 \chapter{Numerical invariants and multiple planes}\label{ch:5}
 \markboth{\qquad 5. \quad Numerical invariants and multiple planes
\hfill}{}

 \section{The Zeuthen--Segre invariant}\label{sec:5.1}
 \markright{\hfill The Zeuthen--Segre invariant \quad 5.1\quad}
 \index{Zeuthen--Segre invariant}
 In the previous chapters we have introduced the invariants of surfaces
that are defined in relation with the canonical system. In doing this we
followed a pattern that appears as a natural extension of the procedure
that leads to the definition of the genus of a curve. But the extension
can be done in different ways, and so it is possible to define other
numerical characters of surfaces, that also have an invariant meaning,
and that can be usefully exploited in the further developments of the
theory. Nevertheless these characters are not essentially new, since
they can be expressed in terms of the numerical genus $p_a$ and the
linear genus $p^{(1)}$, as we shall explain later.

Consider on the surface $F$ a linear pencil of irreducible curves of
genus $\pi\ge0$, having some number $n$ of (simple or multiple) effective
base points, and denote by $\de$ the number of curves of the pencil with
a double point away from the base points.

{\em The expression
 \[
 I=\de-n-4\pi,
 \]
involving the number of base points, the genus, and the number of
curves with a double point of a linear pencil, does not depend on the
arbitrary choice of the pencil and therefore it is a (relative)
invariant of the surface.}

The invariant meaning of this character was recognized for the first
time by Zeuthen\footnote{H. G. Zeuthen, Etudes g\'eom\'etriques etc.
{\it Math.\ Ann.}, IV, 1871.} by considering a pencil of plane sections
of a surface. C.~Segre\footnote{C.~Segre, Intorno ad un carattere delle
superficie ecc., {\it Atti Acc.\ Torino}, 1896.} has proved directly the
equality of the characters equally defined in relations with two
arbitrary pencils of the surface. And so $I$ is usually referred to as
to the Zeuthen--Segre invariant.

We explain Segre's proof, which consists in comparing two pencils
$|C_1|$ and $|C_2|$ on the surface, in no special relation one to the
other.

The first pencil has a certain number $n_1$ of base points, that we
assume to be simple and distinct one from another; the second pencil
also has a certain number $n_2$ of base points, different from the
previous ones and also simple and distinct one from another. We shall
denote by $\pi_1$ and $\pi_2$ respectively the genera of the generic curves
of the two pencils, and by $s$ the number of intersections $(C_1C_2)$. And
we shall consider the curve $L$ which is the locus of the contact points
of $C_1$ and $C_2$, namely the Jacobian of the two pencils.

A curve $C_1$ meets $L$ at the points of the Jacobian group of the
series $g^1_s$ cut out on $C_1$ by $|C_2|$, thus at $2s+2\pi_1-2$ points
away from the $n_1$ base points that also belong to $L$. Similarly a $C_2$
cuts $L$ at $2s+2\pi_2-2$ variable points.


Now on the curve $L$, whose genus we denote by $\pi$, we have two
$\infty^1$ series, that we shall denote by $g_1$ and $g_2$, of degrees
$2s+2\pi_1-2$ and $2s+2\pi_2-2$ respectively. Let us consider the Jacobian
group of $g_1$, consisting of
 \[
 4s+4(\pi_1-1)+2\pi-2
 \]
points.

Let us remark that the double points of the series $g_1$ on $L$ are given
by:
 \begin{enumerate}
 \renewcommand{\labelenumi}{(\arabic{enumi})}
 \item the $\de_1$ double points of the pencil $|C_1|$; indeed each of
these points is double for the group cut out on $L$ by the corresponding
$C_1$;

 \item the $\tau$ points where a $C_1$ and a $C_2$ have a triple contact,
because at such points there are two infinitely near points of a $C_1$
in common with $L$;

 \item and finally, the $n_2$ base points of the pencil $C_2$; indeed
each of these points appears as a fixed point for the series cut out by
the $C_2$ on the curve $C_1$ which passes through it, hence it has
multiplicity two in the Jacobian of this series, and therefore it is a
point of simple contact of $C_1$ and $L$.
 \end{enumerate}

Now we have
 \[
 4s+4(\pi_1-1)+2\pi-2=\de_1+\tau+n_2.
 \]

Similarly, interchanging the role of $|C_1|$ and $|C_2|$ one finds
 \[
 4s+4(\pi_2-1)+2\pi-2=\de_2+\tau+n_1.
 \]

Subtracting one deduces
 \[
 \de_1-n_1-4\pi_1=\de_2-n_2-4\pi_2. \qed
 \]

 \begin{Remark} We have assumed the base points of each of the two
pencils $|C_1|$ and $|C_2|$ to be simple and distinct, but we
can get rid of these restrictions.

Assume for instance that a base point $O$ of $|C_2|$ is a multiple point
of order $r$ greater than one, with variable tangents. Then it is easy to
show that this point has multiplicity $2r-1$ for the Jacobian $L$; indeed
one checks that $L$ completed with two curves $\Cbar_1$ and $\Cbar_2$ of
the two pencils is the Jacobian of the net containing the two pencils
 \[
 \Cbar_2+|C_1| \quad\hbox{and}\quad \Cbar_1+|C_2|,
 \]
thus it has multiplicity $3r-1$ at $O$; subtracting $C_2$, it follows that
$L$ has multiplicity $2r-1$ at $O$. Now when one looks, in the previous
argument, for the multiplicity of $O$ in the group of points section of
$L$ with the $C_1$ passing through $O$, one finds that this multiplicity
is $2r$, thus that $O$ belongs $2r$ times to the Jacobian group of the
$g^1_s$ on $L$. The point $O$ accordingly always appears with
multiplicity one in the computation of the degree of the Jacobian series
of $g_1$.
 \end{Remark}

Concerning the case in which one of the two pencils has some infinitely
near base points, we can say, arguing by continuity, that our formula
always holds, provided that one keeps track of the curves of the pencil
that acquire a double point (lowering the genus $\pi$) at a base point
of the pencil, or at a point infinitely near to it. It suffices to refer
to the case in which the pencil $|C_1|$ has a simple base point where
the curves of the pencil touch each other: then the pencil contains a
curve with a double point at $O$ and another curve, infinitely near to
it, that has a double point infinitely near $O$: these two curves must
appear in the computation of $\de$. Therefore $\de$ will be equal to the
number of curves of the pencil with a double point awayfrom the base points
plus 2.

After this, {\em we can compute the equivalence of a curve $C_1$ with an
\hbox{$r$-tuple} point, {\em awayfrom the base points,} by showing that in
general it gives the same contribution as $(r-1)^2$ curves with a double
point.} Let us compare the pencil $|C_1|$ containing the curve with the
$r$-tuple point with another generic pencil $|C_2|$. First of all, as we
have already remarked, the $r$-tuple point $P$ will be $(r-1)$-tuple for
the Jacobian of the two pencils $|C_1|$ and $|C_2|$. And on every branch
of this Jacobian the point $P$ appears as a $(r-1)$-tuple point for the
series cut out by the $C_2$; so that, repeating the previous argument,
$P$ will be counted $(r-1)^2$ times in the computation of the invariant
$I$.

It is possible that, instead of a curve with a multiple point, there
is in a pencil $|C|$ a curve containing a whole component which is
double or multiple: for instance a curve
 \[
 C^{(0)}=2\chi+\psi,
 \]
where $\chi$ has some genus $\rho\ge0$ and the residual curve $\psi$
(which may also be missing) meets $\chi$ at a certain number $i\ge0$ of
points. It is not difficult to compute the equivalence of this curve
$2\chi+\psi$, that is, the number of curves $C$ with a double point that are
absorbed by it.
This computation is worked out in a paper by Castelnuovo and
Enriques\footnote{Castelnuovo--Enriques, Sopra alcune questioni
fondamentali ecc., {\it Ann.\ di Mat.}, 1901.}. Here we shall explain it
in the simplest case, in which there no complication arising for the
presence of infinitely near base points. Thus we make the assumption
that the pencil has no base point. Actually this is the case in which
later on it will occur to us to apply the formula to the irrational
pencils.

To start with, by comparing the linear pencil $|C|=|C_1|$ with another
generic pencil $|C_2|$, one remarks that $\chi$ is a component of the
Jacobian $L$ of the two pencils, and that the residual curve $L_1$
passes through the $i$ common points of $\chi$ and $\psi$ that are
triple for $C^{(0)}$. With an argument analogous to the one used in
general case, one has to compute the degree of the series cut out on
$L_1$ by the curves of the pencils $|C_1|$ and $|C_2|$.

The degree of the series $g^1_{m_1}$ cut out on $L_1$ by the curves $C_1$ is
 \[
 m_1=2s+2\pi_1-2 \quad\hbox{with}\quad s=(C_1C_2),
 \]
since that is the number of the $C_2$ that touch a generic $C_1$.

In order to determine the degree of the series $g^1_{m_2}$ cut out on
$L_1$ by the $C_2$, one has to keep in mind that the Jacobian group of
the series $g^1_s$ cut out on a generic $C_2$ by the $C_1$ contains also
the group of the points of intersection of $C_2$ with $\chi$. Denoting
the number of these points $(C_2\chi)$ by $\si$, one has that the curves
$C_1$ that touch properly a generic $C_2$ are
 \[
 m_2=2s+2\pi_2-2-\si,
 \]
and this is also the intersection number $(C_2L)$. Now, denoting by $P$
the genus of the Jacobian $L_1$, the Jacobian group of the series
$g^1_{m_1}$ on $L_1$ consists of $2m_1+2P-2$ points, and this group
consists of:

 \begin{enumerate}
 \renewcommand{\labelenumi}{(\arabic{enumi})}
 \item the $\debar$ double points of the remaining nodal curves that
belong to the pencil $|C_1|$;

 \item the $\tau$ points of contact of order 3 of a $C_1$ and a $C_2$;

 \item the $n_2$ base points of the pencil $|C_2|$;

 \item the $i$ common points of $\chi$ and $\psi$, each to be counted
twice, since it is triple for the group cut out by $C^{(0)}=2\chi+\psi$
on $L_1$;

 \item the $2\si+2\rho-2$ double points of the series cut out on $\chi$
by the pencil $|C_2|$, since $L_1$ passes through these points, that are
double for the group cut out by $C^{(0)}$.
 \end{enumerate}

Thus we have
 \[
 2m_1+2P-2=\debar+\tau+n_2+2i+2\si+2\rho-2.
 \]

On the other hand, for the series cut out on $L_1$ by the curves $C_2$ one
has simply
 \[
 2m_2+2P-2=\de_2+\tau,
 \]
since the pencil $|C_1|$ has no base points.

Eliminating $2P-\tau-2$ from these two relations and substituting the
values of $m_1$ and $m_2$ one obtains
 \[
 \debar-4\pi_1+2\rho-2+2i=\de_2-n_2-4\pi_2.
 \]

But if the pencil $|C_1|$ had only curves with isolated double points,
whose number we denote by $\de_1$, we should have
 \[
 I=\de_1-4\pi_1.
 \]

Comparing expressions it follows that {\em a curve $C^{(0)}$ with a
double component of genus $\rho$ intersecting the residual curve at $i$
points must be counted as
 \[
 \de_0=2\rho-2+2i
 \]
curves with a double point} in the computation of the Zeuthen--Segre
invariant, at least under the assumption that {\em the pencil has no base
points.}

The previous argument can be extended to the case in which the pencil
$|C|=|C_1|$ contains a curve
 \[
 C^{(0)}=h\chi+\psi,
 \]
containing a multiple component with multiplicity $h$ (greater than 1),
without double points, meeting the residual curve $\psi$ at $i$ points.

In this case one has
 \[
 \de_0=(h-1)(2r-2)+hi.
 \]

More generally, assume that the pencil $|C|$ contains a reducible
curve
 \[
 C^{(0)}=h_1C_1+h_2C_2+\cdots+h_tC_t
 \]
consisting of $t$ simple or multiple components $\chi$ ($h_r\ge1$), and
that the curves $\chi_r$ and $\chi_s$ have $i_{rs}$ points in common. The
equivalence of this curve in the computation of the Zeuthen--Segre
invariant is given by the {\em Godeaux formula}\footnote{L.~Godeaux, Sur
le calcul de l'invariant de Zeuthen--Segre. M\'emoires de la Soci\'et\'e
des Sciences du Hainaut, t.~66, 1920. See Bull. de l'Acad\'emie Royale
de Belgique, 13 oct. 1934. See also Campedelli, Sul computo
dell'invariante di Zeuthen--Segre per una superficie algebrica; Ancora
sul computo ecc., {\it Rendic.\ Acc.\ Lincei}, s.~6, t.~XIX, 1934.}
 \[
 \de_0=\sum (h_r-1)(2\rho_r-2)+\sum i_{rs}(h_r+h_s-1).
 \]

This formula can be applied to all cases, provided that one takes
into account two modifications:

 \begin{enumerate}
 \renewcommand{\labelenumi}{(\arabic{enumi})}
 \item If one of the curves, $\chi_1$, has a double or multiple point,
then $\de_0$ must be increased accordingly: in the case of a double
point, taking $\rho_1$ to be the effective genus, one has to add $2h_1-1$
to $\de_0$; on the other hand, if $\rho_1$ represents the virtual genus
of $\chi_1$, then $\de_0$ has to be increased only by one.

 \item In the case in which the pencil $|C|$ has infinitely near base
points on a multiple component $\chi_1$ of $C^{(0)}$, it is possible to
eliminate these points with a transformation of the surface that sends
them into exceptional curves: this is equivalent to adding one or several
neighbourhoods of base points to the $\chi_r$. {\em With this convention
the Godeaux formula holds also in the case of infinitely near base
points.}
 \end{enumerate}

 \section{Irrational pencil}\label{sec:5.2}
 \markright{\hfill Irrational pencil \quad 5.2\quad}
 The Zeuthen--Segre invariant of a surface $F$ can be computed not only
for the linear pencils, but also for the irrational pencils belonging
to $F$.

Precisely, {\em the invariant $I$ can be expressed in relation with a
pencil of genus $\rho>0$ consisting of curves of genus $\pi$ and
containing $\De$ curves with a double point by the formula
 \[
 I=\De+4(\rho-1)(\pi-1)-4\footnote{See Castelnuovo--Enriques, Ibidem.}.
 \]}

This formula can be proved directly, by comparing again two pencils
$|C_1|$ and $|C_2|$ and replacing one of them with the irrational pencil
$\{C\}$ that we are considering here. But it is also possible to reduce
the proof to the case of two linear pencils, by replacing $\{C\}$ with a
linear pencil consisting of the groups of $n$ curves $C$ of a $g^1_n$,
where $n$ is a large enough integer, equal for instance to $\rho+1$.
Indeed the fact that one of the two pencils consists of reducible curves
does not affect the arguments developed in the previous section.

Thus the groups of $n$ curves $C$ of the aforementioned $g^1_n$ form a linear
pencil of curves, of genus:
 \[
 \Pi=n\pi-n+1
 \]
without base points; and in this pencil there will be:

 \begin{enumerate}
 \renewcommand{\labelenumi}{(\arabic{enumi})}
 \item $\De$ curves containing a $C$ with a double point and

 \item $2n+2\rho-2$ curves containing a $C$ of genus $\pi$ counted twice,
each equi\-valent to $2\pi-2$ curves with a double point; thus we have
 \[
 I=\De+(\pi-2)(2n+2\rho-2)-4\Pi=\De+4(\rho-1)(\pi-1)-4. \qed
 \]
\end{enumerate}

 \section{Expressing the Zeuthen--Segre invariant in terms of the
genera}\label{sec:5.3}
 \markright{\hfill The Zeuthen--Segre invariant in terms of the
genera \quad 5.3\quad}
 We have said that {\em the Zeuthen--Segre invariant $I$ is a relative
invariant of the surface $F$;} in fact {\em it increases by one when a
point $O$ of $F$ is transformed into an exceptional curve.} This
statement can be verified in two ways: if one takes a point $O$ of $F$
that is not a base point and is simple for a curve of the pencil $|C|$,
then transforming $O$ into an exceptional curve increases by one the
number $\de$ of curves with a double point; on the other hand if $O$ is
a base point of the pencil $|C|$, then the transformation decreases the
number of base points by one.

Let us remark now that a transformation of the surface $F$ that replaces
a point $O$ with an exceptional curve affects the invariant $I$ and the
linear genus $p^{(1)}$ in opposite ways, since we know that the latter
decreases by one. Therefore the sum
 \[
 I+p^{(1)}
 \]
is an absolute invariant of the surface. But this is not a new
invariant, since Noether has shown that it can be expressed in terms of
the numerical genus by of the formula\footnote{Noether, Ibidem, {\it
Math.\ Ann.}, VIII.}
 \[
 I+p^{(1)}=12 p_a+9.
 \]

This expression is related to a set of formulas involving several
invariant numerical characters of the surfaces. We shall prove it here
following the way indicated by T. Bonnesen\footnote{P. Bonnesen, Sur les
s\'eries lin\'eaires triplement infinies de courbes alg\'ebriques sur une
surface alg\'ebrique. Bull. de l'Acad\'emie Royale des Sciences et des
Lettres de Danemarque, 1906, n. 4.}.

Let us refer to a simple irreducible $\infty^3$ linear system of curves
$|C|$, that we may regard as plane sections of a surface $F$ with a
double curve $D$; for the sake of simplicity, we shall assume that $d$ is
irreducible and has only $t$ triple points that are also triple for $F$.
The invariant $I$ of the surface $F$ can be computed with respect to a
pencil of plane sections $C$, and it is given by:
 \begin{equation}
I=\de-n-4\pi,
\label{eq:5.1}
 \end{equation}
where $n$ is the {\em order} of $F$, $\pi$ the genus of the $C$ and $\de$
denotes the {\em class} of the surface $F$. The class of $F$ is also
given by the effective degree of the system $|C_j|$ cut out on $F$ by
the polar surfaces $\Fie_{n-2}$; this system has the $\tau$ cuspidal
points of $F$ on $D$ as simple base points, so that $\de+\tau$ equals the
degree of
 \[
 |C_j|=|3C+K|
 \]
computed without taking into account the $\tau$ stated base points. Thus
introducing the linear genus $p^{(1)}$ (genus of the impure canonical
curves $K$) we have
 \begin{equation}
 \de+\tau=3n+12(\pi-1)+p^{(1)}-1.
 \label{eq:5.2}
 \end{equation}
Now there are some characters (projective characters of the surface $F$)
that are related to the $\infty^3$ system $|C|$ and in particular the
following:

 \begin{enumerate}
 \item[$(\al)$] the degree $d$ of the double curve $D$, that depends on
the degree $n$ and on the genus $\pi$ of $|C|$:
 \begin{equation}
 d=\frac{(n-1)(n-2)}{2}-\pi;
\label{eq5.3}
 \end{equation}

 \item[$(\be)$] the genus $\rho$ of the double curve $D$;

 \item[$(\ga)$] the number $t$ of triple points of $D$, that are also
triple for the surface $F$; moreover, denoting by $\De$ the curve that
corresponds to the double curve $D$ on a nonsingular model $F'$ of the
surface $F$:

 \item[$(\de)$] the (effective) genus $\Pi$ of the curve of the neutral
pairs of $|C|$, that is, the curve $\De$ which is represented on the
double curve $D$ with $\tau$ branch points at the cuspidal points. Note
that the curve $\De$ has $t$ triples of double points corresponding to
the $t$ triple points of $D$, so that the {\em virtual genus} of $\De$ is
 \[
 \Pi+3\tau.
 \]
 \end{enumerate}

In addition to (\ref{eq:5.1}) and (\ref{eq:5.2}), there are other
relations involving the aforementioned characters. {From} these
relations it follows that besides the two characters $n$ and $\pi$ of
the system $|C|$, only two characters of the surface assume independent
values, and are thus up to a point arbitrary.

First of all we remark that the covariant curve $\De$ of the $\infty^3$
system $|C|$ is a partial curve of the system $|(n-4)C|$ on the surface
$F'$ (a transform of $F$ where the double curve $D$ is resolved); more
precisely, adding a canonical curve $K$ to $\De$ one obtains
 \[
 |\De+K|=|(n-4)C|.
 \]

Therefore the intersections of $\De$ with the curves of the adjoint
system $|\De+K|$ system give a canonical group on the curve $\De$ of
virtual genus $\Pi+3\tau$, and thus we have
 \begin{equation}
 2(\Pi+3t)-2=2(n-4)d,
 \label{eq5.4}
 \end{equation}
where $d$ is expressed by the formula (\ref{eq:5.2}).

On the other hand, since the double curve $D$ is in 2-to-1 correspondence
with $\De$ and there are $\tau$ branch points on it, by a well known
formula of Zeuthen's we have
 \begin{equation}
 2(2\rho-2)+\tau=2\Pi-2.
 \label{eq5.5}
 \end{equation}

Finally, we may obtain a sixth relation, by computing the number of
intersections of a $C_j$, the Jacobian of a net in $|C|$, with $\De$, or,
what is the same, with $D$. To do this, we remark that
 \[
 |C_j|=|3C+K| \quad\hbox{and}\quad |\De+K|=|(n-4)C|;
 \]
as a consequence the number of the intersections $(C_jD)=(C_jD)$ is given
by
 \[
 (3C+K)[(n-4)C-K]=2n^2-7n+15+2\pi(n-7).
 \]

But this same number $(C_jD)$ can be also computed directly on the
surface $F$, by looking for the number $a$ of the intersections of $D$
with the second polar $\Fie_{n-2}$ of a generic point $O$; indeed the
$C_j$ cut by the polar of $O$ meets the double curve $D$: at the $\tau$
cuspidal points and at the $a$ common points of $D$ and $\Fie_{n-2}$
(away from the triple points of $D$); at each of these points the tangent
plane to a branch of $F$ passes through $O$. Thus we have
 \[
 (C_jD)=a+t,
 \quad\hbox{where}\quad
 a=(n-2)d-3t.
 \]

Therefore we may write
 \begin{equation}
 2n^2+n+14+2\pi(n-7)-p^{(1)}=(n-2)d-3t.
 \label{eq:5.6}
 \end{equation}

It is convenient to add another relation involving $p_a$ to the six
relations that we have just written: one has
 \begin{eqnarray*}
 p_a&=&\frac{(n-1)(n-2)(n-3)}{6} \\
 && \quad -(n-4)\left[\frac{(n-1)(n-2)}{2}-\pi\right]+2t+\rho-1 \\
 &=&\frac{1}{6}(n-1)(n-2)(9-2n)+\pi(n-4)+2t+\rho-1.
 \end{eqnarray*}

We have found seven equations involving the eleven characters:
$n,\pi,\de,I,d$, $\rho,\Pi,\tau,t,p_a$ and $p^{(1)}$.

Of these characters, $n$ and $\pi$ can take arbitrary values, up to a point, since they depend on the choice of a pencil $|C|$ on the surface
$F$; therefore it is a priori clear that the seven characters
 \[
 \de,\quad I,\quad \rho,\quad \Pi,\quad \tau,\quad t,
 \]
can be expressed as functions of $\pi$ and $n$ and of the invariants
$p_a$ and $p^{(1)}$ of the surface. In particular one has to express $I$
as a function of $p_a$ and $p^{(1)}$: a priori one may think that this
function also depends on $\pi$ and $n$, but in fact it has to be
independent of them, since it is an invariant character of $F$.

Thus one only has to perform the computation, and since (\ref{eq:5.1})
and (\ref{eq:5.2}) already contain an explicit expression of $I$ and $d$,
one has to solve five linear equations, in which $n$, $\pi$, $p_a$ and
$p^{(1)}$ appear as parameters. Thus one obtains:
 \begin{eqnarray*}
\de &=& n+4\pi-p^{(1)}+12 p_a+9\\
\rho &=& \frac{1}{2}(n^2-7n+48)+(n-12)\pi+9p_a-2p^{(1)}\\
\Pi &=& n^2-6n+2(n-10)\pi+12 p_a-3p^{(1)}+36\\
t &=& \frac{1}{6}(n^3-6n^2+26n-78)-(n-8)\pi-4p_a+p^{(1)}\\
\tau &=& 2n+8\pi-12p_a+2p^{(1)}-22.
 \end{eqnarray*}

{From} the first formula one deduces Noether's expression of the
Zeuthen--Segre invariant:
 \[
 I=12 p_a-p^{(1)}+9.
 \]

 \begin{NOTE} The four formulas for $\rho$, $\Pi$, $t$ and $\tau$
express four relative invariants constructed with the characters of an
$\infty^3$ system $|C|$ belonging to the surface. The last formula in
particular is worthy of note, since it involves $n$ and $\pi$ linearly:
it defines the invariant
 \[
 2n+8\pi-\tau=12p_a-2p^{(1)}+22.
 \]

Combining this formula with the first one, one obtains another very
simple relative invariant\footnote{See Bonnesen, Ibidem.}, that is,
 \[
 2\de-\tau=36p_a-4p^{(1)}+40.
 \]
This invariant is constructed with two characters of an $\infty^3$
linear system $|C|$ on the surface; $\de$ denotes, in general, the number
of curves $C$ of a pencil that have a double point (away from the base
points) and $t$ denotes the number of pencils contained in the
$\infty^3$ system $|C|$ that have a double base point (away from the base
points of $|C|$).
 \end{NOTE}

 \begin{Remark} The expression of the Zeuthen--Segre invariant in terms
of the numerical genus and of the linear genus suggests another way of
finding {\em the linear genus of the plane and of the ruled surfaces},
that we computed in Chapter~II, Section~\ref{sec:2.6}.

To start with one may compute the invariant $I$ of the plane, by
considering a pencil of conics, for which one has:
 \begin{itemize}
 \item the genus $\pi$=0,
 \item the number of base points $n=4$,
 \item and the number of curves with a double point $\de=3$.
 \end{itemize}

We have
 \[
 I=\de-n-4\pi=-1;
 \]
therefore from the formula
 \[
 I=12p_a-p^{(1)}+9
 \]
and since $p_a=0$, one obtains
 \[
 p^{(1)}=10.
 \]

Now we turn to the irrational ruled surfaces of genus $p_a=-p$ (with
$p>0$). Here one may compute the value of $I$ in relation to the pencil
of generating lines, which is of genus $p$ and for which
 \[
 \pi=0,\quad \De=0, \quad n=0.
 \]

We have
 \[
 I=\De+4(p-1)(\pi-1)-4=-4p.
 \]

Now from Noether's formula one deduces that
 \[
 12p_a-p^{(1)}+9=-4p
 \]
and since $p_a=-p$, then
 \[
 p^{(1)}=-8(p-1)+1.
 \]
The same result can be found by considering a linear pencil of plane
sections instead of the irrational pencils of the generating lines.
 \end{Remark}

 \section{Cuspidal curves of a net}\label{sec:5.4}
 \markright{\hfill Cuspidal curves of a net \quad 5.4\quad}
 We turn again to the consideration of the $\infty^3$ linear system
$|C|$ and to the characters of it that we have described in the previous
section. Using the formulas we have proved we may determine the
characters of a net of curves $|C|$, and in particular the number $\chi$
of the curves of a net having a cusp.

To do this, let us consider the surface $F$ with the curves $|C|$ as
plane sections. The locus of points of $F$ for which the tangent plane
section has a cusp is the {\em parabolic curve} $C_h$ of $F$, the section
of $F$ with the Hessian surface $H$. Thus the $\chi$ plane sections with
a cusp passing through a point $O$ of the space correspond to the
intersections of $C_h$ with the curve $C_j$, the Jacobian of the net cut
out by the planes through $O$, awayfrom the cuspidal points of $F$. These are
simple points of $C_j$, and double for the parabolic curve $C_h$, as we
shall see.

Therefore in order to compute $\chi$ we have to:

 \begin{enumerate}
 \renewcommand{\labelenumi}{(\arabic{enumi})}
 \item determine the behaviour of the Hessian at the nodal double curve
of the surface $F$;

 \item then find the multiplicity that curve $C_h$ has at the $\tau$
cuspidal points of $F$.
 \end{enumerate}

The first computation can be performed by simply replacing the given
surface with another one, that we still temporarily denote by $F$, that
has a biplanar double point at the origin, with $x$ and $y$ as tangent
planes; in homogeneous coordinates $x,y,z,u$, we write:
 \[
 F=u^{n-2}xy+u^{n-3}\fie_3(x,y,z)+\cdots+\fie_n(x,y,z).
 \]
 The Hessian of $F$ is
 \[
 H=\left|
 \renewcommand{\arraystretch}{2.4}
 \begin{array}{cccc}
\displaystyle\frac{\partial^2F}{\partial x^2} & \displaystyle\frac{\partial^2F}{\partial x\partial
y} &
\displaystyle\frac{\partial^2F}{\partial x\partial z} & \displaystyle\frac{\partial^2F}{\partial
x\partial u} \\
\displaystyle\frac{\partial^2F}{\partial y\partial x} & \displaystyle\frac{\partial^2F}{\partial
y^2} &
\displaystyle\frac{\partial^2F}{\partial y\partial z} & \displaystyle\frac{\partial^2F}{\partial
y\partial u} \\
\displaystyle\frac{\partial^2F}{\partial z\partial x} & \displaystyle\frac{\partial^2F}{\partial
z\partial y} &
\displaystyle\frac{\partial^2F}{\partial z^2} & \displaystyle\frac{\partial 2F}{\partial z\partial
u} \\
\displaystyle\frac{\partial^2F}{\partial u\partial x} & \displaystyle\frac{\partial^2F}{\partial
u\partial y} &
\displaystyle\frac{\partial^2F}{\partial u\partial z} & \displaystyle\frac{\partial^2F}{\partial
u^2}
 \end{array}
 \right|=0
 \]

Making the computation, one finds that the behaviour of the Hessian at
the origin of coordinates $(x=y=z=0,u=1)$ is determined by the Hessian
of the approximating surface of order three, so $H$ has a quadruple
point at a biplanar point of $F$. Now, going back to our surface $F$
with a nodal curve, one deduces that its Hessian $H$ of degree $4n-8$
passes through the stated nodal curve four times. Therefore the
parabolic curve $C_h$ belongs to the linear system
 \[
 |4K+8C|,
 \]
where, as usual, $K$ denotes the canonical curves of the surface.

Now we see that the $\tau$ cuspidal points of $F$ (simple for the $C_j$)
are double for the parabolic curve $C_h$. Indeed, considering a
transformed surface on which the nodal curve of $F$ is resolved, on such
a surface a cuspidal point of $F$ gives rise to a double base point of a
pencil of $|C|$. Hence in that pencil there are two curves with a cusp.

Now the number of cuspidal curves of a net contained in $|C|$ is
given by
 \[
 \chi=(C_j\cdot C_h)-2\tau.
 \]

But, since
 \[
 |C_j|=|3C+K|,\quad |C_h|=|8C+4K|,
 \]
one has
 \[
 (C_j\cdot C_h)=24n+20(2\pi-2-n)+4p^{(1)}-4,
 \]
and recalling the expression for $\tau$
 \[
 \tau=2n+8\pi-12 p_a+2p^{(1)}-22
 \]
from the previous section, it follows that
 \[
 \chi=24(p_a+\pi).
 \]

Thus we arrive at the relation discovered by Zeuthen: {\em the number of
cuspidal curves in a net of genus $\pi$ is in general $24(p_a+\pi)$.}

To be precise, we have proved this formula for a net of curves $C$ that
is contained in an $\infty^3$ system. But it is easy to extend the above
argument considering an $\infty^3$ system $|L+C|$ that contains $|C|$.
And so the formula can be verified in every case\footnote{The proof of
the formula given in the text can also be explained in a slightly
different form, without using the properties of the Hessian. To do this,
it is convenient to associate to the surface $F$ its dual $F'$, which is
the transform of $F$ obtained by taking the Jacobians $C_j$ of the nets
of $|C|$ as plane sections. Now if one constructs the dual surface of
$F'$ one find $F$ back. This means that the Jacobian curves of the nets
of $C_j$ on $F$, say the $C_{jj}$, coincide with the plane sections $C$,
apart from a fixed component, that one proves to be the parabolic curve
$C_h$.

Indeed consider the polars with respect to $F$ of the points of a plane
$\al$, which form a net. It is clear that the plane section $C$ of $\al$
belongs to the Jacobian $C_{jj}$ of this net, because the polar of a
point $P$ of $C$ intersects $F$ in a curve that passes doubly through
$P$. Moreover, if one takes a point $A$ on the parabolic curve $C_h$,
the tangent plane at $A$ meets $\al$ along a line $a$, and the polars of
the points of $\al$ intersect $F$ along the curves of a pencil that has a
base point with fixed tangent at $A$. Among the curves of this pencil
there is one with a double point at $A$; so the curve $C_h$ is a part of
the second Jacobian $C_{jj}$.

Thus from the relation
 \[
 |C_jj|=|C_h+C|
 \]
it follows that
 \[
 |C_h|=|8C+4K|.
 \]
}.

 \section{Historical note}\label{sec:5.5}
 \markright{\hfill Historical note \quad 5.5\quad}
 The relation between the number of cuspidal curves of a net and the
numerical genus was discovered by H.G.~Zeuthen\footnote{H.G.~Zeuthen,
Etudes g\'eom\'etriques etc., {\it Math.\ Ann.}, Band~IV, 1871. See the
memoir of Noether's, Ibidem, in {\it Math.\ Ann.}, VIII.}. Assuming the
invariance of the numerical genus (a fact known by Zeuthen and Noether,
although under some restriction on the singularities of the surface),
Zeuthen proved that the number $c-24\pi$, computed for a net (of plane
sections), is independent of the net and is therefore an intrinsic
character of the surface. F.~Severi gave a direct proof of this theorem
in 1902, with a very elementary argument\footnote{F. Severi, Il genere
aritmetico ed il genere lineare in relazione alle reti di curve
tracciate sopra una superficie algebrica. {\it Atti Acc.\ Torino},
Vol.~XXXVII, 1902.}.

The proof described in this text can be found in Bonnesen,
Ibidem\footnote{See also M. PANNELLI, Sui sistemi lineari triplamente
infiniti di curve tracciati sopra una superficie algebrica. {\it
Rendic.\ Circ.\ Mat.\ di Palermo}, t.~XX, 1905. The remark about the
Jacobian of a net of Jacobians, that we have made in one of the previous
footnotes, was also made by E.~Caporali.}.

We cannot conclude this historical note, that completes the information
given in the previous sections, without recalling a circle of ideas in
which one looks for the functional geometric interpretation of the
enumerative formulas that we have found. It is known that this point of
view was introduced in the theory of curves with the theorem of
Painlev\'e--Castelnuovo (giving the interpretation of Zeuthen's formula
for $[1,n]$ correspondences) and with the consideration of the Jacobian
series of a linear series, that Enriques recognized as the simplest way
of defining the canonical series and of proving its invariance.

In order to find an analogous interpretation for the enumerative
geometry on surfaces, Severi tried, in a series of memoirs since 1932,
to construct a theory of the series of groups of points, that he called
of series of equivalence, that should be a natural extension of the
theory of linear series on a curve.

For a large number of references on the subject see F.~Severi's book:
{\it Serie, sistemi di equivalenza e corrispondenze algebriche sulle
variet\`a algebriche}, Roma, ed.\ Cremonese, 1942.

 \section{Characters of a net and multiple planes}\label{sec:5.6}
 \markright{\hfill Characters of a net and multiple planes \quad
5.6\quad}
 With the formulas we proved in the previous sections we may also
express the characters of a net of curves $|C|$ in terms of two of
characters, e.g., the genus $\pi$ and the degree $n$, and of the invariants
of the surface: $p_a$ and $p^{(1)}$.

Let $|C|$ be a base point free net of irreducible curves (of genus $\pi$ and degree $n$) on the surface $F$; in general we may attach to it the
following numerical characters:

 \begin{enumerate}
 \renewcommand{\labelenumi}{(\arabic{enumi})}
 \item the number $N$ of {\em intersections of the curves $C$ with the
Jacobian $C_j$ of the net:}
 \[
 N=2n+2\pi-2;
 \]

 \item the number $\de$ of {\em curves $C$ of a pencil having a double
point;} this number, as we know, is expressed by
 \[
 \de=n+4\pi+12 p_a-p^{(1)}+9;
 \]

 \item the {\em genus} of the above Jacobian $C_j$ ($=3C+K$)
 \[
 P=p^{(1)}+9\pi-9;
 \]

 \item the number $k$ of {\em contact points of order 3} of the $C$ of
the net (base points of a pencil of osculating $C$);

 \item the number $i$ of {\em curves $C$ of the net $|C|$ having a cusp;}

 \item the number $d$ of the {\em double contact points} of the $C$ of
the net (base points of a pencil of bitangent $C$);

 \item the number $t$ of the $C$ of the net having two double points.
 \end{enumerate}

We may assume that the net of the $C$ is cut out on $F$ by the star of
planes through a point $O$, which in general will be a multiple point of
the surface and will correspond to an irreducible curve $L$ (adding which
it to $|C|$ one obtains the plane sections system).

Then $F$ is projected from $O$ onto a $n$-tuple multiple plane, the
points of which correspond to groups of $n$ points of $F$, belonging to
the involution $I$ defined by the net $|C|$. And the above characters
acquire the following meaning:

 \begin{enumerate}
 \renewcommand{\labelenumi}{(\arabic{enumi})}
 \item $N$ is the {\em order of the branch curve} $D$ of the $n$-tuple
plane (curve of points of the plane corresponding to groups of $I$ with
two coincident points);

 \item $P$ is the {\em genus of the branch curve $D$;}

 \item $\de$ is the {\em class} of the branch curve $D$;

 \item $k$ is the number of lines through $O$ having a contact of order
three with the surface $F$, and therefore it is the {\em number of cusps
of $D$;}

 \item $i$ is the number of cuspidal plane sections of $F$ through $O$,
and therefore it is the {\em number of flexes of the branch curve $D$;}

 \item $d$ is the number of lines through $O$ that are bitangent to $F$
and therefore it is the {\em number of nodes of $D$;}

 \item $t$ is the number of planes through $O$ bitangent to $F$, and so
it is the {\em number of double tangents} (with distinct contact points)
of $D$.
 \end{enumerate}

These characters are related by the Pl\"ucker formulas for the curve $D$,
that in general is irreducible and has no other singular points and no
other exceptional tangents.

In this way one can prove the following relations:
 \[
(A)\quad
 \left\{
 \renewcommand{\arraystretch}{1.3}
 \begin{array}{rcl}
 N &=& 2n+2\pi-2 \\
 P &=& p^{(1)}+9\pi-9 \\
 \de &=& n+4\pi+12 p_a-p^{(1)}+9 \\
 k &=& 3(n+6\pi+p^{(1)}-4 p_a-11) \\
 i &=& 24(\pi+p_a) \\
 d &=& 2\Big[(n+\pi)^2-5n-17\pi-2p^{(1)}+6 p_a+24\Big] \\[6pt]
 t &=& \frac{1}{2}\Big[(n+4\pi+12 p_a-p^{(1)})^2 \\
 && \qquad  +15n-6\pi+132 p_a-17p^{(1)}+74\Big]
 \end{array}
 \right.
 \]

{\em These formulas in general express the characters of an irreducible
base point free net $|C|$ on the surface, or else the Pl\"ucker
characters of the branch curve $D$ of the corresponding multiple plane,
in terms of the genus $\pi$ and of the degree $n$ of $|C|$ and of the
invariants $p_a$ and $p^{(1)}$ of the surface.}

Conversely these invariants are expressed in terms of the characters
of $D$ by the formulas:
 \[
(B)\quad
 \left\{
 \renewcommand{\arraystretch}{1.3}
 \begin{array}{rcl}
 p^{(1)} &=& P-9\pi+9 \\
 p_a &=& \frac{i}{24}-\pi \\
 &&\kern-1cm \hbox{or else} \\
 4p_a &=& n+P-3\pi-\frac{k}{3}-2\\
 (\pi &=& \frac{N}{2}-n+1).\\
 \end{array}
 \right.
 \]

\begin{Remark} The fourth and fifth formulas among the (A) are especially
important, as they compute two invariant characters of the net $|C|$
that depend linearly on $n$ and $\pi$, namely $3(n+6\pi-k)$ and
$24\pi-i$.

But the latter of these two formulas is particularly important, as it
involves the absolute invariant $p_a$. As we have said, these relation
between $p_a$ and the number of cuspidal curves of a net was discovered
by G.H.~Zeuthen. And F.~Severi showed in a direct and elementary way
that the expression $24\pi-i$ is equal for all base point free nets of
the surface $F$. This gives a new proof of the invariance of $p_a$.
\end{Remark}

 \section[$(1,n)$ correspondence between two surfaces: double
planes]{$(1,n)$ correspondence between two surfaces:\\ double
planes}\label{sec:5.7}
 \markright{\hfill $(1,n)$ correspondence between two surfaces: double
planes \quad 5.7\quad}
 Let us consider a $(1,n)$ correspondence between two surfaces $f$ and
$F$: a given point of $F$ corresponds to {\em one} point of $f$,
whereas the points of $f$ correspond to the groups of an {\em involution}
$I$ of degree $n$ on $F$ (every point of $F$ belongs to {\em one} group
of $I$). For the sake of simplicity we assume also that {\em there are no
fundamental points}, that is, points of $f$ or $F$ which in the
correspondence are associated to a curve on the other surface.

Given an irreducible linear system $|c|$ on $f$, there is a linear
system $|C|$ on $F$ corresponding to it and belonging to the involution
$I$. The Jacobian $|c_j|$ of a net contained in $|c|$ is related in the
correspondence with a curve of $F$ contained in the Jacobian of the
corresponding net of the homologous $|C|$. But in order to obtain the
whole Jacobian of this net it is necessary to add the branch curve
$D$, that corresponds to the branch curve on $f$; indeed a point of $d$
is a base point of a pencil of $|C|$ contained in the aforementioned net,
which has a fixed tangent, and therefore contains a $C$ with a double
point. Now recalling the definition of the canonical curves
($|K|=|C_j-3C|$) one deduces the theorem:

{\em Let us consider a $(1,n)$ correspondence between the surfaces $f$
and $F$; then canonical curves of $F$ are obtained by adding the curve
homologous to the branch curve to the curves of $F$ corresponding to the
canonical curves of $f$\footnote{See Enriques, Ricerche etc.}.}

This statement has to be modified when {\em there is a fundamental point}
on $f$, corresponding to a curve on $F$ (or to a curve and a group of
points); {\em this fundamental curve must be added to the images of the
canonical curves of $f$ and to the branch curve,} in order to make
a canonical curve of $F$.

The theorem we have just stated holds even if the {\em canonical curves
of $f$ are virtual.} We may apply it to the problem of determining the
canonical and pluricanonical curves of a double plane.

Let $F$ be a surface with a rational involution $I$ of order two, such
that $F$ is represented on the double plane
 \[
 z^2=f_{2n}(x,y)
 \]
where the branch curve $f_{2n}$ of degree $2n$ may have singular points,
the multiplicity of which has to be computed following the conventions of
Chapter~III, Section~\ref{sec:3.7}. Consider the curve $\fie_m$ of the
double plane belonging to a linear system with no base points away from
the singular points of $f_{2n}$; the adjoints of the corresponding curve
of $\fie_m$ on $F$ are obtained by adding the branch curve homologous to
$f_{2n}$ to the transforms of the $\fie_{m-3}$ adjoint to $f_m$ in the
plane. If one considers a biadjoint $\fie_{m-6}$ instead of an adjoint
$\fie_{m-3}$, then one has to add the branch curve counted twice
to the transform of this curve, and so on.

The virtual canonical curves $K$ on the plane are defined as differences
 \[
 |K|=|\fie_{m-3}-fm|,
 \]
hence one finds canonical curves on $F$ by adding the $K$ to the branch
curve homologous to $f_{2n}$; thus adding these curves to the curves
homologous to the $\fie_m$ one obtains the adjoints of the $\fie_m$.

Let us apply this result by taking, instead of the $\fie_m$, the branch
curve $f_{2n}$ and its biadjoints $f_{2n-6}$. In order to have the
biadjoints on $F$ of the branch curve counted twice, that corresponds to
$f_{2n}$, it suffices to add this curve counted twice to the transforms
of the $f_{2n-6}$. Subtracting the branch curve counted twice from the
system thus obtained, one has the bicanonical curves of $F$; among these
there are those belonging to the involution $I$, having the
$f_{2n-6}$ as images.

Thus: {\em the biadjoint curves of the branch curve of the double plane
are images of bicanonical curves} belonging to the involution $I$. This
result agrees with what we saw in Chapter~III, Section~\ref{sec:3.7}:
the canonical curves of the surface are represented by the $f_{n-3}$
belonging to the half of the biadjoint system of the branch curve.

The argument can be extended to pluricanonical curves: {\em the
tricanonical curves of the double plane are obtained by adding to the
branch curve the third adjoints of this curve,} and so on.

But let us go back to the bicanonical curves, in order to remark
that, we can see also in another way that {\em one obtains bicanonical
curves of the double plane by adding the branch curve to the adjoints
of the curves $f_{n-3}$, images of the canonical curves;} it suffices to
recall that the bicanonical system cuts out groups of the canonical
series on the curves of the canonical system.

Now remark that the bicanonical system is preserved by the involution
$I$, whereas its curves in general do not belong to this involution.
Thus we must regard this system as a linear space of dimension $P-1$
(where $P$ denotes the bigenus of $F$) on which $I$ induces a projective
involution with two subspaces of fixed points $S_r$ and $S_t$, where
 \[
 P-1=r+t+1.
 \]

One of the two spaces, say $S_r$, corresponds to the system of
$f_{2n-6}$, biadjoints of $f_{2n}$, while the other space $S_t$
corresponds to the system consisting of the branch curve $f_{2n}$ taken
together with the adjoints $f_{n-6}$ of the images of the canonical
curves. Thus, denoting by $\pi=t+1$ the genus of the $f_{n-3}$, one has
 \[
 P-1=r+\pi+1.
 \]

{\em This formula expresses the bigenus of the double plane} in terms of
the genus $r+1$ of the adjoint curves $f_{2n-3}$ of the branch curve and
of the genus $\pi$ of the curves that are one half of the biadjoints
$f_{2n-6}$, that is, of the genus of the involution of order two that
$I$ induces on a canonical curve.

These results can be easily extended to the plurigenera. We shall just
remark that if all the plurigenera vanish, then there are no adjoints of
the branch curve of any order, and so the curve is birationally
equivalent either to a curve of degree $2n$ with a $(2n-2)$-tuple point
or to a quartic or to a sextic with two infinitely near triple points,
or to a specialization of one of these\footnote{Theorem of
Castelnuovo--Enriques, 1900. See Enriques--Conforto, {\it Le superficie
razionali}, \S27, page~489.}.

Therefore we can state: {\em A double plane having all the genera equal
to zero either is rational or it can be transformed into a hyperelliptic
ruled surface.}

 \section{Correspondence formulas}\label{sec:5.8}
 \markright{\hfill Correspondence formulas \quad 5.8\quad}
 A first consequence of the theorem given in the previous section,
about the transformation of the canonical curves in a $(1,n)$
correspondence, is that the genus $P_g$ of $F$ cannot be, in any case,
smaller than the genus of $f$:
 \[
 P_g\ge p_g.
 \]

Furthermore the same theorem immediately leads to a relation between
the linear genera $P^{(1)}$ and $p^{(1)}$ of the two surfaces.

It is convenient to make first a remark concerning the branch curve $D$
of the correspondence on $f$. In general this curve will have a certain
number of nodes that correspond on $F$ to groups of $n$ points having two
distinct coincident pairs. In addition, $D$ will have a certain number
of cusps; each of these corresponds to a group of $n$ points, of which
three coincide. Now according to a well-known theorem of Zeuthen, the
curve corresponding to a $n$-tuple canonical curve on $f$ that meets the
branch curve $D$ at $s$ points has genus
 \[
 n(p^{(1)}-1)+\frac{s}{2}+1.
 \]

This curve meets the branch curve $D$ at $s$ points, so that
denoting by $\rho$ the effective genus\footnote{We assume that the only
singularities of $d$ are the nodes and the cusps that have the meaning
explained earlier with respect to our correspondence.} of $D$, one has
 \[
 P^{(1)}=n (p^{(1)}-1)+\rho+3s /2.
 \]

{\em This relation connects the linear genera of the two surfaces in
$(1,n)$ correspondence}\footnote{Enriques, Ricerche etc., 1893.}. One has
to set $\rho=1$ and $s=0$ in it when {\em there is no branch curve} (that
is, it is of degree zero on a nonsingular $f$).

Together with the previous formula, one can write another relation
involving the numerical genera $P_a$ and $p_a$ of $F$ and $f$\footnote{See
Severi, Sulle relazioni che legano i caratteri invarianti di due
superficie algebriche. {\it Rendic.\ Ist. Lombardo}, 1903.}.

To do that, we take a pencil $|c|$ of curves on $f$ and the
corresponding pencil $|C|$ of curves on $F$. We compute the Zeuthen--Segre
invariants $i$ and $I$ of the two surfaces using the characters of these
two pencils and write the relation that connects them.

Thus denote by $m$ the number of base points of $|c|$, by $\pi$ the genus
of the $c$, and by $\de$ the number of $c$ having a double point; we
have
 \[
 i=\de-m-4\pi.
 \]

Now the number $M$ of base points of the pencil $|C|$, and the genus
$\Pi$ of the $C$ are given respectively by
 \[
 M=mn \quad\hbox{and} \quad P=n (\pi-1)+1+\frac{t}{2},
 \]
where $t$ denotes the number of intersections of a curve $c$ with the
branch curve $D$ on $f$. As for the number $\De$ of curves $C$ with a
double point, a first set of these is given by the $\de$ curves
corresponding to the $c$ with a double point, each of which has be
counted $n$ times, since it has $n$ double points. But in general there
will be other $\De-n\de$ curves obtained from the $c$ that touch the
curve $D$, except those having an improper contact with $D$, that is,
those passing through a cusp of $d$, which corresponds to a point of $F$
where three of the $n$ corresponding points come together.

Thus, denoting by $\rho$ the effective genus of the branch curve $D$, and
by $\chi$ the number of its cusps, one has
 \[
 \De=n\de+2t+2\rho-2-\chi,
 \]
thus the Zeuthen--Segre invariant of $F$ is expressed by
 \[
 I=\De-M-4\Pi=n\de+2\rho-2-\chi-nm-4n (\pi-1)-4.
 \]

Now recalling Noether's relation
 \begin{eqnarray*}
 i &=& 12 p_a-p^{(1)}+9 \\
 I &=& 12P_a-P^{(1)}+9,
 \end{eqnarray*}
one deduces
 \[
 24(P_a+1)=24n (p_a+1)+6\rho-6+3s-2\chi.
 \]

{\em This relation connects the numerical genera of the two surfaces $f$
and $F$ in $(1,n)$ correspondence,} under the simplifying assumption that
{\em there are no fundamental points} of the correspondence on $f$. Here
again, {if there is no branch curve,} one has to set
 \[
 \rho=1,\quad s=0.
 \]

The formulas relating the numerical genera and the linear genera of two
surfaces $f$ and $F$ in $(1,n)$ correspondence can also be written in a
different form, introducing the degree $\nu$ of the curve $D$ instead of
the number $s$ that denotes the number of intersections of the branch
curve $D$ of $f$ with the canonical curves $k$: this is a virtual degree
expressed in terms of $s$ and of the virtual genus $\rho+\ep+\chi$ of
$d$, where $\ep$ denotes the number of nodes of $d$ and
$\chi$ the number of cusps. We have
 \[
 s=2(\rho+\ep+\chi)-2-\nu
 \]
and so {\em the formulas relating the characters of $f$ and $F$ become}
 \begin{eqnarray*}
 P^{(1)} &=& n(p^{(1)}-1)+4\rho+3(\ep+\chi)-\frac{3}{2}\nu-3 \\
 24(P_a+1) &=& 24n (p_a+1)+12\rho-12+6\ep+4\chi-3.
 \end{eqnarray*}

The first formula can be also deduced by comparing the degrees of the
canonical system on $f$ and $F$. We recall that one obtains a canonical
curve $K$ of $F$ by adding the branch curve $D$ to the transform of
a $k$ of $f$ that meets it at $s$ points; denoting by $x$ the degree of
$D$, we have
 \[
 P^{(1)}-1=n (p^{(1)}-1)+x+2s.
 \]

This formula agrees with the previous one, as one sees by computing
the value of $x$ which is
 \[
 x=\frac{\nu}{2}-(\ep+\chi).
 \]

In particular, if on $f$ there is a branch curve or one of its
components of degree $n$, which is completely free of double points, the
branch curve corresponding to it on $F$ has degree $\nu/2$. This
especially is clear in the case of a $(1,2)$ correspondence, $n=2$, where
in general the branch curve has no double points: this curve of degree
$\nu$ corresponds to a double curve of degree $2\nu$, the component of
which has therefore degree $\nu/2$.

It is interesting to compare the formulas we obtained here with the
formulas yielding the numerical genus and the linear genus of multiple
planes given in Section~\ref{sec:5.6}. We should find again the same
formulas, bearing in mind that the (relative) linear genus of the plane
is
$p^{(1)}=10$\footnote{See Chapter~II, Section~\ref{sec:2.6}; Chapter~V,
Section~\ref{sec:5.3}.}, while its geometric genus is $p_a=0$.

To do that, consider again the notation of formulas (B) of
Section~\ref{sec:5.6}, thus replacing $\rho$ with $P$, $\chi$ with $k$,
$\ep$ with $d$ and $\nu$ with
$N^2$, where
 \[
 N=2n+2\pi-2.
 \]

With this substitutions, our formulas yield
 \[
 P^{(1)}=9n+4P+3(d+k)-\frac{3}{2}N^2-3
 \]
and since
 \[
 P=\frac{(N-1)(N-2)}{2}-(d+k),
 \]
one deduces
 \[
 P^{(1)}=P-9\pi+9;
 \]
which is the first formula of (B).

{\em We have thus found again the formulas that give the characters of a
multiple plane.}

\begin{EXAMPLES} We use again the notation of this section and apply the
formulas thus obtained to some interesting examples.

Consider the {\em Kummer surface}, that is, the singular surface of the
quadric complex of lines in space, that is defined as a surface $f_4$ of
degree 4 with 16 double points\footnote{See Enriques--Chisini, Lezioni
etc., Libro~VI, \S40, Vol.~IV, page~246.}. It is known that $f_4$
coincides with its dual surface and has 16 bitangent planes; each of
these touches the surface along a conic containing 6 double points. This
follows immediately from the representation of $f_4$ on a double plane
with reducible branch sextic consisting of 6 lines; such a
representation is obtained by projecting $f_4$ from one of its double
points. Now consider 3 bitangent planes of $f_4$ passing through a
double point $A$; the pairwise intersections of these planes are three
lines, joining $A$ with three double points $B,C,D$; hence it follows
that the tetrahedron $ABCD$, whose vertices are double for $f_4$,
contains simply the remaining 12 double points of $f_4$, three on each
face.

Now denoting by $T(x,y,z)=0$ the equation of this tetrahedron, we
construct in the space $S_4(xyzu)$ the surface $F$ of equation
 \[
 u^2=T(x, y, z),\quad (f_4(x,y,z)=0),
 \]
 which is represented on the double surface $f_4$.

It is important to remark that there is no branch curve of order greater
than zero on this double surface. But there are 16 infinitesimal branch
curves, consisting of the neighbourhoods of the 16 double points,
through which $T$ passes an odd number of times. {From} this it follows
that our double $f_4$ is irreducible; on the other hand this also
follows from Comessatti's criterion\footnote{We recall that according to
Comessatti's criterion the necessary and sufficient condition for an
irreducible curve $C$, tangent to the branch curve $D_{2m}$ at all the
intersection points, to be the image of a reducible curve is that the
group of the points of contact belong to the series cut out on $C$ by
the curves $D_m$ that are halves of the branch curve.}, by remarking that
on every quartic plane section of $f_4$ there are 8 apparent critical
points occurring at the contact points of a quadrilateral, which do not
belong to a conic. In the $(1,2)$ correspondence between $f_4$ and $F$,
the system of plane sections of $f_4$, that has genus $\pi=3$ and degree
$n=4$, corresponds to a linear system of genus 5 and degree
$8=(2\cdot5-2)$ on $F$. One deduces that $F$ has, as well as the double
$f_4$, genus
 \[
 p_g=1
 \]
and has, as the double $f_4$, a canonical curve of degree zero, hence
all its plurigenera are equal to one.

Thus we a priori know that the (absolute) linear genus of $F$ is
$P^{(1)}=1$. But let us compute it by means of our formula:
 \[
 P^{(1)}=n(p^{(1)}-1)+4\rho+3(\ep+\chi)-\frac{3}{2}\nu-3.
 \]

To do this, remark that the branch curve $D$ on $f_4$ consists
of the 16 neighbourhoods of the 16 double points; each of these is an
infinitesimal curve of genus zero and degree $-2$ (with $\ep=\chi=0$).
Thus we find
 \[
 P^{(1)}=-4\cdot 5+\frac{3}{2}\cdot32-3=-15.
 \]

This paradoxical conclusion is clarified if one considers that a priori
each branch curve of genus zero and degree $-2$ corresponds to a curve of
genus zero and degree $-1$, which is exceptional, and which in our case
occurs as the neighbourhood of a simple point of $F$. To construct the
canonical curve of $F$, we have added the transform of the branch curve
to the transform of the canonical curve of degree zero of $f_4$; as a
consequence we have computed $P^{(1)}$ regarding the neighbourhoods of
the 16 simple points as exceptional curves. In other words, we have
calculated the linear genus of the transformed surface of $F$ with 16
exceptional curves. Eliminating these curves we find, as we should,
 \[
 P^{(1)}=1.
 \]

Then computing the numerical genus of $F$, we have
 \[
 24(P_a+1)=96-12\cdot15-12+3\cdot32=0
 \]
that is,
 \[
 P_a=-1.
 \]
\end{EXAMPLES}

\begin{Remark} As F.~Klein has shown\footnote{See Enriques--Chisini,
Ibidem}, the Kummer surface $f_4$ can be represented as an involution on
the surface $\Fie$ whose elements are the pairs of points of a curve $C$
of genus 2; precisely, regard as conjugated on $C$ the pairs of points
that are mutually residual with respect to a $g^2_4$. In this way one
defines an involution $I$ on $F$ that has 16 points of coincidence,
corresponding to the 16 pairs of $C$ that are obtained by bisecting the
$g^2_4$. Now, since $I$ is represented by $f_4$, one has that $\Fie$ is
represented by this {\em double Kummer surface, with $16$ branch points,
that occur at the $16$ double points.} Therefore one deduces that the
surface $\Fie$ is equivalent to the surface $F$ whose equation we have
written above. And thus {\em the hyperelliptic surface representing the
set of pairs of points of a curve of genus $2$ has genera
 \[
 p_g=1,p_a=-1
 \]
and all plurigenera equal to 1.} We shall come back again to this
surface later on\footnote{For the construction of the hyperelliptic
surface as a double Kummer surface with branch curve defined by the
tetrahedron $T$, as we have explained before, see Enriques--Severi,
M\'emoire sur les surfaces hyperelliptiques. Acta Math., 1909.}.
\end{Remark}

Another interesting example for our theory is the
surface\index{Enriques|surface} of genus $p_a=p_g=0$ and of bigenus
$P=1$, consisting of the sextic $f_6$ that passes doubly through the
edges of a tetrahedron $T$ (see Chapter~II, Section~\ref{sec:2.8}). We
construct the surface $F$ defined by the equations
 \[
u^2=T(x,y,z),\quad f_6(x,y,z)=0,
 \]
which is represented on the double $f_6$. Since $T$ and $f_6$ have the 6
edges of the tetrahedron $T$ counted twice in common, there is no branch
curve of order greater than zero on $f_6$. But there are no infinitesimal
branch curves either, namely no branch points; a priori these might
occur only at the vertices of $T$; but one of these vertices on $f_6$
comes from the coalescence of three simple points, whose infinitesimal
neighbourhoods belong to the plane faces of the tetrahedron passing
through $A$. Now since $A$ is 4-tuple for the section of $f_6$ with a face
$a$, the points on a that are infinitely near to $A$ will not be branch
points of our double $f_6$. Moreover we remark that the curves $C$ plane
sections of $f_6$ are irreducible plane curves (Comessatti's criterion),
hence they represent curves $C$ of genus $\pi=7$ and degree
$N=2\cdot6=2\pi-2$ on $F$. Since the double of $|C|$ coincides with the
double of the adjoint system:
 \[
 |2C'|=|2C|,
 \]
 one deduces that on $F$,
 \[
 |C'|=|C|
 \]
 hence the geometric genus of $F$ is
 \[
 P_g=1.
 \]
$F$ has a canonical curve of degree zero, so its linear genus is
 \[
 P^{(1)}=1
 \]
and all its plurigenera $P_i=1$.

We compare the characters of $F$ with those of the double surface $f_6$
that has
 \[
 p_a=p_g=0,\quad p^{(1)}=1
 \]
(see Chapter~II, 8). Our formulas give us
 \[
 \renewcommand{\arraystretch}{1.5}
 \begin{array}{c}
 n=2,\quad \rho=1,\quad \nu=0,\quad \ep=\chi=0, \\
 P^{(1)}=2(p^{(1)}-1)+4\rho-3=1
 \end{array}
 \]
and
 \[
 24(P_a+1)=2\cdot24(p_a+1)=48,
 \]
whence
 \[
 P_a=1.
 \]

{\em The sextic $f_6$ passing doubly through the edges of a tetrahedron
-- once desingularized -- is a regular surface of genus zero and bigenus
one with a bicanonical curve of degree zero, can be regarded as a double
surface with all the genera equal to 1 and with canonical curve of degree
zero\footnote{See F.~Enriques, Un'osservazione relativa alle superficie
di bigenere uno. {\it Rendic.\ Acc.\ Bologna}, 1908.}.}

 \section{The existence theorem}\label{sec:5.9}
 \markright{\hfill The existence theorem \quad 5.9\quad}
 Let us consider again the multiple planes. Whenever a surface $F$ with
given genera $p_a$ and $p^{(1)}$ is represented on a multiple plane of
some order $n$ by means of a net of curves of genus $\pi$ and degree
$n$,, we have learned how to compute the characters of the branch curve
of the $n$-tuple plane in terms of $n$, $\pi$, $p_a$ and $p^{(1)}$ (see
Section~\ref{sec:5.8}).

The formulas we obtained already show that the branch curve of a
$n$-tuple plane, for $n>2$, cannot be arbitrarily taken. Actually, given
any plane curve
 \[
 f(x,y)=0
 \]
one can construct the (cyclic) $n$-tuple plane
 \[
z^n=f(x,y)
 \]
whose branch curve is precisely $f=0$.

But the branch curve of this $n$-tuple plane is not only the irreducible
curve $f$, but the curve $f$ counted $n-1$ times plus the line at
infinity, counted one or several times, if the degree $m$ of $f$ is not a
multiple of $n$\footnote{To see this it suffices to consider an $n$-tuple
line or better to a $n$-sheeted Riemann surface, that is constructed by
taking $m$ branch points in the complex plane and associating to each of
these points the cyclic substitution $(123\cdots n)$ of the branches. A
loop surrounding all the $m$ branch points gives rise to a permutation
of the branches expressed by $(123\cdots n)^m$, and this is the
substitution corresponding to a loop around infinity, that therefore
turns out to be a branch point, unless the above said substitution is
the identity, i.e., unless $m$ is a multiple of $n$.}. Thus the branch
curve of a cyclic $n$-tuple plane is not at all an arbitrarily given
irreducible curve.

Since the branch curve of a $n$-tuple plane cannot be given arbitrarily
for $n>2$, the problem arises of {\em determining the conditions for a
plane curve $f(x,y)=0$ to be the branch curve of a $n$-tuple plane.}
These conditions lead to an {\em existence theorem} for multiple planes,
or for the algebraic functions of two variables. This theorem has a
meaning which is analogous, but under a certain respect different, from
the existence theorem for the $n$-tuple lines or for the algebraic
functions of one variable, given by Riemann\footnote{See, for instance,
Enriques--Chisini, Lezioni, Libro~V, Chapter~III, \S33 (Vol.~III,
page~355).}. Indeed in Riemann's case the group of the branch points can
be given arbitrarily, and in correspondence with it one has a finite
number of classes of algebraic functions (or of $n$-tuple lines), each
depending on the assigned substitutions of the branches with respect to
a system of loops surrounding the branch points; here instead, as we
have said, the question is to determine the particular conditions that
the branch curve has to satisfy for the existence of the $n$-tuple
plane. When these conditions are satisfied, there will be, at least in
general, only one class\footnote{See: O.~Chisini, Sulla identit\`a
birazionale delle funzioni algebriche di due variabili dotate di una
medesima curva di diramazione. {\it Rendic.\ Ist. Lombardo di Scienze e
Lettere}, 1944.} of surfaces represented on the plane, namely a unique
$n$-tuple plane, corresponding to one, well determined, system of
substitutions with respect to the loops surrounding the branch curve,
that we shall describe later.

Nevertheless, the algebraic meaning of the question is analogous to
case of the construction of an algebraic function of one variable with
prescribed branch points. When the surface $F(x,y,z)=0$ is projected from
the point at infinity $Z$ of the $z$-axis onto the plane $z=0$, in
general one has in this plane a curve which is the locus of the critical
points of $z(x,y)$, defined by the vanishing of the discriminant of this
function
 \[
 \De(x,y)=0
 \]
 that is the resultant of the two equations
 \[
 F(x,y,z)=0,\quad \frac{\partial F}{\partial z}=0.
 \]

But this curve $\De=0$ in general breaks up into two parts,
 \[
 \De=\De_1\cdot(\De_2)^2,
 \]
one of which, $\De_2=0$, is an apparent critical curve, projection of the
double curve of $F$, while the other component, $\De_1=0$ (the so-called
{\em principal part of the discriminant}) is the actual branch curve of
the algebraic function $z(x,y)$. Our problem consists to {\em assign
the conditions in order that a curve be the branch curve of an algebraic
function of two variables} and thus to {\em construct this algebraic
function with given branch curve}. The formulation of this problem can
be made more precise by assuming that the plane curve $f$, that one wants
to be the branch curve of a $n$-tuple plane, is a curve of even degree
$2m$\footnote{If the degree of $f$ were odd, one would necessarily have
to add the line at infinity to it.}, projection from a point $O$ of the
branch curve $C$ of a surface $F$; so $f$ has a certain number $d$
of nodes and a certain number of cusps, corresponding to the apparent
double points of the aforementioned curve $C$: in addition to these
double points, $f$ can have nodes and cusps corresponding to isolated
double points of $F$ (conical points and biplanar points respectively);
but these double points of $f$ can be left out of consideration by
regarding them as virtually nonexistent.

Now let us consider in the plane of the given curve $f$ a pencil of
lines through a point that, for the sake of simplicity, we may assume
not to be on $f$, say for instance the pencil $y=tx$ through the origin
$O$. For a generic value of $t$, one has an $n$-tuple line on which there
are $2m$ simple branch points; in the representing plane of the complex
variable $x$ one takes $2m$ loops issuing from the point $O$ and surrounding
these $2m$ points; one can assign arbitrarily $2m$ transpositions of the
branches, corresponding to the above-mentioned loops, with the condition
that the group generated by them is transitive and that their product is
the identity; in this way one finds a finite number of irreducible
curves, in general birationally distinct, that are represented on the
$n$-tuple line $y=tx$ and are all of genus
 \[
 \pi=\frac{m}{2}-n+1;
 \]
 these curves are defined up to birational transformations, that is, they
correspond to well determined, and in general distinct, classes of
curves or algebraic functions.

The number $r$ of the above curves $C_1,C_2,\dots,C_r$ has been
determined by Hurwitz, but it is not necessary to know it here. It is
important instead to remember that, when we let $t$ vary along a closed
path in the plane $\tau$ of the complex variable $t$, the curves
$C_1,C_2,\dots,C_r$ (or the classes of curves to which they belong) will
in general be permuted: this happens first of all because the $2m$
branch points intersections of the line $y=tx$ with $f$ are permuted,
and also because the system of loops, by means of which the $n$-sheeted
Riemann surface $C_1$ is constructed, can be continuously transformed in
a new system of loops, that can be reduced to the first one by suitably
changing the corresponding transpositions. Now if $f$ is the branch
curve of a $n$-tuple plane, projection of a rationally determined
surface $F$, one of the above curves, say $C_1$, must be a section of
$F$. Hence it must be rationally determined and therefore its Riemann
surface must not be interchanged with those of $C_2,\dots,C_r$ but must
be transformed again into $C_1$ whenever $t$ varies along a closed loop
in the plane of this complex variable.

The main question is to establish the {\em conditions for the invariance
of (the class determined by) $C_1$, or rather of its Riemann surface,
with respect to all the possible closed paths along which $t$ can vary
in the plane $\tau$}. These conditions will certainly be necessary
conditions for the existence of a $n$-tuple plane with $f$ as branch
curve.

Let us narrow down the meaning of our research. We start by choosing an
initial value of $t$, say $t=0$, and assigning an ordered system of
loops in the plane of the complex variable $x$, issuing from the point
$O$ and going around the branch points $A_1,A_2,\dots,A_{2m}$
(intersections of $f$ with the line $y=0$); then, according to a given
labelling $1,2,\dots,n$ of the branches of the algebraic function $z(x)$,
we keep track of the transpositions that correspond to the above loops
$OA_i$, thus defining the $n$-sheeted Riemann surface that represents
our curve $C_1$ (the one for which we are investigating the invariance
conditions with respect to the closed paths along which $t$ varies in
the plane $\tau$).

If we fix the initial Riemann surface, among those representing the
multiple lines $y=tx$, we let $t$ vary continuously (starting at the
point $t=0$) along a closed path in the plane $\tau$ of this variable; as
$t$ varies in this way, $C_1$ also varies continuously, and its Riemann
surface can be still constructed with sheets covering the same plane of
the complex variable $x$, since (in the language of algebraic geometry)
the multiple line $y=tx$ -- having the intersections with $f$ as branch
points -- can be replaced in general with its projection on the $x$-axis
from the point at infinity of the $y$-axis.

Now, following a continuous variation of $t$ in the plane $\tau$, which
avoids the positions of $t$ where two points $A_r$ and $A_s$ come
together, we may see the points $A_1,A_2,\dots,A_{2m}$ move continuously
in the plane of the complex variable $x$ without ever coming together,
and describe arbitrary (closed or open) paths. Moreover one may let the
loops $OA_i$ around them vary, still continuously, so that they keep
their ordering and are simple open paths, not crossing each other.

Then consider, in the plane $\tau$, a closed path along which $t$ moves,
starting at the origin $t=0$. If this loop does not surround any critical
point corresponding to the fact that two points $A_i$ come together, it
can be continuously deformed into the constant loop, therefore the
ordered system of loops and the corresponding substitutions that define
the Riemann surface $C_1$ have to be preserved along this loop: thus the
above cycle does not give rise to any condition for the invariance of
$C_1$.

Therefore the conditions for the invariance of $C_1$ only depend on the
elementary loops of the plane $\tau$ going along a critical point $t$,
corresponding to which two of the points $A_i$ come together.

Now these loops may be of three different types:

 \begin{enumerate}
 \renewcommand{\labelenumi}{(\arabic{enumi})}
 \item loops surrounding a point $T$, corresponding to a simple tangent of
the curve $f$ which passes through the point $O$ that does not lie on $f$,

 \item loops surrounding a point $D$, corresponding to a line joining $O$
with a node of $f$; and

 \item loops surrounding a point $Q$, corresponding to a line joining $O$
with a cusp of $f$.
 \end{enumerate}

In each of these three cases, it is easy to determine the invariance
conditions for the Riemann surface $C_1$, if one starts from the simple
situation in which the two points $A_1A_2$ that come together at the
critical point ($T$ or $D$ or $Q$) belong to loops $OA_1$ and $OA_2$
contiguous and {\em honestly near}: by this we mean that in the limit
(when $A_1$ and $A_2$ come together) the two paths $OA_1$ and $OA_2$,
having the common endpoint $A_1=A_2$, bound a surface which does not
contain any other point $A_i$. So these two paths can get infinitely
near without crossing any other loop of the variable Riemann surface,
hence without changing the substitutions of the branches of $z(x)$
corresponding to them.

Let us examine the three cases defined above:

 \begin{enumerate}
 \renewcommand{\labelenumi}{(\arabic{enumi})}
 \item If $A_1$ and $A_2$ come together at a point $T$, then on the limit
multiple line $OT$ the two branch points $A_1$ and $A_2$ disappear,
giving rise to one apparent critical point that is the image of a double
point of $C_1$ (and when $t$ goes around the loop they are
interchanged); this means that the two transpositions corresponding to
the two (honestly near) points $A_1$ and $A_2$ must be the same.
Conversely, if this happens, e.g., if the same transposition $(12)$
corresponds to both $OA_1$ and $OA_2$, then the Riemann surface $C_1$ is
invariant with respect to the loop surrounding $T$ in the plane $\tau$.

 \item If $A_1$ and $A_2$ come together at a point $D$, then on the limit
multiple line $OD$ the two branch points $A_1$ and $A_2$ coincide at a
point that is the image of two distinct points of $C_1$. This means that
the transpositions corresponding to the two (honestly near) loops $OA_1$
and $OA_2$ must act on two distinct pairs of branches, that is, they must
commute and be in general disjoint like $(12)$ and
$(34)$.\footnote{Instead, they will be the same, and therefore they will
again commute, if the point $D$ is a virtually nonexistent node of
$f$.} Conversely, if this happens, then the Riemann surface $C_1$
is invariant with respect to the loop surrounding $D$ in the plane
$\tau$.

 \item Finally if $A_1$ and $A_2$ come together at a point $Q$, then on the
limit multiple line $OQ$ the two branch points $A_1$ and $A_2$ coincide
at the point $Q$ and they are interchanged when $t$ goes around the loop;
but the point $Q$ becomes the image of a flex of $C_1$, so that the
transpositions corresponding to $A_1$ and $A_2$ are such that their
product has to be a cyclic substitution of order three, hence they act
on two pairs of branches that have a point in common, namely that are
linked like $(12)$ and $(23)$.
 \end{enumerate}

Conditions (1), (2), and (3) ensure that the Riemann surface $C_1$ is
invariant with respect to the loops surrounding the points $T,D$ and $Q$
in the plane $\tau$ when the two points $A_r,A_s$ that come together are
the end points of loops $OA_r$ and $OA_s$ that become honestly near. The
case in which $OA_r$ and $OA_s$ do not become honestly near can be
reduced to the previous one: indeed in this case it happens that the two
aforementioned paths having the common endpoint $A_r=A_s$ form a closed
path that bounds a surface containing some other points $A_i$; but it is
always possible to let the line $OA_r$ become infinitely near to the
line $OA_s$ by letting it vary continuously and cross the above-said
points $A_i$: this means that, as a consequence of this reduction, also
the transposition corresponding to the loop $OA_i$ is
modified\footnote{See, e.g., Enriques--Chisini, Lezioni, Libro~V, \S3,
Vol.~III, page~18.}.

We can summarize the result obtained by saying that: {\em the conditions
for the invariance of the Riemann surface $C_1$,when one considers it as
varying in the pencil of $n$-tuple lines $y=tx$ with branch points on the
curve $f$, correspond to the loops, described by $t$ in the plane $\tau$,
respectively surrounding the critical points $T,D$ and $Q$ {\em (points
that correspond to the proper tangents and to the lines joining $O$ with
the nodes and the cusps of the branch curve $f$);} these conditions are
expressed by saying that the transpositions corresponding to the branch
points $A_1$ and $A_2$ that come together at one of those critical points
must be respectively the same or must commute (and be in general
disjoint) or must be linked, when the system of loops that defines the
Riemann surface is transformed in such a way that the loops surrounding
the above mentioned points, $OA_1$ and $OA_2$, become honestly near.}

The conditions stated above are necessary conditions in order that
the curve $f$, with nodes and cusps, is the branch curve of an $n$-tuple
plane ($n>2$). We want to show that they are also {\em sufficient} for
the existence of such an \hbox{$n$-tuple} plane.

To do this, consider the pencil of planes through the $z$-axis,
whose intersections with the plane $(z=0)$ of the curve $f$ are the lines
$y=tz$. We want to construct in each of these planes a well defined
curve, that, by projection from the point at infinity on the $z$-axis,
call it $Z$, gives rise to the $n$-tuple line $C_1$ with branch points at
the intersections $A_1,A_2,\dots,A_{2m}$ with the curve $f$. $C_1$, as we
have said before, is defined as a Riemann surface in an invariant way
with respect to the closed paths of $t$ in the plane; thus one can
construct an irreducible projective image of $C_1$, that is, a curve
$C^{(n+h)}$, of some large enough order $n+h$, passing $h$ times through
the point $Z$ and touching the lines $ZA_i$. One can also impose that
such a curve $C^{(n+h)}$ passes through $n$ distinct points
$O_1,O_2,\dots,O_n$ of the line $OZ$ and that moreover it satisfies a
certain number of linear conditions, depending on $h$. Thus one finds a
finite number of irreducible curves $C^{(n+h)}$, and among these a
finite number $s$ of curves represented by the Riemann Surface $C_1$,
corresponding to $s$ algebraic functions
$z_1(x),z_2(x),\dots,z_s(x)$ branched in the same way\footnote{See
Enriques--Chisini, Lezioni. Libro~V, \S33, Vol.~III, page~355.}; then
the sum
 \[
z_(x)=z_1(x)+z_2(x)+\cdots+z_s(x),
 \]
also corresponds to an irreducible curve $C$ represented on the Riemann
surface $C_1$ and rationally defined. Letting this $C$ vary with the
parameter $t$, we shall obtain a surface $F$ represented $n$-tuply on
the plane $z=0$ and with $f$ as branch curve. It is obvious that $f$ is
contained in the branch curve of the $n$-tuple plane obtained by
projection of the surface $F$ from the point $Z$; nevertheless one may
doubt that a certain number of branch lines through $O$ should be added
to it: but if it happened, then some plane through the line $OZ$ should
touch $F$ along a curve, and this is impossible if the points
$O_1,O_2,\dots,O_n$ are distinct, as we have assumed.

In conclusion: {\em The conditions for the existence of an $n$-tuple
plane ($n>2$) with assigned branch curve $f(x,y)=0$, with nodes and
cusps {\em (and not passing through the origin $O=[0,0]$)} are expressed
by the invariance conditions of one of the Riemann surfaces represented
on the multiple line $y=tx$, with respect to the closed paths of the
parameter $t$ in the plane of its complex variable, surrounding the
points corresponding to the tangents of $f$ through $O$ or to the lines
joining $O$ with the nodes or the cusps of $f$.}

It is essential to remark that the conditions required for the existence
of a multiple plane with some branch curve $f$ are {\em qualitative
conditions,} that cannot vary when $f$ varies continuously. Thus the
existence theorem that we have established gives rise to the following:

 \begin{Corollary} Given a continuous system of curves $f$ with nodes and
cusps the plane, if one of these curves is the branch curve of an
$n$-tuple plane, then the same holds for all the curves of the system.
 \end{Corollary}

 \section{Complements: limit forms of the branch curve according to
Chisini}\label{sec:5.10}
 \markright{\hfill Complements: limit of the branch curve
according to Chisini \quad 5.10\quad}
 The existence theorem for multiple planes proved in the previous section
with its corollary, has been established by F.~Enriques in the paper
Sulla costruzione delle funzioni algebriche di due variabili
possedenti una data curva di diramazione, in {\it Ann.\ di Mat.}, s.~IV,
t.~I (1923).

There are also some interesting developments related to this theorem.
First of all O.~Zariski\footnote{O. Zariski, On the problem of existence
of algebraic functions of two variables possessing a given branch curve.
Amer.\ J. Math., Vol.~51 (1929). On the linear connection index of the
algebraic surface $z^n=f(x,y)$. Proc.\ Mat.\ Acad.\ Sci.\ U.S.A., Vol.~15
(1929). On the irregularity of cyclic multiple planes, {\it Ann.\ of
Math.}, s.~II, Vol.~32 (1931). See Algebraic surfaces, Berlin, Springer,
1935.} has put Enriques' existence conditions in a topological group
theoretic form, from which he deduces in particular that a curve without
cusps cannot be the branch curve of a multiple plane of degree $n>2$,
except in the case of a curve counted $n-1$ times that corresponds to a
cyclic
$n$-tuple plane.

But, both in Enriques' form and in Zariski's form, the existence theorem
for multiple planes gives rise to great difficulties in its concrete
application, since it relies on the solution of the further problem of
determining the branch curves satisfying the given conditions. An
important contribution to this problem has been given by
O.~Chisini\footnote{O. Chisini, Un teorema di esistenza dei piani
multipli. Notes~I and~II, {\it Atti Acc.\ Lincei}, VI, 19 (1934). Un piu'
generale teorema di esistenza dei piani multipli, {\it Atti Acc.\
Lincei}, VI, 27, (1938). Altre curve di diramazione dei piani $n$-pli.
{\it Atti Acc.\ Lincei}, VI, 29 (1939).}, with the investigation of the
{\em limit forms} contained in a continuous system of branch curves with
a certain number of nodes and cusps.

To understand Chisini's research, consider a reducible surface $F$ of
degree $m+h$, with an $h$-tuple point $Z$, that is the union of two parts
$f_1$ and $f_2$ meeting the lines through $Z$ at $n$ and $s$ points
respectively (with $m=n+s$). Here the branch curve of the $m$-tuple
plane, on which $F$ is projected from $Z$ breaks up as the union of the
two branch curves of the $n$-tuple plane and of the $s$-tuple plane
corresponding to $f_1$ and $f_2$, and of the double curve projection of
the common curve of $f_1$ and $f_2$, which becomes an apparent branch
curve. This remark leads to the question whether it is possible to
construct the branch curve of a $m$-tuple plane, with $m=n+s$, by adding
to the branch curves of the $n$-tuple plane and of the $s$-tuple plane a
double curve that has to have a suitable behaviour with respect to them,
so that the curve sum of the three, constructed as we have said,
represents the limit curve of a continuous system of branch curves of
$m$-tuple planes, having a certain number number of nodes and cusps.

What we have said explains the origin of Chisini's idea. Still in
general the search for the double curves to be added to the branch
curves of a \hbox{$n$-tuple} plane and of a $s$-tuple plane gives rise
again to great difficulties. To overcome them, Chisini restricts himself
to the case in which one has, for instance, $s=1$, that is, one has only
the branch curve of a $n$-tuple plane, from which, by adding a suitable
double curve, one wants to obtain the branch curve of a $(n+1)$-tuple
plane. It will suffice to explain Chisini's construction by considering
a simple example.

We want to construct the system of the branch curves of the quadruple
planes that arise by projecting the general surface $F_4$ of degree 4.
These branch curves are $C_{12}$ of order 12 with $d=12$ nodes and $k=24$
cusps.

If we let the surface $F_4$ pass through the centre of projection $Z$,
then the branch curve $C_{12}$ of the quadruple plane degenerates as the
union of the branch curve $C_{10}$ of a triple plane, having $k'=18$
cusps (and no more nodes), and of a line $C_1$ doubly tangent to
$C_{10}$, counted twice. In turn $C_{10}$ can degenerate as the union of
a completely general sextic $C_6$, branch curve of a double plane, and a
conic tangent to it at six points, counted twice. This happens when one
imposes that $F_4$ has a double point at $Z$. Finally if the double point
$Z$ of $F_4$ becomes a triple point, then the branch curve $C_6$ of the
aforesaid double plane degenerates as a cubic counted twice.

Now it is useful to examine what the singularities of the branch
curve become like, following the subsequent degenerations shown above.

To start with, when the aforementioned $C_{12}$ degenerates as
$C_{10}+2C_1$, each of the two contact points $P_1$ and $P_2$ of $C_1$
and $C_2$ appears as the limit of three cusps of $C_{12}$, and each of
the remaining 6 common points of $C_1$ and $C_{10}$ turns out to be the
limit of two nodes of $C_{12}$. The statement can be justified by
referring to the surface $F_4$ passing through the centre of projection
$Z$; one has to take into account that the cusps of $C_{12}$ are
obtained by projecting from $Z$ the intersections of $F_4$ with the
first and the second polar of the point $Z$, and that the nodes of
$C_{12}$ come from the bitangents of $F_4$ passing through $Z$, which
now become the six lines passing through $Z$ and tangent at some other
point to the quartic curve intersection of $F_4$ with the tangent plane
at $Z$, each counted twice.

What we have remarked about the degeneration of the singularities of
$C_{12}$ when it becomes a $C_{10}+2C_1$ still holds for $C_{10}$ that
degenerates as $C_6+2C_2$; this means that each of the 6 points of
contact of $C_2$ and $C_6$ absorbs 3 cusps of $C_{10}$.

We have noticed that it is possible to let the branch curve $C_{12}$ of a
quadruple plane representing a surface $F_4$ degenerate first as a
$C_{10}+2C_1$, then as a $C_6+2C_2+2C_1$, and finally also, if one wants,
as a $2C_3+2C_2+2C_1$, where $C_{10}$ is the branch curve of a triple
plane, bitangent to the line $C_1$, and $C_6$ (branch curve of a double
plane) is tangent to the conic $C_2$ at 6 points. Now this analytic
method of reduction can be reversed, thus leading to a synthetic method
of construction of the multiple planes, that is precisely Chisini's
existence theorem. Indeed the author goes back from the limit forms to
more general forms of multiple planes by recovering the triples of cusps
and the pairs of double points that are absorbed by the limit
singularities. Thus, starting with the double plane with branch sextic
$C_6$ and specializing $C_6$ so that it admits a conic $C_2$ tangent to
it at six points, one will consider $C_6+2C_2$ as a special case of
$C_{10}$, for which the conditions of Enriques' explained in the
previous section are satisfied, therefore it turns out to be the branch
curve of a triple plane. And then, adding again a bitangent $C_1$ to
$C_{10}$, one proves in a similar way that $C_{10}+2C_1$ (or
$C_6+2C_2+2C_1$) is the limit form of $C_{12}$ which is the branch curve
of a quadruple plane, representing a surface $F_4$ with plane sections of
genus 3. It is needless to say that the case that we have described is
just a special example, but rather significant, of a method that enables
the author to prove the existence conditions for a larger family of
multiple planes. Here we do not intend to describe Chisini's more
general developments, but we shall restrict ourselves to explain his
construction method, and also the subtle questions which it poses, by
considering a notable family of surfaces that includes the quartics
$F_4$, and that we shall have to consider again later on in these
lectures: we are talking about the surfaces with genera $p_a=p_g=1$ and
$p^{(1)}=1$ with a canonical curve of degree zero.

According to the formulas of \S6, one obtains surfaces belonging to
the aforementioned family by starting with the $n$-tuple multiple planes
with $n=2\pi-2$, defined by a branch curve $C_{3n}=C_{6\pi-6}$ of degree
 \[
2n+2\pi-2=3n=6\pi-6,
 \]
with $k(\pi)=24(\pi-2)$ cusps and $d(\pi)=18\pi_2-78\pi+84$ nodes:
actually such a $n$-tuple plane (if it exists) has genera
$p_a=p^{(1)}=1$ and also $p_g=1$, since it has a canonical curve of
order zero; and the images of the lines in the plane belong to a linear
system of genus $\pi$ and degree $n=2\pi-2$, and thus of dimension
$\pi$. Thus the $n$-tuple plane arises as a projection, from points not
on the surface, of a surface $F_n$ of order $n=2\pi-2$ of $S_\pi$, with
canonical hyperplane sections of genus $\pi$.

Now we shall try to prove here the existence of $n$-tuple planes
corresponding to the branch curves $C_{3n}$ mentioned above, and thus of
the surfaces $F_n=F_{2\pi-2}$ of $S_\pi$ with canonical sections of genus
$\pi$, for all the values of $\pi=3,4,5$,\dots We have already shown in
the previous discussion how one can construct the branch curve $C_{12}$
of a quadruple plane (representing a surface $F_4$) corresponding to
$\pi=3$, starting with the double plane with a sextic $C_6$ branch curve,
tangent to a conic $C_2$ at six points. Now we shall proceed in a
similar way to construct the sextuple plane corresponding to $\pi=4$,
starting with a $C_{12}$ tangent to a conic $C_2$ at six points: we
shall investigate whether $C_{12}+2C_2$ is the degenerate form of a
$C_{16}$, a branch curve of a quintuple plane, and whether one obtains a
$C_{18}$, a branch curve of a sextuple plane, by adding to it a
bitangent line $C_1$ counted twice. If this happens, the aforementioned
$C_{12}$ has $k(\pi)=k (3)=24$ cusps and $d(\pi)=d(3)=12$ nodes, whereas
the $C_{16}$ has 18 more cusps (3 cusps corresponding to each of the
contact points of $C_{12}$ and $C_2$) and 24 more nodes (2 nodes arising
from each of the 12 simple intersections of $C_{12}$ and $C_2$), hence
it has 42 cusps and 36 nodes in total; then adding to $C_{16}$ the
bitangent $C_1$ counted twice, one obtains a $C_{18}$ with 6 more cusps
and 24 more nodes, and so
 \[
 k(4)=\hbox{48 cusps \quad and\quad}
 d(4)=\hbox{60 nodes}.
 \]

To continue this construction in general for the higher values of $\pi$,
it is convenient to use a fact that will be proved in the following
section, namely that the surfaces $F_n$ (of genera $p_a=p_g=p=1$,
$p^{(1)}=1$) depend on at least $9p-2p^{(1)}+12=19$ essential parameters
or moduli,\index{moduli} with respect to birational transformations.
Therefore given a multiple plane defined by a branch curve
$C_{3n}=C_{6\pi-6}$ with the above characters, since we have a free
parameter, it is possible to specialize the $C_{3n}$ in such a way that
it has a conic $C_2$ tangent at six points. In order to follow Chisini's
method we should ask whether $C_{3n}+2C_2$ can be regarded as the limit
form of a $C_{3n+4}$, having, besides the cusps and nodes of $C_{3n}$,
18 cusps and $12n-24=24(\pi-2)$ nodes corresponding respectively to the
6 contact points of $C_2$ and $C_{3n}$ and to the $6n-12$ further
intersections of these curves; and we should also ask whether the
$C_{3n+4}$ thus constructed are branch curves of $(n+1)$-tuple planes.

If the answer is affirmative (so that the invariance conditions of
the previous section are satisfied) the $C_{3n+4}$ thus constructed turn
out to be branch curves of $(n+1)$-tuple planes. And, as we shall see, by
adding a suitable bitangent line $C_1$ to them, one obtains the branch
curve of a $(n+2)$-tuple plane, representing a surface $F_{n+2}$ of the
space $S^{\pi+1}$, with canonical sections of genus $\pi+1$.

But the question formulated above does not always has a positive answer.
In order that $C_{3n}+2C_2$ be actually a limit form of a $C_{3n+4}$
satisfying our conditions, it is necessary that its singularities can be
deformed to nodes and cusps in the way indicated before, and that the
curve $C_{3n+4}$ thus constructed satisfies the invariance conditions for a
branch curve of a $(n+1)$-tuple plane. Now these conditions are not
satisfied for an arbitrary conic tangent to $C_{3n}$ at six points; but
considering, for instance, to the case $n=4$, they can be satisfied by
reducing $C_{12}$ to Chisini's limit form $C_{12}=2C_2+2C_1+C_6$ and
assuming then that there is another conic, different from $C_2$, tangent
to $C_6$ at six points. Thus it is clear that the difficulty can be
overcome for small values of $n$, but it is not easy to extend the
argument to larger values of $n$.

However, if one could show that the curve $C_{3n}+2C_2$ is the limit form
of the branch curve of a $(n+1)$-tuple plane, next step is to add again a
bitangent line $C_1$ to it, chosen in such a way that the curve
$C_{3n}+2C_2+2C_1$ is a limit form of the branch curve of a $(n+2)$-tuple
plane. But here we do not need to repeat the argument we made the case
of $n=4$ for the choice of the $C_2$ tangent to $C_{12}$ at six points.
The difficulty in the choice of $C_1$ can be overcome at once thanks to
the following simple remark: the genera of the $(n+1)$-tuple plane defined
by the branch curve $C_{3n+4}$ are equal to one and therefore such a
multiple plane represents a $F_{n+2}$ (with $n=2\pi-2$) with canonical
sections of genus $\pi+1$, projected from one of its simple points; this
point corresponds to an exceptional line, bitangent to $C_{3n+4}$, which
is the required $C_1$.

What we have said suffices to explain Chisini's constructive method,
and to test somehow its meaning by examining a concrete example. About
this problem -- that is, the existence and the classification of surfaces
$F_n$, with all genera equal to one -- we shall give a direct solution of
it in \S3 of Chapter~VII, where there is also a historical note on the
subject.

We add some more references to those already mentioned. In the
quotations we made above we referred to $n$-tuple planes with $n>2$, but
there are some special researches in the case of double planes:
F.~Enriques, Sui piani doppi di genere 1, {\it Mem.\ della Societ\`a It.\
delle Scienze}, detta dei XL, Roma, 1896: Sui piani doppi di genere
lineare $p^{(1)}=1$, {\it Rendic.\ Acc.\ Lincei}, 1898. L.~Campedelli,
Sopra i piani doppi con tutti i generi uguali all'unit\`a, {\it Rendic.\
del Sem. Mat.\ di Padova}, Vol.~XI, 1940; Le superficie con i generi
uguali all'unit\`a, rappresentabili in infiniti modi sul piano doppio,
{\it Rendic.\ Sem.\ Mat.\ R. Universit\`a di Roma}, s.~V, Vol.~I, 1940
and La classificazione dei piani doppi con tutti i generi uguali
all'unit\`a, {\it Atti del 2$^o$ Congresso dell'Unione Mat.\ Ital.,
Bologna}, 1940; Sui piani doppi con curva di diramazione dell'ottavo
ordine, {\it Rendic.\ Acc.\ Lincei}, 1932; Sui piani doppi con curva di
diramazione del 10$^o$ ordine, ibidem, 1932; Sopra alcuni piani doppi
notevoli con curva di diramazione del 10$^o$ ordine, ibidem, 1932.

The case of double planes is somehow close to the study of those
particular quadruple planes that depend on Abelian equations, namely on
extracting two square roots: P.~Libois, {\it Rendic.\ Acc.\ Lincei},
1934.

The following papers deal with notable classes of $n$-tuple multiple
planes, with $n>2$: B.~Segre, Sulla caratterizzazione delle curve di
diramazione dei piani multipli generali, {\it Rendic.\ R. Acc.\
d'Italia}, Vol.~I, 1930, G.~Pompilj, Sulla rappresentazione algebrica
dei piani tripli, {\it Rendic.\ Sem.\ Mat.\ R. Universit\`a di Roma},
1939; Osservazioni sui piani tripli, {\it Rendic.\ Ist.\ Lombardo},
1941; Sui piani multipli diedrici, {\it Rendic.\ Sem.\ Mat.\ R.
Universit\`a di Roma}, 1939.

 \section{The moduli of a class of algebraic surfaces}\label{sec:5.11}
 \markright{\hfill The moduli of a class of algebraic surfaces \quad
5.11\quad}
 We have considered so far multiple planes with simple branch curve, on
which in general it is possible to represent an algebraic surface, and
we explained in the previous sections the corresponding existence
theorem. Using these concepts we can now say something about the number
of {\em moduli} or parameters on which depend the surfaces of a given
class varying in a continuous family corresponding to certain given
values of the characters $p$ and $p^{(1)}$. For the sake of simplicity
we assume that the surfaces are regular $p=p_g=p_a$, and that they have
some plurigenus $P_i>0$, so that they can be transformed in such a way
that the exceptional curves can be eliminated and $p^{(1)}$ becomes the
absolute linear genus.

On a surface $F$ satisfying the conditions stated above, let us choose
arbitrarily an irreducible, base point free, linear system $|C|$, of
genus $\pi$ and degree $n\le2\pi-2$, that we assume regular and
nonspecial, thus of dimension
 \[
 r=p+n-\pi+1.
 \]

By means of a generic net contained in $|C|$, $F$ can be represented on a
multiple plane of order $n$, with branch curve $D$ of order $2n+2\pi-2$,
that has a certain number $d$ of nodes and a certain number $k$ of
cusps. These numbers can be expressed in terms of $p$ and $p^{(1)}$
using the formulas of \S6. Now we have seen that, if we let the curve
$D$, of order $2n+2\pi-2$, vary in a continuous system of curves still
having $d$ nodes and $k$ cusps, then this is again the branch curve of a
$n$-tuple plane, corresponding to a surface with the same characters $p$
and
$p^{(1)}$ as $F$. But not all the curves of this system, of dimension
 \[
s\ge(n+\pi-1)(2n+2\pi+1)-d-2k,
 \]
correspond to surfaces birationally distinct from $F$. First, the
$\infty^8$ homo\-graphic transforms of $D$ give rise to the same $F$, and
to the same net of curves $|C|$ on it; secondly, the $\infty^r$ system
$|C|$ contains infinitely may nets, depending on $3r-6$ parameters. Thus
we have at least
 \[
 (n+\pi-1)(2n+2\pi+1)-d-2k-8-(3r-6)
 \]
parameters for the surfaces, that must be regarded as birationally
distinct from $F$. Indeed if $\infty^1$ curves $D$ not corresponding to
nets contained in $|C|$ give rise to surfaces that are birationally
equivalent, this means that $F$ has a continuous series of inequivalent
nets of curves, containing a net of $|C|$, against the assumption that
the surface is regular ($p_a=p_g$).

To be precise, one might object to this argument that a net of $C$, say
$|C|^2$, may appear as the limit of a net of curves $|L|2$, of higher
order and with base points, as well as the net of the lines in the plane
appears as the limit of a net of conics with three base points. But this
doubt can be easily excluded for the surfaces $F$ we are considering, on
which the degree of a linear system cannot exceed twice the genus minus
two, and that therefore can be regarded as free from exceptional curves.
Actually if a net $|L|_2$, of effective genus $\pi$ and effective degree
$n$, has some base point of order $i\ge1$, then the virtual intersection
number of the $L$ with the canonical curves $K$ is $2\pi-2-n-\sum i^2$,
and in the limit they have to become reducible as curves $C$ plus a
fixed component $\theta$, that must intersect the $K$ in a negative
number of points, $-\sum i^2$. Thus this curve $\theta$ has a certain
genus $\rho$ and degree
 \[
 \nu=2\rho-2+\sum i^2>2\rho-2;
 \]
and the presence of such a curve on the surface $F$ contradicts the
assumption we made, that $F$ can be transformed in such a way that it has
no exceptional curves. In order to show the correctness of our
computation (that is, the statement that there are $\infty^{3r-6+8}$
birationally equivalent multiple planes), it is convenient to add here
that $F$ cannot admit a continuous series of birational transformations
into itself; actually this would imply that: either the complete system
$|C|$ belongs to a larger series of inequivalent curves, against the
assumption $p_a=p_g$, or $|C|$ would be transformed into itself, against
the assumption that $F$ does not belong to the family of ruled
surfaces\footnote{Indeed the surfaces that have a continuous family of
projective transformations are rational or ruled. Theorem of
Enriques--Fano. See Enriques--Conforto, {\it Le superficie razionali},
quoted work, Chapter~VI, \S37, page~496.}.

We are now able to write down the number $M$ of moduli on which the
class of surfaces $F$ depends, or at least a lower bound for it. Indeed
we have
 \[
 M\ge (n+\pi-1)(2n+2\pi+1)-d-2k-8-(3r-6)
 \]
 that is,
 \[
 M\ge (n+\pi-1)(2n+2\pi+1)-d-2k-3r-2.
 \]

Let us replace in this formula $d$ and $k$ with their expressions in
terms of $p$ and $p^{(1)}$ given by the formulas of \S6:
 \begin{eqnarray*}
 d &=& 2\Big[(n+\pi)^2-5n-17\pi-2p^{(1)}+6p_a+24\Big]\\
 k &=& 3(n+6\pi+p^{(1)}-4p_a-11);
 \end{eqnarray*}
 moreover we replace $r$ with its value
 \[
 r=p+n-\pi+1.
 \]

Thus we find
 \[
 M\ge 9p-2p^{(1)}+12,
 \]
 or, more precisely,
 \[
 M=9p-2p^{(1)}+12+\om,
 \]
where $\om\ge0$ denotes the superabundance of the system of plane curves
$D$; this means that in the continuous system of the curves $d$ of
degree $2n+2\pi-2$, with $d+k$ double points, the conditions that $k$ of
the double points become cusps are not $k$ independent conditions but
only $k-\om$ conditions.

For the time being we may conclude by asserting that:
 {\em The regular surfaces of some plurigenus $P_i>0$, of geometric
genus $p$ and absolute linear genus $p^{(1)}$, depend, in general, on
 \[
 M=9p-2p^{(1)}+12+\om
 \]
moduli, where $\om\ge0$.}

For surfaces with $p_a<p_g$, the previous computation must be modified
in view of the characteristic property of these surfaces, that we shall
establish later on, that is, the property of containing
$\infty^{p_g-p_a}$ series of inequivalent linear systems. Thus {\em for
irregular surfaces, the number of moduli will be
 \[
 M=10p_a-p_g-2p^{(1)}+12+\om,
 \]
with $\om\ge0$\footnote{With possibly a different meaning of $\om$, the
formula a fortiori holds for surfaces admitting a continuous group of
transformations in themselves.}.}

Let us go back to regular surfaces ($p_a=p_g=p$). For the surfaces
with canonical or bicanonical curve of order zero ($p=1,p^{(1)}=1$, and
$p=0,p^{(1)}=1$), one has respectively
 \[
 9p-2p^{(1)}+12=19
 \]
 and
 \[
9p-2p^{(1)}+12=10,
 \]
 and indeed we shall verify that the number of moduli is precisely, for
the former surfaces
 \[
 M=19,
 \]
 and for the latter
 \[
 M=10;
 \]
 thus
 \[
 \om=0.
 \]
 But in general one has
 \[
 \om>0.
 \]

 \section{Digression on the completeness of\\ the characteristic series
of a complete\\ system of plane curves with nodes and\\
cusps}\label{sec:5.12}
 \markright{\hfill Digression on the completeness of the characteristic
series \quad 5.12\quad}
 To express the computation of the moduli of a surface in a more
meaningful formula, it is convenient to study the meaning of the
character $\om$ appearing in the formula of the previous section. It is
easy to recognize that {\em the superabundance $\om$ of a continuous
system of plane curves $\{D\}$, of order $n$ with $d$ nodes and $k$
cusps, is smaller than or equal to the index of speciality of its
characteristic series on a generic $D$, that is, of the series cut out
on $D$ by the curves of the same order that pass simply through the $d$
nodes and the $k$ cusps, and touch the cuspidal tangents at those
points.}

Indeed let us recall that, given a continuous series $\{D\}$ of plane
curves $D$, of order $n$ with $d$ nodes and $k$ cusps, the curves of
$\{D\}$ infinitely near to a given $D=D_0$ have 2 intersections with it
absorbed by each node and 3 intersections absorbed by each
cusp\footnote{Remark of De Jonqui\`eres and Beck ({\it Math.\ Ann.},
Band~14, 1878). See Enriques--Chisini, Libro~II, Chapter~II (Vol.~I,
page~331, see Vol.~II, page~286).}. This means that the characteristic
series, cut out on $D_0$ by the infinitely near $D$, is contained in the
complete series cut out on $D_0$ by the curves of the same order passing
through its $d$ nodes and its $k$ cusps and tangent to the cuspidal
tangents at these points. Now, if the continuous system $\{D\}$ has
superabundance $\om$, then its dimension is not equal to
 \[
 r=\frac{n(n+3)}{2}-d-2k,
 \]
but rather
 \[
 r+\om=\frac{n(n+3)}{2}-d-2k+\om,
 \]
Thus the dimension of the series $g_m$, of degree $m=n-2d-3k$, cut out on
the curve $D_0$ of genus
 \[
 \pi= \frac{n(n-3)}{2}+1-d-k,
 \]
by the infinitely near $d$ is $\ge r+\om=m-\pi+\om$, that is, this
characteristic series is special, with index of speciality not smaller
than $\om$. At first sight, one would be tempted to say that $\om$ is
exactly equal to the abovementioned index of speciality, i.e., that the
{\em the characteristic series of the complete system $\{D\}$, formed by
the plane curves of order $n$ with $d$ nodes and $k$ cusps, is
complete.} Indeed it is easy to verify that the following {\em
differential property} is satisfied: given a curve $D$ of the system,
the curve infinitely near to $d$ in the pencil determined by $d$ and by
one of its adjoint curves $\fie_n$ touching the cuspidal tangents at the
$k$ cusps, has $d$ nodes and $k$ cusps infinitely near to those of $D$.

But this differential property does not yield that each of the
aforementioned curves with $d$ nodes and $k$ cusps infinitely near to
$D$, belongs to a continuous series of curves having the same number of
nodes and cusps. Indeed let us consider the variety $V$ representing the
elements (curves) of the system $D$. This variety $V$ appears as the
intersection of a variety $W$ of the space $S_R$ of dimension
$R=n(n+3)/2$ representing the system of all plane curves of order $n$
with $d+k$ nodes, with $k$ hypersurfaces or linear branches of
hypersurfaces of the same space $S_R$. And, for instance, it might
happen that these $k$ branches have the same tangent space of dimension
$R-(k-1)$ at each common point although they intersect along a variety
of dimension $R-k$ and not of dimension $R-k+1$. In this case a point of
$V$ (representing a curve of $\{D\}$) would not have $\infty^{r-1}$ but
$\infty^r$ infinitely near points belonging to $V$ (that is, images of
curves of $\{D\}$).

Although the examples that we know suggest to exclude this
possibility, we have not yet succeeded in proving what we believe to be
true, therefore we have to restrict ourselves to make the following:

{\em Working hypothesis:} The complete continuous system of the plane
curves $d$ with $d$ nodes and $k$ cusps has a complete characteristic
series; therefore {\em the superabundance $\om$ of this system $\{D\}$
equals the index of speciality of the characteristic series,} cut out on
$d$ by the infinitely near $D$.

This assumption yields a few remarkable consequences and has some
heuristic value, in relation with the developments that follow.

 \section{The moduli of the regular surfaces of genus
$p>3$}\label{sec:5.13}
 \markright{\hfill The moduli of the regular surfaces of genus
$p>3$ \quad 5.13\quad}
 We have seen how to express in general the number of moduli on which a
class of algebraic surfaces depends (assuming that some plurigenus is
greater than zero, that is, for surfaces not belonging to the family of
ruled surfaces), for surfaces represented as multiple planes with branch
curves $d$ of a certain order $n$, with a certain number of nodes and
cusps. The formula we found in \S11 contains, besides characters such as
superficial genera $p_a,p_g$ and absolute linear genus $p^{(1)}$, also a
numerical, nonnegative, character $\om$, that is, the superabundance of
the continuous system of the $D$, or, according to our working
hypothesis (\S12), the index of speciality of the characteristic series
of $\{D\}$.

Now the meaning we have given for $\om$ admits a different
interpretation which suggests a more significant formula for the number
of moduli of regular surfaces of genus $p>3$.

To do this, let us consider a surface $F=F_n$ of ordinary space, of a
certain order $n$, with normal singularities, and consider the
representation of $F$ on a $n$-tuple plane obtained by projecting
$F$ from a generic point $O$ external to $F$. In this representation the
branch curve $D$ arises as the projection of the Jacobian curve $C_j$ of
the net of plane sections $C$ with planes through $O$. Now the complete
characteristic series on the plane curve $D$ of the continuous system
$\{D\}$, consisting of the curves of the same order with the same number
of nodes and cusps, has the index of speciality precisely equal to
$\om$. We wish to read this series, with speciality $\om$, on the
Jacobian curve $C_j$. To do this, remark that $C_j$ is a section of $F$
with the polar of the point $O$, that is, (denoting by $K$ the canonical
curves of $F$) it is a curve of the linear system $|3C+K|$ cut out on $F$
by its adjoint surfaces $\fie_{n-1}$ of order $n-1$, passing through the
$\tau$ cuspidal points of $F$; we will denote by $G$ the group of these
points, belonging to the double curve. The other curves $C_j$, passing
through the $\tau$ cuspidal points, cut out on $C_j$, away from these
points, a linear series that, as we shall see in a moment, is projected
from $O$ into the series cut out on the branch curve $D$ by the first
polars. Indeed, denoted by $\Cbar_j$ the section of $F$ with the first
polar of a point $\Obar$, the tangent plane to $F$ at a point $P$ common
to $C_j$ and $\Cbar_j$ contains the line $O\Obar$; thus the tangent to
the curve $D$ at the point $P'$, projection of $P$ from $O$, passes
through the intersection of the line $O\Obar$ with the multiple plane.

It follows from this that the characteristic series of $\{D\}$ on the
curve $D$ projection of $C_j$ is the projection of the series cut out on
$C_j$ by the curves of the linear system $|L|=|4C+K|$ passing through $G$
(that is, by the adjoint surfaces $\fie_n$ passing through the $\tau$
points of the group $G$). Thus the character $\om$, that is, the index of
speciality of the aforementioned characteristic series of $\{D\}$, may
also be interpreted as the {\em index of speciality of the series cut out
on the curve $C_j$, which is a curve of $|3C+K|$ passing through $G$, by
the curves of the linear system $|L|=|4C+K|$ also passing through $G$.}

This {\em projective interpretation of the character $\om$} for the
surface $F$, holds for both regular and irregular surfaces. And,
according to it, the computation of the number of moduli on which a
class of algebraic surfaces depends when it varies in the larger
continuous family of surfaces containing it, acquires a meaning that
agrees very well with intuition.

Indeed, consider the continuous system of all surfaces $F$, having
the same order $n$ and a double curve with the same characters, on which
there is the same number $\tau$ of cuspidal points: the surfaces of such a
system $\{F_n\}$ containing $F$ determine on one of them, and in particular on
the given $F$, a {\em characteristic linear system}, consisting of the
intersections of $F$ with the surfaces of the system that are infinitely
near to it. This system (that may a priori be complete or not) is cut
out on $F$ by the adjoint surfaces $\fie_n$ of order $n$, which are
tangent to the cuspidal planes of $F$ at the points of the group G.

Our existence theorem together with the working hypothesis of \S12
yields that the continuous system $F_n$ cuts out the {\em complete
characteristic linear system} on a surface $F$; and conversely, this
{\em is equivalent to the above working hypothesis}, and it leads to the
computation of the number of moduli of the class of $F$ as varying
inside the system $\{F_n\}$. Of course in order to obtain a surface with
{\em general moduli}, it is convenient to start with a projective model
$F$ which one a priori knows to be contained in a continuous family of
surfaces with given characters. For instance one may consider a surface
$F$ corresponding to a regular linear system of large enough degree $n$
and genus $\pi$, for which the inequality $n>\pi-1$ holds; or one may
also consider a canonical surface, that is, a surface having $\infty^3$
canonical curves $K$ as plane sections, except the case of small genus
or some other analogous exception. Let us see what is the outcome of the
computation of the moduli, referring to the two choices of $F$ explained
above and assuming, for the sake of simplicity, the surface to be
regular.

If we start with the first choice, we find again the formula of \S11.
Indeed the question is of computing the dimension $R$ of the
characteristic system of $\{F_n\}$ and then the number $S$ that denotes the
number of parameters of the $F_n$ of the above continuous system that are
birationally equivalent to a given $F$; then one has
 \[
 M=R-S.
 \]

In order to compute $R$, we start by calculating the dimension of the
base point free system $|L|=|4C+K|$, that we may assume to cut out a
complete and nonspecial series on a $C_j$. Then we subtract from this
dimension $\tau-\om$, that is, the number of independent conditions that
the group of the $\tau$ cuspidal points of $F$ imposes on the curves $L$
that contain it. The dimension of $|L|$ is
 \[
 p+4\pi+6n-4,
 \]
and therefore the dimension of the characteristic system of $\{F_n\}$ is
 \[
 R-1=p+4\pi+6n-4-\tau+\om,
 \]
where
 \[
 \tau=2n+8\pi-12p+2p^{(1)}-22,
 \]
so that
 \[
 R=13p+4n-4p-2p^{(1)}+19+\om.
 \]

On the other hand, $S$ is computed by adding 15 (number of parameters
of a space homography) to the number of parameters defining an
$\infty^3$ system inside the complete linear system $|C|$ of dimension
$r$. Thus one obtains
 \[
S=4r-12+15=4r+3
 \]
that is,
 \[
 S=4p+4n-4\pi+7.
 \]
Thus one finds
 \[
 M=R-S=9p-2p^{(1)}+12+\om,
 \]
which is the formula given in \S11.

But, if we take into account the meaning of $\om$ (according to our
working hypothesis of \S12), it is possible to give a more meaningful
formula, showing that for $p>3$ one has in general
 \[
 \om\ge p,\quad\hbox{and}\quad
 M=10p-2p^{(1)}+12+\om' \quad (\hbox{with }\om'\ge0).
 \]

In order to see this, we go back to our previous computation, considering
a surface $F_n$ of degree $n=p^{(1)}-1$ with canonical curves $K$ as
plane sections. The following modifications are necessary here:

 \begin{enumerate}
 \renewcommand{\labelenumi}{(\arabic{enumi})}
 \item the curves $L$ of $|L|=|5K|$ cut out the complete canonical series on
a $C_j=K_j$, thus, if we still denote by $\om$ the index of speciality of the
series cut out on $K_j$ by the curves of $|5K|$ that pass through $G$,
the computation of $R$ gives us again
 \[
 R-1=p+4\pi+6n-4-\tau+\om
 \]
whence it follows that the $\tau$ cuspidal points do not impose
$\tau-\om$ but $\tau-(\om-1)$ independent conditions on the $L$ to
contain them;

 \item the system $|K|$, unlike the nonspecial regular system $|C|$
considered before, has dimension
 \[
 p-1=p+n-\pi\quad
(\hbox{with } n=p^{(1)}-1, \hbox{ and } \pi=p^{(1)}),
 \]
instead of
 \[
 p+n-\pi+1;
 \]
thus the number of surfaces $F_n$ birationally equal to a given one turns
out to be
 \[
 4p+4n-4\pi+3
 \]
 instead of
 \[
 4p+4n-4\pi+7.
 \]
 \end{enumerate}

Therefore one finds that the number of moduli of $F_n$ is given by
 \[
 M=9p-2p^{(1)}+16+\theta
 \]
where $\theta=\om-4$ denotes the index of speciality of the series cut
out on a $K_j$ by the curves of $|5K|$ passing through the group $G$ of
cuspidal points lying on the surface $F$, image of a linear system
chosen inside $|K|$.

Now we can compute $\om$, or at least give a lower bound for it, by
remarking that the group $G$ of cuspidal points is a group of points
lying on the Jacobian curve $K_j$ of a net of curves $K$, and is a
partial intersection with a curve $L$ of the system
 \[
 |L|=|5K|
 \]
that is, the adjoint of
 \[
 |K_j|=|4K|.
 \]

Therefore the group $G_\tau$ is a special group, imposing precisely
$\tau-\theta$ conditions on the curves of $|5K|$ that must contain it,
where $\theta$ is the dimension of the complete series to which $G$
belongs on $K_j$ (theorem of Riemann--Roch).

Now a lower bound for $\theta$ can be found by remarking that the net
whose Jacobian is $K_j$ belongs to $\infty^{p-4}$ linear systems of
dimension 3 of curves $K$ contained in the canonical system. Each of
these $\infty^3$ systems defines a group $G_\tau$ on $K_j$, that is, the
group of double points for $\infty^1$ curves of the system. Thus we have
 \[
 \theta\ge p-4
 \]
and finally
 \[
 M=10p-2p^{(1)}+12+\om' \quad (\hbox{with } \om'\ge0).
 \]

Let us state the result thus obtained, but keeping in mind that it is
based on the working hypothesis of \S12, a proof of which is still
missing:

The number of moduli on which depends a class of regular surfaces of
genus $p>3$, with simple irreducible canonical system ($p^{(1)}>5$), is
most likely to be
 \[
 M\ge10p-2p^{(1)}+12.
 \]

In the next section we shall say how this formula can be extended to
the case of irregular surfaces.

 \section{Historical note and complements}\label{sec:5.14}
 \markright{\hfill Historical note and complements \quad 5.14\quad}
 The first attempt to determine the number $M$ of moduli on which a
class of algebraic surfaces depends is due to
M.~Noether\footnote{M.~Noether, Anzahl der Moduln einer Classe
algebraischer Fl\"achen etc., Sitzungsberichte der Akademie, Berlin,
1888.} (1888) who considered the formula
 \[
 M=10p-2p^{(1)}+12, \quad\hbox{where}\quad (p=p_a=p_g),
 \]
to be true under very mild assumptions; this is precisely the formula
that, according to our working hypothesis of \S12, gives in general a
lower bound for $M$ for regular surfaces with $p>3$, corresponding to
$\om'=0$. Noether arrived at this type of divination by the intuition of
genius. He started with the formula that gives the postulation of a
twisted curve $C$ (with triple points) for the surfaces $F_n$ of order
$n$ that have to contain it as a double curve. If the number of
conditions thus imposed on $F_n$ is negative, $-R$, he conjectures that
$R$ coincides with the number of conditions in order that the curve be
double for some surface $F_n$; thus one finds the number of moduli
corresponding to the $F_n$ having $C$ as a double curve, by computing
the number $N$ of parameters on which $C$ depends:
 \[
 M=N-R.
 \]

The question of the moduli of the surfaces was reconsidered by Enriques
in two papers, of 1908 and 1912\footnote{F. Enriques, Sui moduli delle
superficie algebriche. {\it Rendic.\ Acc.\ Lincei}, June 1908; Sui moduli
di una classe di superficie algebriche e sul teorema d'esistenza per le
funzioni algebriche di due variabili. {\it Atti Acc.\ Torino}, 1912.
See {\it Ann.\ di Mat.}, 1923--24.}. The computation is performed by
considering the continuous system of the surfaces $F_n$ of order $n$ --
in ordinary space -- having a double curve with given characters, and
assuming that the characteristic system, cut out on $F_n$ by the
infinitely near
$F_n$, is complete: the author did not realize at that moment that this
hypothesis contains implicitly an assumption that is not satisfactorily
proved by showing the corresponding differential property; the gap was
pointed out only by Zariski\footnote{O.~Zariski, Algebraic Surfaces.
Berlin, 1935 (see V, 8).} (1935) after that a similar gap had been
discovered by Severi (1921) in the proof of the fundamental theorem
expressing the characteristic property of irregular surfaces (see
Chapter~IX, \S6).

However Enriques, implicitly relying on the hypothesis mentioned
above, found the formula of \S11:
 \[
 M=10pa-p_g-2p^{(1)}+12+\om \quad (\hbox{with }\om\ge0).
 \]
which holds for all regular and irregular surfaces that do not belong
to the family of ruled surfaces (that is, with some plurigenus $P_i>0$);
then he found the more meaningful formula that should hold, in general,
for the regular surfaces (with simple irreducible canonical system) of
genus $p>3$ ($p^{(1)}\ge6$):
 \[
 M=10p-2p^{(1)}+12+\om' \quad (\hbox{with }\om'\ge0);
 \]
as we have said, this formula reduces to Noether's for $\om=0$. In the
second note of 1912, Enriques started from the formula of the moduli to
make some remark on the existence theorem for the multiple planes.
Indeed he pointed out that the conditions in order that a plane curve
with some number of nodes and cusps be the branch curve of a multiple
plane are purely arithmetic in nature, so that one can say that {\em if
a curve $D$ is the branch curve of a $n$-tuple plane, then the same must
happen for all the curves of the largest continuous system having the
same number of nodes and cusps.} The author claimed, on that occasion,
that it should be possible to determine such arithmetic conditions by
studying the monodromy group, as he did later on in the paper in
{\it Ann.\ di Mat.}, 1923, as we have explained here in \S9.

It is now clear that, starting -- as we did before in \S11 -- from the
existence theorem for the multiple planes, one can deduce the first
formula of the moduli that (except for the family of ruled surfaces) is
completely general, and it is therefore immune to Zariski's criticism.
Which affects instead the second more meaningful formula (for
$p>3$), that in the present treatment we prove under the working
hypothesis of \S12.

The question of moduli of the algebraic surfaces was considered again by
B. Segre\footnote{B. Segre, Sui moduli delle superficie algebriche
irregolari. {\it Rendic.\ Acc.\ Lincei}, Vol.~XIX (1934).} in 1934,
especially with the aim of finding a more meaningful formula for the
irregular surfaces. But in this research he still makes use of the
hypothesis (that the characteristic system of a continuous system of
surfaces $F_n$ is complete) that we have reduced to the working
hypothesis of \S12; actually Segre was aware of the difficulty brought
to light subsequently by Zariski's criticism.

Here we remark that, on the basis of the stated hypothesis, the
argument we developed above for the general case extends immediately to
the case of irregular surfaces (if one keeps in mind the characteristic
property of these surfaces that we have already mentioned). In this way
one obtains the formula
 \[
 M=12 p_a-2p_g-2p^{(1)}+12+\om \quad (\hbox{with }\om\ge0).
 \]

But Segre follows another way that leads to another more meaningful
formula, so that already in the case of the regular surfaces he can
write $\om\ge p-1$, and thus
 \[
 M\ge10p-2p^{(1)}+11.
 \]
This formula (apart from the restrictions mentioned above) should hold
for all surfaces with some nonzero plurigenus, with no need of assuming
$p>3$, irreducible canonical curves, and so on.

On the other hand, this result is included in the more general formula
that gives the moduli of the irregular surfaces. Indeed B.~Segre finds
that $\om\ge2p_g-p_a-1$, hence
 \[
 M=9 p_a+p_g-2p^{(1)}+11+\theta.
 \]

We close by stating a formula that has been communicated to us by
Prof. G.~Castelnuovo, who has found the following inequality for the
character $\om$:
 \[
 \om\ge2 p_a+p_g+p^{(1)}-I-3.
 \]
