 \chapter{Regular canonical and pluricanonical surfaces}\label{ch:8}
 \markboth{\qquad 8. \quad Regular canonical and pluricanonical surfaces
\hfill}{}

 \section{Introduction}\label{sec:8.1}
 \markright{\hfill Introduction \quad 8.1\quad}
 Surfaces with genus $p_g>3$ have a (pure) canonical system $|K|$ of
dimension $p_g-1\ge 3$, via which this they are, as a rule, transformed
into {\em canonical surfaces}\index{canonical surface} 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 {\em regular
surfaces}:\index{reulgar!surface} $p_g=p_a=p$.

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 {\em essential exception}\index{elliptic!surface} 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:

 \begin{enumerate}
 \renewcommand{\labelenumi}{(\arabic{enumi})}
 \item the surfaces for which the genus $p$ is $\le3$;

 \item those on which the canonical system, although irreducible and at
least $\infty^3$ ($p>3$), belongs to an involution $I_n$, of order $n>1$,
so that the canonical surface reduces to a multiple ($n$-tuple) surface;

 \item the surfaces, with linear genus $p^{(1)}>1$, for which the (pure)
canonical curves $K$ are always reducible; as we saw in Chapter~IV,
Section~\ref{sec:4.18}, there are indeed examples in which the curves $K$
contain (non-exceptional) 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.
 \end{enumerate}

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
$\infty^{p-1}$ 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,
$\pbar^{(2)}<p^{(2)}=p^{(1)}-1$.

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
$\infty^3$, 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.

 \section{Canonical surfaces of genus $p=4$ and linear genus
$p^{(1)}=6,5$}\label{sec:8.2}
 \markright{\hfill Canonical surfaces of genus $p=4$ and linear genus
$p^{(1)}=6,5$ \quad 8.2\quad}
 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
 \[
 p^{(2)}=p^{(1)}-1\ge 5.
 \]

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
 \[
 p=4,\quad p^{(1)}=6.
 \]

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:

 \begin{enumerate}
 \renewcommand{\labelenumi}{(\alph{enumi})}
 \item 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 $F_7$.

 \item 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
$C'_6(1^32^23^24^25^26)$ 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 $F_5$.

 \item 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 $F_5$.
 \end{enumerate}

We want now to show with the utmost rigour and generality that:
 \begin{proc}
 every surface\index{surface with $p=4$ and $p^{(1)}=6$} $F$ 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 $F_5$.
 \end{proc}

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
$n$ ($\le5$) 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 $p\le1$) 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 $\pi\le p^{(1)}$ ($=6$), the branch curve on the $F_2$
will be a $C_{2\pi+2}$ of order $2\pi+2$. We shall firstly prove that we
certainly do not have
 \[
 \pi<5,
 \]
and thus, a priori, only the following cases are possible:
 \[
 \pi=6, 5.
 \]

Indeed we cannot have $\pi<5$, that is, $\pi\le4$, since then the canonical
curves $K$ should intersect the curves $k$ of genus $\pi$ in residual
groups of the characteristic series $g^2_n=g^2_4$, thus in no more than
2 points, while at the same time we must have $kK>kk=4$. But, if
$\pi=5$, we have on the double $F_2$ a branch curve of order 12; and,
if $\pi=6$, 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 $\infty^3$ 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.

We have then {\em 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 $p^{(1)}=5$, {\em corresponds to the
minimal value of $p^{(1)}$ for $p=4$} (this agrees with the general
inequality $p^{(1)}\ge2p-3$, 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: {\em 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 $b$.

 \section{Surfaces with $p=4$ and $p^{(1)}=7$}\label{sec:8.3}
 \markright{\hfill Surfaces with $p=4$ and $p^{(1)}=7$ \quad 8.3\quad}
 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 {\em 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.

 \begin{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 {\em 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.
\end{NOTE}

Hence the double $F_3$ has a representation on the {\em 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 {\em 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:
 \[
F_3^2+\la F_6=0,
 \]
for $\la=0$ 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 $f_3^2+\la f_6=0$, reduces by continuity (for
$\la=0$) 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 $\la=0$, must reduce to the cubic
envelope $f_3=0$ counted twice (whose class is $2\cdot 6=12$), plus the 18
pencils having as centres the branch points of the given double cubic; and
it is evident\footnote{Cf.\ Enriques--Chisini, Lezioni, Vol.~III.} that
the contact points of the tangents issued from any point $P$ to the $f_6$
end up, for $\la=0$, 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 $f_3^2+\la f_6=0$, 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 $g^1_2$), which lead to canonical double
quadrics representable over the double plane with {\em a branch curve
$C_{10}$ of order 10, endowed with two $[3,3]$ points.}

And indeed these double planes do form {\em 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
$g^1_3$, therefore $|K|$ turns out to belong to a rational involution
$I_3$ of order 3. {\em The triple canonical quadric $Q^3_2$, studied by
G.~Pompilj\footnote{G. Pompilj, Sulle superficie algebriche le cui curve
canoniche posseggono una $g^1_3$, {\it Rendic.\ Ist.\ Lombardo}, 1940.},
is a cone and has a branch curve $C_{18}$ of order 18.}

One can see that {\em the triple canonical quadrics $Q^3_2$ 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 $Q^3_2$, that is, a triple conic $C^3_2$
having 18 branch points. For this, we first observe that the $\infty^{27}$
system of plane sextics cannot contain the $\infty^{29}$ system of the
triple conics of the same genus with 24 branch points. One cannot
nevertheless exclude that among the $\infty^{27}$ $C_6$ there might occur
particular $C^3_2$ with 24 critical points; but if one wants to find
inside this family a $C^3_2$ 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
{\em collinear} apparent critical points, and this is impossible for an
{\em irreducible} $C^3_2$.

 \section{Surfaces with $p=4$ and $p^{(1)}=8$}\label{sec:8.4}
 \markright{\hfill Surfaces with $p=4$ and $p^{(1)}=8$ \quad 8.4\quad}
 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 $\Phi_2$. The curve $C_7$, by a formula we have already
seen\footnote{Cf.\ Chapter~V, Section~\ref{sec:5.3}}, has a triple point
that is double for the adjoint $\Phi_2$, which is therefore a cone. This
agrees with the fact that the $C_7$ is the contact curve of the $\Phi_2$
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 $\Phi_2$ 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 $C'_{14}(O^7O_1^7)$ and the system
of halves of the system $|C'_{14}|$ is a priori a system
$C'_7(O^4O_1^3)$ or a more particular system contained in it. But the
$C'_7(O^4O_1^3)$ which do not have multiplicity greater than 4 at $O$,
are images of curves $C_7$ of $\Phi_2$ which are contact curves of
$\Phi_2$ with some $F_7$. And they cannot be double for an irreducible
$F_7$ since the generators of the cone $\Phi_2$ 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 $\Phi_2$ is $C_7(O^5O_1^2)$; therefore the double
curve passes three times through the vertex of the cone and has genus
$\rho=4$.

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 $\rho=4$, with a
triple point $V$, belonging to the system of halves of the system cut on
$\Phi_2$ 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~\ref{sec:5.13} to be true:
 \[
 M\ge10p-2p^{(1)}+12=36,
 \]
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 $\infty^{26}$ canonical surfaces $F_7$: in fact the $C_7$ over the
quadric cone are $\infty^{17}$ and thus, in view of the $\infty^7$
projective transformations of the cone, they give rise to $\infty^{10}$
projectively distinct curves; on the other hand, the surfaces $F_7$ having one and the same double curve $C_7$ are $\infty^{26}$, since
splitting off the cone we have $\infty^{25}$ 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 $p=4$,\linebreak[2] $p^{(1)}=8$) 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\footnote{A. Franchetta, Su alcuni
esempi di superficie canoniche, Sem.\ Mat.\ di Roma, 1939.}, 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 $\infty^6$)
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
$C_{11}(A^5B_1^3\dots B_9^3D_1\dots D_8)$ where the 18 points $A$, $B$ and
$D$ belong to a fundamental curve of order seven $K_7(A^3B_1^2\dots
B_9^2D_1\dots D_8)$; the $C_{11}$ are curves of genus 8 and the
characteristic series of the $\infty^3$ 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) $\infty^9$ surfaces $F'_7$ with
the same double curve $C_7$, hence -- since there are $\infty^{19}$
$F'_7$ -- we have (at least) $\infty^{10}$ 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 $\infty^{25}$ degenerate $F_7$ formed by the quadric cone
containing $C_7$ and by a surface $F_5$ through the curve $C_7$, we obtain
$\infty^{26}$ canonical surfaces $F_7$ which pass doubly through the
given $C_7$. With this argument we have proved that: {\em there exist
$\infty^{36}$ projectively distinct canonical surfaces of the 7th order
$F_7$, whose double curves are the $\infty^{10}$ 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 $\infty^{26}$ and to possess a
triple point imposes {\em one} condition to them, as it is a priori
apparent from the fact that the $g^3_7$ admitting a neutral triple over a
hyperelliptic curve of genus 4 are defined by the condition of
containing the double of the $g^1_2$ and thus they depend on three
parameters.

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 $\infty^{33}$ 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 $\infty^{34}$ 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 $g^1_2$),
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 $\al$. Such a curve projects from
$O$ down to a curve composed of the triple line $a$, trace of the plane
$\al$, 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 {\em
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) {\em 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 $\ga$ of this, a
tacnode $O$ whose tacnodal plane $\al$, different from $\ga$, 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 $\al$ also counted
twice, and a $F_6$ formed by the plane $\ga$ 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 $\al$.

The $F_6$ thus constructed has in fact a fixed component of the
canonical curves, that is, the line $a$ along which the plane $\al$ touches
the $F_6$ and which thus appears twice in the intersections of the $F_6$
with the $\infty^3$ adjoint quadrics, formed by a variable plane and by
the fixed plane $\al$. Denoting by $\rho=0$ and by $-\nu$ the genus and
the degree of this line $a$, the genus and the degree of $2a$ are
respectively given by:
 \[
 2\cdot0-\nu-1=-(\nu+1)
 \quad\hbox{and}\quad
 -4\nu;
 \]
hence one can compute the degree $\pbar^{(2)}$ and the (virtual) genus
$\pbar^{(1)}$ of the impure canonical system on $F_6$ (the latter is its
relative genus):
 \begin{eqnarray*}
 & \pbar^{(2)}=6-4\nu+4=10-4\nu \\
 & \pbar^{(1)}=8-(\nu+1)+2-1=8-\nu \quad (\hbox{with }p^{(1)}=8);
 \end{eqnarray*}
and from the relation
 \[
\pbar^{(2)}=\pbar^{(1)}-1
 \]
one obtains
 \[
10-4\nu=7-\nu,\quad\nu=1.
 \]

Thus it remains proved a posteriori that the line $a$ is actually an
exceptional line for $F_6$ ($-\nu=-1$), therefore because of its
presence the relative linear genus becomes
 \[
\pbar^{(1)}=7=p^{(1)}-1
 \]
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\footnote{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.} 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 $\al$ depend on 37 parameters; hence the $F_6$
with a double conic, a tacnode and an assigned tacnodal plane are
$\infty^{38}$.

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
$\infty^3$ homologies, we see that: {\em 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 $\ga$ 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 $\ga$; 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 {\em 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 $F_7$.}

 \begin{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 {\em 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
 \[
 p=4,\quad p^{(1)}=8.
 \]

We can now verify that {\em 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
 \[
 F_3^2+\la F_6=0.
 \]

In this pencil we see that, for $\la=0$, 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.
 \end{NOTE}

A second family of multiple canonical surfaces is the family of {\em
double quadrics} which can be reduced to the type of {\em 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 $AA_1D$;
hence
 \[
 p=4 \quad\hbox{and}\quad p^{(1)}=8,
 \]
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 $<p^{(1)}$.

 \section{Surfaces with $p=4$ and $p^{(1)}=9$}\label{sec:8.5}
 \markright{\hfill Surfaces with $p=4$ and $p^{(1)}=9$ \quad 8.5\quad}
 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 $\Phi_3$ having 4 double points
there. One has to consider the $F_9$ passing triply through the 4 double
points of $\Phi_3$, and then take the system of halves of the linear
system of curves cut out by them on the $\Phi_3$.

We appeal to the usual representation of the $\Phi_3$ on a plane, where
the images of the plane sections are the curves $C_3$ passing through
three pairs of infinitely near points, $PP_1,QQ_1$ and $RR_1$, lying on a
conic; then the half system we are looking for is given by the curves of
the 9th order,
 \[
 C_9(P^4P_1 Q^4Q_1 R^4R_1).
 \]

These are $\infty^{21}$ curves $C_{12}$ of the 12th order and of genus 10
on the $\Phi_3$, 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
$\infty^{13}$ surfaces $F_8$, since splitting off the adjoint $\Phi_3$
there remain $\infty^{12}$ surfaces of the 5th order $F_5$ passing through
$C_{12}$. Since the cubic surfaces with 4 double points are
$\infty^{12}$, 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 $\infty^{13}$)
irreducible $F_8$, we will have $\infty^{34}$ 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~\ref{sec:5.13}, 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
$\infty^{13}$ 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 {\em 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 $\Phi_3$ 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 $\infty^3$ 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 $C_{11}(A_1^3\dots
A_{12}^3B_1\dots B_5)$ where the 17 points $A$ and $B$ belong to a
fundamental curve of the 7th order $K_7(A_1^2\dots A_{12}^2B_1\dots B_5)$.

After having thus proved that there do exist irreducible canonical
surfaces $F_8$, we are furthermore lead to admit that the {\em number of
moduli}\index{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.

 \begin{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 {\em a double plane with a
branch curves $C_{14}$ of order 14, with a 6-tuple point $O$ and with eight
4-tuple points $A_1,A_2,\dots,A_8$.} Here the (double) images of the
canonical curves are the quartics through $O^4A_1 A_2\dots A_8$, whence
one has indeed
 \[
 p=4\quad\hbox{and}\quad p^{(1)}=9.
 \]
 \end{NOTE}

It is easy to verify that the stated double planes, or rather the
corresponding $F^2_4$ 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 $F^2_4$ belong, which is certainly
 \[
M\ge 9p-2p^{(1)}+12=30.
 \]

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
{\em a double plane with branch curves $C_{12}$ with four 4-tuple points,
$A_1A_2A_3A_4$, and two $[3,3]$ points: $B_1$ and $B_2$.} Here the images
of the canonical curves $K$ are given by the cubics through
$A_1A_2A_3A_4B_1B_2$; 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.

 \section{Surfaces with $p=4$ and $p^{(1)}>9$}\label{sec:8.6}
 \markright{\hfill Surfaces with $p=4$ and $p^{(1)}>9$ \quad 8.6\quad}
 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 $\Phi_{n-5}$ (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 $\Phi_4$ of order 4, with
 \[
 t=10
 \]
double points, and the matter is to bisect the linear system cut on
the $\Phi_4$ by the surfaces $F_9$ of order 9 which pass triply through
these 10 double points.

The problem which presents itself for $p^{(1)}=10,11$,\dots suggests the
following remarks:

 \begin{enumerate}
 \renewcommand{\labelenumi}{(\arabic{enumi})}
 \item One cannot in general say that the bisection of the system cut
on a $\Phi_{n-5}$ 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.

 \item 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 $F_n$.

 \item {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 $n=10$.

 \item 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
$\Phi_{n-5}$) 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
 \[
 nm+s=p^{(1)}-1.
 \]
 \end{enumerate}

 \section{Canonical surfaces with $p=5$ and $p^{(1)}=9$}\label{sec:8.7}
 \markright{\hfill Canonical surfaces with $p=5$ and $p^{(1)}=9$ \quad
8.7\quad}
 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 $\Phi_3$. When $p=4$, the $\Phi_3$ 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 $\Phi_3$ 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 $\Phi_3$, 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 $\Phi_3$ with surfaces of the 4th order
$F_4$; and each of the $C_{12}$ is effectively double for $\infty^4$
surfaces $F'_8$. Each of these $F'_8$, possessing $\infty^4$ adjoint
surfaces $\Phi_4$ (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 $V^2_3$ and of a variety
$V^4_3$ of the 4th order. In fact, the system of surfaces $\Phi_4$
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 $\Phi_3$ 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
$V^2_3$. Let us add that there are $\infty^{15}$ varieties of the 4th
order $V^4_3$ 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: {\em the surfaces of genus $p=5$ and absolute linear
genus $p^{(1)}=9$ {\em (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 {\em number of
moduli}\index{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 $V^2_3$ (represented on $S_3$ by the
surfaces of the 2nd order through $C$) and of a variety $V^4_3$ 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 $C$.

 \section{Surfaces with $p=5$ and $p^{(1)}=10$}\label{sec:8.8}
 \markright{\hfill Surfaces with $p=5$ and $p^{(1)}=10$ \quad 8.8\quad}
 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 $A_1A_2\dots A_5$, 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 $\Phi_2$ ($=\Phi_{n-4}$) adjoint to the
$F_6$ ($=F_n$), cutting out the canonical curves on $F_6$. Now, the
system of such $\Phi_2$ 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
$A_1A_2\dots A_5$. 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 $A_1A_2\dots A_5$. 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 $\infty^{34}$ 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
 \[
 2\cdot34-2-24=42
 \]
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 $\Phi_4$ 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,
$B_1B_2\dots B_6$, corresponding to the 6 lines through $O$ of the cubic
variety passing through $F_9$ and $O$, and\footnote{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 $O$.}
therefore they belong to a conic\footnote{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 $\Fie_4$ 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
$\infty^{28}$ continuous systems, one of which is indeed formed by the
$\Fie_4$ with 6 double points lying on a conic, which cannot be extended
to a larger system. One can immediately see that there do exist $\Fie_4$
with 6 double points not belonging to a conic, since one can assign 5
double points in general position, $A_1,A_2,\dots,A_5$, and then impose a
double point (not at a prescribed point) to the above system of $\Fie_4$,
of dimension 34-20=14: we obtain in this way $\infty^{28}$ $\Fie_4$ with
6 double points not belonging to a plane. But, on the other hand, if we
impose to a $\Fie_4$ to possess a plane tangent along a conic $C$, the
$\infty^4$ sextics cut out on $\Fie_4$ by the quadrics through $C$ (among
which one finds $\infty^2$ 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 $\Fie_4$. It is now easy to see that imposing a tangent
plane along a conic implies in fact 6 conditions for $\Fie_4$ (these can
be expressed by requiring that $\Fie_4$ contains a conic which is self
residual with respect to the system of plane sections): whence the
$\Fie_4$ with 6 double points lying on a conic form themselves too a
continuous system $\infty^{28}$ of surfaces of the 4th order with 6
double points.

We believe that on a surface $\Fie_4$ with 6 double points not belonging to
a conic, it is not 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 $\Fie_4$ on a double
plane, via a projection from one of its double points.} $C_2$.

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
$\Phi_4$ by the surfaces of the 9th order passing triply through the 6
points $B_1,B_2,\dots,B_6$, which are double for the aforementioned
$\Phi_4$. 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 $\Phi_4$ by the general surfaces of the 4th order passing
through $B_1,B_2,\dots,B_6$, and of the conic $C_2$ along which the
plane of $C_2$ touches $\Phi_4$.

$\Phi_4$ 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 $\pi$, if it is free from base points (at simple points of the
surface). Therefore the system $|C_{16}|$, of degree $64-2\cdot6=52$,
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 $B_1,B_2,\dots,B_6$) at 2 points, we obtain $C_{18}$ of genus
 \[
 \pi=27+0+2-1=28.
 \]

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 $\Phi_4$ with 6 double points
$B_1,B_2,\dots,B_6$ belonging to a conic, $\infty^{28}$ curves $C_{18}$,
in general irreducible, having triple points at $B_1,B_2,\dots,B_6$.
These are a priori ``possible'' double curves for nonnormal canonical
surfaces $F'_8$. Let us assume that, among these $C_{18}$, there are
$\infty^x$ (with $x\le28$) 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 $\infty^5$ irreducible $F'_9$,
since there exist $\infty^4$ $F'_9$ which are reducible into the
$\Phi_4$ and a $F_5$ through it. We therefore have
$\infty^{x+5}$ $F'_9$ with a double curve $C_{18}$ belonging to a given $\Phi_4$ (endowed with 6 double points lying on a conic). But there are
$\infty^{14}$ $\Phi_4$ with the same 6 double points and, by varying these,
we obtain $\infty^{28}$ $\Phi_4$: whence we have $\infty^{28+x+5}$
$F'_9$ in the space $S_3$. Among these there are $\infty^{15}$ which
are projectively equivalent to a given one, and $\infty^{19}$ which are
birationally equivalent, and corresponding to projections of $F_9$ from
$\infty^4$ 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
 \[
 M=42,
 \]
we infer
 \[
 x=28.
 \]
This means that
 \begin{proc}
 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.
 \end{proc}

We end proposing to the researcher some open problems.

 \begin{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.
 \end{Questions}

 \begin{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.
 \end{NOTE}

 \section{Surfaces on a variety of dimension 3 or more}\label{sec:8.9}
 \markright{\hfill Surfaces on a variety of dimension 3 or more \quad
8.9 \quad}
 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:
 \begin{enumerate}
 \renewcommand{\labelenumi}{(\arabic{enumi})}
 \item 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 {\it Math.\ Ann.}, Band~8;
 \item 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;
 \item completely new and unexpected properties which do not have any
analogue in the theory of surfaces\footnote{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.}
 \end{enumerate}

 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
 \[
 |F_j-4F|
 \]
defines the canonical system, effective or virtual, of $V$\footnote{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).}, 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 $\Ga$, 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 $\Phi_{n-5}$ of order
$n-5$, which are simply defined by the condition of passing through the
double surface $\Ga$. Instead, the adjoint $\Phi_{n-4}$ 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
$\Phi$ 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:

 \begin{enumerate}
 \renewcommand{\labelenumi}{(\arabic{enumi})}
 \item 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 $V$;

 \item 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 $V$.
 \end{enumerate}

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
 \[
 2 x+y-1=p^{(1)}
 \quad
 4y=p^{(1)}-1=p^{(2)},
 \]
 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 $p_a$\footnote{Cf.\ Severi, Ibidem.}.

{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
$m+n-5$\footnote{Cf.\ Enriques--Chisini, Ibidem, Libro~V, Chapter~V, \S41
(Vol.~III, page~528).}; we have here an extension of the well-known
theorem of Noether concerning skew curves\footnote{Cf.\ Severi, Su alcune
questioni di postulazione, {\it Rendic.\ Circ.\ Mat.\ di Palermo},
Vol.~XVII, 1903.}.

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
$n_1,n_2,\dots,n_{r-2}$, 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 $n_1+n_2+\cdots+n_{r-2}-r-1$ passing
through $F^x$.

 \section{Examples of canonical surfaces in hyperspace}\label{sec:8.10}
 \markright{\hfill Examples of canonical surfaces in hyperspace \quad
8.10\quad}
 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 $V^4_3$, 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 $n_1,n_2,\dots,n_{r-2}$, when these orders ($\ge2$)
are such that one has
 \[
n_1+n_2+\cdots+n_{r-2}-r-1=1,
 \]
that is,
 \[
n_1+n_2+\cdots+n_{r-2}=r+2.
 \]

Taking into account that we must have $n_i\ge2$, 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 $V^n_3$ of the space $S_r$, with $n=2r-4$, with projectively
canonical curve sections\footnote{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.}, 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 $V^n_3$ of the space $S_{n+1}$ ($n>3$) with
elliptic curve sections\footnote{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. {\it Rendic.\ Acc.\ Lincei}, may-june 1894. Cf.\
{\it Math.\ Ann.}, Band~46.}, 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
 \[
 |(3F)'|=F,
 \]
 so that the surfaces $F$, and also the hyperplanes of the space $S_{n+1}$, cut the intersections of
the $V^n_3$ with a cubic variety along their canonical curves.

Summing up, we have obtained two series of canonical surfaces of genus
$p$ in the space $S_{p-1}$:

 \begin{enumerate}
 \renewcommand{\labelenumi}{(\arabic{enumi})}
 \item 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 $V^{p-2}_3$ having elliptic curve sections; and

 \item 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 $V^{2p-6}_3$ having projectively canonical curve sections.
 \end{enumerate}

We shall see later on other classes of canonical surfaces defined on
3-dimensional 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 variet\`a a tre dimensioni a curve
sezioni canoniche, {\it Atti Acc.\ Lincei}, 1936; Su alcune variet\`a
algebriche a tre dimensioni aventi curve sezioni canoniche in {\it Scritti
matematici offerti a L.~Berzolari}, page~329 (1936); Sulle variet\`a
algebriche a tre dimensioni a curve sezioni canoniche, {\it Mem.\ Acc.\
d'Italia}, 1937.

L.~Godeaux, Sur la construction d'une surface canonique, {\it Bull.\
Soc.\ Math.\ Li\`ege}, 5, (1936); Sur une surface canonique appartenant
\`a la vari\'et\'e de Segre representant les couples de points de deux
plans, {\it Bull. de l'Acad\'emie Belgique}, 22, (1936); Sur les
surfaces alg\'ebriques dont les sections hyperplanes sont des courbes
canoniques, I, II, III, {\it Bull.\ Soc.\ Math.\ Li\`ege}, 6,
(1937); Construction d'une surface canonique du septi\`eme ordre,
{\it Bull. de la Soc.\ des Sciences de Li\`ege}, 1944; Construction
d'une surface canonique du huiti\`eme ordre, {\it Bull.\ des Sciences
Math.}, 1944; Construction d'une surface canonique du neuvi\`eme ordre,
{\it Bull.\ de l'Acad.\ de Belgique}, 1944. P.~Burniat, Note sur
quelques surfaces canoniques et sur des surfaces de genre un, {\it Bull.
de l'Acad. de Belgique}, 5, 22, (1936); Note sur quelques surfaces
canoniques, {\it Acad.\ de Belgique}, 3, 22, (1936); Sur des
hypersurfaces et vari\'et\'es canoniques, {\it Acad.\ de Belgique} (5),
23, (1937).

E. A.~Maxwell, Regular canonical surfaces of genus three and four, Proc.\
Cambridge Phil.\ Soc., 33, (1937).

 \section{Minimal value of the linear genus with respect to the
superficial genus}\label{sec:8.11}
 \markright{\hfill Minimal value of the linear genus with respect to the
superficial genus \quad 8.11\quad}
 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
 \[
 p^{(1)}\ge2p-3,
 \]
or, equivalently,
 \[
 p\le(p^{(1)}+3) /2:
 \]
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 {\it 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
$g^r_n$ on an arbitrary curve is $n\ge2r$\footnote{Cf.\ Enriques--Chisini,
Ibidem, Libro~V, Chapter~I, \S11 (Vol.~III, page~87).}. 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
 \[
 p^{(1)}-1\ge2p-4.
 \]

But the same inequality holds, however reducible the canonical curves $K$ might be, provided the variable part splits into curves of genus
$\pi\ge3$. Firstly, if their variable components $k$ are irreducible,
their degree is $n\le p^{(1)}-1$ (Chapter~IV, Section~\ref{sec:4.18}),
whereas the dimension of $|k|$ is the same of $|K|$, that is, $p-1$; we
have therefore
 \[
n\ge2p-4
 \]
 and a fortiori
 \[
 p^{(1)}-1\ge2p-4
 \quad
 p^{(1)}\ge2p-3.
 \]
 If instead the variable curves $k$ are assumed to be composed with
$p-1$ curves of a pencil of genus $\pi\ge3$, the genus of $|k|$ ($\le
p^{(1)}$) will be
 \[
 \ge(p-1)(\pi-1)+1;
 \]
 we infer
 \[
 p^{(1)}\ge2p-1.
 \]
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 $g^r_n$ on a curve $K$ ($n=p^{(1)}-1$, $r=p-2$) has to have $n=2r$,
and this implies indeed that $K$ is hyperelliptic\footnote{Cf.\
Enriques--Chisini, Ibidem, Libro~V, Chapter~I, \S12, Vol.~III, page~97.}.
If instead the curves $K$ were reducible, the argument could be repeated
for their variable components, which are certainly irreducible
themselves (otherwise $p^{(1)}\ge2p-1$): 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
 \[
 p^{(1)}\ge2p-3+r.
 \]
 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\footnote{Cf.\ Enriques--Chisini, Libro~V, Chapter~II,
\S22, Vol.~III, page~194.}:

 \begin{enumerate}
 \renewcommand{\labelenumi}{(\Roman{enumi})}
 \item net of the lines,

 \item $\infty^5$ system of the conics,

 \item 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 $O$.
 \end{enumerate}

The double planes whose canonical systems have such systems of
rational curves as images are respectively:
  \begin{enumerate}
 \renewcommand{\labelenumi}{(\Roman{enumi})}
 \item the double plane with a branch curve $C_8$ of the 8th order: $p=3$,
$p^{(1)}=3$;

 \item the double plane with a branch curve $C_{10}$ of the 10th order:
$p=7$, $p^{(1)}=9$;

 \item the double plane with a branch curve $C_{2n+6}$, of order $2n+6$,
having a $2n$-tuple point $O$: $p=2n+1$, $p^{(1)}=4n-2$; and

 \item[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 $p=2n$, $p^{(1)}=4n-4$.
 \end{enumerate}

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$'$)\footnote{Cf.\
Enriques, Sopra le superficie algebriche di cui le curve canoniche sono
iperellittiche {\it Rendic.\ Acc.\ Lincei}, s.~V, Vol.~V, page~191
(1896).}.

We may replace Noether's inequality
 \[
 p\le(p^{(1)}+3) /2
 \]
with a stronger one, valid for surfaces with a simple canonical system
($p>3$), which has been pointed out by
Castelnuovo\footnote{G.~Castelnuovo, Osservazioni intorno alla
geometria sopra una superficie. Nota II. ({\it Rendic.\ Ist.\ Lombardo},
s.~II, Vol.~XXIV, 1891). {\it Memorie scelte}, no.~XVIII.}. 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
 \[
 p\le(p^{(1)}+6)/3, \quad p^{(1)}\ge3p-6.
 \]
In order to prove this inequality, it is convenient to start from a
lemma of Bertini concerning the autoresidual $g^r_n$ which belong to a
curve of genus $\pi$. Let us consider, on a curve which contains such a
$g^r_n$, two groups of points $G$ and $G'$ of the series, having in common
a group $\Ga$ 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 $\Ga$, and the group $\Ga$ cannot belong to
a linear series $g^s_{r-1}$ of dimension $s>0$. With these assumptions,
the theorem of Riemann--Roch says that the group G of the $g^r_n$
imposes $n-r$ conditions to the groups of the canonical series in order
to contain it\footnote{Cf.\ e.g., Enriques--Chisini, Ibidem, Libro~V,
Chapter~I, Vol.~III, page~85.}. 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 $G+G'-\Ga$ is less than or equal to
 \[
 n-r+(n-2r+1)=2n-3r+1.
 \]

Now, the canonical groups satisfying these conditions cannot be more
than one, that is, they coincide with $G+G'$, whence the points of $\Ga$
are double; since otherwise $\Ga$ (which is formed by $r-1$ generic
points of the curve, where $r<\pi$) should belong to a series
$g^s_{r-1}$, with
$s>0$. We infer that
 \[
 2n-3r+1\ge\pi-1,
 \]
$\pi-1$ being the number of conditions which determine the group G+G'
inside the canonical series.

But, since the $g^r_n$ is autoresidual,
 \[
n=\pi-1,
 \]
hence
 \[
 \pi-3 r\ge0.
 \]

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
 \[
n=\pi-1=p^{(1)}-1; r=p-2,
 \]
 and therefore
 \[
 p^{(1)}-3p+6\ge0, \qed
 \]

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:

 \begin{enumerate}
 \renewcommand{\labelenumi}{(\arabic{enumi})}
 \item the surface of the 5th order in $S_3$, for which $p=4$ and $p^{(1)}=6$;

 \item 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 $p^{(1)}=9$;

 \item 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:
 \[
 p=7,\quad p^{(1)}=15.
 \]
 \end{enumerate}

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 $V^{p-3}_3$, described by a simply infinite
system of planes in $S_{p-1}$, and of order $p-3$, with a hypersurface
$V^4_{p-2}$, of the 4th order passing through $p-5$ planes of the
$V^{p-3}_3$.

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
$V^{p-3}_3$ with rational hyperplane sections\footnote{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.\ {\it Math.\ Ann.}, Band~46.} is represented on
the space via projection from $p-4$ of its points: the representative
system is formed by $\infty^{p-1}$ 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 $\Phi_{2p-7}$
to the $F_{2p-3}$ split into the $p-4$ planes $rt$ and into a residual
surface $\Phi_{p-3}$, 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, \qed

 \section{Upper bound for the linear genus $p^{(1)}$ in terms of the
superficial genus}\label{sec:8.12}
 \markright{\hfill Upper bound for the linear genus $p^{(1)}$ in terms of the
superficial genus \quad 8.12\quad}
 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\footnote{A.~Rosenblatt, Sur quelques
in\'egalit\'es etc., {\it Comptes Rendus Acad.\ des Sciences}, Paris,
t.~154 (1912).}:
 \[
 p^{(1)}\le16p_a+27.
 \]

More recently, $F$.Severi\footnote{{\it Rendic.\ Circ.\ Mat.\ di Palermo},
t.~LVI, page~79 (1932).}, relying on a transcendental result by HODGE, has
obtained the inequality:
 \[
 p^{(1)}\le2p_g+8p_a+11-\rho,
 \]
where $\rho$ 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
 \[
M=10p-2p^{(1)}+12+\om,
 \]
one has to investigate the upper bound, and possibly the maximum
effectively reached, for the superabundance $\om$. 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.

 \section{Canonical systems belonging to an involution}\label{sec:8.13}
 \markright{\hfill Canonical systems belonging to an involution \quad
8.13\quad}
 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 $n\ge2$,
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
 \[
 p^{(1)}\ge 3p-6.
 \]

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 $C_{2\nu}$ free from
singularities, for which
 \[
 p=n (n-3)/2+1,\quad p^{(1)}=2 (n-3)^2+1
 \]
 and the ratio $p^{(1)}/p$ tends to 4 for $n=\infty$.

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 $\Phi_5$ of the 5th order, having 20 conical double
points. In the pencil
 \[
 \la F_{10}+\Phi_5^2=0
 \]
one will see the surface $F_{10}$ tend by continuity to the $\Phi_5^2$,
on which there is no proper branch curve, but $t=20$ conical points giving
rise to as many infinitesimal branch curves. Thus the quintic $\Phi_5$,
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\footnote{B.
Segre, Sulle superficie algebriche aventi il sistema canonico composto
con un'involuzione, {\it Rendic.\ Acc.\ Lincei}, s.~VI, Vol.~XVI, page~316
(1932).}. 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 $\infty^2$
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).

 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,
 \[
 p=4 \quad\hbox{and}\quad p^{(1)}=7.
 \]

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 $g^1_3$, 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 {\em determining all the surfaces (with $p>2$) whose
canonical curves (irreducible and without base points) contain a $g^1_3$,
which are therefore transformed into triple canonical surfaces with
rational sections}, was recently solved by
G.~Pompilj\footnote{G.~Pompilj, Sulle superficie algebriche le cui curve
canoniche posseggono una $g^1_3$, {\it Rendic.\ Ist.\ Lombardo}, 27 marzo
1941.}. There are three types of canonical surfaces corresponding to
these conditions:
 \begin{enumerate}
 \renewcommand{\labelenumi}{(\arabic{enumi})}
 \item a triple plane with branch curve $C_{12}$ of order 12 (already
mentioned in the {\it Lezioni} by Enriques--Campedelli)\footnote{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 $F_6,F_7,F_8,\dots$ 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 $p^{(1)}=5, 6,\dots$

Letting instead the above canonical surfaces acquire two triple points,
one obtains surfaces with $p=2$ and linear genus $p^{(1)}=2, 3,\dots$,
where the pencil of canonical curves possesses $p^{(1)}-1$ base points.},
which has
 \[
 p=3 \quad\hbox{and}\quad p^{(1)}=4;
 \]

 \item a triple quadric cone with branch curve $C_{18}$ of order 18, for
which
 \[
 p=4 \quad \hbox{and} \quad p^{(1)}=7,
 \]
in this type is included (as a particular case?) the example of B.~Segre;

 \item a triple rational normal cubic cone in $S_4$, with branch curve
$C_{24}$ of order 24 for which
 \[
 p=5 \quad \hbox{and} \quad p^{(1)}=10.
 \]
 \end{enumerate}

These three types are represented analytically by the following
equations:

 \begin{enumerate}
 \renewcommand{\labelenumi}{(\arabic{enumi})}
 \item 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
 \[
4p_4^3+q_6^2=0;
 \]

 \item 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
 \[
 4p_6^3+q_9^2=0;
 \]

 \item 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 $p_8(O^5O_1^2O_2^2)$ and
$q_{12}(O^8O_1^4O_2^4)$; the branch curve is
 \[
 4p_8^3+q_{12}^2=0.
 \]
 \end{enumerate}

We shall not dwell on Pompilj's analysis, based on the theory of triple
planes established by himself\footnote{G.~Pompilj, Sulla rappresentazione
algebrica dei piani tripli. Sem.\ Mat.\ Universit\`a di Roma, s.~4,
Vol.~III. Osservazioni sui piani tripli, {\it Rendic.\ Ist.\ Lombardo}, 27
febbraio 1941.}, which can also be carried on in a different
way\footnote{Cf.\ A.~Franchetta, Sulle superficie le cui curve canoniche
posseggono una $g^1_3$. Boll.\ U.M.I., 1942.}. Rather we would like to
indicate how to justify Castelnuovo's remark to the effect that {\em if
the canonical curves of a surface contain a $g^1_3$, then the
characteristic series is necessarily composed with it.}

In order to prove the assertion it suffices to consider a curve of genus
$\pi>3$, containing a $g^1_3$ and an autoresidual series of dimension
$r\ge1$, inside which we may choose a $g^1_n$, with $n=\pi-1$: we shall
derive a contradiction from the assumption that the $g^1_n$ is not
composed with the $g^1_3$ (away from the fixed points). Indeed, if the
$g^1_n$ is not composed with the $g^1_3$ we can transform the given curve
into a $C_m$ of order $m=n+3$, on which both the $g^1_n$ and the
$g^1_3$ 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.

 \section{Surfaces with linear genus $p^{(1)}=2$: first case
$p=2$}\label{sec:8.14}
 \markright{\hfill Surfaces with linear genus $p^{(1)}=2$: first case
$p=2$ \quad 8.14\quad}
 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:
 \[
 p\le2;
 \]
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:
 \[
 p=2,\quad p=1,\quad p=0.
 \]
And, corresponding to the three values of $p$ above, the bigenus will
respectively be
 \[
P=p+p^{(1)}=4, 3, 2,
 \]
and the trigenus
 \[
P_3=p+3p^{(1)}-2=6, 5, 4.
 \]

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 $\infty^3$ of
bicanonical curves, of genus $3p^{(1)}-2=4$, cutting the $K$ along the
$g^1_2$ 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 $g^1_2$.

Summing up:
 \begin{proc}
 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.
 \end{proc}

 \begin{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 $\theta$: $K=nk+\sum\theta$. But, in any case, the
curves $k$ could not have genus $\pi=0$ or $\pi=1$, 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~\ref{sec:4.18}), will have genus equal to 2.

Now, the $\infty^3$ 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.
 \end{Remark}

 \section{Surfaces with $p^{(1)}=2$ and $p=1$ ($P=3$,
$P_3=5$)}\label{sec:8.15}
 \markright{\hfill Surfaces with $p^{(1)}=2$ and $p=1$ ($P=3$, $P_3=5$)
\quad 8.15\quad}
 Here there is a canonical curve $K$ of genus $p^{(1)}=2$ and $\infty^2$
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 $g^1_2$, would not be complete, as required by the
regularity of the given surface $F$. With this in mind, we have an $\infty^4$ 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 $g^1_3$. {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\footnote{Otherwise $O$ would be a triple point.} 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 $\infty^7$
(here $7=p_a+\pi-1$) 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 $\infty^2$ 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:
 \[
 6=9(p^{(1)}-1)-3=9-3\footnote{The classification of the surfaces
with $p^{(1)}=2$, $p=1,2$ is given in the Note by F.~Enriques, Le
superficie algebriche di genere lineare $p^{(1)}=2$. {\it Rendic.\ Acc.\
Lincei}, February 1897.}.
 \]

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
$\infty^3$ 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 $\infty^7$ system of adjoint cubic
surfaces $\Phi_3$, 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 $\Phi_3$ 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 $\Phi_3$ 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.

\begin{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
 \[
 |2K|=|k_2|+\sum\theta
 \]
having some fixed component $\theta$, and whose variable parts are
irreducible curves $k_2$ (cf.\ Chapter~IV, Section~\ref{sec:4.19}). We
denote by $\pi$ and $n$ the genus and the degree of a curve $k_2$, and
respectively by $\rho$ and $\nu$ the genus and the degree of some $\rho$.
Then the additive character expressing the virtual number of
intersections of the bicanonical curves with the canonical ones is given
by
 \[
 2 p^{(1)}-2=2=2\pi-2-n+\sum (2\rho -2-\nu).
 \]
And, assuming the surface to be free from exceptional curves (thus,
for $\rho=0$, $\nu\le-2$), we have for all the components $\theta$:
 \[
 2\rho-2-\nu\ge0,
 \]
hence
 \[
2\pi-2-n\le2.
 \]
Now, if some $\theta$ does actually exist, in order that $K$, and
therefore also the curves of $|2K|$, are connected with order of
connectivity $\ge2$, the number of intersections of some $k_2$ with the
$\sum\theta$ (residual with respect to the bicanonical system) has to be
 \[
 k_2\cdot\sum\theta=4\pi-4-3n\ge2.
 \]

Whence it follows that
 \[
4\pi-4-2n=(4\pi-4-3n)+n\le4, \quad\hbox{for $n\le2$},
 \]
and therefore $n=2$ (since $n>1$),
 \[
2\pi-2\le n+2=4;
 \]
that is, the net $|k_2|$ consists of curves of genus $\pi\le3$, and of
degree $n=2$. But this conclusion is absurd since (in the most
favourable condition: $\pi=3$) 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.
 \end{Remark}

 \section{Surfaces with $p^{(1)}=2$ and $p=0$ ($P=2$,
$P_3=4$)}\label{sec:8.16}
 \markright{\hfill Surfaces with $p^{(1)}=2$ and $p=0$ ($P=2$,
$P_3=4$) \quad 8.16\quad}
 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 $\Phi_5$, but is nevertheless a double curve for $\infty^4$
biadjoint surfaces $\Phi_{10}$ 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
 \[
 p=0 \quad\hbox{and}\quad p^{(1)}=2,
 \]
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 $C_{10}$:
 \[
4=65-8-5\cdot 9-8;
 \]
instead, the number of moduli of the family of surfaces $F_9$, with
$p=0$ and $p^{(1)}=2$, is certainly
 \[
 M\ge 9p-2 p^{(1)}+12=8.
 \]

 \begin{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 $\theta$, which is
fundamental for the pencil $|K_2|$. Then the tricanonical system, once
deprived of $\theta$, will be the adjoint to the curve $K_2-\theta$,
whence $(K_2-|\theta|)'$ would have the same dimension as $|K_2'|$,
that is, dimension 3: now our aim is to see that this is impossible since
$K_2-\theta$ is a connected curve of genus $<4$ (cf.\ Chapter~IV,
Section~\ref{sec:4.17}). 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 $\theta$ from
$|K_2|$ leads necessarily to a connected curve of genus $\le2$.

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,\dots 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 $\pi<4$; but, if $\pi<3$, then all
the plurigenera of $F$ turn out to be zero; if instead $\pi=3$, then the
bicanonical curves of $F$, which-by assumption-form a pencil, would cut
on a curve $C$ the pairs of a $g^1_2$, and this is incompatible with the
fact that the $C$ contain a $g^1_3$ without fixed points.

With this in mind, since the curves $C$ and $L$ have genus 4, we see that
the characteristic $g^1_3$ 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 $g^1_3$ (which are $\infty^1$) 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 $g^1_3$, thus they are partly
contained in the double system $|2C|$, and similarly in $|2L|$.
 \end{Remark}

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 $\al$: the
images of the plane sections of $F_3$ are the cubics through six points
$A_1A_2\dots A_6$, not belonging to a conic; the curves of the minimum
double system $|2C|$ may be assumed to be represented by the conics of
$\al$; 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 $\al$, 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 $\theta$ on $F$ which is fundamental for the
mentioned system $|K_3|$. Whence we deduce that, subtracting the curve
$\theta$ from such a system, we get the adjoint to $|C|$:
 \[
 |C'|=|K_3'-\theta|, (|K_3-\theta|=|C|),
 \]
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,
 \[
 \pi=4 !
 \]

We have thus carried through the discussion: the triple bicanonical $F_3$ of genus
$p=0$ and $p^{(1)}=2$ do not exist. \qed

 \section{Surfaces with linear genus $p^{(1)}=3$: first
case $p=3$}\label{sec:8.17}
 \markright{\hfill Surfaces with linear genus $p^{(1)}=3$: first
case $p=3$ \quad 8.17\quad}
 \footnote{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 $p^{(1)}=3$, {\it Rendic.\ Acc.\
Lincei}, may 1897. In that note only the simple type for
$p^{(1)}=3$ and $p=2$, of Section~\ref{sec:8.18} is missing.} Let us
examine the (regular) surfaces with linear genus $p^{(1)}=3$. Their
superficial genus
 \[
 p\le(p^{(1)}+3)/2
 \]
cannot exceed the value 3 (cf.\ Section~\ref{sec:8.11}). 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
$\sum\theta$ and in a variable part $|k|$ does not lead to any new case.
Let us denote by $\pi$ and $n$ the characters of the $k$, and by $\rho$
and
$\nu$ those of a $\theta$. Calculating the additive character which
indicates the number of intersections of the canonical curves with
these curves, we have:
 \[
 2\pi-2-n+\sum (2\rho-2-\nu)=2.
 \]

Now, the number of the intersections of a (irreducible) $k$ with the
rest of the canonical curves, $\sum\theta$, is
 \[
 2\pi-2-2 n\ge0,
 \]
and the degree of the system $|k|$ is $n\ge2$, hence
 \[
 2\pi-2-n\ge2.
 \]

On the other hand, if we suppose to have eliminated any exceptional
curve from the surface, we have for each $\theta$
 \[
 2\rho-2-\nu\ge0;
 \]
as a consequence
 \[
 2\pi-2-n=2, 2\rho-2-\nu=0,
 \quad
 n=2 \quad\hbox{and}\pi=3.
 \]

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 $\theta$ turn out not to exist.

 \section{Surfaces with $p^{(1)}=3$ and $p=2$ ($P=5$)}\label{sec:8.18}
 \markright{\hfill Surfaces with $p^{(1)}=3$ and $p=2$ ($P=5$) \quad
8.18\quad}
 We assume that the canonical pencil $|K|$ is irreducible, disregarding
the possible exceptional fixed components corresponding to its base
points. Then also the $\infty^4$ bicanonical curves, cutting on each $K$
the complete canonical series $g^2_4$, 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 $\om=oo'$ counted
four times, by the plane $\ga$ 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 $\om=oo'$ counted twice);
and

 2) by the presence of the double conic $C_2'$ (whose plane $\ga$ 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
$\Phi_7$, $\Phi_7=\om^5\ga^2$, 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 $A_1A_2\dots A_8$. Here the
canonical curves are represented by cubics $C_3$ through $A_1A_2\dots
A_8$, 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
$\infty^4$ bicanonical curves whose images on the plane are curves of
order twelve passing four times through $A_1A_2\dots A_8$, and being
8-tangent to the branch curve $C_{12}$; and these curves represent a
complete $\infty^4$ linear system which is simple. Inside the above
complete system there are $\infty^3$ 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 $\al$, 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 $\al$. Conversely,
assume a quartic $F_4$ is given with a tangent plane $\al$ along a
conic $C$ and let us construct the surface $F=F_{42}$ by extracting the
$\nu$ $\al\be$ on $F_4$, in such a way as to possess a branch quartic
$C_4$, $\be=0$, to which the apparent critical conic $C$ has to be
added. It is easy to see that the $\infty^1$ sections with the planes
passing through the common line of the two planes $\al$ and $\be$ form
double quartics, images of reducible curves as the section with the
plane $\be$, and therefore -- if they are taken simply -- they represent
$\infty^1$ 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 $\be$) 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 $F_4$\footnote{Cf.\
F.~Enriques and F.~Conforto, {\it Le superficie razionali}, page~144.}
(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\footnote{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 $r$.}.

Here the canonical curves are represented by $\infty^1$ conics in a
pencil (each one to be taken as double with 6+2=8 branch points) and
the bicanonical curves by $\infty^4$ quartics with two double base
points and two contacts:
 \[
 p=2,\quad p^{(1)}=3.
 \]

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 $g^2_4$ 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 $g^1_2$ 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.

 \begin{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 $\infty^2$ $K$ and $p>2$.

Then we can assume that the $K$ possess, beyond the $k$, some fixed and
non-exceptional components $\theta$
 \[
 |K|=\sum\theta +|k|.
 \]

If we denote by $\pi$ and $n$ the characters of $|k|$ and by $\rho$ and
$\nu$ those of a $\theta$, we have
 \[
 2\pi-2-n+\sum (2\rho-2-\nu)=p^{(1)}-1=2,
 \]
while $\pi\ge2$.

Moreover, if we refer to a surface without exceptional curves, we
have for each $\theta$
 \[
 2\rho-2-\nu\ge0.
 \]

Since moreover a $k$ has $2\pi-2-2n$ intersections with the residual
curve $\sum\theta$, then:
 \[
 2\pi-2-2 n\ge0.
 \]

The above inequalities allow only three possibilities for the values
of $\pi$ and $n$:

1) $\pi=2,n=0$;

2) $\pi=2,n=1$;

3) $\pi=2,n=2$.

Assumption 1) can immediately ruled out. The adjoint system $|k'|$ to $|k|$ would be an $\infty^2$ system of curves of degree
 \[
0+2\cdot 2+2=6,
 \]
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) ($\pi=2$, $n=1$) 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) ($\pi=3$, $n=2$) 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: $|K|=|k|$.
 \end{Remark}

 \section{Surfaces with $p^{(1)}=3$ and $p=1$ ($P=4$)}\label{sec:8.19}
 \markright{\hfill Surfaces with $p^{(1)}=3$ and $p=1$ ($P=4$) \quad
8.19\quad}
 Assuming the canonical curve $K$ to be irreducible, the $\infty^2$
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 $\Phi_7$, of order 2.
8-9=7, which together with a plane forms a $\Phi_8$ cutting on $F_8$ a
bicanonical curve; actually, the $C_{14}$ is the complete intersection
of the $F_8$ and $\Phi_7$ ($7\cdot 8/4=14$). On the other hand, there
must exist an adjoint quartic $\Phi_4$, which cuts $F_8$, off $C_{14}$,
along the canonical $K$ of order 4 ($4\cdot8-2\cdot14=4$). With this
set-up, it is easy to characterize the curve $C_{14}$: it is, in general,
the double curve of the surface $\Phi_7$ 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 $\Phi_7$ of the 7th order with elliptic sections\footnote{}. In order to construct the surface $F_8$, it suffices to
combine linearly the adjoint quartic $\Phi_4$ to $\Phi_7$, counting it
twice, with the $\Phi_7$ 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 $\Phi_7$) has 12 projective invariants and is
a double curve for $\infty^4$ $F_8$, thus there are $\infty^{16}$
projectively distinct $F_8$: the number of moduli of the $F_8$ equals:
 \[
 M=16=10p-2 p^{(1)}+12.
 \]

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 $g^2_4$, 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
 \[
 \la F_8+F_4^2=0
 \]
we see the generic surface (that is, a $F_8$) to tend by continuity (for
$\la=0$) 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 $\Phi_7$, is an apparent
critical curve.

Now, the number of triple points of the $C_{14}$, double curve of $F_8$
(and also of $\Phi_7$), 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 $\al\Phi_7$, where $\al=0$ is the plane of the branch curve
$K$, and $\Phi_7$ the surface of the 7th order having the $C_{14}$ as
double curve (an apparent critical curve). One can replace this polynomial
$\al\Phi_7$ with another polynomial $\al F_7$, 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 $C_{14}\pm C_{4n}$. 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 $\nu\al F_3$, 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
 \[
4\cdot 7^2-2. 10,
 \]
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 $g^{11}_{14}$, 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
$g^7_{10}$, by subtracting also this curve we shall have an $\infty^3$
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 $\nu$
$\al F_3$, where $\al$ 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 $\infty^6$ 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 $g^{11}_{14}$ 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 $g^2_4$ 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 $F^2_4$ 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 $F^2_4$ 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 $F^2_4$, 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 $F^2_4$ 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: $A_1A_2A_3A_4$. Indeed on this double plane there is a
canonical curve $K$ of genus $p^{(1)}=3$, whose image is the conic through
$OA_1A_2A_3A_4$, and there is a system of bicanonical curves,
represented doubly by $\infty^3$ quartics $C_4$ with $O$ as a double point
and having a contact at $A_1A_2A_3A_4$. 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
 \[
13 < 9p-2 p^{(1)}+12=15.
 \]

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.
 \section{Surfaces with $p=0$ and $p^{(1)}=3$}\label{sec:8.20}
 \markright{\hfill Surfaces with $p=0$ and $p^{(1)}=3$ \quad 8.20\quad}
 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
 \[
 3<9p-2p^{(1)}+12=6.
 \]
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 $g^2_8$
composed with the $g^1_2$ belonging to the $K_2$, and therefore it would be
contained in a complete $g^4_8$; but (if the surface has to be regular)
this yields P=6 instead of P=3.

 \section{Simple and multiple pluricanonical surfaces}\label{sec:8.21}
 \markright{\hfill Simple and multiple pluricanonical surfaces \quad
8.21\quad}
 In the previous sections we have met several types of surfaces which
give rise to double bicanonical surfaces, in particular the following
general types:

 \begin{enumerate}
 \renewcommand{\labelenumi}{(\Roman{enumi})}
 \item For $p=3$ and $p^{(1)}=3$, the double plane with branch curve $C_8$ of
order 8;

 \item 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;

 \item 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.
 \end{enumerate}

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~\ref{sec:4.19}), 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 $\infty^2$, 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 $\infty^2$ lines of the plane;

2) the $\infty^5$ 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 $O_1O_2\dots O_s$,
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~\ref{sec:8.15}.

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~\ref{sec:8.17}.

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 $h_1h_2\dots h_s$ 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 $C_n+\sum
h_iOO_i$, of order $n+d=n+\sum h_i$, 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 $C_n+\sum h_i\cdot OO_i$), it is necessary to have near
to $O$ another $(2m-5)$-tuple point of $C_{2m}$; but one has
 \[
 2m-5+2m-5\le2 m,
 \]
whence
 \[
 m=5,\quad 2m=10
 \]
or (in particular)
 \[
 m=4, \quad 2m=8;
 \]
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 $\le9(p^{(1)}-1)$, 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~\ref{sec:8.16} for the
case $p^{(1)}=2$).
