).

(17)See for instance Enriques–Chisini, Lezioni, Libro V, Chapter III, §27 (Vol. III, page 262).

(18)See Enriques–Chisini, Ibidem, Chapter IV, §39 (Vol. III, page 446).

After thus having constructed the two pencils (C) and (K) on F, we easily determine the characters of the surface. Indeed the computation of the Zeuthen–Segre invariant for (K) gives us

Δ = 13 − 12p_{a} − p^{(1)} ≥ 0

and so

p_{a} ≥ −1 (p^{(1)} ≥ 1).

On the other hand, the existence of an elliptic pencil (K) and of a pencil of genus ρ yields that (there is an ∞^{ρ+1} series of inequivalent curves on F, and therefore that one has)

p_{g} − p_{a} ≥ ρ + 1 p_{g} ≥ ρ.

Finally an argument that we have already used before leads us to establish that the canonical curves of F – which for ρ > 0 are formed by 2ρ − 2 curves C – must give a series contained in the canonical series g_{2ρ−2}^{ρ−1} in the pencil (C); by comparing we get

p_{g} = ρ, p_{a} = −1. Q.E.D.

Remark From what we have said we deduce that on an elliptic surface F of determinant n > 1 there is an involution I_{n} formed by the groups of n points common to the C and the K, and it is generated by an Abelian group Γ_{n} of (permutable) birational transformations, and precisely by the transformations of the group Γ of the surface which leave the C and the K fixed. If the groups of points of I_{n} are supposed to be elements (points) of a new surface f, we obtain an elliptic surface of determinant 1, on which F is represented as a multiple, and precisely as an n-tuple, surface; if p_{g} = 0, then this improper elliptic surface is nothing but an elliptic surface, or, if we like it better, a cubic cylinder f(x, y) = 0.

It is worth pointing out here that the elliptic surfaces f of determinant 1 are in general defined as the surfaces representing the ∞^{2} variety of the pairs of points belonging to an elliptic curve C and to a curve K of genus π (≥ 0).

The construction of the elliptic surfaces of genus p_{g} = 0 10.13 389

Starting with these surfaces one can construct the elliptic surfaces with determinant n > 1, by introducing algebraic irrationalities by means of which the group Γ_{n} can be solved. Concerning this, we can say that, a priori, either only one radical or two non-superimposed radicals are needed, according to whether the group Γ_{n} is cyclic or Abelian with a basis of order two. In fact Γ_{n}, that is a finite group of transformations of an elliptic curve C into itself, cannot have a base of order > 2^{(19)}. In next section, where we shall limit ourselves in particular to the hypothesis p_{g} = 0, and again, more precisely, in Section ??, we explain the meaning of this construction, which yields the existence of the elliptic surfaces, and the conditions under which it holds.

10.13 Construction of the elliptic surfaces of genus p_{g} = 0

In order to construct the (proper) elliptic surfaces F of genus p_{g} = 0, with any determinant n > 1, we start with a cubic cylinder f(x, y) = 0, that – as we have said – is a model of the (improper) elliptic surface of genus p_{g} = 0 and determinant 1.

The surface F can be represented on an n-tuple cylinder f, by means of the two pencils of irreducible curves |C| and (K) characterizing it; the branch curve is formed by a certain number of normal plane sections c, so that the images of the curves K are the generators k of the cylinder, and the (trajectories of the ∞^{1} group of transformations of F) C correspond to the normal plane sections c of the cylinder itself. The branch curve of the n-tuple cylinder consists of a certain number of curves c, corresponding to multiple curves of the type C = sC_{s}.

So the representation of the elliptic surface F on the n-tuple cylinder f must satisfy the following conditions:

(1) that the curve K defined by a n-tuple k is irreducible and contains an involution g_{n} either of cyclic type or of (properly) Abelian type with base 2;

(2) that the curves C corresponding to the normal plane sections c of f, which are n-tuple curves without branch points, are generally irreducible.

We notice indeed that, if condition 2) is not satisfied, then the surface F, represented by the n-tuple cylinder, is not an (elliptic) surface with determinant n any more, but it has a determinant < n, and it turns out to be, in general, of genus p_{g} > 0. This happens, for instance, for the surface

(19)See Enriques–Chisini, Lezioni, Libro V, Chapter III, §28 (Vol. III, page 271) and Chapter IV, §39 (Vol. III, page 446–451).

u^{2} = (z − a)(z − b)(z − e)(z − d)ψ^{2}(x, y) f(x, y) = 0

which is an irreducible surface F represented on the double cylinder, with the branch curve formed by the plane sections z = a, z = b, z = d, z = e.

In fact a generic normal section:

z = c(c = const.)

corresponds to a pair of distinct elliptic curves on F:

u = ±\sqrt{(c − a)(c − b)(c − d)(c − e)}ψ,

so that the irreducible curves C part of the images of the c form a pencil (no more linear, but elliptic) of unisecant curves of the K; and therefore the geometric genus of F is:

p_{g} = 1.

It is convenient to recall here the criteria in order that an n-tuple elliptic curve without branch points is irreducible^{(20)}. There are two possible cases: the cyclic case and the Abelian case (with basis two). In the first case the n-tuple curve can be constructed, starting for instance with a cubic curve f(x, y) = 0, by extracting an nth root of a polynomial φ(x, y) = 0 on it,

9u^{n} = φ(x, y) 0f(x, y) = 0;

in order that the multiple curve has no branch points, it is necessary that the critical points

f = 0, φ = 0,

be apparent critical points, and therefore that the φ, taken for instance of order n, has three n-fold contact points with f. Therefore the irreducibility of the mentioned multiple curve depends on the fact that “the series g_{3}^{2} containing the triple of the apparent critical points is different from the series cut out on f by the lines of the plane”.

In the Abelian case (in which n = pq) the construction of the n-tuple curve requires the extraction of two not superimposed roots of index p and q of two polynomials φ(x, y) and ψ(x, y), which we may assume of order p and q: in order that the branch curve has no branch points, the curves φ and ψ must have three points of contact with f of orders p − 1 and q − 1 respectively, that therefore form two triples of apparent critical points; the irreducibility of the multiple curve yields that these triples do not belong to the g_{3}^{2} cut out on f by the lines of the plane, and moreover that they satisfy a certain dissimilarity condition.

(20)See Enriques–Chisini, Lezioni, Libro V, Chapter IV, §39 (Vol. III, page 446).

Let us develop this construction in the cyclic case, which one encounters in particular when the determinant is a prime number. So we have that the involution I_{n} on the surface F, formed by the groups CK, is generated by a cyclic group of transformations Γ_{n}, of order n.

Starting with the n-tuple cubic cylinder

f(x, y) = 0,

we fix the normal plane branch sections

z = a_{1}, z = a_{2}, … z = a_{t};

the n points of the group G_{n} corresponding to a generic point of f can be given by the values of a radical

u = ^{n} \sqrt{(z − a_{1})^{r_{1}}(z − a_{2})^{r_{2}}...(z − a_{t})^{r_{t}}φ(x, y)}

satisfying the following conditions:

(1) for the exponents r_{i} < n one has

r_{1} + r_{2} + ⋯ + r_{t} ≡ 0 mod n,

since otherwise there would be a branch point at infinity on the generators of the cylinder f.

(2) If h_{i} is the greatest common divisor of n and r_{i}, so that

n = h_{i}s_{i}, r_{i} = h_{i}ρ_{i}

then the following factor appears in the expression of u

n \sqrt{(z − a_{i})^{r_{i}}} = ^{s_{i}} \sqrt{ε_{i}(z − a_{i})^{r_{i}}},

where ε_{i} is a h_{i}th root of unity.

So on the n-tuple generator of the cylinder f, the point

z = a_{i}

is a branch point corresponding on the homologous K to a group G_{n} (of the g_{n} cut out by the C) formed by h_{i} groups of s_{i} coinciding points. In other words, the point z = a_{i} is a branch point of order s_{i} for the multiple k, with

s_{i} = n/h_{i},

and therefore we have

∑(n/s)(s − 1) = 2n + 2π − 2

(the summation is taken over all values of s = s_{1}, s_{2}, …, s_{t}), where π denotes the genus of the K.

(3) Each one of the numbers s_{i} (i = 1, 2, …, t) is a divisor of the lowest common multiple of the remaining ones.

This property is a general consequence of the Abelian character of Γ_{n}. In fact since on the multiple line k there are branch points at infinity, the product of the cyclic substitutions on the sheets of the algebraic function u(xyz), in correspondence with the branch points a_{i}, has to be equal to the identity:

S_{t} ⋯ S_{2}S_{1} = 1

so that, for instance

S_{1} − 1 = S_{t} = … = S_{3}S_{2},

that is,

S_{1}s_{1} − 1 = S_{t} = … = S_{3}S_{2}.

But the substitution S_{1}s_{1} is cyclic with period s_{1}, whereas the product S_{t} ⋯ S_{3}S_{2} is cyclic with period a divisor of the lowest common multiple of s_{2}, s_{3}, …, s_{t}^{(21)}, since

S_{t}^{s_{t}} ⋯ S_{3}^{s_{3}}S_{2}^{s_{2}} = (S_{t} ⋯ S_{3}S_{2})^{s_{2}s_{3}···s_{t}} = 1.

4) Since the n-tuple cylinder has no branch generators, the polynomial φ(x, y) appearing in the expression of u under the radical must represent a cylinder having an n-fold contact with f along any common generator. And in order that the curves C represented by z = const. be irreducible, the group of the apparent critical generators must not be equivalent to a multiple of the triples of generators cut out by the planes

αx + βy + γ = 0.

Surfaces with all P_{i} = 0: characterization of the ruled surfaces 10.14 393

Moreover we may take φ of order n, so that φ = 0 touches f = 0 with an n-fold contact along three generators (not lying in a plane). Indeed we can easily see that, if one lets φ vary continuously, then the surface F that one obtains is always birational to a given one, since the curves C and the K, represented by the plane sections z = const. and by the generators of f respectively, are preserved. So if we take, instead of the cylinder φ, a cylinder ψ of order nh, with h > 1, we may replace ψ(x, y) with a polynomial which is reducible into a simple factor φ of order n and into a n-tuple factor φ_{1}^{n} with φ_{1} of order h − 1:

(21)See for instance L. Bianchi, Lezioni sulla teoria dei gruppi di sostituzioni E delle equazioni algebriche secondo Galois, (Pisa, 1900), Chapter III, §30.

ψ = φ_{1}^{n}φ;

thus we have

u = φ_{1} ^{n} \sqrt{φ(x, y) · (z − a_{1})^{r_{1}}(z − a_{2})^{r_{2}} ⋯ (z − a_{t})^{r_{t}}},

whence – by changing u into uφ_{1} – we obtain precisely the expression we wanted.

In conclusion: the typical elliptic surface with determinant n, belonging to the cyclic case, is the surface

u = ^{n} \sqrt{φ(x, y)(z − a_{1})^{r_{1}}(z − a_{2})^{r_{2}} ⋯ (z − a_{t})^{r_{t}}φ(x, y)} f(x, y) = 0

where the exponents r_{i} satisfy the conditions mentioned above and φ(x, y) = 0 is a cylinder of order n touching f with an n-fold contact along three generators which do not lie in a plane.

The construction of the surfaces F as n-tuple covers of a cubic cylinder f(x, y) = 0 can be repeated with few modifications in the case in which the group is Abelian (with base 2). The only difference is that, instead of having only one radical, we have now two non-superimposed radicals in the expression of u. We do not dwell on this construction, which we shall develop later on (Section ??) for the family of surfaces F for which the curves K′, as well as the C, are elliptic. We only want to point out here that again, as in the cyclic case, one has the equality

∑[n(s − 1)]/s = 2n − 2π − 2,

and each one of the numbers s_{i} has always to divide the lowest common multiple of the remaining ones.

10.14 Surfaces with all plurigenera equal to zero: characterization of the ruled surfaces

We have proved that “the irregular surfaces with geometric genus p_{g} = 0, which do not belong to the family of the ruled surfaces, are elliptic surfaces with arithmetic genus p_{a} = −1”. Now we want to determine the plurigenera of these elliptic surfaces and to verify that these characters cannot be all zero, as it happens for the ruled surfaces (rational or not). Thus, if we recall the rationality conditions of Castelnuovo, given by the vanishing of the arithmetic genus and of the bigenus (Chapter VI, Section ??): p_{a} = P_{2} = 0, then the family of ruled surfaces is defined by the vanishing of the plurigenera, and more precisely by the following theorem: The necessary and sufficient condition in order that a surface is birational to a ruled surface is that both its quadrigenus and its sextigenus vanish:

P_{4} = P_{6} = 0.

Let us consider a (proper) elliptic surface F, of genus p_{g} = 0 and determinant n ≥ 2, represented on the n-tuple cubic cylinder f(x, y) = 0 in the above described way, in relation to the two characteristic pencils (K) and |C|. We may determine the virtual canonical curve of F by adding to the transform of the canonical curve of the cylinder f the curve corresponding to the branch curve on this multiple cylinder (see Chapter V, Section ??).

Now the virtual canonical curve of f can be obtained by subtracting the linear system 2c, formed by two normal plane sections of the cylinder, from its adjoint system, consisting of the three lines cut out by the plane at infinity; and since these lines are the images of exceptional curves on F, we only have to add to the transform of the branch curve the virtual curve −2C, corresponding to −2c. So, if we denote by z = a_{i} the components of the branch curve, each corresponding to a multiple curve C = s_{i}C_{i}, then the pure canonical curve of F is given by

D − 2C = ∑(s − 1)C_{s} − 2C.

Since the curves C homologous to the c are irreducible, we immediately see that the subtraction appearing in the above formula is not possible, and so we prove once more that the irreducibility of the mentioned curves C implies that the genus of F is p_{g} = 0.

In fact the linear system |D|, defined by D = ∑(s − 1)C_{s}, has dimension x = 0 and therefore it cannot contain the system |C|, let alone the system |2C|. To justify this assertion, let us notice that, since the curve d is formed by elliptic components of degree 0, a priori it has to define a linear system composed with the curves of the pencil |C| and with curves whose multiples lie in that pencil:

|D| = | ∑(s − 1)C_{s}| = |xC + ∑ h_{s}C_{s}|.

Whence we deduce

|xC| = | ∑ k_{s}C_{s}|,

with

0 ≤ k_{s} < s − 1.

It is clear that such a relation cannot hold for x > 0 (because the generic curves c are irreducible), since this would contradict the reducibility principle, inasmuch as a variable (irreducible and therefore connected) curve of (C) would degenerate in parts C_{s} which are not connected to one another.

Now from the above formula we can derive the expression of the pluricanonical curves and therefore the computation of the plurigenera of F.

In general the m-canonical curves are given by

|m ∑(s − 1)C_{s} − 2mC| = |m(t − 2)C − m ∑ C_{s}|,

where the summation is taken over all the values of s corresponding to the t multiple curves of the pencil |C|:

s = s_{1}, s_{2}, …, s_{t}.

If we denote by x = P_{m−1} the dimension of the complete m-canonical system, then this system consists of x variable (irreducible) curves C and of some fixed components C_{s}, each to be counted a certain number of times

h < s.

So we have

|m(t − 2)C − m ∑ C_{s}| = |xC + ∑ hC_{s}|,

with 0 < h < s, the summation being extended to the values

s = s_{1}, s_{2}, …, s_{t}

and to the corresponding values of

h = h_{1}, h_{2}, …, h_{t}.

Now, since sC_{s} = C, from the above relation we deduce

|(m(t − 2) − x)C| = | ∑((m + h)/s)C| :

this means that the fractions are integers:

(m + h)/s = ρ,

and each of this integers (ρ = ρ_{1}, ρ_{2}, …, ρ_{t}) expresses by excess the value of the fraction m/s, that is, it equals

[(m + s − 1)/s],

the greatest integer smaller than (m + s − 1)/s.

In conclusion we have

x = m(t − 2) − ∑ \frac{m + s − 1}{s},

and therefore the plurigenus of order m (≥ 2) is given by the formula

P_{m} = 1 + m(t − 2) − ∑ \frac{m + s − 1}{s},

where one has to take P_{m} = 0 when the expression becomes negative.

Let us set here m = 2; taking into account that – for s = s_{1}, s_{2}, …, s_{t} – we always have

s ≥ 2,

we have

[(s + 1)/s] = 1, ∑[(s + 1)/s] = t,

and therefore the bigenus is:

P_{2} = t − 3.

Now, for t > 3 we certainly have P_{2} > 0 and therefore also P_{4} > 0, P_{6} > 0, etc. If instead P_{2} = 0, since the hypothesis t = 2 has to be excluded because it yields that F is ruled, then we have

t = 3.

But, if we denote by n the determinant of F, then we have the equality

∑ n(s − 1) = 2n + 2π − 2,

whence (since s_{i} ≥ 0, π ≥ 1)

∑(1 − (1/s)) ≥ 1

that is,

t − ∑(1/s) ≥ 1 ∑(1/s) ≤ t − 1.

So for t = 3 we have

∑(1/s) = (1/s_{1}) + (1/s_{2}) + (1/s_{3}) ≤ 1.

Let us assume that the (integral and positive) numbers s_{1}, s_{2}, s_{3} are in non-decreasing order:

s_{3} ≥ s_{2} ≥ s_{1};

the above inequality gives us either

s_{1} = 2, s_{2} = 3, s_{3} ≥ 6

or

s_{1} = 2, s_{3} ≥ s_{2} ≥ 4,

or

s_{1} ≥ 3, s_{3} ≥ s_{2} ≥ 3,

or finally

s_{3} ≥ s_{2} ≥ s_{1} ≥ 4.

In the first and the second case:

[(s_{1} + 2)/s_{1}] = 2, [(s_{2} + 2)/s_{2}] = [(s_{3} + 2)/s_{3}] = 1

thus

P_{3} = 0,

whereas in the third and fourth case:

[(s_{1} + 2)/s_{1}] = [(s_{2} + 2)/s_{2}] = [(s_{3} + 2)/s_{3}] = 1,

and

P_{3} = 1.

On the other hand, if we compute the quadrigenus

P_{4} = 5 − ∑[(s + 3)/s],

then we have:

in the first and third case

P_{4} = 0,

in the second case

P_{4} = 1

and in the fourth case

P_{4} = 2.

And for the sextigenus

P_{6} = 7 − ∑[(s + 5)/s]

we find:

in the first and third case

P_{6} = 1,

in the second case

P_{6} = 0

in the fourth case

P_{6} = 2.

For our purposes it is sufficient to notice that when P_{2} = P_{4} = 0 (first and third case in the above discussion) then

P_{6} ≥ 1.

In conclusion for an elliptic surface, which is not birational to a ruled surface, at least one of the two plurigenera P_{4} or P_{6} is different from zero:

P_{4} + P_{6} ≥ 1, Q.E.D.

Remark The conditions for a surface to be ruled can also be expressed by the vanishing of the genus of order 12:

P_{12} = 0.

In fact for the elliptic surfaces with P_{4} ≥ 1 or P_{6} ≥ 1, one certainly has

P_{12} ≥ 1.

It is also worth remarking that one has

P_{12} = 1

for the elliptic surfaces (with a virtual canonical curve of order zero) possessing an elliptic pencil of elliptic curves K, and, by contrast

P_{12} > 1

for the elliptic surfaces possessing an elliptic pencil of curves K of genus π > 1, whose virtual canonical curves intersect the K at 2π − 2 points.

Indeed the difference between the two types of surfaces F consists in this: in the case of the K of genus π = 1 one has the equality

2n = ∑((s − 1)/s)n

that is,

t − 2 = ∑(1/s),  (10.3)

whereas in the case in which the K have genus π > 1 the following formula

2n + 2π − 2 = ∑ \frac{s − 1}{2} n

holds, which yields the inequality

t − 2 > ∑ \frac{1}{s} .  (10.4)

It follows immediately that in case 1) one has

P_{12} = 1,

and not P_{12} > 1, not only by the formula for P_{12}, but also by noticing that on the surface F, possessing two pencils of elliptic curves, there cannot be (pure) pluricanonical curves of order > 0.

Now let us prove that, if the inequality 2) holds, then one necessarily has

P_{12} > 1

and not P_{12} = 1. This proof cannot be simply deduced from the formula for P_{12}, but one also has to take into account the existence condition for the elliptic surfaces (related to the Abelian character of the group Γ_{n}) which we have already pointed out in Section ??, that is, that “each one of the numbers s_{i} has to divide the lowest common multiple of the remaining ones”^{(22)}.

Let us write down the formula

P_{12} = 12t − 23 − ∑ \frac{s + 11}{s} ,

and let us discuss the cases that can occur for t > 4, for t = 4 and for t = 3. For t > 4, that is, t ≥ 5, one has

[\frac{s_{i} + 11}{s_{i}}] ≤ 6

thus we certainly have

∑ [\frac{s_{i} + 11}{s_{i}}] ≤ 24,

(22)This is in fact the way the inequality P_{12} > 1 is established in of the by Enriques– Campedelli, Lezioni, §15 (Sem. Mat. di Roma, 1934). But in the original Note by Enriques, Sulla classificazione delle superficie algebriche ecc., Rendic. Acc. Lincei, febbraio 1914) the matter is not well explained, so that one could be induced to believe that the result could be simply deduced from the general formula that gives P_{12}. Such a deduction would be wrong, as was pointed out by H. Geppert in his Bericht die Klassification der algebraischen Flächen (Jahresbericht der Mathematische Vereinigung, 1931); indeed, by taking, for instance, s = 2, s_{2} = s_{3} = 5, we find P_{12} = 1.

Therefore Geppert proposes to consider P_{24} = 1 instead of P_{12} = 1. But, as a matter of fact, for the elliptic surfaces with pluricanonical curves of order > 0 one already has P_{12} > 1. The case of exception which was pointed out does not correspond to an existing elliptic surface, because 2 is not a divisor of 5 .

P_{12} ≥ 12t − 47 ≥ 12(t − 4) + 1 > 1.

For t = 4 the minimum value for P_{12} is:

P_{12} = 12(t − 4) + 1 = 1,

and it is attained only when the four numbers s_{i} are all equal to 2:

s_{1} = s_{2} = s_{3} = s_{4} = 2,

so that the equality (1) holds, because this is the only case in which for each of the s_{i} we have

[\frac{s_{i} + 11}{s_{i}}] = 6;

by contrast, if one of these numbers is greater than 2, as the inequality 2) requires, for instance if

s_{4} ≥ 3,

then we have

[\frac{s_{4} + 11}{s_{4}}] ≤ 4

thus

P_{12} ≥ 3.

Finally, let t = 3 and let us write all the integral solutions of the inequality

(1/s_{1}) + (1/s_{2}) + (1/s_{3}) < 1 (s_{1} ≤ s_{2} ≤ s_{3}),

that moreover satisfy the condition that each one of the s_{i} divides the lowest common multiple of the remaining ones. We have

\{ \begin{matrix}s_{1} = 2, s_{2} = 4, s_{3} = 8, \\ s_{1} = 2, s_{2} = 5, s_{3} = 10, \\ s_{1} = 2, s_{2} ≥ 6, s_{3} ≥ s_{2}, s_{1} = 3, s_{2} ≥ 3, s_{3} > 3, \\ s_{1} > 4, s_{2} ≥ s_{1}, s_{3} ≥ s_{2}. \end{matrix}

In all these case we have

[\frac{s + 11}{s}] < 12 P_{12} > 1. Q.E.D.

10.15 A historical note

The classification of the irregular surfaces of genus p_{g} = 0 (p_{a} < 0), as well as the proof that those which are not birational to ruled surfaces are elliptic surfaces containing an elliptic pencil of curves K of genus π ≥ 1 having an ∞^{1} group of transformations into themselves and another linear pencil of elliptic curves C, and the characterization of the ruled surfaces by means of the vanishing of the quadrigenus and the sextigenus, form the subject of F. Enriques’ memoir Sulle superficie algebriche di genere geometrico zero, Rendic. Circ. Mat. di Palermo, 1905. In any case (apart from some minor lack of accuracy in a few arguments) the proof of these results was affected by a gap, concerning the lemma II of §§6–8; in fact the existence of another elliptic curve essentially distinct from the parabicanonical C was simply deduced from the fact that the curves of the system {K′′}, the second adjoint system to a K, passing through a bicanonical group of K, must form an ∞^{1} system of index > 1: as a matter of fact one cannot exclude a priori that the reducible curves of this system containing K as a part consist of a single component to be counted with a certain multiplicity.

In order to fill this gap, the author found the argument relying on the construction of curves cutting out on the K groups of the semiquadricanonical, or, in general, a submultiple of a multiple of the canonical series. This procedure was treated by him in a Note of the Acc. dei Lincei, in 1930^{(23)} and then by Enriques–Campedelli, in Lezioni sulla classificazione delle superficie algebriche particolarmente di genere zero (Rendic. Sem. Mat. R. Università di Roma, 1934).

But in the new proof still remains a delicate weak point. In fact it is not possible to assume a priori, as we temporarily did with the hypothesis II of the §§6–8, that if a linear series is given or added to the rationality field of a curve K, then a single group of the series can be rationally determined. Luckily enough we were able to overcome the difficulty, by proving that the hypothesis II is verified for some multiple of the aforementioned series, so that we can justify the construction procedure for the essentially distinct parapluricanonical curves.

We still have to add a remark concerning the case in which our surfaces possess an elliptic pencil of (birationally distinct) elliptic curves; in order to make the construction of the trajectories C rigorous, or rather to prove that these are elliptic too, one has to exclude the possibility that some of these curves K can degenerate to multiple curves (thus proving that for p_{a} = −1 there are no paraelliptic surfaces): this point has been made precise by relying on the analysis of the singularity arising from the crossing of two branch curves (Chisini) and can be found in Enriques–Campedelli, Lezioni of 1934 (§12).

Finally, as far as the effective construction of the elliptic surfaces is concerned, in Enriques’ original Memoir of 1905 only the cyclic case was considered. The first examples concerning the Abelian case (with basis two) have been brought into light by G. Bagnera and M. De Franchis in their paper^{(24)}.

(23)Enriques, Sopra le superficie algebriche trasformabili in rigate. (Luglio, 1930).

(24)Sopra le superficie algebriche che hanno le coordinate del punto generico esprimibili con

Surfaces with pluricanonical curves of order zero 10.16 402

The general construction of all the elliptic surfaces (which can be reduced to the resolution of Abelian equations with a group of basis two) was finally developed by O. Chisini in the Notes of 1921^{(25)}.

10.16 Surfaces with pluricanonical curves of order zero: the types with elliptic curves K with general modulus

In Section ??, and in particular in the Remark at the end, we proved that:

The properly elliptic surfaces with genera p_{g} = 0 and p_{a} = −1 can be distinguished from the ruled surfaces by the 12-genus, since they have

P_{12} > 0,

instead of

P_{12} = 0.

More precisely the elliptic surfaces with p_{g} = 0 and p_{a} = −1 give rise to two families:

(1) surfaces with virtual canonical curve of order zero, characterized by the property of possessing an elliptic pencil of curves K of genus π = 1;

(2) surfaces with virtual canonical curve of order 2π − 2, containing an elliptic pencil of curves K of genus π > 1.

The former (on which any pure linear system of genus p has the degree n = 2p − 2) have

P_{12} = 1,

whereas the latter (for which n < 2p − 2) possess linear systems of dimension r > 0 of pluricanonical curves (of genus p^{(1)} = 1) and have

P_{12} ≥ 2.

Now we want to analyze more closely the classification of the surfaces of class (1) for which P_{12} = 1, showing that these give rise to few well determined types, the equations of which can be written down.

funzioni meromorfe quadruplamente periodiche di due parametri, Rendic. Acc. dei Lincei, 1907, 1.

(25)Le superficie ellittiche il cui determinante e’ un numero composto, Rendic. Acc. Lincei, 1921, 2.

Let us refer to the elliptic surfaces of the family 1); it is convenient to distinguish between the cases in which:

Either the elliptic curves K (which are birationally equivalent to one another) have general modulus, or the curves K are harmonic, or they are equianharmonic.

Let us start here with the general case: we shall prove that the surface F has either determinant n = 2 (the cyclic case) or n = 4 (the Abelian case).

In order to do this, we have to consider the involution g_{n}^{1} cut out on an elliptic curve K by the trajectories C. This involution is generated by an Abelian group and, since it is rational (and of genus 1), it has to contain some involutory transformation I of the second kind (g_{2}^{1}). Now, if the group Γ_{n} is not equal to the group Γ_{2} formed by I and by the identity, (since the products of two transformations I are transformations π of the first kind) then Γ_{n} must contain as many transformations π as I, say m of each type: thus n = 2m. More precisely the transformations π that are permutable with a given I are involutory, and so the Abelian group Γ_{n} generated by the aforementioned π and I is the dihedral group Γ_{4} formed by two involutions I, one involution π and the identity.

Therefore our surface F with a pencil of elliptic curves K with general modulus either has determinant n = 2 or n = 4, and the corresponding g_{n}^{1} defined on a K belongs to one of the following two types:

(a) n = 2, g_{n}^{1} = g_{2}^{1},

(b) n = 4, g_{n}^{1} = g_{4}^{1} is Abelian and dihedral, generated by two permutable transformations of the second type, whose product is a γ_{2}^{1} of the first series.

In the former case we may take as a model of the surface F of determinant n = 2 the surface defined by the equations:

I_{a} : u = \sqrt{(z − a_{1})(z − a_{2})(z − a_{3})(z − a_{4})φ(x, y)} f(x, y) = 0,

where f is a cubic cylinder and φ is a quadric cylinder having a double contact with it along three generators not lying in a plane.

In the latter case, starting with the corresponding representation of the quadruple line k (u = 0), we have to construct the elliptic curve K with a dihedral g_{4}^{1} in the plane (uz).

This construction can be explained either by using the general theory of the Abelian equations^{(26)}, or, as we shall do here, specifically relying on the properties of the elliptic curves^{(27)}.

(26)See for instance L. Bianchi, Ibidem, Chapter IV, §76.

(27)See Enriques–Chisini, Lezioni, Libro V, Chapter IV, §39 (Vol. III).

Let us remark that our dihedral g_{4}^{1} (on K) has 8 double points which fall into two quadruples: a quadruple of double points of the former g_{2}^{1} formed by two pairs of conjugate points of the latter

A_{1}A_{2} and B_{1}B_{2},

and a quadruple of double points of the latter g_{2}^{1} formed by two pairs of the former

D_{1}D_{2} and E_{1}E_{2}.

Therefore the 4-tuple line k has four (double) branch points:

abde,

and the rationality field of K may be defined by setting

u = \sqrt{φ(z − a)(z − b)} ± \sqrt{ψ(z − d)(z − e)},

(with φ and ψ constant with respect to z and u) so that the two branch points a and b correspond to the pairs of double points A_{1}A_{2} and B_{1}B_{2}, and the points d and e correspond to the pairs of double points D_{1}D_{2} and E_{1}E_{2}.

The proof that the equation written above actually represents the elliptic K corresponding to our quadruple k may be obtained by remarking that the curve that we have represented possesses a g_{4}^{1} with 8 double points; and anyway-by eliminating the radicals-we obtain an equation of degree 4 in z and u and of degree 2 in z, that defines a quartic with a double point at the point at infinity of the z axis and another double point infinitely near to it. Let us add that the two transformations of the second type, that is, the series g_{2}^{1} each having 4 double points, that compose the dihedral g_{4}^{1}, can be represented by the equations

\begin{matrix}u′ = [φ(z − a)(z − b) − ψ(z − e)(z − d)]/u \\ u′′ = [ψ(z − d)(z − e) − φ(z − a)(z − b)]/u, \end{matrix}

as one may deduce from the identity:

\frac{φ(z − a)(z − b) − ψ(z − d)(z − e)}{\sqrt{φ(z − a)(z − b)} + \sqrt{φ(z − d)(z − e)}} =

= \sqrt{φ(z − a)(z − b)} − \sqrt{φ(z − d)(z − e)}.

Therefore the equations of the elliptic surface F with determinant n = 4, with K of general modulus, can be reduced to the type:

I_{b} u = \sqrt{(z − a)(z − b) · φ(x, y)} + \sqrt{(z − d)(z − e) · ψ(x, y)} f(x, y) = 0,

where f is a cubic cylinder and φ and ψ are two quadric cylinders touching f along two triples of generators, not lying in a plane and not equivalent to one another.

Continuation: The harmonic case 10.17 405

10.17 Continuation: The harmonic case

In the harmonic case, besides the g_{2}^{1} and the g_{4}^{1} of the general case, one can have on the elliptic curve K a cyclic or Abelian g_{n}^{1}, whose groups of n points are transformed into themselves by a singular transformation of order four. In fact the harmonic elliptic curves possess, besides the ordinary transformations of the first and of the second type, π and I, two ∞^{1} continuous series of singular transformations of order 4, ω and τ; and the transformations of the four types are related in the following way^{(28)}:

\begin{matrix}πω = ω_{1} & ω_{1}ω = I & Iω = τ & τω = π \\ πI = I_{1} & I_{1}I = π & ωI = τ & τI = ω \\ πτ = τ_{1} & τ_{1}τ = I & Iτ = ω & ωτ = π \\ π_{1}π = π_{2} & Iπ = I_{1} & ωπ = ω_{1} & τπ = τ_{1}. \end{matrix}

The various types of cyclic or Abelian g_{n}^{1} belonging to an elliptic harmonic curve K can be classified by taking into account the following remark: if the related group Γ_{n} is not generated only by ordinary transformations (corresponding to K with general modulus), then it must contain an invariant subgroup Γ_{4} generated by a singular transformation ω or τ. Now this Γ_{4} gives rise to a g_{4}^{1} with two quadruple points A and B and a pair of double points D_{1} and D_{2} (forming a group of the g_{4}^{1}), that leads to the representation of K on a quadruple line k (u = 0) with two quadruple branch points: z = a and z = b, and a double branch point: z = d. If the g_{n}^{1} does not consist of this g_{4}^{1} (that is, if n > 4), then the group Γ_{n} has to contain the transformations corresponding to the linear transformations of the quadruple line fixing the point d and permuting the two points A and B: these transformations generate an Abelian Γ_{8}, formed by 4 singular transformations of order 4, two ω and two τ, by 2 involutions of the second type I, and by 2 transformations π of the first type: an involution γ_{2}^{1} and the identity.

In conclusion the g_{n}^{1} on the elliptic harmonic curve K, provided that it is not generated by ordinary transformations only, belongs to one of the following two types:

(a) n = 4, a cyclic g_{4}^{1} generated by a singular transformation of order 4: ω or τ; and

(b) n = 8, a g_{8}^{1} generated by an Abelian group Γ_{8} containing 4 singular cyclic transformations of order 4 (two ω and two τ) and 4 ordinary transformations: two I and two π (that is, a γ_{2}^{1} and the identity).

(28)See for instance Enriques–Chisini, Lezioni, Libro V, Chapter III, §27 (Vol. III, page 260).

In the first case the elliptic K represented on the quadruple k, with the branch points a, b and d, can be realized on the (zu) plane by writing, for instance:

u = ^{4} \sqrt{φ(z − a)(z − b)(z − d)^{2}}

with φ constant (with respect to z and u).

The equation that we have just written corresponds to a certain choice of the substitutions on the sheets of the algebraic function u(z), in relation to the branch curves z = a, b, d; if, by contrast, we take the inverse substitutions then we find the similar type:

u = ^{4} \sqrt{φ(z − a)^{3}(z − b)^{3}(z − d)^{2}}

which is birationally distinct from the first one.

So the elliptic harmonic surfaces with determinant n = 4, cyclic case, can be reduced to the type represented by the equations:

II_{a} u = ^{4} \sqrt{(z − a)(z − b)(z − d)^{2} · φ(x, y)} f(x, y) = 0,

or to a similar type: where f is a cubic cylinder and φ a cylinder of order four touching f with a fourfold contact along three generators not lying in a plane. The similar types can be distinguished from one another, as we have said, by the choice of the substitutions on the sheets of u, and, although birationally distinct, they still have the same characters.

In the properly Abelian case we can still represent in the (zu) plane a type of elliptic curve K containing a g_{8}^{1} with two groups, each formed by two quadruple points, and a group formed by four double points: in order to do this we have to construct an 8-tuple line with three branch points

z = a, z = b, z = d

such that the first two correspond to a cyclic substitution of order 4 on the sheets, whereas the third one corresponds to a simple substitution of period 2.

This construction can be made according to the general principles of the theory of the Abelian equations^{(29)}. On the other hand the formula that one obtains in this way can be justified a posteriori^{(30)} by looking at the substitutions on the sheets corresponding to the branch points, as we mentioned above.

(29)See, for instance, L. Bianchi, Lezioni sulla teoria dei gruppi di sostituzioni e delle equazioni algebriche secondo Galois, (Pisa, 1900) Chapter III, §30.

(30)See O. Chisini, Le superficie ellittiche il cui determinante e’ un numero composto.

Rendic. Acc. Lincei, s. V, Vol. XXX, 1921, 2.

Let us say that the curve K can be reduced to the type represented by the equations

u = \sqrt{(z − b)(z − d)fie} + ^{4} \sqrt{(z − a)(z − b)^{3}psi}

or to a similar type (which differs from this by a different choice of the aforementioned substitutions): here φ and ψ denote constants (with respect to u and z).

To justify this assertion, let us denote the 8 sheets of u by means of two indices:

u_{11}u_{12}u_{13}u_{14} u_{21}u_{22}u_{23}u_{24},

where the first index corresponds to the two values +√ and −√ of the quadratic radical entering in the expression of u, and the second to the four values + ^{4} √, i ^{4} √, − ^{4} √, −i ^{4} √ of the radical of order 4.

The branch point z = a corresponds the substitution on the sheets

S_{a} = (u_{11}u_{12}u_{13}u_{14})(u_{21}u_{22}u_{23}u_{24}).

And similarly the point z = b corresponds to a cyclic substitution of order 4, that is,

S_{b} = (u_{11}u_{24}u_{13}u_{22})(u_{12}u_{21}u_{14}u_{23}).

The point z = d corresponds, by contrast, to a substitution of period 2:

S_{d} = (u_{11}u_{21})(u_{12}u_{22})(u_{13}u_{23})(u_{14}u_{24}).

So we obtain three substitutions that, by multiplication, generate an Abelian group of order 8, having the structure of the Γ_{8} we mentioned above.

Let us add that the aforementioned substitutions satisfy the existence condition^{(31)} for the algebraic function u(z), (which is unbranched at the point z = ∞), that is, that

S_{d}S_{b}S_{a} = 1;

and just in order to satisfy this condition we had to assign the exponent 3 (instead of 1) to the term z − b appearing under the fourth order radical.

In this way the function u(z) gives rise to the required elliptic curve K with an Abelian g_{8}^{1}, and moreover it gives the type of such a curve, although we can also have another similar type, not birational to the first one, given by

u = \sqrt{(z − a)(z − b)^{3} · φ} + ^{4} \sqrt{(z − a)(z − d) · ψ}^{(32)};

page 429).

(31)See, for instance, Enriques–Chisini, Lezioni, Libro V, Chapter IV, §38 (Vol. III, (32)From the projective point of vew, the curve K representing the function u(z) has a tangent line (parallel to the u axis) with two fourfold contacts in correspondence with the point z = a, a tacnode where two linear branches have a fourfold contact (which therefore yields two quadruple points for the g_{8}^{1}) in correspondence with z = b, and finally a quadruple tangent line with four simple contacts in correspondence with z = d.

other types which seem to occur can be reduced to the types mentioned here with simple exchanges of the names of the branch curves.

It is worth adding that the two substitutions on the sheets of the function u in the neighbourhood of the branch points z = a and z = b can be extended to birational transformations, of the same order 4, of the curve K whose equation

θ(zu) = 0

can be obtained by eliminating the radicals appearing in the expression of u; and the same can be said for the period two substitution corresponding to the point z = d.

In fact one can prove, for instance, that u_{12} is a rational function of u_{11} and of z; this follows from the consideration of the two equations satisfied by a given value of z:

θ(uz) = 0, θ(u′z) = 0

and of their resultant

R(uu′) = 0 :

this, together with θ(uz) = 0, defines u′(uz), as a rational function.

Now we can write the equations of an elliptic surface, harmonic type, with determinant n = 8:

II_{b} u = ^{2} \sqrt{(z − a)(z − b)^{3} · φ(x, y)} + ^{4} \sqrt{(z − b)(z − d) · ψ(x, y)}, f(x, y) = 0

where f represents a cubic cylinder and φ and ψ may be assumed to be two cylinders, one of order 2 and the other of order 4, the former tangent to f along 3 generators and the latter tangent to it with a fourfold contact along three other generators. In order that the surface F is actually of determinant n = 8 (irreducibility of the curves C corresponding to z = const. on it) one needs to suppose that the two triples of generators of contact of f and φ and of f and ψ do not lie in planes, so that for z = const. the two curves

u = ^{2} \sqrt{(z − a)(z − b)^{3} · φ(x, y)}, f(x, y) = 0

and

u = ^{4} \sqrt{(z − b)(z − d) · ψ(x, y)}, f(x, y) = 0,

are irreducible, and moreover the two curves

u = ^{2} \sqrt{(z − a)(z − b)^{3} · φ(x, y)} and u = ^{4} \sqrt{(z − b)(z − d) · ψ(x, y)}

are birationally distinct^{(33)}: this yields that the triple g_{1}g_{2}g_{3} of the generators of fourfold contact of ψ with f is not similar to the triple of generators g′_{1}g′_{2}g′_{3} of contact of φ with f, that is, the triple g_{1}g_{2}g_{3}, counted twice, forms a sextuple which is not equivalent to the sextuple obtained by adding to g′_{1}g′_{2}g′_{3} three generators of f lying in a plane.

(33)See Enriques–Chisini, Lezioni, Libro V, Chapter IV, §33 (Vol. III, page 452).

The equianharmonic case 10.18 409

10.18 The equianharmonic case

In the equianharmonic case, if the elliptic curve K contains a g_{n}^{1} not generated by simple ordinary transformations (n = 2, 4), then it has to be generated by a cyclic or Abelian group Γ_{n} of transformations (with basis 2) containing some singular transformation of order 3 or 6. In fact the elliptic equianharmonic curves possess, besides the ordinary transformations of first and second kind, π and I, two ∞^{1} continuous series of singular cyclic transformations of order 3, ω and τ, and two more series of cyclic singular transformations of order 6, σ and ρ^{(34)}. And the transformations of the various types are related in the following way:

\{ \begin{matrix}πσ = σ_{1} & σ_{1}σ = τ & τσ = I & Iσ = ω & ωσ = ρ & ρσ = π \\ πτ = τ_{1} & τ_{1}τ = ω & ωτ = π & στ = I & Iτ = ρ & ρτ = σ \\ πI = I_{1} & I_{1}I = π & σI = ω & ωI = σ & τI = ρ & ρI = τ \\ πω = ω_{1} & ω_{1}ω = τ & πω = π & σω = ρ & ρω = I & Iω = σ \\ πρ = ρ_{1} & ρ_{1}ρ = ω & ωρ = I & Iρ = τ & τρ = σ & σρ = π. \end{matrix}

Now for the Abelian g_{n} not generated by ordinary transformations on the equianharmonic curve K, three cases are possible:

(a) n = 3, a cyclic g_{3}^{1} generated by a singular transformation of order 3, ω or τ;

(b) n = 6, a cyclic g_{6}^{1} generated by a singular transformation of order 6, σ and ρ ;

(c) n = 9, an Abelian g_{9}^{1} (with basis two) whose group contains three permutable singular cyclic transformations of order 3, say ωω_{1}ω_{2}, three more cyclic transformations as above τ = ω^{2}, τ_{1} = ω_{1}^{2}, τ_{2} = ω_{2}^{2}, and finally three ordinary transformations of the first kind π, π^{2}, π^{3} = 1.

This classification can be easily justified starting from the remark that the group Γ_{n} has to contain an invariant cyclic subgroup Γ_{3}, generated by a singular transformation ω or τ. Since Γ_{3} generates a cyclic g_{3}^{1} with three triple points, we are led to examine the transformations of K corresponding to the linear transformations preserving a triple line of the type

u^{3} = (z − a)(z − b)(z − d);

and these are:

(1) the involutions, for instance (ab)(dd), having one of the points of the triple (abd) as a double point and permuting the other two points; and (2) the cyclic linear transformations (abd) or (adb).

(34)Enriques–Chisini, Lezioni, Libro V, Chapter III, §27 (Vol. III, page 260–263).

Now by adding a transformation of type (1) to Γ_{3} one obtains a cyclic Γ_{6} (with a sextuple point, two triple points and three double points) that is not contained in a larger Abelian group; whereas the addition of a transformation (abd) of type (2) yields an Abelian Γ_{9}, containing three singular transformations of order three, which in turn is not contained in a larger Abelian group.

We can easily construct the equianharmonic curves K corresponding to the aforementioned Abelian g_{n}^{1}, for

n = 3, 6, 9,

and accordingly the corresponding elliptic surfaces.

We have as a model of equianharmonic elliptic surfaces with determinant n = 3 (the cyclic case) the surfaces:

III_{a} u = ^{3} \sqrt{(z − a)(z − b)(z − d) · φ(x, y)} f(x, y) = 0,

where f is a cubic cylinder and φ is another cylinder of the same order, touching f with a triple contact along three generators not lying in a plane.

And we have as the model of equianharmonic elliptic surfaces with determinant n = 6 (the cyclic case) the surfaces:

III_{a} u = ^{3} \sqrt{(z − a)(z − b)^{2}(z − d)^{3} · φ(x, y)} f(x, y) = 0,

where f is a cubic cylinder and φ is another cylinder of order 6, touching f with a sixfold contact along three generators not lying in a plane.

In the properly Abelian case, in which one deals with equianharmonic elliptic surfaces F of determinant n = 9, a model of F can be constructed in a way similar to the one used in the harmonic case (n = 8).

In order to do this, we need to construct the equianharmonic elliptic curve K of the plane (uz) represented on a multiple line of order 9 (u = 0), the image of our Abelian g_{9}^{1}, generated by two singular permutable transformations of order 3. As the type of this curve we can take

u = ^{3} \sqrt{(z − a)^{2}(z − d) · φ} + ^{3} \sqrt{(z − b)(z − d)^{2} · ψ}

with φ and ψ constant.

To prove that the u(z) thus defined actually corresponds to an equianharmonic elliptic curve containing the aforementioned Abelian g_{9}^{1}, it is convenient to compute the substitutions on the sheets of u corresponding to the branch points

z = a, z = b, z = d.

Summary 10.19 411

To this end, let us denote the sheets by means of two indices, the former assuming the values 1, 2, 3 in correspondence with the values of the first cubic radical

3 √, ε ^{3} √, ε^{2} ^{3} √ (where ε = e^{2πi/3})

appearing in the expression of u(z), while the latter index accordingly assumes three values in correspondence to those of the latter cubic radical. We have, in correspondence with the three branch points, the following substitutions:

for z = a

S_{a} = (u_{11}u_{31}u_{21})(u_{12}u_{32}u_{22})(u_{13}u_{33}u_{23});

for z = b

S_{b} = (u_{11}u_{12}u_{13})(u_{31}u_{32}u_{33})(u_{21}u_{22}u_{23});

for z = d

S_{d} = (u_{11}u_{23}u_{32})(u_{12}u_{21}u_{33})(u_{13}u_{22}u_{31}).

The third substitution is the inverse of the product of the other two, so that the existence condition for the existence of the multiple line u = 0, not branched at the point at infinity z = ∞, is satisfied^{(35)}:

S_{d}S_{b}S_{a} = 1.

Thus one sees that the aforesaid substitutions generate an Abelian group Γ_{9} giving rise to an, equally Abelian, g_{9}^{1} on an equianharmonic elliptic curve.

So one eventually proves that the equianharmonic elliptic surfaces with determinant n = 9 (the Abelian case) give rise to the type

III_{c} u = ^{3} \sqrt{(z − a)^{2}(z − d) · φ(x, y)}+\sqrt{(z − b)(z − d)^{2}psi(x, y)} f(x, y) = 0,

where f is a cubic cylinder and φ and ψ can be also assumed to be cubic cylinders osculating f along two triples of inequivalent generators not lying in a plane.

10.19 Summary

We can summarize the analysis of the elliptic surfaces of genus p_{g} = 0 with virtual canonical curve of order zero by stating the following theorem:

The elliptic surfaces of genus p_{g} = 0 and p_{a} = −1 with virtual canonical curve of order zero, characterized by the value P_{12} = 1, that is, the surfaces with p_{a} = −1 possessing an elliptic pencil of elliptic curves K and another linear pencil of elliptic curves C, fall into seven families with determinant n = 2, 4; 4, 8; 3, 6, 9, and are represented by the equations of one of the types I_{a}, I_{b}, II_{a}, II_{b}, III_{a}, III_{b}, III_{c}, or by a similar type.

(35)See Enriques–Chisini, Lezioni, Libro V, Chapter IV, §38 (Vol. III, page 429).

The types I_{a} and I_{b} (respectively the cyclic type and the Abelian type) correspond to elliptic curves C with general modulus; II_{a} and II_{b} (the cyclic type and the Abelian type) to harmonic curves C, and finally III_{a} and III_{b} to a cyclic type with equianharmonic C and III_{c} to an Abelian equianharmonic type.

The aforesaid elliptic surfaces can also be distinguished, to a certain extent, by means of the values of the plurigenera smaller than P_{12}, that can be computed with the formula of Section ??. The surfaces I_{a} and I_{b} (determinant 2 and 4) have the characters:

P_{2} = 1, P_{4} = 1, P_{12} = 1;

the surfaces of the harmonic type II_{a} and II_{b} (determinant 4 and 8) have:

(P_{2} = 0), P_{4} = 1, P_{6} = 0, P_{12} = 1;

and finally the surfaces of the equianharmonic type III_{a} (determinant 3) have

P_{2} = 0, P_{3} = 1, P_{4} = 1, P_{12} = 1,

while III_{b} and III_{c} (determinant 6 and 9) have

P_{2} = 0, P_{3} = 0, P_{4} = 1, P_{12} = 1.

Note The elliptic surfaces with virtual canonical curve of order zero were already noticed by Enriques in his study of the surfaces of genus zero (1905); they appear as particular hyperelliptic surfaces (more precisely as involutions on the Jacobi surface corresponding to a degenerate curve of genus two), and from this point of view they have been considered by G. Bagnera and M. De Franchis in the memoir of 1907 Sopra le superficie algebriche che hanno le coordinate esprimibili con funzioni meromorfe quadruplamente periodiche di due parametri^{(36)}.

These authors discover that such surfaces give rise to a finite number of types and succeed in classifying them by giving their hyperelliptic parametric representations, corresponding to our 7 typical equations.

Independently of Bagnera and De Franchis, but after them, also Enriques and Severi found this result again in a different way; their starting point is still the parametric representation of our surfaces with hyperelliptic functions^{(37)}.

(36)Rendic. Acc. Lincei, s. V, Vol. XVI, 19071.

(37)F. Enriques and F. Severi, Intorno alle superficie iperellittiche irregolari, Rendic. Acc.

Lincei, s. V, Vol. XVII, 19081 (cf. by the same authors, ibidem, Vol. XVI, 1907, 1). Mémoire sur les surfaces hyperelliptiques, Acta Math., Vol. 32 and 33 (1909). See especially the §§56 and 57.

It seems to us particularly remarkable that in this treatment there is the idea of constructing the hyperelliptic or properly Abelian surface as a multiple elliptic surface without branch curves, thus computing the characters defining it (p_{a} = −1, p_{g} = 1, P_{4} = 1 according to a theorem of Picard–Enriques). For instance, if we refer to the simplest elliptic surface, the one of type I_{a}, then one has a F_{m} of a certain order m in S_{3}, possessing a biadjoint surface φ_{2n−8} and one obtains a proper hyperelliptic surface (p_{a} = −1, p_{g} = 1) by extracting the ^{2} \sqrt{φ_{2n−8}} on F_{m}.

The present treatment of the question, that is, the algebraic classification of our elliptic surfaces (which, in the constructive part, uses the paper of O. Chisini quoted above), was the object of a Note by Enriques, Sulle superficie ellittiche di genere zero^{(38)} of 1934, and then it has been treated in the Lezioni by Enriques–Campedelli of the Sem. Mat. di Roma (1934).

(38)Rendic. Acc. Lincei, s. VI, Vol. XIX, the 18th of February 1934.

The general classification of surfaces

11.1 Introduction

The reader who has followed the development of this treatise, if – as it often happens to the mathematician – he likes the general truths best, may have formed the impression that the author has given too much room to examples and particular cases, being somehow guided by the feeling of curiosity of the naturalist, who gathers in a museum the most different types of animals, or plants or minerals.

But, as the museum can give an idea of the richness of forms of life and therefore suggests the consideration of general problems in biology, also the collection of examples in this field of mathematics assumes an essential significance from the heuristic or historical-constructive point of view of the science. We can clarify its value by repeating the words with which G. Castelnuovo gave an account of the efforts, that we made together, to dissipate the darkness that we had met at the beginnings of the theory of surfaces^{(1)}. “We had constructed, in an abstract sense I mean, a large number of models of surfaces in our space or in higher dimensional spaces; and we had distributed those models, so to speak, in two show-cases. One contained the regular surfaces, for which everything proceeded as in the best of all possible worlds; analogy made it possible to extend to these the most important properties of the plane curves. But when we tried to verify these properties for the surfaces of the other show-case, the irregular ones, then the troubles started, and we found exceptions of every kind. Finally, an assiduous study of our models led us to predict some properties that had to be true, with suitable modifications, for the surfaces of both show-cases; then we tested these properties with the construction of new models. If they stood the test then, as the last step, we tried to find their logic justification”.

(1)La geometria algebrica e la scuola italiana, Lecture at the International Congress of the Mathematicians, Bologna, september 1928.

But now we can say more: the determination of particular families or classes of surfaces, over which we lingered, does not simply give mere examples, whose meaning is then surpassed by the inductive development of the science; in fact these are brought together in a synthesis that can be said to be the main purpose of this algebro-geometric theory; namely the general classification of the algebraic surfaces.

To this aim, it is convenient to notice that algebraic geometry is, on the whole, a higher form of arithmetic, where the criterion itself of generality has only a relative meaning. Here there is a very deep remark, which goes back to Poncelet and Noether: from a purely logical point of view we cannot say, for instance, that the family of surfaces which are not birational to a ruled surface is larger than the one of the ruled surfaces. If one proceeds to the classification of surfaces (either in the ordinary space or in a hyperspace) according to their order n, the case of surfaces which are not birational to ruled surface, whose hyperplane sections have genus π with

n ≤ 2π − 2

appears to be more general; but if, by contrast, we proceed to the classification of the surfaces according to the genus π of their hyperplane sections, then the more general case appears to be the one in which the order is

n > 2π − 2,

that is, the case corresponding to surfaces birational to ruled surfaces.

We do not want to draw paradoxical consequences from what we have said. Thus we will go on considering the study of surfaces not belonging to the family of the ruled surfaces as more general than the one of the ruled surfaces. But this is a value judgement, not an absolute fact. In giving it we weigh in our hand – so to speak – the problems concerning the two types of surfaces, the richness of the cases to which they give rise, the possibility of relating the two families from more meaningful points of view. From this we can draw this lesson: that every family or class of surfaces possesses its own characters that distinguish it from the others and it is an object of study equally worth to be pursued; and the definition of such a family by the integral invariant characters has often a general interest. For example the definition of the ruled surfaces by the vanishing of the quadrigenus and the sextigenus gives, rather than a particular result concerning this family, a significant theorem for the general theory; telling us of the existence of canonical or pluricanonical curves (of order ≥ 0) on any surface which is not birational to a ruled surface.

Surfaces with a continuous series of transformations 11.2 416

Our classification makes a synthesis of the results of previous analyses and succeeds in distinguishing 4 families of surfaces, which have essentially different properties and are defined by the value of P_{12} and – subordinately – of p^{(1)}. These are:

(A) the ruled surfaces (existence of linear systems of genus π and degree n > 2π−2, infinitely many exceptional curves that cannot be eliminated, and also – as we shall see – continuous series of birational transformations which do not form a group of finite dimension);

(B) the surfaces (without exceptional curves) with pluricanonical curves of order zero (pure linear systems of degree n = 2π − 2);

(C) the surfaces (n < 2π − 2 with π > 1) having canonical or pluricanonical curves composed with the elliptic curves of a pencil (p^{(1)} = 1);

(D) the surfaces (n < 2π − 2) with p^{(1)} = 1, that give rise to canonical or pluricanonical surfaces offering a model whose projective properties reflect the invariant properties of the class of surfaces.

Each one of these families is an object of study interesting in itself, but the relevance of the characteristic properties distinguishing each of them from the others has a significance for the general theory, as we pointed out above.

Now, to let the classification acquire its proper meaning, it is convenient to extend some of the remarks that we made in the case p_{g} = 0 to the surfaces with p_{g} > 0; and to this aim we need first of all to make a deeper study of the surfaces admitting a continuous series of birational transformations into themselves.

11.2 Surfaces admitting a continuous series of birational transformations into themselves: the cases leading to ruled surfaces

We know that the surfaces birational to ruled surfaces possess continuous series of birational transformations into themselves; in particular on a rational surface, or rather on the plane, one knows^{(2)} the types of continuous groups of Cremona transformations; and on the other hand one also has continuous series of transformations, like the quadratic transformations, which do not generate continuous groups of finite dimension. And such series can also be found on a ruled surface of genus p > 0; where however one has series of transformations acting on the surface in a intransitive way, if p > 1.

(2)Enriques, 1893. See Enriques–Chisini, Lezioni. Libro V, Chapter II, §28 (Vol. III, page 200).

With some additional restriction, one can prove the converse of the above remarks, that is, the possess of a continuous system of birational transformations enables us to characterize the family of ruled surfaces. This can be done from four points of view:

(1) any surface possessing an ∞^{1} rational group of transformations into itself belongs to the family of ruled surfaces;

(2) any surface possessing a (transcendental or algebraic) continuous group of transformations into itself, fixing a linear system |L| of curves, at least ∞^{1}, and acting transitively on it^{(3)}, belongs to the family of ruled surfaces;

(3) any surface F possessing a continuous group of dimension r ≥ 2 of transformations into itself, acting intransitively on the points of F, or also a continuous group of dimension r ≥ 3 acting transitively on it, belongs to the family of ruled surfaces;

(4) any surface possessing a continuous series of transformations into itself, that does not generate a group of finite dimension, belongs to the family of ruled surfaces.

The theorem (1) follows directly from the theorem of Noether–Enriques (1898)^{(4)} concerning the surfaces with a pencil of genus p ≥ 0 of rational curves, because the trajectories of the ∞^{1} rational group form in fact a pencil of rational curves. But it should be pointed out that the result we are mentioning was obtained by Painlevé before Enriques’ extension to the case p > 0 of Noether’s theorem for p = 0: indeed the fact that the rational curves C of a pencil are trajectories of an ∞^{1} group makes it easy to construct a unisecant curve to the C, by means of which the C can be rationally transformed into lines.

The theorem 2) states, on the whole, that “a surface transformed into itself by a continuous group of linear transformations is either rational or ruled”; the theorem was given in this projective form, with some restriction, by Enriques (1893), and it was proved in general.by Fano (1895, 96, 97)^{(5)}.

It is however convenient to give a new proof of this theorem, which also takes care of the case of a group Γ fixing a linear system |L|, at least ∞^{2}, belonging to an involution, and the case of a Γ fixing a linear pencil |L|; and more than that, the proof that we propose here just reduces the general theorem to the latter case.

(3)We mean that it permutes the elements (curves) of |L| in infinitely many ways.

(4)See for instance Conforto, Le superficie razionali, Libro II, Chapter I, §§4, 5, 6 (pages 245, 260, 262).

(5)See Conforto, Le superficie razionali, Libro II, Chapter II, §37 (page 496).

We start from (Picard’s) remark that “a continuous group of birational transformations of an algebraic surface into itself is always contained in an algebraic continuous group”^{(6)}.

If the surface F admits an algebraic continuous group of transformations into itself, fixing a linear pencil |L| of curves and acting transitively on it, we may assume that the curves L are permuted by Γ in the same way as by an ∞^{1} group, since otherwise we may replace Γ with the subgroup of its transformations fixing one or two particular curves L.

In this setting we need to distinguish between the following assumptions:

(a) the group Γ is ∞^{1}; then its trajectories, which admit ∞^{1} transformations into themselves, are curves K (a priori rational or elliptic)^{(7)} on which the curves L cut out an invariant rational involution, therefore they are rational^{(8)}: it follows that F is birational to a ruled surface.

(b) The group Γ is ∞^{1} at least, and it acts transitively on the points of the surface. Then the curves of the pencil |L|, which are transformed into themselves by ∞^{1} transformations of Γ, are either rational or elliptic.

If they are rational, then F is birational to a ruled surface. If instead they are elliptic, then let us consider the ∞^{1} elliptic series {D} formed by the images of a curve D, chosen in a completely general way on F: through any point P of F there is a certain number i of curves d of {D}, and P may vary on a curve K in such a way that the groups of i curves d passing through it are equivalent; hence the curves K thus defined must form a base point free elliptic pencil (K)^{(9)}.

Therefore, among the transformations of our group Γ that permute the elements (curves) of (K), there are ∞^{1} transformations forming a subgroup Γ′ and having the K as their trajectories. By SCWARTZ’s theorem, these K are a priori either rational or elliptic, but the latter case can be excluded; let us in fact remark that the transformations of Γ′ have to fix the linear series g_{n}^{1} cut out on a K by the curves L of |L|. Thus we deduce that F is birational to a ruled surface. Q.E.D.

(6)Indeed the conditions that a surface F is transformed into itself by birational transformations of a given order are algebraic conditions, and therefore they define a mixed or continuous algebraic group. If the group is mixed it contains a continuous group.

(7)SCHWARTZ’s theorem. See Enriques–Chisini, Lezioni, Libro V, Chapter III, §31 (Vol. III, page 309).

(8)The curves admitting infinitely many transformations into themselves fixing a g^{1} can be transformed, with a multiple of such a series, into curves (of a certain space) with infinitely many projective transformations, and are therefore rational. See, for instance, Enriques–Chisini, Ibidem, Libro V, Chapter III.

(9)See Chapter IX, Section ??.

After having established theorem 2) in the case in which the linear system |L| fixed by the group Γ is a pencil, it is easy to extend the proof to the case in which |L| is an ∞^{r} linear system, with r > 1. In fact the transformations of Γ (that can be assumed to be a continuous algebraic group) act on the elements (curves) of |L| as linear transformations of an S_{r}, also forming an algebraic group Γ′, and the subgroups of dimension ≥ 1 formed by all the transformations of Γ′ permutable with a given ω are algebraic, too. Now such a subgroup, let us call it \overline{Γ}′, fixes the points fixed by ω, and therefore it has also a fixed line not formed by fixed points: this line corresponds to a linear pencil of curves L which is transitively invariant for \overline{Γ}′. Therefore we fall again in the case we have treated before.

Theorem 3)^{(10)} can be proved as follows.

Let us start by assuming that the ∞^{r} group Γ of transformations of F acts intransitively on the surface, that is, that it fixes the curves C of a pencil on it. If Γ is (as we may always assume) an ∞^{r} algebraic group with r ≥ 2, then we can get an ∞^{1} algebraic subgroup Γ′ of Γ, by imposing on the transformations of Γ that they fix r − 1 generic points of F: say O, O′,…

Now the group Γ′ – which is an ∞^{1} algebraic series of elements with ∞^{1} transformations into itself – is a priori either rational or elliptic. But it cannot be elliptic, because it contains an invariant series g_{n}^{1}, of a certain order n > 1, corresponding to the series of the points of F infinitely near to O. Thus it is rational and (according to Theorem (1) F is birational to a ruled surface.

To tell the truth, the above argument fails if the transformations of Γ′ fix all the points of F infinitely near to O; but in that case we can repeat the same argument for the successive neighbourhood of a point O_{1} near to O in the first order neighbourhood, and similarly – if needed – for the first neighbourhood of a point O_{s} near to O in the sth order neighbourhood, not entirely consisting of fixed points of Γ′.

In an analogous way one proves the second case of the theorem, where one has an algebraic group Γ at least ∞^{3}, acting transitively on the points of F. If we refer to the simplest hypothesis, in which Γ has dimension 3, then we obtain an ∞^{1} algebraic subgroup Γ′ by imposing on the transformations of Γ that they fix a point O, and – as in the previous case – we see that Γ′ is a rational group.

If by contrast Γ has dimension r > 3, then we similarly obtain an ∞^{1} rational subgroup Γ′ of it by imposing on the transformations of Γ that they fix a point O of F, and then another point O_{1} infinitely near to O, etc.

(10)See Castelnuovo–Enriques, Sur les surfaces algébriques admettant un groupe continu de transformations birationelles en elles-mêmes. (Comptes rendus, Paris, the 29th of July,

Elliptic and hyperelliptic surfaces 11.3 420

Finally we prove theorem 4), which answers a subtle question raised by E. Picard^{(11)}.

If the surface F admits a continuous series Γ – say ∞^{1}-of transformations into itself, which does not generate a group (of finite dimension), then this means that the products of the transformations of Γ, taken r by r, form in general an ∞^{r} series of transformations Γ_{r}, where r is susceptible of becoming arbitrarily large. Therefore the transformations of Γ_{r} transform a curve C, chosen on F in the most general way, in the curves C_{r} of a continuous system {C_{r}} of dimension r. If the curve C belongs to a base point free linear system, of genus π and degree n, then the system {C_{r}} is formed by linear systems |C_{r}| of the same genus and degree, with a certain number of base points of multiplicity i (for i = 1, 2, …); and the degree and genus of a |C_{r}| are respectively given by

N_{r} = n + ∑ h_{i}^{2}

and

Π_{r} = π + ∑ \frac{h_{i}(h_{i} − 1)}{2}

Therefore we have

N_{r} − 2Π_{r} = n − 2π + ∑ h_{i}.

But, since the degree of {C_{r}} (which is a priori ≥ r − 1) grows with r as much as we wish, we can choose r in such a way that

∑ h_{i}^{2} > (2π − n)^{2}

thus

∑ h_{i} ≥ \sqrt{∑ h_{i}^{2}} > 2π − n.

So it is apparent that the surface F possesses curves of genus Π_{r} and degree N_{r} > 2Π_{r−2} and therefore^{(12)} it is birational to a ruled surface. Q.E.D.

11.3 Elliptic and hyperelliptic surfaces

The analysis of the previous section allows us to “determine all the surfaces F, not birational to a ruled surface, admitting a continuous group of transformations into themselves”.

First, we may assume this group Γ (not contained in a larger group) to be algebraic; therefore the group gives rise to one of the following cases:

(11)Castelnuovo–Enriques, Sopra alcune questioni fondamentali nella teoria delle superficie algebriche, Ann. di Mat., 1901 (n. 19).

(12)See Chapter X, Section ??.

(1) either Γ acts intransitively on the points of the surface F, and then (Section ??) it is an ∞^{1} elliptic group fixing the elliptic curves C of a pencil, which has a certain genus ρ ≥ 0;

(2) or Γ is an ∞^{2} group of transformations, acting transitively on the points of F.

Case 1 In the first case the surface F admitting an ∞^{1} algebraic group of birational transformations is an elliptic surface, with the arithmetic genus p_{a} = −1, the geometric genus p_{g} = ρ, and the linear genus p^{(1)} = 1.

In fact the ∞^{1} transformations of Γ map a curve L, generically chosen on F, to the ∞^{1} curves of an elliptic series {L}; there is a certain number i of curves L going through a given point P, and P can be allowed to move along a curve K in such a way that the group of the aforementioned i curves L always stays in the same equivalence class; therefore the curves K form an elliptic pencil (K) on F, and the curves L cut out equivalent groups on the curves of this pencil: for this reason the curves K are essentially distinct from the trajectories C of Γ.

Now the surface F, containing a pencil of genus ρ of elliptic curves C and an elliptic pencil of curves K (of genus > 0), is elliptic (Chapter X, Section ??) and it has the aforesaid characters, which can be easily computed.

In order to do this, we compute the number Δ of the curves of (K) with a double point, expressing it by means of the Zeuthen–Segre invariant. Since (K) is an elliptic pencil, we find

Δ − 4 = 12p_{a} − p^{(1)} + 9.

And since the curves of (K) are interchanged by the transformations of Γ, that do not fix any of them, we find:

Δ = 0;

thus

12p_{a} + p^{(1)} + 13 = 0, p_{a} = −1, p^{(1)} = 1.

Finally, if ρ = 0 then we have p_{g} = 0, and if ρ > 0 then the groups of the g_{2ρ−2}^{ρ−1} in the pencil (C) give canonical curves of F (Chapter X, Section ??); therefore in any case one has

p_{g} = ρ.

Note The simplest example of elliptic surface of genus p_{g} > 0 is given by the surfaces F of determinant 1 representing the pairs of point of an elliptic curve and of a curve of genus p_{g}, which on the other hand can be represented on a multiple cubic cylinder (see Chapter X, Section ??). The elliptic surfaces of genus p_{g} > 0 and of determinant n > 1 can be constructed by starting with a multiple F of determinant 1, or also with a multiple cubic cylinder. On the whole one can repeat the arguments developed in the case p_{g} = 0 with a few modifications.

It is therefore apparent that the elliptic surface with p_{g} > 0 give rise to infinitely many families corresponding to the following integral characters, which are, to a certain extent, independent:

(a) the geometric genus p_{g}, genus of the pencil of elliptic curves (K);

(b) the determinant n, number of the intersections of a curve K with a curve of the elliptic pencil (C) (formed by the unisecant curves of the K);

(c) the number of the multiple elliptic curves (K = sK) belonging to the pencil (K) and the corresponding multiplicities (s_{i}): as for p_{g} one can relate these numbers to the plurigenera of the surface^{(13)}.

(d) the genus π of the curves of the elliptic pencil (C):^{(14)}

π = 1 + n(p_{g} − 1) + ∑[(s_{i} − 1)n]/2s_{i}.

Case 2 After having completed the discussion of Case 1, we now examine case 2), in which we deal with surfaces admitting an ∞^{2} continuous group Γ of transformations into itself.

To start with we notice that, if the transformations of Γ are not permutable, then the transformations which are permutable with a given one form an ∞^{1} algebraic subgroup of Γ, and therefore we obtain a particular case of 1).

Therefore we may assume Γ to be an ∞^{2} group, transitive on F, formed by permutable transformations. And moreover we may assume that it is transitive without exceptions on the surface F without exceptional curves: otherwise one has on F at least a fixed point O or a fixed curve χ and, by fixing a point near to O, or a point of the curve χ, we would find an ∞^{1} algebraic subgroup of Γ with a fixed point, thus F would be birational to a ruled surface (Section ??).

Finally it is easy to see that the group Γ acts on the points of F in a simply transitive manner, that is, we cannot have a finite number n > 1 of transformations of Γ in which a point A of the surface corresponds to one and the same point A′. Indeed a contradiction can be drawn from this hypothesis in the following way.

(13)

(14)Here there are interesting problems concerning the classification: for instance “to determine the elliptic surfaces corresponding to the first values of π”. For π = 2, p_{g} = 0, we quote the Memoir by L. Campedelli, Intorno alle superficie ellittiche con un fascio di curve di genere due. Rendic. Sem. Mat. di Padova, 1935.

Assuming that the hypothesis is verified, let us consider, together with F, the surface F′ (which gives the parametric representation of the group Γ according to S. Lie) whose points correspond, without exceptions, to the elements (transformations) of Γ; since the transformations of Γ act on one another by multiplication, Γ acts on F′ in a simply transitive way: F′ is therefore a hyperelliptic surface, that is, an Abelian surface (Chapter IX, Section ??). On the other hand there is a (1, n) correspondence between F and F′, in which the points of F correspond to the groups of an involution I_{n} on F′; thus F′ is represented on the n-tuple F without branch points. Whence it follows that the involution I_{n} on F′ is generated by a finite Abelian group G_{n} of transformations of Γ, which is invariant for the ∞^{2} transformations of Γ itself; and therefore the n transformations of Γ relating two groups of the involution, correspond to only one transformation of F relating the two homologous points A and A′, against the hypothesis we started from, that there were n transformations of Γ sending A to A′.

In conclusion we may say that the surfaces F, not birational to ruled surfaces, admitting an ∞^{2} group of birational transformations (case 2) are hyperelliptic surfaces, that is, two dimensional Abelian varieties, according to the definition of Chapter IX, Section ??.

We have already computed the genus of these surfaces (Chapter IX, Section ??), that is

p_{g} = 1

and it is easy to find the other characters.

First of all the hyperelliptic surface F (without exceptional curves) cannot have any canonical curve K of order > 0.

In fact the ∞^{2} group of transformations of F, fixing K, would not carry its points to other points on the surface, against the transitivity of Γ. It is convenient to notice here that F cannot a priori have a canonical system of dimension p_{g} − 1 > 0; in fact, this linear system would be fixed by the transformations of Γ and thus F itself would be birational to a ruled surface (Section ??).

Now, since the surface F has a canonical curve of order zero, its (absolute) linear genus is

p^{(1)} = 1.

We are left with the computation of the arithmetic genus p_{a}. To do this, it is convenient to prove that the irregularity of F is

q = p_{g} − p_{a} = 2,

so that

p_{a} = −1.

We first prove that “the Jacobi variety V_{p}, corresponding to a curve C of genus p, which clearly has irregularity ≥ 0, has exactly the irregularity p and not greater, that is, we cannot have on V_{p} an ∞^{r} series of inequivalent V_{p−1}, with r > p”.

This proposition follows from the third equivalence criterion (Chapter III, Section ??) which can be easily extended from the surfaces to the higher dimensional varieties. According to this criterion we may say that two inequivalent varieties V_{p−1} belonging to V_{p} must cut out inequivalent varieties V_{p−2} on the V_{p−1} corresponding to the series of the groups of p points of C with a fixed point A_{1}, and accordingly inequivalent varieties V_{p−3} on the V_{p−2} corresponding to the groups of p points of C with two fixed points A_{1} and A_{2}, etc.

Therefore the property that we have stated can be proved by showing that the V_{2}, corresponding to the series of groups of p points of C with p − 2 fixed points A_{1}A_{2} ⋯ A_{p−2}, that is, the surface representing the pairs of points of the curve C of genus p, has exactly the irregularity p and not > p.

In order to prove this, it suffices to remark that an ∞^{r} series of inequivalent curves on such a surface gives rise to an ∞^{r} series of inequivalent groups of points on C, thus we certainly have

r ≤ p. Q.E.D.

Once we have established the proposition on the irregularity of a Jacobi variety V_{p}, we deduce that “a q-dimensional Abelian variety V_{q} imprimitively contained in the Jacobi variety V_{p} (with q < p), that is, fixed by an ∞^{q} subgroup of the ∞^{p} group of V_{p}, also has irregularity q”.

Since a V_{q−1} of V_{q} is transformed by the corresponding Abelian group Γ_{q} in the V_{q−1} of an ∞^{q} series, the irregularity of V_{q} is certainly ≥ q, and it suffices to prove that it cannot be > q.

Indeed if a V_{p} contains such a V_{q}, then the group Γ_{p} of the ∞^{p} permutable transformations of V_{p} contains an ∞^{q} algebraic subgroup Γ_{q} fixing the varieties V of an ∞^{p−q} series of imprimitivity; then Γ_{p} contains also an algebraic ∞^{p−q} subgroup Γ_{p−q}, fixing the varieties V_{p−q} of an ∞^{q} series^{(15)}.

Characterization of elliptic and hyperelliptic surfaces by genera 11.4 425

Now, if one of the aforementioned V_{q} in V_{p} has an ∞^{r} continuous system of inequivalent V_{q−1}, then these generate, by means of the ∞^{p−q} transformations of Γ_{p−q}, a continuous system of dimension r + p-q of inequivalent V_{p−1}, thus we have

(15)This theorem (already stated in Chapter IX, Section ??) follows, as we have said, from the geometric interpretation, due to Castelnuovo, of a theorem by Picard–Poincaré concerning the reducible Abelian integrals on a curve. See Enriques–Chisini, Lezioni L. VI, Chapter III, §38 (Vol. IV, pages 237–241).

r + p − q ≤ p, r ≤ q.

So: the V_{q} contained in V_{p} has the irregularity q and not > q.

In particular for q = 2, an Abelian V_{2}, that is, a hyperelliptic surface F imprimitively belonging to a Jacobi V_{p} (p ≥ 2) and transitively transformed into itself by ∞^{2} transformations of the corresponding Γ_{p}, has the irregularity

p_{g} − p_{a} = 2

and therefore

p_{a} = −1.

But (Chapter X, Section ??), any hyperelliptic surface F′ can always be reduced to such a simple or multiple F, imprimitively contained in V_{p}, and if one has a multiple F according to a certain number n > 1, then in any case this has no branch curve; therefore we can compute the arithmetic genus of F′ using Severi’s formula that relates the arithmetic genera p_{a} = −1 and p′_{a} of the two surfaces F and F′ in a (1, n) correspondence (Chapter V, Section ??); since there is no branch curve of the multiple F, this formula can be written as

24(p′_{a} + 1) = 24n(p_{a} + 1),

and therefore it gives

p′_{a} = p_{a} = −1.

In conclusion a hyperelliptic surface has the characters

p_{g} = 1, p^{(1)} = 1, p_{a} = −1.

Moreover, since it has only one canonical curve of order zero, all its plurigenera are equal to one.

11.4 Characterization of elliptic and hyperelliptic surfaces by means of the values of the genera

Conversely the values of the genera (p_{a}, p_{g}, p^{(1)}, P_{4}) that we have found for the elliptic and hyperelliptic surfaces enable us to characterize these surfaces (not birational to ruled surfaces) and therefore the whole family of surfaces possessing a continuous group of birational transformations into themselves.

Firstly we prove that a surface F with the genera

p_{a} = −1, p_{g} > 1, p^{(1)} = 1,

is an elliptic surface.

In fact the ∞^{p_{g}−1} canonical curves of F are composed with the elliptic curves K of a pencil (K), of a certain genus ρ; and a priori the ∞^{q} (q = p_{g}−p_{a}) inequivalent curves L of an ∞^{q} complete system constructed on F either cut out ∞^{q} equivalent groups on the K or they cut out groups belonging to a series formed by ∞^{1} linear series: in the first case the curves L of {L} are obtained from one another by sum and subtraction of curves K of (K), whence we deduce

ρ = q = p_{g} + 1;

in the second case the system {L} is formed by ∞^{1} continuous systems of dimension q − 1, and each of these can be deduced from one of the curves of {L} by summing and subtracting curves K of (K), so that

ρ = p_{g}

So-as we have done many times before-we may construct on F, besides (K), another irrational, elliptic, pencil of directrices of the K. Accordingly the surface F turns out to be elliptic (Chapter X, Section ??).

Now we shall see that in fact only the latter case occurs, and not the former.

To see this, we compute the Zeuthen–Segre invariant of F by using the pencil of elliptic curves K: we have

Δ − 4 = 12p_{a} − p^{(1)} + 9 = −4

and therefore Δ = 0;

whence it follows that the K of (K) never acquire double points and thus they are all birationally equivalent to one another; in (K) there are a certain number of multiple elliptic curves with the same modulus

s_{1}K_{1}, s_{2}K_{2}, …

where

s_{i} > 1.

In this setting, with the procedure that we have explained in Chapter X, Section ?? (in the case of a pencil of genus ρ = 1) we may construct a base point free pencil (which is a priori either rational or elliptic) of directrices C, of genus π > ρ. The canonical curves of F cut out the canonical series g_{2π−2} (but not in general the complete series) on a C. Since the K cut out an involution of genus ρ on C, whose intersections with the K_{i} are s_{i}-tuple points, the canonical curves of F are formed by groups of the g_{2ρ−2}^{ρ−1} in the pencil (K) of genus ρ, plus the curves (s_{i} − 1)K_{i}. Whence we deduce that the genus of F is

p_{g} = ρ. Q.E.D.

Also for p_{g} = 0 and p_{a} = −1, p^{(1)} = 1, as well as for p_{g} > 1 and p^{(1)} = 1, the surface F is elliptic (see Chapter X, Section ??), except that it can be an improper elliptic surface, that is, an elliptic ruled surface (P_{12} = 0)^{(16)}.

But the case p_{g} = 1 gives rise to an exception: the surfaces whose genera are

p_{a} = −1, p_{g} = 1, p^{(1)} = 1,

form two families:

(1) surfaces with elliptic canonical curve of order greater than zero;

(2) surfaces with pure canonical curve of order zero.

The surfaces of the first family are elliptic, that is, they admit an ∞^{1} elliptic group of transformations into themselves, whereas the surfaces of the latter family are hyperelliptic, and in general not elliptic: this means that they admit an ∞^{2} Abelian group Γ of transformations into themselves, but only for particular moduli there is in Γ an ∞^{1} (elliptic) algebraic subgroup.

In order to justify what we have said about Cases 1 and 2, it is convenient to prove a

Lemma on the surfaces with irregularity q = 2: “a surface F of irregularity q = p_{g} − p_{a} = 2, not containing a pencil of genus two of curves, can be represented on a simple or multiple hyperelliptic surface”.

In order to prove this lemma, let us consider on F a (simple) continuous system {C}, formed by ∞^{2} inequivalent curves, and let us consider the elements (curves) of {C} as “points” of a surface Φ: this is a hyperelliptic surface (that is, the Picard variety of F). There is now a correspondence between the surfaces F and Φ, in which the curves L of an ∞^{2} system {L} on Φ correspond to the points of F (that is, the image of a point is the ∞^{1} system of curves C passing through it). We claim that the ∞^{2} curves L on Φ cannot be equivalent. Otherwise the ∞^{1} curves C of {C} passing through a point O of F would cut out equivalent groups on another C = \overline{C}, and, by the third equivalence criterion (Chapter III, Section ??) they would be equivalent.

In order to justify this assertion, it is convenient to consider the series of groups of points G, on the surface F, cut out on a particular C = \overline{C}, not passing through O, by the ∞^{1} curves C through O; if we denote the index of this series by i, then we have i groups of G having a point P of \overline{C} in common and i groups G having another point P′ of the same curve in common; and, by virtue of a criterion by Severi–Castelnuovo^{(17)}, it is necessary and sufficient to show that the two sets of groups G thus defined on \overline{C}, P(G) and P′(G), are equivalent. To prove this, let us consider the curve \overline{L} on Φ corresponding to the point O (or rather to the series C(O) of the curves C through O) and the point A corresponding to the curve \overline{C}: if, as we assume, the ∞^{1} curves L through A are equivalent on Φ, then they cut out equivalent groups of points on \overline{L}; so in particular the groups cut out on the aforementioned curve by the two curves L – say L_{P} and L_{P′} – corresponding to the two points P and P′ of F are equivalent. But – going back to the surface F – this means that the two groups of curves C through O and P and through O and P′, that cut out the groups of points P(G) and P′(G) on \overline{C}, are equivalent within the series of the C through O and so they belong to an ∞^{1} rational series of groups of curves C; if we recall that a rational series of groups of points on a curve is always formed by equivalent groups^{(18)}, then we deduce that P (G) and P′(G), cut out on \overline{C} by these groups of C, are equivalent on the aforementioned curve \overline{C}.

(16)See Chapter X, Section ??.

Now that we have proved that the ∞^{2} curves L of Φ (corresponding to the points of F) cannot be all equivalent, we still have to examine the hypothesis that any L of {L} belongs to an ∞^{1} series of equivalent curves L; in this case we shall see that the surface F contains a pencil of genus 2. Indeed under this assumption we find on F a pencil of curves K whose points correspond to equivalent curves L, and we can see that the C of a series C(O), passing through any given point O on the surface cut out equivalent groups on a K, and therefore the same happens for all the curves C: to prove this, it suffices to repeat the previous argument, showing that the groups of curves common to C(O)C(P) and to C(O)C(P′) are equivalent (inside the series C(O)), and therefore that also the groups of points P (G) and P′(G) cut out on K by the C of C(O) are equivalent on the curve K.

In this way we verify that the ∞^{2} inequivalent curves C of F cut out equivalent groups on any K; hence the ∞^{2} continuous system {C} can be constructed from a linear system |C|, or from a curve C, by summing and subtracting curves K of the pencil (K): therefore this turns out to be of genus (ρ ≥ 2, thus) p_{g} − p_{a} = 2.

So the statement of the Lemma is justified.

Now we can add that: the surfaces of irregularity q = 2, with p_{g} > 0, containing a pencil of genus two of curves (of genus π > 0) have p_{a} ≥ 0.

In fact let F be a surface with arithmetic genus p_{a} = −1 containing a pencil of genus two of curves K; we prove that one has p_{g} > 1.

(17)See Enriques–Chisini, Lezioni Libro V, Chapter IV, §42 (Vol. III, page 483).

(18)See Enriques–Chisini, Lezioni, Libro V, Chapter I, §10, Nota (Vol. III, page 78).

Computing the Zeuthen–Segre invariant with the pencil (K), we find

Δ − 4 + 8(π − 1) = 12p_{a} + p^{(1)} + 9 = −3 − p^{(1)}

and so

p^{(1)} = 1, Δ = 0, π = 1;

therefore the elliptic curves of the pencil (K) are birationally equivalent to one another. Then (as in Chapter X, Section ??) we can construct a base point free pencil of directrices curves C of the K (which a priori could either be rational or elliptic, but it is elliptic, as we shall see): so^{(19)} the canonical curves of F are composed with canonical groups of the K of (K) and, possibly, also with fixed parts (more precisely by (s−1)C_{s} where sC_{s} = C is a multiple s-tuple curve of the pencil (C)); whence

p_{g} ≥ 2. Q.E.D.

By the lemma on the surfaces with irregularity q = 2 and the remark that the surfaces with q = 2 and a pencil of genus two have p_{g} ≥ 2, hence p_{a} ≥ 0, we can say that a surface F with p_{a} = −1 and p_{g} = 1 can be represented on a simple or multiple hyperelliptic surface Φ, having a certain branch curve D. The curve of F corresponding to d is a canonical curve; thus, if the linear genus of F is p^{(1)} = 1, one of the following two cases, corresponding to the above Cases 1 and 2, must occur:

(A) either d is an elliptic curve of order greater than zero, or

(B) d is a curve of order zero.

(1) In the first case the ∞^{2} transformations of the hyperelliptic surface Φ, belonging to the continuous group Γ, send the curve d (of degree zero) to the curves of a pencil, and each curve d of this pencil (or each irreducible component of D) is invariant for an ∞^{1} elliptic subgroup of Γ. Whence it follows that the multiple surface Φ admits an ∞^{1} elliptic group of transformations, hence it is an elliptic surface. Q.E.D.

(2) If, by contrast, there is no branch curve on the multiple surface Φ (without exceptional curves), then F is an Abelian surface, as well as Φ, hence it is hyperelliptic (Chapter IX, Section ??); so we are in case (2).

We may distinguish between the two cases 1) and 2) by remarking that, in case 1) (since p_{g} = 1 and not p_{g} > 1) the elliptic curves of the pencil (D) of Φ, or rather their irreducible components, correspond to curves K of an elliptic pencil of elliptic curves on F, in which the number of curves having a double point is Δ = 0.

(19)See the remark to Chapter IX, Section ??.

Therefore in this pencil (K), formed by birationally equivalent curves, there is at most a finite number of multiple curves D_{i} each to be counted s_{i} (> 1) times, and the particular curve d giving the branch curve of the multiple Φ is formed by a certain number r of these curves:

D = ∑(s_{i} − 1)D_{i}.

Whence it follows that the canonical curve, corresponding to D, in the pencil (K) on F, is

∑[(s_{i} − 1)/s_{i}]K.

So, if r > 1, we have on F a bicanonical system of dimension ≥ r − 1:

|rK + ∑[(s_{i} − 2)/s_{i}]K|,

having as fixed components the curves corresponding to the D_{i} of multiplicity s_{i} > 2, each counted s_{i} − 2 times; in any case, if r = 1, then there is a quadricanonical system, at least ∞^{1}, whose variable components are pairs of curves K, forming a g_{2}^{1} inside the elliptic (K).

In conclusion, in case 1), the elliptic surfaces F (p_{a} = −1, p_{g} = 1, p^{(1)} = 1) has the quadrigenus

P_{4} > 1,

while the hyperelliptic surface (with the same p_{a} = −1, p_{g} = 1 and p^{(1)} = 1, but with canonical curve of order zero) has all plurigenera equal to one, and, in particular, the quadrigenus:

P_{4} = 1.

Let us summarize the results that we have obtained:

The nonruled surfaces (P_{4} + P_{6} > 0 or P_{12} > 0), admitting a continuous group of birational transformations into themselves, are globally characterized by the values of the genera:

p_{a} = −1, p^{(1)} = 1;

they are always either elliptic surfaces (with an ∞^{1} group) and p_{g} > 1, or p_{g} = 1 and P_{4} > 1; or hyperelliptic surfaces (with an ∞^{2} group) if

p_{g} = P_{4} = 1.

In general the hyperelliptic surfaces are not also elliptic, that is, their ∞^{2} group does not contain in general any ∞^{1} algebraic (elliptic) subgroup. This can be seen by considering the Jacobi surface corresponding to a curve of genus two (the surface F that represents the pairs of points of C), or similarly the Abelian involutions on F or the multiple F′ without branch curve.

The hyperelliptic F constructed this way contains no elliptic curves, unless C either contains an elliptic involution γ_{n}^{1} of a certain order n > 1, or it degenerates into a pair of elliptic curves; in the particular case in which F contains an elliptic curve C, this is carried by the transformations of the group Γ of F into the curves of an elliptic pencil (C) and (according to the Picard–Poincaré theorem on reducible integrals, geometrically interpreted by Castelnuovo) there is another elliptic pencil (K) of elliptic curves on F: these can be constructed as directrices of the pencil (C) in the way we have indicated in Chapter X, Section ??. Therefore F can be considered an elliptic surface in two different ways, since it admits two ∞^{1} elliptic groups of transformations, subgroups of the ∞^{2} group Γ, whose trajectories are the C and the K. Let us finally add that such a twice elliptic F, of determinant n > 1 (if the C and the K meet at n points), can be represented on that particular hyperelliptic surface (also twice elliptic of determinant one) which is the Jacobi surface corresponding to the degenerate curve of genus two formed by two elliptic curves; such a surface has to be taken as multiple of order n.

Remark The conditions that we have used to define the family of elliptic and hyperelliptic surfaces (p_{a} = −1, p^{(1)} = 1) are superfluous: the condition p^{(1)} = 1 is a consequence of p_{a} = −1. We shall prove this for the elliptic surfaces with any p_{g} in Section ??; here we shall justify the assertion only for the surfaces with p_{g} = 1, that is, essentially for the hyperelliptic case.

Let us start from the conclusion we have found above that: “the surfaces F with the genera p_{a} = −1 and p_{g} = 1 can be represented on a, simple or multiple, hyperelliptic surface Φ”. If we have a multiple Φ, then there is a branch curve d on it, of genus p^{(1)} equal to the linear genus of F, and we must ascertain that p^{(1)} > 1 yields

p_{a} ≥ 0.

Let us prove this in the simplest case, in which the branch curve d of Φ is a simple curve, of genus ρ = p^{(1)}, having a certain number δ of nodes and a certain number τ of cusps, so that we may apply Severi’s formula^{(20)} giving the relation between the arithmetic genera p′_{a} and p_{a} of the two surfaces Φ and F in a (1, n) correspondence (see Chapter V, Section ??). This formula gives us^{(21)}

(20)So, in order to complete the proof, one has to extend this formula to the case in which the branch curve is however composed of multiple parts.

We know a priori that the claim holds thanks to a result (partly relying on the theory of simple integrals of the first kind belonging to a surface), which has been kindly communicated to us by G. Castelnuovo. It is the following: for the irregular surfaces which are not birational to ruled surfaces, one has

I + 4 = 13 + 12p_{a} − p^{(1)} ≥ 0 where I is the Zeuthen–Segre invariant. In fact for p_{a} < 0 (that is, p_{a} = −1), this yields p^{(1)} = 1.

24(p_{a} + 1) ≥ 24n(p′_{a} + 1) + 6(ρ − 1) − 2τ,

where

p′_{a} = −1, ρ > 1,

that is,

24(p_{a} + 1) ≥ 6(ρ − 1) − 2τ.

But the curve d belongs to a system, at least ∞^{2}, on the hyperelliptic surface Φ, and it has a characteristic series (cut out by the infinitely near curves) of degree

2δ + 3τ ≤ 2(ρ + δ + τ − 1).

Therefore one has

τ ≤ 2(ρ − 1)

and

24(p_{a} + 1) > 0,

that is,

p_{a} ≥ 0. Q.E.D.

Summarizing: The surfaces with genera

p_{a} = −1 and p_{g} = 1

have the linear genus

p^{(1)} = 1

and, if they have a canonical curve of order zero (as happens for P_{4} = 1), then they are hyperelliptic surfaces.

Note The analysis of the nonruled surfaces admitting a continuous group of transformations into themselves starts with the study of hyperelliptic surfaces made by Picard from the transcendental point of view^{(22)}.

A brilliant application of the theory of simple integrals of total differentials introduced by him, leads the author to the characterization of the surfaces of the aforementiond type, possessing an ∞^{2} group of permutable transformations, by means of:

(21)The case in which the strict inequality holds corresponds to the hypothesis in which there are on d some fundamental points for the (1, n) correspondence.

(22)E. Picard, Mémoire sur la théorie des fonctions algébriques de deux variables indépendantes. J. de Math., 1889. (Chapter III) See the Vol. III of the Traité by Picard and Simart, Paris 1906.

Surfaces with the genera p_{g} = 1 and p^{(1)} = 1 11.5 433

(1) the existence of two simple integrals of the first kind;

(2) the existence of a double integral of the first kind (that, according to Clebsch and Noether, means p_{g} = 1);

(3) the nonexistence of a (pure) canonical curve of order greater than zero.

The condition 3) (pointed out by Picard, who, deceived by a mistake of Noether, believed it to be superfluous) yields the inversion of the two simple integrals and therefore the parametric representation of the surface with (hyperelliptic) four times periodic functions of two independent variables.

From this result Enriques^{(23)} drew the characterization of the hyperelliptic surfaces by means of the values p_{a} = −1, p_{g} = P_{4} = 1 of the genera.

Later on Severi^{(24)} translated the proof of Picard’s theorem in a geometric form, deducing the existence of the ∞^{2} group of transformations directly from the conditions p_{a} = −1, p_{g} = 1 and canonical curve of order zero, as we have also done here.

The case of surfaces with an ∞^{1} elliptic group, already met by Picard, has been exhausted, from the transcendental point of view, by Painlevé’s analysis^{(25)}. He defines the elliptic surfaces by means of a parametric representation with elliptic functions of one parameter and algebraic functions of another, and he succeeds then in classifying all nonruled surfaces possessing a continuous group of birational transformations into themselves. The geometric translation of this result and the classification of the whole family of surfaces by means of the values of the genera was obtained by Enriques in the quoted Memoir of 1905^{(26)}.

11.5 Surfaces with genera p_{g} = 1 and p^{(1)} = 1

The results that we have established in the previous sections enable us to classify now the surfaces of geometric genus p_{g} = 1 and linear genus p^{(1)} = 1. These surfaces can be arranged in three families:

(1) Regular surfaces of genus p_{a} = p_{g} = 1 with canonical curve of order zero (characterized by the value of the bigenus P_{2} = 1), which give rise to infinitely many classes of surfaces (with 19 moduli) with canonical hyperplane sections of genus π and order n = 2π − 2, in the space S_{π} (π = 2, 3, …) (See Chapter VII, Section ??).

(23)Sulle superficie algebriche che ammettono un gruppo continuo di trasformazioni birazionali in se stesse. Rendic. Circ. Mat. di Palermo, 1905.

(24)Sulle superficie algebriche che ammettono un gruppo continuo permutabile a due parametri di trasformazioni birazionali, Atti Ist. Veneto, 1908.

(25)P. Painlevé, Leçons sur la théorie analytique des équations différentielles. Stockholm, 1895 (page 270).

(26)Sulle superficie algebriche che ammettono un gruppo continuo ecc., and partially already in the previous Memoir, also published in Rendic. Circ. Mat. di Palermo: Sulle superficie di genere zero (1905).

(2) Hyperelliptic surfaces with genera p_{a} = −1 and p_{g} = 1, also possessing a canonical curve of order zero (characterized by the value of the quadrigenus P_{4} = 1), which give rise again to infinitely many classes of normal surfaces with canonical hyperplane sections of genus π and order n = 2π − 2 in the space S_{π−2} (as we shall see each of these classes depends on three moduli).

(3) Surfaces possessing a (rational or elliptic) pencil of elliptic curves: this property is shared by the surfaces of types (1) and (2) only for particular values of their moduli.

In order to prove that “the three families of surfaces (1), (2) and (3) exhaust the surfaces with genera p_{g} = 1 and p^{(1)} = 1” it is convenient on the one hand to recall some remarks already made before in this treatise and on the other hand to point out a few facts we have not remarked yet. We mean the following propositions:

(a) the surfaces with geometric genus p_{g} = 1 have the arithmetic genus

p_{a} ≥ −1,

that is,

either p_{a} = 1 or p_{a} = 0 or p_{a} = −1;

(b) the (regular) surfaces with p_{g} = p_{a} = 1 and canonical curve of order zero are defined by the value of the bigenus P_{2} = 1 (Chapter VI, Section ??); by contrast, the surfaces of this family having the linear genus p^{(1)} = 1 and possessing a (pure) canonical curve of order greater than zero have the bigenus P_{2} > 1 and contain a (linear) pencil of elliptic curves, that are either bicanonical curves or components of bicanonical curves;

(c) the surfaces with p_{g} = 1 and with a canonical curve of order zero have the arithmetic genus either p_{a} = 1 or p_{a} = −1 (Chapter IX, Section ??) and, if p_{a} = −1, then they have also the linear genus p^{(1)} = 1, and they are either hyperelliptic surfaces or elliptic surfaces, with a pencil of elliptic curves, according to the value of the quadrigenus P_{4} = 1 or P_{4} > 1 (Section ??);

(d) the surfaces for which p_{g} = 1, p_{a} = 0 and p^{(1)} = 1 (which by c) must contain a canonical curve of order greater than zero) possess an elliptic pencil of paracanonical curves, and therefore have the bigenus P_{2} > 1.

The propositions b) and c) just recall facts we have already seen; therefore it suffices to prove a) and d), stating something new.

Since the proposition a) has been already established for the elliptic and hyperelliptic surfaces, it suffices to prove it under the hypothesis the surface F (of genus p_{g} = 1) does not possess a continuous group of transformations into itself (later on, in Section ??, this proposition will appear as a particular case of a more general theorem to the effect that the arithmetic genus of nonruled surfaces always is p_{a} ≥ −1).

Let F be a surface of genus p_{g} = 1, which does not admit a continuous group of transformations, and let q = p_{g} − p_{a} be its irregularity: we want to deduce a contradiction from the hypothesis

q > 2,

that is,

p_{a} < −1.

To do this, we assume q > 2 and consider the Picard variety V_{q} corresponding to an ∞^{q} continuous system of inequivalent curves C on F. As we have seen (Chapter IX, Section ??), if F does not contain an irrational pencil of curves K, then an ∞^{1} series of curves C of {C} leads to the construction of a, simple or multiple, surface identical with F inside V_{q}. And more than that, since we have in mind to prove that p_{g} ≥ q, we may as well assume that we have in V_{q} a simple surface identical with F: in fact in the case of a multiple surface we recall the remark of Section ?? and see that, if this surface admits a continuous group of transformations into itself, whereas F does not, then p_{a} ≥ 0, and accordingly p_{g} ≥ q for F itself.

Thus let us have in V_{q} a surface identical with F (denoted again by F) which does not admit a continuous group of transformations into itself; by means of the ∞^{q} transformations of V_{q} our F is transformed into an ∞^{q} series, say {F }, of analogous surfaces. For simplicity we assume first of all q = 3. Since V_{q} = V_{3} has the geometric genus equal to one, the ∞^{2} characteristic system of F, cut out on a given F by the infinitely near surfaces, is formed by special curves (and in fact canonical curves, since they are residual with respect to the intersection with the canonical variety of V_{3}); thus one finds

p_{g} ≥ 3

for F.

The argument can be extended to the case q > 3. In this case we may take an ∞^{q−3} series of surfaces inside the series {F }: this forms a variety V_{q−1} and we may assume in general that such a variety is contained in a linear system |V_{q−1}| of dimension r (as large as we want). Then we construct the ∞^{r+q} continuous system {V_{q−1}} generated by |V_{q−1}| via the ∞^{q} transformations of V_{q}. This continuous system has an ∞^{r+q−1} characteristic linear system of special or canonical varieties, cut out on a V_{q−1} by the infinitely near varieties; therefore these V_{q−1} have the genus

P ≥ r + q.

Now we compute the parameters on which the V_{q−1} of {V_{q−1}} containing an F depend: there are ∞^{q} surfaces F′ and ∞^{r+q} V_{q−1} each containing ∞^{q−3} surfaces F′, thus the dimension of the system {F } is (in the worst case):

r + 2q − 3 − x = q

where x denotes the dimension of the system of the V_{q−1} containing an F: x = r + q − 3. Whence it follows that, in a given V_{q}, the characteristic system of {V_{q−1}} cuts out on a F contained in V_{q} a (special or canonical) linear system whose dimension is at least:

r + q − 1 − (x − 1) = 3.

And therefore we deduce that the genus of F is

p_{g} ≥ 4.

In conclusion we have proved that a surface F of genus p_{g} < 3, which does not contain an irrational pencil of curves, cannot have the irregularity

q > 2;

in particular an F of genus p_{g} = 1 must have the arithmetic genus

p_{a} ≥ −1.

The conclusion can be extended to the hypothesis that F contains an irrational pencil of curves (C), since the computation of the Zeuthen–Segre invariant by means of this pencil gives

Δ − 4 ≥ 12p_{a} − p^{(1)} + 9. 12(p_{a} + 1) − p^{(1)} ≤ 0, p_{a} ≥ −1

and for p_{a} = −1, p^{(1)} = 1.

Theorem d) can be proved as follows.

Let us consider an (irreducible) linear system |C|, of a certain genus π, on F.

The adjoint system |C′| has the dimension

p_{a} + π − 1 = π − 1,

and cuts out on a C a series g_{2π−2}^{π−2} of deficiency one, contained in the canonical series. Moreover |C′| is contained in an ∞^{1} series of paradjoint linear systems |\overline{C}′|, which also cut out a series g_{2π−2}^{π−2} on C. Therefore there is an elliptic paracanonical curve defined as the difference

\overline{C}′ − C,

and the analogous paracanonical curves K form an elliptic pencil on F: let us add that any curve 2K belongs to a linear system, at least ∞^{1}, and therefore P_{2} > 1. Q.E.D.

Remark An example of surfaces with the genera p_{a} = 0, p_{g} = 1, p^{(1)} = 1, and possessing an elliptic pencil of elliptic curves, is given by the simplest paraelliptic surfaces, that we have met in Chapter X, Section ??.

It could be interesting to study and classify the surfaces with p_{a} = 0 and p_{g} = 1, having the linear genus p^{(1)} = 1, or also p^{(1)} = 2,…

The results we have found can be summarized by saying: The nonruled surfaces (P_{12} > 0) possessing a canonical or pluricanonical curve of order zero (and therefore base point free linear systems of genus π and degree n = 2π−2) are defined by the condition

P_{12} = 1,

which yields

p^{(1)} = 1

and they give rise to the following families:

(1) p_{g} = 0 and p_{a} = 0, regular surfaces with bicanonical curve of order zero (P_{2} = 1), represented by the surface of order 6 possessing with multiplicity two through the edges of a tetrahedron (Chapter VII, Section ??);

(2) p_{g} = 0 and p_{a} = −1, elliptic surfaces of the types I_{a} I_{b} II_{a} II_{b} III_{a} III_{b} III_{c} (Chapter X, Sections 16–19);

(3) p_{g} = 1 and p_{a} = 1 (P_{2} = 1), regular surfaces with all the genera equal to one (Chapter VII, Section ??);

(4) p_{g} = 1 and p_{a} = −1 (P_{4} = 1) hyperelliptic surfaces (Sections 4 and 6 of this chapter).

A note on the geometric theory of hyperelliptic surfaces 11.6 438

11.6 A note on the geometric theory of hyperelliptic surfaces

Among the families with P_{12} = 1, listed with the numbers 1), 2), 3), 4) at the end of the previous section, one may distinguish: on the one hand the surfaces of genus p_{g} = 1, of the types 3) and 4) (p_{g} = 1, p_{a} = 1 and p_{a} = −1), and on the other hand the surfaces of genus p_{g} = 0 (the regular surfaces with P_{12} = 1, and the elliptic ones) of the types 1) and 2). The latter types form a finite number of families each having a certain number of moduli: and precisely 10 moduli those of type 1), 2 moduli or 1 modulus those of type 2). By contrast the surfaces 3) and 4) give rise to infinitely many families distinguished by the values of (at least) one arithmetic character, each having a certain number of moduli.

What we just said has been proved for the surfaces with all the genera equal to one (type 3)) in Chapter VII, Section ??; but it is also true for the hyperelliptic surfaces of type 4), as we shall explain now.

From the transcendental point of view, the classification of the hyperelliptic surfaces stems from the parametric representation of these surfaces by means of four times periodic functions of two independent variables^{(27)}. Their primitive normal periods can be reduced to the form

\begin{matrix}1 & 0 & g & h \\ 0 & 1/δ & h & g′, \end{matrix}

where δ is an integral character (the so called divisor of the corresponding surfaces). Now for δ = 1 one has the standard surface, that is, the Jacobi surface F_{j}, representing the variety of the pairs of points of a curve of genus two; by contrast, for δ > 1 one has a multiple F_{j} of order δ without branch curve, or also an involution of order δ on the surface F_{j}.

The classification of the hyperelliptic surfaces obtained in this way has an algebro-geometric meaning, and therefore the problem arises of justifying it from this point of view. We say briefly how this can be done.

To start with we can determine the characters of the Jacobi surface F_{j}:

p_{g} = 1, p_{a} = −1

and canonical curve of order zero (thus P_{4} = 1). This can be obtained by representing F_{j} on the double Kummer surface of order 4 with 16 double points, as we have seen in Chapter V, Section ??^{(28)}. This representation is based on the knowledge of the (involutory) transformations of the 2nd kind of the variety of the pairs of points of a curve of genus two; these are obtained by associating the coresidual pairs with respect to a given g_{4}^{2} of C: such an involution is birationally equivalent to a Kummer surface of order four, having sixteen double points corresponding to the sixteen pairs of points of C coming from the division by two of the g_{4}^{2}.

(27)See for instance Enriques–Chisini, Lezioni Libro VI, Chapter III, §39 (Vol. IV, page 242).

(28)See also Enriques–Chisini, Lezioni, L. V, Chapter III, §40 (Vol. IV, page 24).

Now the products of these involutions I on the Jacobi surface F_{j} give rise to a continuous Abelian group Γ; and there are cyclic transformations in Γ generating finite groups G_{δ} contained in it: such a G_{δ} (let for instance G_{δ} be cyclic) gives rise to an involution I_{δ} of order δ on F_{j}, without double points, corresponding in general to a hyperelliptic surface F_{δ} distinct from F_{j}. In this way one obtains the hyperelliptic surfaces of divisor δ > 1. And we can see that these can also be produced as multiple F_{j} of order δ, without branch curve; to do this we need to see that it is possible to construct with the groups of an involution I_{δ} another involution of higher order, say rδ, which is birationally identical with F_{j}.

In what we have said there is a sketch of a geometric theory of the hyperelliptic surfaces, based on the study of cyclic transformations belonging to the group Γ of the Jacobi surface F_{j}: such a study appears as an extension of the study of the cyclic involutions on the elliptic curves, developed in Enriques–Chisini, Lezioni^{(29)}.

But the consideration of the hyperelliptic surfaces from the geometric point of view also confronts us with other interesting problems.

Indeed we have proved (Chapter IX, Section ??) that an Abelian variety V_{q} can be regarded as identical with a simple or multiple variety imprimitively contained in a Jacobi variety V_{p} corresponding to a curve of genus p ≥ q; but it is not true in general that one can always take p = q, because the number of moduli on which V_{q} depends is greater than the number 3q − 3 of moduli of the curves of genus q^{(30)}. Now the transcendental theory shows that the hyperelliptic surfaces (with a given divisor) depend on three moduli, and therefore they can all be reduced to the type of a simple or multiple Jacobi surface F_{j}, or also to the (fixed point free) involutions on the F_{j}.

This can be seen from the geometric point of view, in a way that we want to briefly sketch. On a hyperelliptic surface F (for which p_{a} = −1) a curve of genus π belongs, in general, to a linear system of dimension π − 2, in which there is a finite number of curves C with π − 2 double points, and therefore of genus two.

By means of the transformations of the second kind of F, belonging to the group Γ, we therefore obtain an ∞^{2} continuous system {C} of curves of genus two on F, birationally equivalent to one another, whose three moduli give the moduli of F. Now a C of this system is invariant for a single (involutory) transformation of the second kind^{(31)} of F, inducing the g_{2}^{1} on C; therefore the group Γ is birationally equivalent both to the surface F (regarded as a set of points) and to the system {C} regarded as an ∞^{2} variety of elements (curves C). Whence it follows that F is birationally equivalent to the ∞^{2} series of groups of points cut out by the curves of {C} on a particular C: if the groups of this series are not equivalent to one another, then F is identical with the Jacobi surface of C; otherwise, if for any group of this series there is a certain number δ of groups equivalent to it in the series, then the hyperelliptic surface F is birationally identical with an involution I_{δ} of order δ on the Jacobi surface.

(29)Libro V, Chapter III, §§27–28 (Vol. III, pages 251–298).

(30)See Enriques–Chisini, Lezioni, Libro V, Chapter III, §33 (Vol. III, page 355) and Libro VI, Chapter III, §35 (Vol. IV, page 216).

The theorem we have established takes an interesting geometric significance in relation with the various constructions of the hyperelliptic surfaces that one can give. Indeed one can construct a hyperelliptic surface starting with a Jacobi variety corresponding to a curve C_{p} of genus p. If on such a C_{p} there are two Abelian integrals which are reducible to four periods (according to Castelnuovo’s geometric interpretation of the Picard–Poincaré theorem)^{(32)} then the corresponding Jacobi variety V_{p} contains an ∞^{2} algebraic subgroup of the group Γ of the transformations of the first kind, and therefore one has an Abelian variety (that is, a hyperelliptic surface) imprimitively contained in V_{p}. The previous theorem tells us that the hyperelliptic surfaces constructed in this way can also be found as simple or multiple Jacobi surfaces, or also as involutions on the Jacobi surfaces corresponding to a curve of genus p = 2.

It is convenient to illustrate this property by referring to the simplest case, in which we deal with the hyperelliptic surfaces imprimitively contained in a Jacobi variety V_{3}, corresponding to a curve C_{3} of genus 3. Here the group Γ of the transformations of the first kind of V_{3} contains two complementary algebraic subgroups: an ∞^{2} Γ_{2}, giving rise to an imprimitivity series formed by ∞^{1} hyperelliptic surfaces F_{2}, and an ∞^{1} Γ_{1} giving rise to ∞^{2} elliptic curves C_{1}; the surfaces F_{2} and the curves C_{1} meet one another at a certain number of points. And it is easy to prove that each C_{1} is a unirational transform of C_{3}, that is, it corresponds to an elliptic involution γ_{n}^{1} (of a certain order n) belonging to C_{3}.

Therefore one can construct a hyperelliptic surface F_{2}, imprimitively contained in a Jacobi variety V_{3}, corresponding to a curve C_{3} of genus 3 possessing an elliptic involution γ_{n}^{1}; to do this, it suffices to construct the curve C_{3} corresponding to a multiple cubic of multiplicity n, with an arbitrary choice of the four branch points and of the corresponding substitutions on the sheets^{(33)}. We notice that the C_{3} containing an elliptic γ_{n}^{1} depend on four moduli: the absolute invariant of the K and the distances of 3 branch points from the remaining point on the elliptic curve K.

(31)If we refer to the parametric representation of the group G, then a transformation of the second kind corresponds to the transformation sending a π of G in the inverse π^{−1}. It is easy to see that, even if the curve C of genus two has singular transformations into itself, there cannot be more than one transformation of the second kind of F leaving it fixed.

(32)See Enriques–Chisini, Lezioni, Libro VI, Chapter III, §38 (Vol. IV, page 234).

The absolute invariant of K corresponds to the modulus of the C_{1} on the V_{3}, and the remaining 3 moduli belong to the complementary surfaces F_{2}.

Let us add that the Jacobi’s V_{3} we have just constructed contains, in general, an involution given by the groups of points cut out on the C_{1} by the F_{2}: the \overline{V}_{3} representing these groups, which is a unirational transform of V_{3}, is the Jacobi variety of a degenerate curve of genus 3 formed by an elliptic curve and a curve of genus two; so then (according to the theorem we have just treated), the hyperelliptic surfaces F_{2}, forming an imprimitivity system for the group of the transformations of the first kind of the V_{3}, turn out to be birationally equivalent to an involution on one of the analogous \overline{F}_{2}, which is the Jacobi surface corresponding to the aforementioned curve of genus two.

We think that the short account we have just given suffice to indicate the interest, the beauty and the richness of the properties of the hyperelliptic surfaces.

We only add that the study of the irregular hyperelliptic involutions on a hyperelliptic surface F (and in particular on a Jacobi surface F_{j}) can be naturally extended to the study of the other irregular elliptic involutions or regular involutions with the genera equal to one that may belong to F; these lead to surfaces still having a parametric representation by hyperelliptic functions of rank r > 1^{(34)}.

The irregular elliptic involutions (belonging to F with special moduli) give rise to elliptic surfaces with p_{a} = −1, p_{g} = 0 and P_{12} = 1, possessing a virtual canonical curve of order zero, as already mentioned in Chapter X, Sections 16–19.

The general classification of the algebraic surfaces 11.7 442

The regular involutions of order 2 belonging to a hyperelliptic surface of divisor δ ≥ 1 lead in general to surfaces of rank r = 2 (that are hyperelliptic in an extended sense), and more precisely: for δ = 1 to the Kummer surfaces of order four with 16 double points (Chapter V, Section ??); and for δ > 1 to a series of analogous surfaces of order n = 2π − 2 with canonical hyperplane sections of genus π in the space S_{π} (π = 4, 5, 6, …), with sixteen double points. But for special moduli there can also be singular regular involutions on an F, generated by a finite group of birational transformations of F; and, looking at this way of generating them (as proved by Enriques–Severi), they have been completely classified in the memoirs by Enriques and Severi^{(35)} quoted above, and by Bagnera and De Franchis^{(36)}, which complement each other.

(33)See the existence theorem for the multiple curves in Enriques–Chisini, Lezioni, Libro V, Chapter IV, §38 (Vol. III, page 427).

(34)Where each point of the surface corresponds to r points in the prismatoid of the periods.

(35)Mémoire sur les surfaces hyperelliptiques, Acta Math., t. 32 and 33 (1908).

Finally we recall that this circle of ideas can be extended from the hyperelliptic surfaces to the regular surfaces with all the genera equal to one; on these it is also possible to find regular involutions of the same genus p_{g} = 1 (or with p_{g} = 0 and bigenus P_{2} = 1), generated by a finite group of birational transformations. In relation to this topic we want to mention the interesting papers of the Belgian school, and particularly of Godeaux.

11.7 The general classification of algebraic surfaces

We are now in a position to put together a synthesis^{(37)} of the main results of the theory of surfaces, giving a general classification of the algebraic surfaces, according to the value of the 12-genus P_{12}: in what follows, this classification will be made more precise, as we shall see in Sections 8–9.

We distinguish between the algebraic surfaces according to the value of the P_{12} and – in the second instance – of the (absolute) linear genus p^{(1)}.

We have four large families of surfaces:

(A) for P_{12} = 0, the surfaces birational to ruled surfaces (rational or otherwise);

(B) for P_{12} = 1, the surfaces with effective or virtual canonical curve of order zero;

(C) for P_{12} > 1 and p^{(1)} = 1, the surfaces containing infinitely many linear systems of canonical or pluricanonical curves composed with the elliptic curves of a pencil;

(D) for P_{12} > 1 and p^{(1)} > 1, the surfaces admitting canonical or pluricanonical surfaces as a projective model.

It is convenient to recall and explain the fundamental and characteristic properties that distinguish these four families of surfaces:

(36)Le superficie algebriche le quali ammettono una rappresentazione parametrica mediante funzioni iperellittiche di due argomenti, Mem. Soc. It. delle Scienze (detta dei XL), 1908.

(37)See Enriques, Sulla classificazione delle superficie algebriche in particolare di genere lineare p^{(1)} = 1, Rendic. Acc. Lincei, Febbraio 1914.

(A) P_{12} = 0 Surfaces birational to ruled surfaces (rational or otherwise): these possess linear systems of curves of genus π and degree n > 2π − 2, an infinite number of exceptional curves that cannot be eliminated, ∞^{1} rational groups of transformations and continuous series of birational transformations which do not form a group. They can be divided into:

(A′) rational (regular) surfaces, characterized by the conditions

p_{a} = P_{2} = 0,

and

(A′′) irrational ruled surfaces, for which

p_{a} < 0;

the ruled surfaces of genus p = −p_{a} > 1 are characterized (as we shall see in Section ??) by the simple condition

p_{a} < −1,

whereas the ruled surfaces of genus p = 1 are distinguished from the properly elliptic surfaces with p_{a} = −1 by the conditions P_{4} = P_{6} = P_{12} = 0).

(B) P_{12} = 1, p^{(1)} = 1 (p_{a} ≥ −1) Surfaces having a canonical or pluricanonical curve of order zero. For any base point free linear system of genus π and degree n on the surface without exceptional curves, one has

n = 2π − 2;

thus one in general obtains as projective images of the surfaces of this type surfaces F_{n} with plane or hyperplane sections of genus π and order n = 2π−2, in a space S_{r} with r = π, π − 1 or π − 2 dimensions (the three values of r correspond to the three cases: p_{a} = p_{g} = 1, p_{a} = p_{g} = 0, and p_{a} = −1).

The surfaces F_{n} of the family (B) divide in two categories:

(B′) surfaces with (plane or hyperplane) canonical sections, having geometric genus p_{g} = 1 and arithmetic genus p_{a} = 1 or p_{a} = −1, that is, regular surfaces with all the genera equal to one (Chapter VII, Section ??) and hyperelliptic surfaces; (B′′) surfaces whose sections are not canonical curves, but curves representing linear series coming from the division of the s-canonical series by s (for s = 2, 3, 4, 6).

These are the regular surfaces of genus p_{g} = p_{a} = 0 and P_{2} = 1 (P_{3} = 0) which reduce to the sextic surfaces F_{6} passing with multiplicity two through the edges of a tetrahedron (Chapter VII, Section ??) and the elliptic surfaces with p_{a} = −1 and p_{g} = 0 containing a linear pencil of elliptic curves, which we have divided in the seven types I_{a}, I_{b}, II_{a}, II_{b}, III_{a}, I_{b}, III_{c} in Chapter X, Sections 16–19.

The difference between the surfaces (B′) and B′′) is this: while the (B′′) give rise to a finite number of types, the (B′) give rise to infinitely many families of surfaces, depending on (at least) one integer character, which is susceptible of assuming infinitely many values.

This has been proved for the regular surfaces with all the genera equal to one in Chapter VII, Section ?? (where we also saw that each family depends on 19 moduli), but it is also true for the hyperelliptic surfaces; in that case, together with the Jacobi surface F_{j} corresponding to a curve of genus two, there are the infinitely many other surfaces corresponding to involutions of order δ on F_{j}, or more precisely to multiple covers of the F_{j} of order δ without branch curve: each of these families depends on three moduli (see Section ??).

By contrast the surfaces (B′) give rise, as we have said, to a finite number, and precisely to 8 families, which are clearly characterized, and having: the first family (regular surfaces) 10 moduli (Chapter VII, Section ??) and the others (elliptic surfaces) 2 moduli (the I_{a} and I_{b}) or 1 modulus (the remaining families, of the harmonic or equianharmonic type).

Let us add that the surfaces (B′) always contain pencils of elliptic curves, whereas the surfaces (B′′) in general do not contain such pencils; therefore they have a closer similarity with the surfaces of type (C).

(C) P_{12} > 1 and p^{(1)} = 1 Surfaces possessing a (rational or irrational) pencil of elliptic curves K, components of canonical or pluricanonical curves. Together with the (B′′), these surfaces form the vast family of surfaces containing a pencil of elliptic curves.

These surfaces can be classified, as we have seen in the regular case (Chapter VII, Section ??), according to some integral characters, that is:

(1) the arithmetic genus p_{a};

(2) the determinant n, that is, the order of the smallest group of points that can be rationally determined on a K;

(3) the number of the curves of the pencil that reduce to a multiple curve; and moreover, in the irregular case (4) the genus ρ of the pencil, which is

ρ = p_{g} − p_{a} (p_{a} ≥ 0)

in general (Remark, Chapter IX, Section ??), and

ρ = p_{g} − p_{a} − 1

in the case that the surface contains, besides (K), another elliptic pencil of curves; this case yields p_{a} = −1 and leads to the elliptic surfaces.

For the surfaces C) that are regular, we have seen that their families, depending on characters mentioned above, can be also characterized by the values of the plurigenera (Chapter VII, Section ??).

It seems that these results also extend, to a great extent, to the irregular surfaces: p_{a} < p_{g}. In any case this is an object of study, which we recommend to the attention of the young researchers.

(D) P_{12}, p^{(1)} > 1 These surfaces have to be thought of as forming the most general type; in contrast with the previous cases (A), (B) and (C), these give rise, for given values of the characters p_{a}, p_{g}, p^{(1)}, to a finite number of families, each containing a continuous system of classes, depending on a certain number of moduli. In fact the surfaces (D) correspond to a type of canonical or pluricanonical surfaces, of order p^{(1)} − 1 or a multiple of p^{(1)} − 1, respectively, which give a projective birational model of the surfaces themselves; the classification of the surfaces of given order, with some other given characters, can only lead to a finite number of arithmetically distinct types.

We have already had the opportunity of studying the general problem of the classification of the surfaces with p^{(1)} > 1 in the regular case (Chapter VII). The considerations we have developed extend, to a great extent, to the case in which one drops the hypothesis p_{a} = p_{g}; in any event this is an object of research for those interested. It is worth remarking that when one proceeds to the construction of canonical or pluricanonical surfaces with given p_{g} and p^{(1)} one usually meets regular surfaces, and the possibility of irregular surfaces only appears as an exception; thus the canonical surfaces with p_{a} ≥ 4, which are constructed in the first cases p^{(1)} = 5, 6, 7, … (Chapter VIII), turn out to be regular, and also regular appear to be the tricanonical surfaces for p^{(1)} = 2 and the bicanonical ones for p^{(1)} = 3, which are constructed in correspondence with p_{a} > 0 following the procedure of Chapter VIII, Section ??–??^{(38)}.

(38)Some considerations which can be completed with the theorems on the simple integrals belonging to a surface lead us to state what follows: in general the surfaces of irregularity

Surfaces with negative arithmetic genus 11.8 446

It is worth adding here that, in force of an inequality which we shall talk about in the next section, all the surfaces (D) (with p^{(1)} > 1) have the arithmetic genus p_{a} ≥ 0.

11.8 Surfaces with negative arithmetic genus

The classification we have treated in the previous section can be made more precise on the basis of an inequality between the irregularity and the arithmetic genus, that so far has not been proved in an algebro-geometric way, and which therefore we need to take from the transcendental theory (see Note, page ??). The following theorem holds: For any surface of genus p_{g} and irregularity q = p_{g} − p_{a} which does not contain an irrational pencil of curves, one has

p_{g} > 2(q − 2)

or also

p_{g} < 2p_{a} + 4.

So: if p_{g} ≥ 2p_{a} + 4, then the surface contains an irrational pencil of curves (C).

The previous inequality is certainly satisfied if the arithmetic genus is

p_{a} < −1 (p_{g} ≥ 0).

Therefore we can say that the surfaces F of arithmetic genus p_{a} < −1 contain an irrational pencil (C): a pencil of genus p > 0 formed by curves C of a certain genus π. But it is easy to see that π = 0, and therefore F can be transformed into a ruled surface (of genus p = −p_{a}).

To see this, we argue by contradiction: if π > 0, then F can be reduced to a surface without exceptional curves, with absolute linear genus p^{(1)} ≥ 1; thus – by computing its Zeuthen–Segre invariant in relation to (C) – we find (as in Chapter X, Section ??)

Δ − 4 = 12p_{a} − p^{(1)} + 9 Δ ≥ 0 p_{a} ≥ −1!

Therefore we conclude that: the surfaces of arithmetic genus p_{a} < −1 are all birational to ruled surfaces (of genus p = −p_{a}): in the theorem (Chapter X, Section ??) which gives p_{a} = −p and p_{g} = 0 as conditions for a surface to be ruled, the condition p_{g} = 0 is superfluous, since it follows from the other as a consequence.

q ≥ 3 and p_{a} ≥ 0, whose canonical system is not composed with the curves of a pencil, have the linear genus p^{(1)} ≥ 6p_{a} + 4; if q = 2 and p_{a} ≥ 0, then p^{(1)} ≥ 3p_{a} + 4.

Therefore we would be left with the problem of examining the surfaces of irregularity q = 1, with an elliptic pencil of curves (this was kindly communicated to us by G. Castelnuovo, November 1942).

Let p_{a} = −1. If the surface has geometric genus

p_{g} > 1,

then the inequality

p_{g} ≥ 2p_{a} + 4

is still satisfied and therefore F possesses an irrational pencil of curves C (of genus π > 0). But, in relation to this pencil one finds the Zeuthen–Segre invariant

Δ − 4 = 12p_{a} − p^{(1)} + 9 p_{a} = −1, Δ ≥ 0

and therefore

p^{(1)} = 1.

So the condition p^{(1)} = 1, which – in addition to p_{a} = −1 and p_{g} > 1 – characterizes the elliptic ruled surfaces (Chapter XI, Section ??) is a consequence of the other two conditions. Since for p_{g} = 0 and p_{a} = −1 one still has p^{(1)} = 1 and F turns out to be a (proper or improper) elliptic surface, we may state that: the surfaces with arithmetic genus p_{a} = −1 and geometric genus p_{g} ≠ 1 are elliptic surfaces (either proper elliptic surfaces or ruled surfaces of genus one).

Let us finally examine the case:

p_{a} = −1, p_{g} = 1.

In this case we have seen (Chapter X, Section ??) that either F is hyperelliptic (with P_{4} = 1) or it can be reduced to a multiple hyperelliptic surface Φ, with an elliptic branch curves, and then it is an elliptic surface (with P_{4} > 1).

So we can complete the theorems established in Chapter X, Section ??, in the sense that we have already announced there, by saying that:

The elliptic and hyperelliptic surfaces are globally characterized by the condition

p_{a} = −1

(together with the conditions excluding the improper elliptic surfaces, birational to ruled surfaces: P_{4} + P_{6} = 0 or P_{12} = 0).

The hyperelliptic surfaces (which only for particular moduli are also – in a double way – elliptic) can be distinguished from the elliptic ones by the conditions

p_{g} = P_{4} = 1

(instead of

p_{g} = 1, P_{4} > 1).

We may summarize the results we have obtained by stating the theorem: The surfaces with negative arithmetic genus (p_{a} < 0) are:

(1) either birational to (irrational) ruled surfaces,

(2) or elliptic or hyperelliptic surfaces (in the latter case: p_{g} = 1).

Or also:

The negative arithmetic genus defines the whole family of irrational surfaces admitting a continuous group of birational transformations.

Note The fundamental inequality p_{g} < 2p_{a} + 4, for the surfaces which do not contain an irrational pencil of curves, follows – as we have said – from considerations of the transcendental theory. Let us firstly say that the genus p_{g} of a surface F is the number of the linearly independent double integrals of the first kind belonging to F, and, on the other hand, the irregularity is the number of the simple integrals of the first kind of F; moreover we recall that Noether pointed the way to “construct a double integral of the first kind starting with two simple integrals of the first kind which are not one a function of the other”.

Now the fact that the surface F possesses two simple integrals, I_{1} and I_{2}, one a function of the other, means that F contains an irrational pencil of algebraic curves C, on which I_{1} and I_{2} are constant.

This remark occurred for the first time to Castelnuovo, in relation to the surfaces of genus p_{g} = 0 (see Chapter IX, Section ??); later (1905), for the surfaces of any genus p_{g}, to De Franchis^{(39)}, Castelnuovo^{(40)} and Enriques^{(41)}, who discovered in this way the appropriate tool for the classification of the surfaces of genus p_{a} < 0.

The relation between the genus p_{g} and the irregularity q, or between p_{g} and p_{a}, has been clarified and proved in the simplest way by Castelnuovo with the aforementioned inequality

p_{g} < 2p_{a} + 4,

Summary of the above classification 11.9 449

holding for the surfaces F which do not contain an irrational pencil of curves. Meanwhile, on the basis of this relation, De Franchis stated that the surfaces F of genus p_{a} < −1 possess an irrational pencil of curves^{(42)} while Castelnuovo and Enriques – making use of an argument already employed in the hypothesis p_{g} = 0 (Chapter X, Section ??) – saw that these surfaces are birational to ruled surfaces of genus p = −p_{a} (> 1).

(39)Sulle superficie algebriche le quali contengono un fascio irrazionale di curve, Rendic.

Circ. Mat. di Palermo (April 25th, 1905).

(40)Sulle superficie aventi il genere aritmetico negativo. Ibidem, May 14th, 1905 (Memorie scelte, XXVII, page 501).

(41)Sulle superficie algebriche che ammettono un gruppo continuo di trasformazioni ecc., (Ibidem, May 14th, 1905).

(42)Let us also add that De Franchis (clarifying a previous result of 1904) proves that “every irregular double plane possesses an elliptic or hyperelliptic pencil of curves”.

Finally Enriques made a deeper analysis of the surfaces for which p_{a} = −1, succeeding in proving that they are either elliptic or hyperelliptic surfaces.

The developments we have just mentioned have been followed by some important papers by A. Rosenblatt^{(43)} and A. Comessatti^{(44)} that led to the classification of the surfaces for which

p_{g} ≥ 2p_{a} + 4.

Already Rosenblatt (1913) saw that there are only few types of these. Comessatti (1919 and 1922) takes up again Castelnuovo’s analysis, dealing with the more general problem of “determining which relations there are between some functions of two variables when their mutual Jacobians satisfy a certain number of linear homogeneous equations with constant coefficients”.

Whence one, in particular, draws the classification of the algebraic surfaces for which

p_{g} ≥ 2p_{a} + 4,

proving that they belong to the following types:

(1) ruled surfaces of genus p > 1;

(2) elliptic surfaces with p_{g} > 1 and p_{a} = −1;

(3) surfaces representing the pairs of points of two curves of genus 2 and π: p_{g} = 2π and p_{a} = π − 2;

(4) particular surfaces with linear genus p^{(1)} = 1 and p_{a} ≥ 0.

It would be desirable to give an algebro-geometric proof of these results, as well as of Castelnuovo’s inequality

11.9 Summary of the above classification

The theorems concerning the surfaces with negative arithmetic genus that we have just treated enable us to give a more precise general classification of the surfaces; in particular they imply that for p^{(1)} > 1 one must have p_{a} ≥ 0.

This deduction has a meaning of general interest: for the surfaces of type D) (P_{12} > 1, p^{(1)} > 1), since p_{a} ≥ 0, one obtains a bicanonical system of dimension P_{2} − 1 = p_{a} + p^{(1)} − 1, and therefore – for p^{(1)} > 3 – one has P_{2} − 1 ≥ 3. So, except for an objection about the reducibility of the bicanonical curves, which has to be critically examined^{(45)}, the surfaces for which p^{(1)} > 3 have as their projective model a bicanonical surface (that, in some particular cases, can be multiple).

(43)Sur les surfaces irrégulières etc. Rendic. Circ. Mat. di Palermo, 1913.

(44)Rendic. Acc. Lincei, 1919 and Intorno alle superficie algebriche irregolari ecc., Rendic.

Circ. Mat. di Palermo, t. XLVI, 1922.

And it is therefore apparent the special interest of having a complete classification of the surfaces with p^{(1)} = 2 and p^{(1)} = 3, to which we refer in the study we have treated in Chapter VIII, Section ??–??; and even more if one happens to find that these surfaces have (except possibly some cases) the property of regularity, which in our analysis has been assumed to be true a priori^{(46)}.

Let us now summarize the more precise results we have obtained in the general classification of the algebraic surfaces, using also some formula whose meaning has been explained before, in Section ??.

(45)As we did for the regular surfaces in Chapter IV, SectionS 18–20.

(46)See the footnote at the end of Section ?? of this Chapter.

P_{12} = 0: ruled surfaces n > 2π − 2, non-eliminable exceptional curves, continuous series of transformations not forming a group.

\{ \begin{matrix}p_{a} = 0, rational surfaces \\ (p_{a} = P_{2} = 0) \\ p_{a} = −1, elliptic ruled surfaces \\ (p_{a} = −1,P_{4} = P_{6} = 0) \\ p_{a} < −1, ruled surfaces of \\ genus p = −p_{a} > 1. \end{matrix}

P_{12} = 1: virtual canonical curve of order 0, n = 2π − 2 (p^{(1)} = 1)

\{ \begin{matrix}p_{g} = 1 \\ p_{g} = 0 \end{matrix}

p_{a} = 1, infinitely many families of surfaces (p_{a} = P_{2} = 1) p_{a} = −1, infinitely many families of hyperelliptic surfaces of divisor δ = 1, 2, 3, … (p_{a} = −1, p_{g} = P_{4} = 1).

p_{a} = 0, surfaces of order 6 passing doubly through edges of tetrahedron (p_{a} = P_{3} = 0, P_{2} = 1).

p_{a} = −1, elliptic surfaces I_{a}, I_{b}, II_{a}, II_{b},

III_{a}, III_{b}, III_{c} (P_{2} = 0, 1,

P_{4} = 0, 1, P_{3} = 0, 1, P_{4} = 0, 1).

P_{12} > 1 (n < 2π − 2) \{ \begin{matrix}p^{(1)} = 1 \\ p^{(1)} > 1 \end{matrix}

p_{a} = −1, elliptic surfaces with a pencil of genus p_{g} of elliptic curves (p_{a} = −1, P_{4} ≠ 1)

p_{a} ≥ 0, surfaces whose canonical and pluricanonical curves are composed with elliptic curves of a pencil of genus p_{g} − p_{a}: depend on more arbitrary integral characters

p_{a} ≥ 0, surfaces which admit a projective canonical (p_{g} > 3) or bicanonical (p^{(1)} > 3) model, etc.: a finite number of types for a given value of p^{(1)}.

At this point we would like to be allowed to stop for a while, as when one climbs a mountain and likes to rest for a moment on the peak he as reached, whence admiring the view of the Nature which is offered to the sight.

Fifty years ago the study of these theories started in Italy, after they had been sketched by the genius of a precursor (Max Noether); at that time, joking on the difficulties and the exceptions that we met at every step, we used to say that while algebraic curves (whose theory was already harmoniously settled) had been created by God, the surfaces were, by contrast, the works of the Devil.

It is now apparent that God liked to create for the surfaces a more harmonious order, where a wonderful beauty shines; and He wanted in them – as we say with the Poet^{(47)} –

del creator suo spirito piu’ vasta orma stampar.

The richness of properties and the beauty, which were hidden for such a long time but are now apparent, should not be a reason of vain pride for the Italian school of geometry or for the foreign mathematicians who contributed to discover them; they rather have to rouse a feeling of reverence for the wonderful order of the mathematical objects, that the thought encounters and almost collects, as the living species, from Mother Nature; so we have to nourish the faith of the young scholars that, behind the difficulties, the exceptions, the apparent incongruities, there is actually in this world of objects a divine harmony, on which the joint efforts of the researchers will succeed to shed a clearer and clearer light.

(47)Note of the translators: “to impress a stronger footprint of his creating spirit”, from “Il 5 Maggio”, by Alessandro Manzoni

Index

adjoint surfaces, 63 Jacobian curves, 39 numerical genus, 95 Riemann–Roch, 95 surface

with normal singularities, 26