 \chapter{Classification of surfaces of linear genus
$p^{(1)}=1$}\label{ch:7}
 \markboth{\qquad 7. \quad Classification of surfaces with $p^{(1)}=1$
\hfill}{}

 \section{Surfaces of genus zero and bigenus one, with bicanonical
curve of order zero}\label{sec:7.1}
 \markright{\hfill Surfaces of genus zero and bigenus one \quad
7.1\quad}
 We have seen (Chapter~VI, Section~\ref{sec:6.2}) 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:
 \begin{enumerate}
 \renewcommand{\labelenumi}{(\arabic{enumi})}
 \item every curve $C$ of genus $\pi$ on $F$ has the degree
 \[
 n=2\pi-2;
 \]

 \item the dimension of the complete linear system $|C|$ (virtually
free of base points) is
 \[
r\ge\pi-1;
 \]

 \item for every irreducible linear system $|C|$, of genus $\pi>1$, one
has exactly
 \[
r=\pi-1
 \]
namely the system is regular;

 \item 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
 \[
 |L|=|2C|=|2C'|;
 \]

 \item the surface $F$ may contain {\em elliptic
curves},\index{elliptic!curve} namely irreducible curves of virtual
genus $\pi=1$, thus of degree $n=0$. These may either be {\em 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;

 \item a curve $\theta$, 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 $\theta$ with the curves C/2;

 \item an irreducible curve $\theta$ of (virtual) genus zero, thus of
degree $-2$, on the surface $F$, has no adjoint curves.

In fact, if an adjoint curve $\theta'$ existed, then it would meet
$\theta$ at $-2$ points, and therefore it would coincide with $\theta$ ;
but $\theta=\theta'$ yields $p=1$, contrary to the hypothesis.

In other words the existence of $\theta'$ implies that $\theta$ has
genus $>0$;

 \item given a pencil of elliptic curves $|C|$ on the surface $F$,
there is no irreducible curve $\theta$ of genus 0 which is a bisecant of
the pencil.

In fact the system $C+\theta$, which would be a pencil of curves of
virtual genus $\rho=2$, should have an adjoint system $|L|$ of dimension
1, containing the composite curve $\theta+C/2+(C/2)'$; 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
 \[
\frac{C}{2} L=\frac{C}{2} \theta+\frac{C}{2} \frac{C}{2}
+\frac{C}{2}\left(\frac{C}{2}\right)'
 \]
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 $\infty^1$ of them which are not reduced to
$\theta$ 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 $\theta'$ adjoint to $\theta$: but the
existence of $\theta'$ would imply that the genus of $\theta$ is $>0$,
contradicting the assumption.
 \end{enumerate}

We will take into account assertions (1--8).

If $|C|$ is an irreducible linear system, of dimension $\pi-1$, let us
consider the double system $|L|$ which has degree $4n=8\pi-8$, genus
$4\pi-3$, and dimension $4\pi-4$. Inside the system $|L|$ there are two
$\infty^{2\pi-2}$ subsystems, that we call $\Si$ and $\Si'$, the first
made up of pairs of curves $C$, and the second of pairs of curves $C'$.
Each one has index
 \[
 i=\frac{1}{2} \left( \renewcommand{\arraycolsep}{.2em}\begin{array}{c}
 2\pi-2\\
 \pi-1
 \end{array} \right).
 \]

If we denote by $S_{4\pi-4}$ the space whose points represent the
elements (curves) of the system $|L|$, then we have inside this space
two varieties of dimension $2\pi-2$ and order $i$, which represent $\Si$
and $\Si'$, 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 $\pi>1$, a system which is partially contained in it and has
genus strictly less than $\pi$.

More precisely we can find a curve composed of two $C$:
 \[
 C=C_1+C_2 \quad\hbox{and}\quad C=C_3+C_4,
 \]
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
 \[
 C_1+C_3=C',\quad C_2+C_4=C',
 \]
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 $\ge1$.

We distinguish two cases:
 \begin{enumerate}
 \renewcommand{\labelenumi}{(\alph{enumi})}
 \item one of the two curves, $C_1$ say, has genus $\pi_1>1$.

 \item both $C_1$ and $C_2$ has genus 1.
 \end{enumerate}

In case (a) we can say that $C_1$ belongs to a complete linear system
$|C_1|$ of dimension $\pi_1-1\ge1$. Since this system is partially
contained in $|C|$, it has dimension $\pi_1-1<\pi-1$, thus $\pi_1<\pi$.

Furthermore, the variable parts of the curves $C_1$ must be irreducible
and again of genus $\rho>1$ (with $\rho<\pi$). 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 $C_1$:
 \[
 |C_1|=\theta+|sK|.
 \]

In order that the genus of $C_1$ be $\pi_1>1$ it is necessary that
$\theta$, or one of its components $\theta_1$, 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 $\le1$). Since the number of these
intersections is even, thus greater than or equal to two, by adding
$\theta_1$ to $|sK|$ one certainly succeeds in enlarging the system
$|sK|$ if $\theta_1$ has genus 1, and even if $\theta_1$ 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 $\pi$ 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
 \[
 1\le s\le\pi-1.
 \]

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
$s\le\pi-1$ 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, $C_1=\theta+\theta_1$ and similarly $C_2=\theta+\theta_2$; then
$C_1$ and $C_2$ have the common component $\theta$, but, in the
arguments which are based on the use of the virtual characters, the
curves $\theta+\theta_1$ and $\theta+\theta_2$ behave ``as if'' they
were irreducible with
 \[
 s=(\theta+\theta_1,\theta+\theta_2)
 \]
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 $\infty^1$ linear system $|C|$ of
genus 2, or of a pair $C_1$ and $C_2$ of elliptic curves with $s$
($\ge1$) 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 $\theta$, 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 $s'\ge1$) points,
and thus also a pair of elliptic curves with {\em 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 $\pi=s+1>2$.
Therefore among the pairs of curves $2C$, one has in general
 \[
 i=\frac{1}{4} \left( \renewcommand{\arraycolsep}{.2em}\begin{array}{c}
 2\pi-2\\
 \pi-1
 \end{array} \right)^2 \ge9.
 \]
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 $\Si$ and $\Si'$ representing the pairs of
curves $C$ and $C'$ have some other common point that is {\em distinct}
from the two points corresponding to the pairs
 \[
 C_1+C_2,\quad C_1'+C_2'
 \]
 and
 \[
 C_1+C_2',\quad C_2+C_1';
 \]
 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 $\Si$ and $\Si'$.

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
$\infty^1$ 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~\ref{sec:5.7}.

We may therefore conclude by saying
 \begin{proc}
 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.
 \end{proc}

This {\em family} of surfaces depends on 10 moduli.\index{number of
moduli} In fact the construction of the $D_6$ with 5 assigned double
points, and therefore of the $D_8$, depends on
 \[
\frac{6\cdot9}{2}-5\cdot3=12
 \]
parameters, so, adding the 6 constants which depend on the choice of
the singular points of the $D_6$, one has $\infty^{18}$ curves $D_8$.
{From} the 18 parameters on which they depend, we have to subtract 8,
corresponding to the $\infty^8$ 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 $\infty^{13}$ sextics $F_6$ passing doubly
through the edges of a given tetrahedron (we wrote the equation of
these in Chapter~III, Section~\ref{sec:3.11}), and that there are
$\infty^3$ 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\footnote{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
$\infty^1$ 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.}.

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
$A_1A_2A_3A_4$, 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 $\infty^5$ curves $D_6$ of the 6th order passing
doubly through $O$ and having at $P$ and $Q$ two tacnodes with tacnodal
tangent lines $p=PO$ 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 $p=PO$ 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
$F$.

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 $\infty^3$ 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 $\infty^3$ 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 $\infty^3$ 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\footnote{F.~Enriques, Sopra le
superficie algebriche di bigenere uno, Mem.\ Soc.\ It.\ delle Science,
detta dei XL, 1906.}:
 \begin{proc}
 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.
 \end{proc}

 \begin{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. 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 $O$.
 \end{Remark}

 \section{Surfaces with all the genera equal to one}\label{sec:7.2}
 \markright{\hfill Surfaces with all the genera equal to one \quad
7.2\quad}
 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 $\pi$ without base points has degree
$n=2\pi-2$ and is self-adjoint; therefore, for an irreducible system
$|C|$ the characteristic series is the canonical series.
 \begin{proc}
On the surface $F$ (with $p=P=1$), a connected curve, reducible or
irreducible, of virtual genus $\pi\ge0$ belongs to a complete linear
system of dimension $r=\pi$ (see Chapter~IV, Section~\ref{sec:4.17}).
 \end{proc}
By contrast a disconnected curve belongs, in general, to a linear
system of dimension greater than its genus $\pi$; in particular: {\em
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 $\theta$ 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 $\theta$ 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 $\pi>1$
which belong to it. To start with we may suppose the genus of such a
minimal system $|C|$ to be $\pi=2$, and then $\pi=3$, $\pi=4,\dots$

The hypothesis $\pi=2$ 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 $\pi=3$ 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 $\pi=4$ 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 $\infty^{14}$ 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 $\infty^5$
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
$\infty^4$ self-adjoint system of curves of genus 4, whence
 \[
 p=p_a=p_g=1,\quad P=1.
 \]

For $\pi=5$ 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 $\pi$, 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:
 \begin{proc}
 The regular surfaces with genus $p=1$ and bigenus $P=1$ {\em (with
canonical curve of order zero)} may be classified by means of the
minimum value of the genus $\pi$ $(>1)$ 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 $n=2\pi-2$,
with canonical hyperplane sections, belonging to the space $S_\pi$.

There are families of surfaces $F_n$ corresponding to all the values of
$\pi=2,3,4,5,\dots$, 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 $\pi$ are in general distinct one from the
other with respect to the birational transformations.
 \end{proc}
The actual existence of surfaces $F_n$ for all the integral values of
$\pi$, or rather for all the even values of $n=2\pi-2$, may be proved
by observing that there exist at least $\infty^{18}$ 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 $\infty^{19}$ $F_n$, there are $\infty^{18}$ 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 $x-\pi+4$ conditions, if the
existence of such a hyperplane yields, as a consequence, the existence
of $\infty^x$ 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
$\infty^{\pi-3}$ system of similar hyperplanes, precisely the
hyperplanes through the plane of the conic, and then one has $x=\pi-3$.
Therefore, since $x-\pi+4=1$, 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 $\pi+1$ and dimension $\pi+1$,
which has $C$ as a fundamental curve; thus this system yields the
existence of an $F_{n+2}$ of $S_{\pi+1}$ 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\footnote{As we see, the above argument is suggested by the
attempt to construct surfaces $F_n$ with $n=2\pi-2$ inductively
(passing from $\pi$ to $\pi+1$) by means of Chisini's constructive
process for multiple planes (see Chapter~V, Section~\ref{sec:5.10}).
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.}.

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~\ref{sec:5.11}) 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.

However our assertions gives rise to certain critical doubts of a
delicate nature. We need to prove that:
 \begin{enumerate}
 \renewcommand{\labelenumi}{(\arabic{enumi})}
 \item the $F_n$ which (according to Chapter~V, Section~\ref{sec:5.11})
have at least 19 moduli, actually have 19 moduli and no more;

 \item 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.
 \end{enumerate}

These assertions can be justified by counting the number of projective
invariants which belong to a rational surface with canonical hyperplane
sections of genus $\pi$, having a triple point and lying in a space
$S_\pi$. These surfaces can be represented on the plane by an
$\infty^\pi$ linear system of genus $\pi$ and degree $2\pi-2$, having a
fundamental cubic curve on which there must be 12 base points of the
system forming a $G_{12}$\footnote{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, {\it Le superficie razionali},
Libro~I, \S41, page~184 and \S46, page~220. In this book (see also
\S46, 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 $\pi$, has degree $n>2\pi-2$; therefore all the systems
$|C|$ with canonical characteristic series fall in the statement of
page~184.}; 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 $S_\pi$ 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 $n=2\pi-2$) corresponding to different values of $\pi$ $(>1)$,
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 {\em on a surface $F_n$ of $S_\pi$
{\em (corresponding to the minimum value $\pi$ 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.}

Now we wonder whether it is possible to find, among the $\infty^\pi$
hyperplanes of $S_\pi$, an $m$-tangent hyperplane to $F_n$ such that its
section with $F_n$ is reducible into two curves of genus $\pi_1$ and
$\pi_2$ with $m$ common points: $\pi=\pi_1+\pi_2+m-1$. If such a
hyperplane exists, then there are $\infty^x$ such hyperplanes, with
$x=\pi_1+\pi_2$; therefore the number of parameters on which the
hyperplanes we look for depend is $\pi-m-x=-1$: 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 $\Phi_4$ with a triple point $O$: the monoid, which
contains 12 lines $a_1a_2a_3\dots a_{12}$ 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 $A_1A_2\dots
A_{12}$ 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 $\Phi_4$, an irreducible, complete linear system of
curves $C$ on $F_4$ becomes in the limit a linear system of curves $\Cbar$,
which are complete intersections of $\Phi_4$ with surfaces of a certain
order $d$, not containing the triple point $O$. So, if $\pi$ is the genus of
the $\Cbar$, and therefore also of the $C$, the dimension of $|C|$ is $\pi$ and its degree
$2\pi-2$, hence the curves $C$ on $F_4$ must have $2\pi-2$ common points
with the intersections with the surfaces $F_d$ of order $d$, which are
also of genus $\pi$ and degree $2\pi-2$; since a $C$ cannot contain a
linear series $g_{2\pi-2}$ of dimension $\pi$, 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 $\Phi_4$, a system $|C|$ of
genus $\pi$ and degree $2\pi-2$ belonging to $F_4$ will a priori tend
to an $\infty^\pi$ system of connected limit curves which will be
formed by the intersections of $\Phi_4$ with surfaces passing a certain
number of times through $O$, plus the lines $a_i$, counted $h_i$ times
respectively (with $h_i\ge0$), plus the neighbourhood of $O$ counted $s$
($\ge0$) times. Therefore the images of such curves on the plane on
which $\Phi_4$ is represented will, a priori, be curves of a given order
$n$, belonging to a complete linear system
 \[
 |C_n|=C_m(A_1^{r_1}A_2^{r_2}\dots A_{12}^{r_{12}})+sC_3+\sum h_iA_i,
 \]
where the curves $C_m$ (of order $m\le n$) belong to an irreducible
$\infty^\pi$ linear system. Since the $C_m$ have genus $\pi$ and degree
$2\pi-2$, we have
 \begin{eqnarray*}
& n^2-\sum (r_i+s-h_i)^2 = 2\pi-2 & \\
& n(n-3)-\sum (r_i+s-h_i)(r_i+s-h_i-1) = 2\pi-2, &
 \end{eqnarray*}
 and therefore
 \[
3n=\sum (r_i+s-h_i).
 \]

This means that for the linear system $|C_n|$ the cubic $C_3$ is {\em 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:

\paragraph{(1)} $C_3$ is also actually a fundamental curve for the
complete system $|C_m|$, namely
 \[
 |C_n|=|C_m|, \quad s=0, \quad h_i=0.
 \]

Under this assumption we easily prove that $|C_n|$ is a multiple of the
system $C_4(A_1A_2\dots A_{12})$ and therefore the corresponding system on
$\Phi_4$ 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:
 \begin{eqnarray*}
 \sum r_iA_i &\equiv& nG \\
 \sum A_i &\equiv& 4G;
 \end{eqnarray*}
multiplying the second relation by $r_1$ and subtracting the first one
from it, we find:
 \[
 \sum_{i=2,\dots,12}(r_i-r_1)\equiv (n-4r_1)G;
 \]
 since the group of the 11 points $A_2A_3\dots A_{12}$ can be arbitrarily
chosen, the latter relation cannot hold unless we have $n=4r_1$ and
 \[
r_1=r_2=\cdots=r_{12}=r;
 \]
thus the curves $C_n$ have order $4r$ and belong to the system $|rC_4|$. \qed

If, by contrast, assumption (1) is not satisfied, then we have two
possibilities which we have to exclude.

\paragraph{(2)} First, that the $C_n$ do not contain the curve $C_3$,
but only the neighbourhood of some point $A_i$:
 \[
s=0,\quad h_i>0.
 \]

Now, if $C_3$ is a fundamental curve for $|C_m|$, this irreducible
linear system is a multiple, of a certain order
$r=r_1=r_2=\cdots=r_{12}$, 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 $\Phi_4$.

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.

\paragraph{(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 $\sum h_iA_i$ only a certain number $t$ of the
points $A_i$ may appear, and each $h_i\le s$ times. If we denote by
$l=C_mC_3$ the number of variable intersections of a $C_m$ with the
cubic $C_3$, we have now
 \[
l+ts-3s\ge0.
 \]

So we have to distinguish between two a priori possible cases: either
$l+t\ge 4$, in which case the enlargement of the linear system
$C_m+C_3+\sum A_i$ follows; or $l+t<4$, which yields $s\le 1$; 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 $d=1,2$,\dots

 \begin{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 $\rho$ and degree $\nu$; thus one has
 \[
 n=s\nu=2\pi-2=s(2\rho-2).
 \]

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
 \[
 n=s\nu \quad\hbox{and}\quad 2\pi-2=s(2\rho-2),
 \]
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.
 \end{Remark}

 \section{A historical note}\label{sec:7.3}
 \markright{\hfill A historical note \quad 7.3\quad}
 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\footnote{See F.~Enriques,
Ricerche ecc., III, 6.} (1893). In 1896, Enriques\footnote{F.~Enriques,
Sui piani doppi di genere 1. Mem.\ della Soc.\ It.\ delle Scienze
(detta dei XL), 1896.} 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\footnote{F.~Enriques, Le superficie di genere
uno, {\it Rendic.\ Acc.\ di Bologna}, 1908.} and Severi\footnote{F.
Severi, Le superficie algebriche con curva canonica d'ordine zero, {\it
Atti Ist. Veneto}, Vol.~LXV, 2, 1909}, independently succeeded in
proving the existence of families of surfaces $F_n$ of $S_\pi$, of even
order $n=2\pi-2$, for all the values of $\pi$. Both took as a starting
point the count of moduli made by Enriques (1908, see Chapter~V,
Section~\ref{sec:5.11}). 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).

Severi gives an algebro-geometric proof of the existence of the $F_n$
(with $n=2\pi-2$) for any value of $\pi$, 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 {\em 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 {\it
Lezioni} by Enriques--Campedelli (page~311). For example one obtains
some $F_n$ with
$n=2\pi-2$, where $\pi$ is of the form $\pi=2+3s$, by starting from a
double plane with a sextic branch curve which is the envelope of an
$\infty^1$ 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 $\pi=2+3s$\footnote{If $\pi$ is of the form $\pi=3+3s$, one can
makes a similar construction starting with an $F_4$ which contains a
line; and if $\pi=4+3s$, one may start from a particular
$F_4$ which contains a plane cubic.}.

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\footnote{A.~Franchetta, Sulle superficie regolari di
genere uno con curva canonica d'ordine zero. Ist.\ Lombardo, 1942.}, 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 {\em 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 $n\ge4$) 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\footnote{Enriques and Severi, M\'emoire sur les surfaces
hyperelliptiques. Acta Math., Vol.~XXXII, no.~27.}, 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.

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,
{\it Abhandlungen der Akademie der Wissenschaften}, Berlin, 1883.

K.~Rohn, Die Raumcurven auf der Flaechen vierter Ordnung, {\it Berichten
der koenigl.\ Sachs.\ Gesellshaft der Wissenschaften}, Leipzig, 1897.

F. Severi, Le superficie algebriche con curva canonica d'ordine
zero, {\it Atti Ist. Veneto}, LXV, 2, 1909.

G. Fano, Sulle variet\`a algebriche che sono intersezioni complete di
piu' forme. {\it Rendic.\ Acc.\ Torino}, Vol.~XLIV, 1909.

S. Lefschetz, On certain numerical Invariants of algebraic
Varieties, {\it Trans.\ Amer.\ Math.\ Soc.}, Vol.~22, 1921);
Concerning Noether's theorem on curves traced on nonsingular
surfaces, {\it Bull. Amer. Math.\ Soc.}, Vol.~29, 1923.

A. Franchetta, Sulle superficie regolari di genere uno con curva
canonica d'ordine zero, {\it Rendic.\ Ist.\ Lombardo}, 18 giugno 1942.

B. D'Orgeval, {\it Sur les surfaces alg\'ebriques dont tous le genres
sont 1}, Gauthier-Villars, Paris, 1945.

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 $n=2\pi-2$, with canonical hyperplane sections in an
$S_\pi$, 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\footnote{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, {\it Rendic.\ Acc.\ Lincei},
febbraio-marzo, 1932.} 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 $S_\pi$ form a unique family.

 \section{Surfaces of linear genus $p^{(1)}=1$ with
$P>1$}\label{sec:7.4}
 \markright{\hfill Surfaces of linear genus $p^{(1)}=1$ with
$P>1$ \quad 7.4\quad}
 Let $F$ be a surface of (absolute) linear genus $p^{(1)}=1$ and
geometric genus $p>1$. The $\infty^{p-1}$ 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: {\em 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~\ref{sec:6.2}) that $p>1$ yields
 \[
P\ge 2.
 \]
 and $p=0$ implies:
 \[
P\ge1,\quad P_6\ge 2.
 \]

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:

{\em 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 $n=2\pi-2$, with canonical hyperplane sections in $S_\pi$,
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 {\em two invariant
characters} belonging to them, which are {\em capable of taking
arbitrary integral values}, namely:
 \begin{enumerate}
 \renewcommand{\labelenumi}{(\arabic{enumi})}
 \item 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 $I=S-4=12p+8$ ($=12p_a-p^{(1)}+9$).

 \item and the {\em 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.
 \end{enumerate}

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 {\em 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\footnote{See F.~Enriques, Sulle superficie algebriche con
un fascio di curve ellittiche {\it 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.}.

Therefore an $\infty^1$ 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 {\em determinant} $d>1$, can be transformed in such a way that
the $C$ become {\em 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 $L+C,L+2C,\dots,L+sC$, 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 $D_{2n}$.

In order to better understand our surfaces of linear genus $p^{(1)}=1$
(regular, irrational, surfaces with $P>p\ge1$), 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 $\theta$ which is unisecant the $C$. By means of
an $\infty^3$ linear system contained in $|3\theta+sC|$, $F$ 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:
 \begin{enumerate}
 \renewcommand{\labelenumi}{\Roman{enumi}.}
 \item $D_{2n}$ with a point $O$ of multiplicity $2n-3$, without other
singularities affecting the genus; and

 \item $D_{2n}$ 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 $B$.
 \end{enumerate}

The double planes of Type~I have genus
 \[
 p=n-2,
 \]
 since they have $\infty^{p-1}$ 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 $\infty^{n-4}$ canonical
curves, represented by the groups of $p-1=n-4$ lines through $O$, plus
the exceptional branch line to be added to them\footnote{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).}; therefore the genus is
 \[
 p=n-3.
 \]

Similarly the $i$-genus is
 \[
P_i=i(p-1)+1,
 \]

Thus we conclude that:

{\em The surfaces $F$ of absolute linear genus $p^{(1)}=1$ and of
determinant $d=1$, with $P\ge1$, can be represented as double planes of
Type~I}, with branch curve $D_{2n}$ having a point $O$ of multiplicity
$2n-3$, {\em 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 $\infty^6$ 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.

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 $\infty^6$ 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~\ref{sec:5.11}, 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: {\em 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
 \[
n\ge4,\quad p>1,
 \]
 this number is always
 \[
 \ge 9p+10\quad (=9p-p^{(1)}+12).
 \]

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 $\infty^6$ 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 $(\theta+hC)$; 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 {\em
irreducible types}:
 \begin{enumerate}
 \item[I$'$.] $D_{2n}$ with a point of multiplicity $2n-4$ and without
other singularities affecting the genus:
 \[
 p=n-2;
 \]

 \item[II$'$.] $D_{2n}$ with a point of multiplicity $2n-4$ and a
4-tuple point
 \[
 p=n-3;
 \]

 \item[III.] $D_{2n}$ with a point $O$ of multiplicity $2n-4$ and $s$
($\ge1$)
$[3,3]$ points: $A_1A_2\dots A_s$. 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:
 \[
 p=n-2-s.
 \]
 \end{enumerate}

{\em The surfaces $F$ of linear genus $p^{(1)}=1$ and determinant $d=2$}
(for which $P>p\ge1$) give rise, for each value of the genus $p$, {\em
not only to the double planes of Type~I$'$ and~II$'$ for which
 \[
 p=n-2,\quad n-3,
 \]
 and
 \[
 P=2(p-1),
 \]
but also to infinitely many families of double planes of Type~III,
which depend on another integer $s=1,2,3$,\dots:
 \[
 p=n-2-s,
 \]
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 $OA_1,\dots,OA_s$ (the images of fixed curves $C/2$), or
better, the neighbourhoods of the $[3,3]$ points $A_1\dots A_s$ (see
Chapter~III, Section~\ref{sec:3.8}, \ref{sec:3.11} and Chapter~V,
Section~\ref{sec:5.7}); 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 {\em 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 {\em 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
$A_1\dots A_s$, with tacnodal lines $OA_1\dots OA_s$, depends on
$5(2n-s)-6-6s$ parameters, while the group of the $s+1$ points
$O,A_1,\dots,A_s$ 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 $d$:

{\em 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 $1<r\le d$);
each of these characters, $s_r$, denotes the number of elliptic curves
$C/r$, which, counted $r$ times, belong to the pencil $|C|$.}

{\em The various families of surfaces $F^d$ {\em 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 $i$th 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 $r$th
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
{\em 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:

{\em 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 {\em two surfaces
$F^1$ and $F^d$ have the same Zeuthen--Segre invariant and the same
genus $p_a$.}

 \begin{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
$p^{(1)}=1$, {\it Rendic.\ Acc.\ Lincei}, 1$^o$ sem.\ 1898, Sulla
classificazione delle superficie algebriche e particolarmente sulle
superficie di genere lineare $p^{(1)}=1$, {\it 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~\ref{sec:9.2} and~\ref{sec:9.3})).

Precisely, in the note Intorno alle superficie algebriche di genere
lineare $p^{(1)}=1$\footnote{{\it Rendic.\ Acc.\ di Bologna}, dic.\
1906.}, Enriques proved that: {\em the irregular surfaces of linear
genus $p^{(1)}=1$ and arithmetic genus $p_a\ge0$ ($p_g>p_a$) 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 $p_g-p_a$.

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 $\infty^1$ and $\infty^2$ system of birational transformations into
themselves'' (Chapter~XI).
 \end{NOTE}
