 \chapter[The numerical genus and the Riemann--Roch theorem for
surfaces]{The numerical genus and the Riemann--Roch theorem\\ for
surfaces}\label{ch:4}
 \markboth{\qquad 4. \quad The numerical genus and the Riemann--Roch
theorem\hfill}{}
 \index{Riemann--Roch}\index{numerical genus}

 \section{Introduction}\label{sec:4.1}
 \markright{\hfill Introduction \quad 4.1\quad}
In order to understand the meaning of what follows it is worth
recalling the problems in the theory of curves that have their
natural extension in this setting\footnote{See, e.g.,
Enriques--Chisini, Ibidem, Libro~V, Vol.~III.}.

After having defined the linear series on a curve and the elementary
operations concerning them, and after having proved the invariance of
the canonical series, one is faced with the question of computing the
dimension $r$ of the complete series determined on a curve of genus
$p$ by a group of $n$ points. The first answer that one obtains
here, that is, $r\ge n-p$, follows by computing the dimension of the
system of adjoint curves of large enough order to the plane curve
$C$ of order $m$ and genus $p$. Since the system of these adjoints
$\fie_{m-3+s}$ always cuts out on $C$ a complete series (for any
$s$, positive or negative), and since for large enough $s$ the
double points of $C$ impose
 \[
 d=\frac{(m-1)(m-2)}{2}-p.
 \]
independent linear conditions on the curves of order $m-3+s$ that
must contain them, the series of degree $n=2p-2+sm$ cut out on $C$
by these adjoints turns out to be of dimension
 \[
 n-p.
 \]
Therefore it is clear that any series cut out on $C$ by the
$\fie_{m-3+s}$ passing through a group $G$ of points imposing
independent conditions on them has dimension equal to the degree
minus the genus $p$. These results are then made more precise by
proving that the canonical series, of degree $2p-2$, cut out on $C$
by the adjoints $\fie_{m-3}$, always has dimension $p-1$: this
follows by computing the number of conditions that the double points
of $C$ impose on the curves of order $m-3$ that have to contain
them, these conditions being independent. Finally, the theorem of
Riemann--Roch actually answers in the most general setting the
question of computing the dimension of the complete linear series
determined on $C$ by a group of $n$ points: denoting by $i$ ($\ge0$)
the index of speciality of the group, that is, the number of
linearly independent canonical groups that contain it, the dimension
of the stated series is precisely
 \[
 r=n-p+i.
 \]
Now, as we want to extend this type of questions to surfaces, we shall
first of all be led to investigate what is, in the case of a surface
$F$ of order $n$ of ordinary space, the dimension of the linear system
cut out by the adjoint surfaces $\Phi_{m-3+s}$ for $s$ large enough,
and to relate the result obtained with the computation of the
superficial genus of $F$, for which we shall thus obtain a virtual
expression. Following this pattern, we shall be led to compute the
dimension of the linear systems adjoint to some other linear systems,
thus establishing a ``theorem of Riemann--Roch for adjoint systems'',
and then to examine the case of an arbitrary linear system, not given as
the adjoint of another one.

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

\section{Conditions imposed on a surface that contains a curve:
postulation formulas}\label{sec:4.2}
 \markright{\hfill Conditions imposed on a surface: postulation
formulas \quad 4.2\quad}
 Let us consider an (irreducible) twisted curve $C$ of a certain
order
$m$ and genus $p$, without multiple points, in ordinary space; the
surfaces $\Phi$ of arbitrary order $n$ cut out on $C$ a linear series
$g^r_{mn}$ of degree $mn$, whose dimension we denote by $r$. If the
series is complete and nonspecial, one can write
 \[
 r=mn-p.
 \]
Under these assumptions one immediately finds the postulation of the
curve $C$ with respect to the surfaces of order $n$, that is, the number of
independent linear conditions that $C$ imposes on the $\Phi$ that have
to contain it; in fact the number of these conditions will be
 \[
 r+1=mn-p+1.
 \]
Our question would thus be answered in the simplest way. But two causes
may modify the result mentioned above in opposite directions: the
series $g^r_{mn}$ may be special, and therefore of dimension $r>mn-p$,
and it may be deficient, that is, it may have smaller dimension than the
complete series. Because of the former circumstance the postulation of
$C$ for the $\Phi$ may turn out to be $>mn-p+1$, and because of the
latter it may be smaller. But one of the two perturbing causes can be
certainly removed by referring to surfaces $\Phi$ of large enough order
$n$; indeed it suffices to choose $n$ so that
 \[
 mn>2p-2.
 \]
Therefore one can say that for the surfaces $\Phi$ of large enough order $n$,
the twisted curve $C$ of order $m$ and genus $p$, has postulation $\le
mn-p+1$; this means that there are {\em at least}
 \[
 \frac{(n+3)(n+2)(n+1)}{6}-mn+p-1
 \]
linearly independent surfaces $\Phi$ passing through $C$.

One can make the result thus obtained more precise, by proving that
for $n$ large enough, $n\ge m-2$, the above series $g^r_{mn}$ turns
out to be complete\footnote{The proof that follows is due to
G.~Castelnuovo, {\em Sui multipli di una serie lineare di gruppi di
punti appartenente ad una curva algebrica.} {\it Rendic.\ Circ.\
Mat.\ di Palermo}, t.~VII, 1893. Reprinted in {\it Memorie scelte},
page~95.}. To do this, one projects the curve $C$ from a generic
point $O$ of the space to a plane: one thus has a curve of order $m$
with
$d=(m-1)(m-2)/2-p$ double points, on which the adjoint curves of
order $\ge m-2$ cut out a complete and nonspecial series. Whence it
follows that the cones of order $m-2$ (or larger) having the vertex
at $O$ and containing the $d$ chords of $C$ passing through $O$ cut
out on $C$ a complete series, away from the intersection points of
these chords with the curve. A fortiori the surfaces of order $n\ge
m-2$ passing through the same points cut out a complete series on
$C$ away from these points. This series becomes larger if one
extends the system of surfaces mentioned before by considering all
surfaces $\Fie_n$ of order $n$ of the space. Now, in order to show
that these cut out on $C$ a complete series it suffices to see that
the group $G$ of the $2d$ points of intersection of
$C$ with its chords through $O$ imposes independent linear conditions on
the $\Fie_n$, so that adding the points of $G$ leads from a complete
(nonspecial) series of dimension
 \[
 mn-2d-p
 \]
to a complete series of dimension
 \[
 mn-p.
 \]
To see this, we refer for the sake of simplicity to the case $n=m-2$; it
suffices to construct a surface of order $m-2$ that passes through
$2d-1$ points of $G$ and not through the remaining one; one such a
surface will be obtained by adding to a cone of order $m-3$ that passes
through $d-1$ chords of $C$ passing through $O$, and not through the
remaining one, a plane that passes through one of the two intersection
points of the remaining chord, but not through the other. The existence
of the aforementioned cone of order $m-3$ follows from the well known
theorem that the double points of a plane curve of order $m$ (the
projection of $C$) impose independent conditions on the curves of order
$m-3$ that have to be adjoint to it.

Therefore we conclude: {\em an irreducible twisted curve of order $m$
and genus $p$, without multiple points, imposes
 \[
 mn-p+1
 \]
linear conditions on the surfaces of order $n\ge m-2$ that must contain
it.}

The computation of the postulation of a curve described above, can be
extended with slight modifications to the case of twisted curves with
multiple points, and even to reducible curves. Let us start by
considering the case, most relevant to us, of a {\em curve} $C$ (of
order $m$ and genus $p$) having {\em a certain number $t$ of given
triple points}, with tangents not lying in the same plane, and that we
still wish to assume irreducible. Here one shows that the $t$ stated
points impose $4t$ linearly independent conditions on the surfaces
$\Phi$ of large enough order $n$ for which they must be double, and
then that the $\Phi$, passing doubly through the $t$ triple points of
$C$, cut out on $C$, away from these, a (nonspecial) complete series,
so that {\em the postulation of the curve $C$ for the $\Phi$ of large
enough order $n$ turns out to be}
 \[
 4t+nm-6t-p+1=nm-2t-p+1.
 \]
The first assertion is self-evident, since to prescribe a double point
at a fixed position gives four linear conditions that the surface must
satisfy; the latter assertion can be shown by the same argument as for
the curves $C$ without multiple points.

In a similar way one obtains the postulation of $C$ in the case when
$C$ {\em has a certain number $\tau$ of (nodal) double points}: in this
case it is better to impose first that $\Phi$ pass also through these
$\tau$ points, which gives $\tau$ linear conditions, and lowers the
number of intersections with $C$ by $2\tau$; so the {\em postulation}
of $C$ for the $F$ of order $n$ turns out to be
 \[
 \tau+nm-2t-2\tau-p+1=nm-2t-\tau-p+1.
 \]
Finally we wish to extend this formula to the case in which the curve
$C$ splits as the union of several parts, of order $m_1,m_2,\dots$ and
genus $p_1,p_2,\dots$ respectively.

For the sake of simplicity, we refer to the case in which one has only
two components $C_1$ and $C_2$, that are only allowed to have normal
crossings, that is, double points of the reducible curve $C=C_1+C_2$, or
may also be connected at some of the triple points of the curve $C$
that are double for one of the components and simple for the other.
Now, we remark that the surfaces $\Phi$ of large enough order $n$,
passing doubly through the triple points of $C$ and simply through its
double points, cut out a (nonspecial) complete linear series on both
$C_1$ and $C_2$; indeed, let us drop the conditions imposed on the
$\Phi$ requiring that they pass through the double or triple points of
$C_2$; the $\Fiebar$ thus obtained cut out a complete series on $C_1$;
but we may always assume that there is a system of surfaces $\Fiebar$
passing through a group of this series, and so large that it is always
possible to satisfy the omitted conditions and therefore to find a
$\Phi$ that passes through that group. But we can say more: not only is
each of the linear series cut out by the $\Phi$ on $C_1$ and $C_2$
complete, but also the series that one has on the reducible curve turns
out to be complete, allowing us to associate to an arbitrary group of
the series on $C_1$ a group on $C_2$; in other words the $\Phi$ passing
through a group of the series on $C_1$ and through another group of the
series on $C_2$, both taken arbitrarily, do not have as a consequence
to contain one of the two curves. To see this, by the same argument we
made in the case of an irreducible $C$, we can reduce to the plane, and
invoke the result that ``on a reducible plane curve, the adjoint curves
of sufficiently large order always cut out a complete series''.

We have thus established that the postulation of the curve $C$ for the
surfaces $\Phi$ of large enough order, passing as we said through the
intersection points of the components of $C$, is equal to the sum of the
postulations of its components. It remains only to compute these
postulations. In order to do this we denote by $m_1$ and $m_2$ the
orders and by $p_1$ and $p_2$ the genera of the above components. If
one does not take into account the assigned fixed points of the $\Phi$,
the degrees of the two series cut out on $C_1$ and $C_2$ would be
$nm_1$ and $nm_2$ respectively and the dimensions would be $nm_1-p_1$
and $nm_2-p_2$. But a double point of $C=C_1+C_2$ lowers the dimension
of the system of the $\Phi$ by 1, and it lowers the sum of the degrees
of the two series cut out on $C_1$ and $C_2$ by 2 in total, irrespective
of whether this point is a double point for one of the irreducible
components of $C$ or a simple point for both. Similarly, requiring a
triple point of the reducible curve $C$ to be double for the $\Phi$
lowers the dimension of the system of the $\Phi$ by 4, and it lowers
the sum of the degrees of the two series cut out on $C_1$ and $C_2$ by 6
in total, whether it is a double point for one and a simple point for
the other, or a triple point for one of them. In conclusion, one sees
in this way that the $t$ triple points of the curve $C$ and its $\tau$
double points have the same influence as in the case of an irreducible
curve. So the postulation of $C$ will be expressed by the same formula
written before, involving the $t$ triple points of the curve $C$ and
its $\tau$ double points, provided that one makes the modifications
that we now indicate. Instead of the series of degree $x=mn-6t-2\tau$
cut out on the irreducible curve $C$ by the $\Phi$, one has to consider
the two series cut out on $C_1$ and $C_2$, whose degrees we denote by
$x_1$ and $x_2$; and, instead of the dimension $x-p$ of that series,
one has the sum of the dimensions of the two given series on $C_1$ and
$C_2$, that is,
 \[
 x_1-p_1+x_2-p_2.
 \]
But, while in the previous case the postulation of $C$ was given by
$x-p+1$, now this postulation is obtained by adding the postulations of
$C_1$ and $C_2$ and is therefore
 \[
 x_1-p_1+1+x_2-p_2+1=x-(p_1+p_2)+2.
 \]
In conclusion the postulation of the curve $C=C_1+C_2$ for the surfaces
of large enough order $n$ is expressed by
 \[
 mn-2t-\tau-(p_1+p_2)+2
 \]
This formula can be reduced to the one we have written for the
irreducible $C$ if, instead of the genera $p_1$ and $p_2$ of the
components $C_1$ and $C_2$ of $C$, one introduces the genus $p$ of the
reducible curve, computed, according to Noether, under the assumption
that the components have no intersection points\footnote{These points
have been removed by imposing them as base points of the $\Phi$.}:
 \[
 p=p_1+p_2-1.
 \]
Let us state the general result that one obtains in the case of {\em a
curve of order $m$ consisting of $h$ irreducible curves of genera
$p_1,p_2,\dots,p_h$, which has $t$ triple points (with generic tangents)
and $\tau$ double points: the postulation of the curve for the surfaces
of large enough order $n$ is
 \[
 nm-2t-\tau-p+1,
 \]
where
 \[
 p=p_1+p_2+\cdots+p_h-h+1
 \]
denotes the genus of the reducible curve under the assumption that its
parts are not connected.}

 \section{Dimension of the systems of adjoint surfaces to a given
one}\label{sec:4.3}
 \markright{\hfill Dimension of adjoint systems \quad 4.3\quad}
 Let us consider a surface $F$ of a certain order $n$ in ordinary
space, with normal singularities: that is, a nodal double curve with
triple points with noncoplanar tangents, and, if one likes, also
with double points\footnote{A priori one may assume that the given
surface $F$ has a double curve without self-crossings: $\tau=0$ (see
{\em Introduction}, etc., Section~\ref{sec:4.5}). Still it may be useful
to have the formulas also in the case
$\tau>0$.}. We compute the number $N_m$ of linearly independent surfaces
adjoint to $F$, of large enough order $m=n-4+s$. We still denote these
surfaces by $\Phi_m$ and by $d$\/ the order of the double curve of $F$,
by $\rho$ its genus, by $t$ the number of triple points and by $\tau$
the number of double points. We have that the number $N_m$ of the
$\Phi_m$ will be
 \[
 N_m=\binom{m+3}{3}-(dm-2t-\tau-\rho+1);
 \]
thus the difference $N_m-N_{m-1}$ will be given by
 \[
 N_m-N_{m-1}=\binom{m+2}{2}-d.
 \]
The meaning of this relation is that {\em the adjoint surfaces $\Phi_m$
of large enough order cut out on a plane the complete linear system of
the adjoint curves $\fie_m$ of the section of $F$.}

In fact the dimension $N_{m-1}-1$ of the system $|\Phi_{m-1}|$ can be
deduced from the dimension $N_{m-1}$ of the system of the $\Phi_m$, by
subtracting a plane from these; and splitting off the plane yields
precisely $r+1$ linear conditions, where $r$ is the dimension of the
system cut out by the $\Phi_m$. Then in this case one finds
 \[
 r+1=\binom{m+2}{2}-d, \quad\hbox{that is,}\quad r=\frac{m(m+3)}{2}-d,
 \]
which is the dimension of the complete system of the adjoints $\fie_m$
to the plane section of $F$.

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

In order to prove this, we start with a value $m$ of the order such
that the actual number of $\Phi_m$ is equal to the virtual number $N_m$, and
such that $m\ge n-3$.

The number of $\Phi_{m-1}$ adjoint to $F$ can be deduced from the number
of $\Phi_m$ by subtracting one plus the dimension $r$ of the system cut
out on a plane by the $\Phi_m$. If this system is the complete system
of the $\fie_m$ adjoint to the plane section, then the actual number of
the $\Phi_{m-1}$ is again equal to the virtual number; however, if there
is a difference, then the system of the $\fie_m$ sections of the
$\Phi_m$ on a plane can no longer be complete, and then
 \[
 r<\frac{m(m+3)}{2}-d;
 \]
hence the actual number of the $\Phi_{m-1}$ turns out to be larger than
$N_{m-1}$. This argument can be applied as long as $m\ge n-3$, since
under this assumption the complete system of the adjoint curves of the
section of $F$, of order $n$, is regular, that is, it has dimension
$m(m+3)/2-d$ and not larger, since the double points of the plane curve
section of $F$ impose independent conditions on the curves $\fie_m$
that have to pass through them. On the other hand if one takes $m=n-4$,
the system of the $\fie_{n-4}$ adjoint to the section of $F$ turns out
to be superabundant, so that the previous argument is no longer
applicable.

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

Because of what we have seen above one can say that: the numerical
genus of the surface $F$ is in any case smaller than or equal to the
geometric genus:
 \[
 p_a\le p_g.
 \]
We shall soon prove the invariance of the numerical genus with respect
to birational transformations of the surface $F$. In the meantime, we
write the value of this character by means of the postulation formulas
established above:
 \[
 p_a=\binom{n-1}{3}-(n-4)d+2t+\tau+\rho-1.
 \]
As a consequence the number of adjoint surfaces $\Phi_{n-4+s}$ of $F$ that
are linearly independent can be expressed as follows:
 \begin{eqnarray*}
 N_{n-4+s}&=&\binom{n-1+s}{3}-(n-4+s)d+2t+\tau+\rho-1=\\
 &=&p_a+\binom{n-1+s}{3}-\binom{n-1}{3}-sd-1.
 \end{eqnarray*}
Therefore the linear system consisting of the intersections of these
adjoint $\Phi_{n-4+s}$ with $F$ will in general have dimension:
 \[
 N_{n-4+s}-1-\binom{s-1}{3}=p_a+\binom{n-1+s}{3}
 -\binom{n-1}{3}-\binom{s-1}{3}-sd-1,
 \]
where $\binom{s-1}{3}$ (note that this vanishes for $s=1,2,3$) is the
number of linearly independent surfaces of order $s-4$ ($\ge0$), or
the dimension of the system of surfaces $\Phi_{n-4+s}$ having the same
intersection with the surface $F$ of order $n$. The dimension of the
system of sections of the $\Phi_{n-4+s}$ written above can be
expressed in a very simple form, by introducing the genus $\pi_s$ of
the curves cut out on $F$ by the surfaces of order $s$. First of all we
compute the $\pi_s$ in terms of the genus $\pi$ ($=\pi_1$) of the
plane sections of $F$; we have
 \[
 \pi_s=s \pi+\frac{s(s-1)}{2} n-s+1,
 \]
and, substituting the value of $\pi$
 \[
 \pi=\frac{(n-1)(n-2)}{2}-d,
 \]
gives
 \begin{eqnarray*}
 \pi_s&=&s\left[\frac{(n-1)(n-2)}{2}-d\right]+\frac{s(s-1)}{2} n-s+1=\\
 &=&\frac{s(n-1)(n-2)}{2}+\frac{s(s-1)}{2} n-s+1-sd.
 \end{eqnarray*}
But we have
 \[
 \frac{s(n-1)(n-2)}{2}-\frac{s(s-1)}{2}n-s+1=
 \binom{n-1+s}{3}-\binom{n-1}{3}-\binom{s-1}{3},
 \]
and therefore
 \[
 N_{n-4+s}-1-\binom{s-1}{3}=p_a+\pi_s-1.
 \]
We conclude that {\em the linear system cut out on $F$ by the surfaces
$\Phi_{n-4+s}$ of order $n-4+s$, adjoint to the system $|C_s|$ cut out
by the surfaces of order $s$ has dimension
 \[
 p_a+\pi_s-1
 \]
for large enough $s$,} where $\pi_s$ denotes the genus of $|C_s|$, the
$s$-tuple of the system of plane sections.

 \section{The numerical genus}\label{sec:4.4}
 \markright{\hfill The numerical genus \quad 4.4\quad}
 We consider always a surface $F$ of ordinary space, having a certain
order $n$ and normal singularities. We propose to compute the
dimension of the adjoint system $|C'|$ of an irreducible system $|C|$
belonging to $F$. We start by recalling that the adjoint curves $C'$
cut out canonical groups on a curve $C$; subtracting $C$ from $|C'|$
one gets the canonical system $|K|$ of dimension $p_g-1$, and there
will thus be $\infty^{p_g}$ curves $C'$ passing through the canonical
group formed by the intersection of an adjoint $C'$ with a curve $C$.
{From} this it follows that in all cases the dimension of the system
$|C'|$ is given by
 \[
 r=p_g+\pi-1-\de(C),
 \]
where $\de(C)$ denotes the deficiency of the linear series cut out by
the $C'$ on one of the $C$; in particular, if $|C'|$ cuts out on $C$ the
complete canonical series $g^{\pi-1}_{2\pi-2}$, then
$\de(C)=0$ and
 \[
 r=p_g+\pi-1.
 \]
The deficiency $\de(C)$ depends a priori on the choice of the system
$|C|$; but we can prove a fundamental inequality obtained by
considering the sum of the irreducible (both at least
$\infty^1$)\footnote{In the case of two pencils it is understood that
they are distinct, so that also $|C+L|$ turns out to be irreducible.}
linear systems $|C|$ and $|L|$.

Indeed one has
 \[
 \de(C)\le\de(C+L).
 \]
Assume for simplicity that the systems $|C|$ and $|L|$ have no base
points on the surface $F$. We start with the basic relation
 \[
 |(L+C)'|=|L+C'|=|L'+C|.
 \]
By this relation the system $|C'|$ is obtained from $|(L+C)'|$, by
subtracting the curve $L$, which yields a number of linear conditions,
given by one plus the dimension of the series cut out on $L$ by
$|(L+C)'|$.

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

In this setting we denote by $\pi$ the genus of $|C|$, by $\rho$ the
genus of $|L|$, and by $i$ the number of intersections $(LC)$. Then
the genus of $|L+C|$ is
 \[
 \pi'=\pi+\rho+i-1,
 \]
and the dimension of $|(L+C)'|$ is thus
 \[
 p_g+\pi'-1-\de(L+C)=p_g+\pi+\rho+i-2-\de(L+C).
 \]
On the other hand, the dimension of $|C'|$ is
 \[
 p_g+\pi-1-\de(C).
 \]
As we have said, the difference between the two dimensions is one
plus the dimension of the nonspecial series $g$ cut out on $L$ by
$|(L+C)'|$, plus one. If the series $g$ is complete and thus of
dimension
 \[
 2\rho-2+i-\rho=\rho+i-2,
 \]
we certainly have
 \[
 \de(C)=\de(L+C);
 \]
if, by contrast, $g$ is deficient, its deficiency expresses the
difference $\de(L+C)-\de(C)$. In any case one has:
 \[
 \de(C)\le\de(L+C).
 \]

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

We remark in addition that the hypothesis that we have made, that is,
that $|L|$ belongs to an irreducible system, at least $\infty^1$, is
stronger than necessary: actually, $L$ could also be an isolated
curve, made up of several parts $L_1,L_2,\dots$, provided that none
of these is fundamental for the system $|C|$ and that when one adds
$L_1,L_2,\dots$ successively to $|C|$ one obtains irreducible systems
of larger dimension each time. In fact one can subtract $L$ from
$|(L+C)'|$ by subtracting in turn its parts $L_1,L_2,\dots$, and if
these are not fundamental for $|C|$, then the series cut out on them,
whose dimensions we need to know, are always nonspecial, since they
contain the canonical series.

Now we recall that we have computed the dimension of the system cut
out on $F$ by the adjoint surfaces for $s$ large enough; here it will be
convenient to denote by $|L_s|$ this system, which is the adjoint of
the $s$-tuple system of plane sections. Denoting by $\pi_s$ the genus
of $|L_s|$, we have found that the dimension of the system $|L_s'|$ is
 \[
 p_a+\pi_s-1.
 \]
On the other hand, this dimension can be expressed in terms of the
geometric genus $p_g$ and the deficiency of the series cut out on
$L_s$ by $L_s'$, and is
 \[
 p_g+\pi_s-1-\de(L_s).
 \]
Whence one deduces that
 \[
 \de(L_s)=p_g-p_a,
 \]
where as we have said, $s$ is a large enough number.

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

We thus deduce that
 \[
 \de(C)\le\de(L_s).
 \]
This conclusion can be easily extended to the case in which $|C|$ has
base points, with certain multiplicities $i_1, i_2,\dots$ In fact,
referring to a large enough value of $s$, we may always impose that
the system $|L_s|$ has the same base points as $|C|$ with the same
multiplicities, thus constructing a system $|\Lbar_s|$ that contains
$|C|$, and even such that the residual system $|M|=|\Lbar_s-C|$ is an
irreducible linear system. Now one can assume that the base points of
$|\Lbar_s|$ impose $i(i-1)/2$ independent linear conditions on the
adjoint curves $L_s'$, that have to pass through a $i$-tuple point
with multiplicity $i-1$. Thus we shall have a linear system
$|\Lbar_s|$ with the same base points as $|C|$, for which one has
again
 \[
 \de(\Lbar_s) \le p_g-p_a.
 \]
Whence one deduces as before that
 \[
 \de(C)\le\de(\Lbar s).
 \]
In this way we have proved that: {\em on the surface $F$, the
deficiency of the (canonical) series cut out by the adjoint system to
an irreducible linear system $|C|$ on a generic $C$ has a maximum
value. This value is an intrinsic character of the surface, equal to
the difference $p_g-p_a$ between its geometric and numerical genus.}
This maximum is attained by a class of linear systems belonging to
$F$, including in any case the {\em large enough systems}, that is, the
systems such that they contain a multiple of large enough order of
another irreducible system. Indeed this class enjoys, by definition,
the property that ``any irreducible linear system {\em larger} than
another system belonging to the class (that is, partially containing it),
belongs again to the class''.

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

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

Nevertheless, it will be convenient to distinguish between the two
cases by calling {\em regular} surfaces those for which $p_a=p_g$, and
{\em irregular} (of irregularity $p_g-p_a$) those for which $p_a<p_g$.

For either type of surface, the result obtained contains the
extension (from the series on a curve to the systems on a surface) of
the theorem of Riemann--Roch, but so far only for those linear
systems $|C'|$ that are adjoint to another irreducible linear systems
$|C|$ (of genus $\pi$ and degree $n$). The dimension of $|C'|$, or at
least a lower bound for it, is computed in terms of the genus of $C$
by the formula:
 \[
 r'\ge p_a+\pi-1.
 \]
But it is easy to express $r'$ also in terms of the characters of the
system $|C'|$, that is, its genus $\pi'$ and degree $n'$. In fact,
denoting by $p^{(1)}$ the (relative) linear genus of the surface $F$,
and assuming, for the sake of simplicity, that $|C|$ and $|C'|=|C+K|$
are base point free on $F$, we have
 \begin{eqnarray*}
 \pi'&=&\pi+p^{(1)}+2\pi-2-n-1=3\pi-3-n+p^{(1)},\\
 n'&=&n+(p^{(1)}-1)+4\pi-4-2n=4\pi-5-n+p^{(1)},
 \end{eqnarray*}
and thus
 \[
 \pi-1=n'-\pi'+1.
 \]
Thus the dimension of the system $|C'|$ will be
 \[
 r'\ge p_a+n '-\pi'+1;
 \]
and similarly, if also $|C|$ is the adjoint of another irreducible
linear system, then its dimension will be
 \[
 r\ge p_a+n-\pi+1.
 \]

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

 \section{Complements}\label{sec:4.5}
 \markright{\hfill Complements \quad 4.5\quad}
 In the above sections we have referred to a surface $F$ of ordinary
space, with normal singularities: under this assumption we have
determined the postulation of the adjoint surfaces, which are
required to pass simply through the double curve of $F$. But there
is no difficulty in extending our considerations, considering also
surfaces with higher order singularities. If one wants to follow the
pattern of the previous sections, then one is led, as a first step,
to extend the postulation formula, determining (according to
Noether) the number of conditions that a curve, of order $m$ and
genus $p$, imposes on the surfaces of large enough order $n$ that
have to contain it with multiplicity $s$, that is,\footnote{See also
L.~Campedelli, Sulla postulazione di una curva $i$-pla, {\it Rendic.\
Circ.\ Mat.\ di Palermo}, 55, 1931.}:
 \[
 \frac{s(s+1)}{6} [(3n-4s+4)m-(2s+1)(p-1)].\footnote{To be modified
in case of presence of multiple points of the curve.}
 \]
And similarly, we must compute the number of conditions imposed on
the surfaces of given order by requiring that they have an isolated
multiple point, that is, for an $s$-tuple point
 \[
 \binom{s+2}{3}.
 \]
But, as the singularities of $F$ become more complicated, the
computation of the postulation, on which the dimension of the systems
of adjoint surfaces depends, becomes more difficult, as is already
apparent from what we have said concerning the virtual influence on
the genus of the proper multiple points of $F$. It is thus useful to
point out a faster way, independent of the nature of the
singularities of the surface $F$. We shall always denote by $n$ the
order of $F$, by $\pi$ the genus of its plane sections, by $n_s$ and
$\pi_s$ the characters of the $s$-tuple system, cut out on $F$ by
the surfaces of order $s$.

We start with the following remark: if one assumes that the
arithmetic genus $p_a$ has been determined (for a transform of $F$ with
normal singularities), we know a priori the dimension of the systems
of adjoint surfaces of $F$ and we may say that the adjoint surfaces
$\Phi_{m-3+s}$, for $s$ large enough, cut out on $F$ a complete
linear system of dimension
 \[
 p_a+\pi_s-1,
 \]
and that this in turn cuts out a complete series of degree
$2\pi-2+sn$ on the plane section $C$ of $F$. Therefore the
$\Phi_{m-3+s}$ cut out on every plane the complete system of adjoint
curves of order $n-3+s$ of the plane section $C$.

Now one sees that {\em the dimensions of the linear systems of the
adjoint surfaces $\Phi_{m-3+s}$ to $F$, for $s$ large enough, form
an arithmetic progression of the third order whose ratio is
 \[
 \pi+sn+\frac{s(s-3)}{2}.
 \]}
This progression, extended for the smaller values of $s$, defines in any
case the {\em virtual expression} of the number of linearly
independent adjoint surfaces of given order and in particular the
{\em numerical genus $p_a$.}

In this sense one can say that the above arithmetic progression,
which is by the way a corollary of the postulation formulas,
replaces the actual knowledge of these
formulas\footnote{Independently of the fact that the base group
assigned to the $\Phi$ consists of the singularities of a surface $F$
with respect to which the $\Phi$ must be adjoint, one can show that
the numbers expressing the dimension of the system of surfaces $\Phi$
of large enough order required to pass with certain multiplicities
through the points and the curves of an assigned base group, always
give rise to this type of third order arithmetic progression. The
knowledge of this progression suffices, in many cases, to replace
the actual knowledge of the postulation formula. See G.~Castelnuovo,
{\em Alcuni risultati sui sistemi lineari di curve appartenenti ad
una superficie algebrica.} {\it Mem.\ Soc.\ It.\ delle Scienze}
(detta dei XL), s.~III, t.~10, 1896. Reprinted in {\it Memorie
scelte}, page~335.}.

We resume the consideration of the systems of adjoint surfaces
$\Phi_{m-3+s}$ of $F$.

We denote by $h\ge0$ the smallest number such that for $s\ge h$ these
systems cut out on a plane all the adjoint curves of the same order;
this means that, provided that $h\ne0$, the system of
$\Phi_{n-3+h-1}$ (which is of order greater than or equal to $n-3$)
cuts out on a plane an incomplete system of adjoint curves, with
deficiency $\de_{h-1}>0$. For $s<h-1$ the systems of $\Phi_{m-3+s}$
(for $s=0,1,\dots,h-2$) cut out on the same plane complete or
incomplete systems of adjoint curves, that a priori we may assume to
have deficiencies
 \[
 d_0\ge0,\quad d_1\ge0,\dots,\quad\de_{h-2}\ge0.
 \]
Now we have seen that, corresponding to the deficiency $\de_{h-1}$,
and in contrast to its virtual expression, the dimension of the
system of adjoint surfaces of order $n-3+h-2$ increases precisely by
$\de_{h-1}$, and the same holds for the dimension of the system of
curves cut out by these on $F$, which will be
 \[
 p_a+\pi_{h-2}-1+\de_{h-1}.
 \]
Similarly if $\de_{h-2}>0$, then the dimension of the system cut out
by the $\Phi_{n-3+h-3}$ increases in the same way by $\de_{h-2}$, and
is equal to
 \[
 p_a+\pi_{h-3}+\de_{h-2}+\de_{h-1}.
 \]
Finally, the dimension of the canonical system, cut out by the
$\Phi_{n-4}$, will turn out to be
 \[
 p_g-1=p_a-1+\de_0+\de_1+\cdots+\de_{h-1}.
 \]
In other words: the irregularity $p_g-p_a$ of the surface equals the
sum of the deficiencies of the systems of curves cut out on a plane by
the adjoint surfaces of $F$, of order
 \[
 n-3+s \quad \hbox{for}\quad s=1, 2,\dots
 \]
We can make the previous statements slightly more precise, by proving
that {\em if the adjoint surfaces of order greater than or equal to
$n-2$ cut out on a plane the complete linear system of adjoint
curves of the same order, then the same happens for all the linear
systems of adjoint surfaces of larger order.} (We shall explain
later how the statement extends for $s=0$, that is, to the case that
the adjoint surfaces of order $n-3$ cut out on a plane the complete
system of adjoints of that order).

To prove the theorem we have stated, we can pass from the systems of
surfaces $\Phi$ to the systems of curves they cut out on $F$, and
then to the series cut out by these systems on a plane section $C$
of $F$. And it suffices to prove that: ``if the curves of the system
$|sC+C'|$ cut out on $C$ the complete series for $s\ge1$, then also
the series cut out on $C$ by the curves of $|(s+1)C+C'|$ is
certainly complete''.

We start by assuming that $s>1$ and -- for the sake of
simplicity -- we set $s=2$. The assumption is thus that the curves of
$|2C+C'|$ cut out on $C$ the complete series $g$ of degree
$2\pi-2+2n$ determined by the adjoint curves $\fie_{n-1}$. And
obviously it suffices to prove that the minimal sum of this series
$g$ and of that cut out by the lines is necessarily complete.

In conclusion, the question is to prove the following result
(following Castelnuovo):

\paragraph{Lemma on the completeness of the sum of two series
belonging to a curve.}

{\em On a curve of genus $\pi$, to a complete and nonspecial series
$g_m$ of a certain degree $m$, we add a base point free series
$g_n$, at least $\infty^1$, which is contained in $g_m$, and with
respect to which the residual series is nonspecial; then the
minimal sum of $g_m$ and $g_n$ is necessarily complete.}

In fact let us add to $g_m$, which is of dimension $m-\pi$, two
groups $G$ and $G'$ in $g_n$, that we fix throughout: thus we shall
have two linear series of degree $m+n$ and dimension $m-\pi$, that
have in common the series consisting of $2n$ fixed points given by
the group $G+G'$ and the groups of $|g_m-g_n|$, which is a series of
degree $m-n$, assumed to be nonspecial, and thus of dimension
 \[
 m-n-\pi.
 \]
So our two series, that have the groups $G$ and $G'$ as fixed, will be
contained in a series of dimension at least
 \[
 m+n-\pi.
 \]
which therefore turns out to be complete. \qed

This lemma can be applied immediately to the case $h\ge2$. But, with
a slight modification, the argument can be extended to $h=1$ (but
not to
$h=0$). Indeed one has to prove that if one adds to the series
$g_{2\pi-2+n}$ cut out on $C$ by the adjoint curves $\fie_{n-2}$
the series cut out by the lines, then one always obtains, as the
minimal sum, the complete series cut out by the $\fie_{n-1}$.

In fact let us add to $g_{2\pi-2+n}$ two fixed groups $G$ and $G'$ cut
out by two lines $a$ and $a'$ of the plane; we shall have two linear
series of dimension $\pi-2+n$, that have in common the canonical
$g^{\pi-1}_{2\pi-2}$ (not of dimension $\pi-2$, but of dimension
$\pi-1$): the minimal sum of the two series with fixed groups $G$
and $G'$ is now a series of degree $2\pi-2+2n$ and of dimension
$\pi-2+n-1$ instead of $\pi-2+n$. This series, which is incomplete,
is cut out on $C$ by the system of all $\fie_{n-1}$ that pass
through the common point of the lines $a$ and $a'$. Now the minimal
linear system that contains two systems of $\fie_{n-1}$ with a
different base point is the complete system of all the $\fie_{n-1}$.
So it follows that the minimal sum of the series $g_{2\pi-2+n}$ and
of the series cut out by the lines of the plane is necessarily the
complete series cut out by the $\fie_{n-1}$.

 \section{Remarks on regularity conditions for a
surface}\label{sec:4.6}
 \markright{\hfill Remarks on the regularity conditions for a
surface \quad 4.6\quad}
 Let $F$ be a surface of order $n$ of ordinary space, that for the
sake of simplicity we wish to assume with normal singularities. We
have seen that, if the $\Phi_{n-2}$ of order $n-2$ adjoint to it cut
out on a plane the complete system of the adjoint curves of the
section of $F$, then the same happens for the adjoint surfaces
$\Phi_{n-2+r}$ of order $>n-2$; thus in this case the irregularity
$p_g-p_a$ of the surface is equal to the deficiency of the
(canonical) series cut out on the plane sections $C$ of $F$ by the
$\Phi_{n-3}$. In particular, if the $\Phi_{n-3}$ cut out the complete
canonical series on the above $C$, then the irregularity is
$p_g-p_a=0$, that is, the surface $F$ is regular.

But the regularity criterion, thus established by Enriques, can be
reduced to a simpler one.

{\em The surface $F$ of order $n$ of ordinary space is regular
($p_g=p_a$) if the surfaces $\Phi_{n-3}$ of order $n-3$ adjoint to
it cut out the complete canonical series on a plane section.}

We shall briefly sketch the proof of this theorem given by
Castelnuovo\footnote{Alcune propriet\`a fondamentali dei sistemi
lineari di curve tracciati sopra una superficie algebrica, {\it
Ann.\ di Mat.}, II, 25 (1897), reprinted in {\it Memorie scelte},
XXIII.}. To start with it is useful to prove the following result:

 \begin{Lemma}
If a group of $m<n$ points of our surface $F$, chosen among the $n$
points cut out by a line $a$, belongs to a special series $g^h_m$ of
dimension $h\ge1$, on a plane section $C$ through $a$, then the same
happens for every other section of $F$ with any plane through $a$.
 \end{Lemma}

This is an immediate consequence of the assumption that the
$\Phi_{n-3}$ cut out the complete canonical series on the plane
sections $C$ of $F$. In fact, by the theorem of Riemann--Roch, a
group of the aforementioned $g^h_m$ on $C$ imposes precisely $m-h$
conditions on the canonical groups of $C$, that is, also on the $C'$
that have to contain it: but the number of such conditions is
independent of the choice of a particular $C$ of the pencil cut out
on $F$ by the planes through $a$.

Using the above lemma we are able to see that {\em if the
$\Phi_{n-3}$ cut out the complete canonical series on the plane
sections $C$ of $F$, then the characteristic series of the complete
system $|C|$ is complete.}

We know that the complete system $|C|$ can be constructed on $F$ via
the linear system of all the adjoints $\Phi_{n-2}$ that pass through
the section $C'$ of a $\Phi_{n-3}$ (the residue theorem). So it
suffices to show that these $\Phi_{n-2}$ through a $C'$ cut out on a
plane the complete series of the curves $\fie_{n-2}$ of order $n-2$
adjoint to the section of $F$.

To this purpose one should try to construct a surface $\Phi_{n-2}$
adjoint to $F$, as the locus of a curve $\fie_{n-2}$ of the same
order $n-2$ adjoint to the section $C$ of a plane $a$ varying in a
pencil, having a generic line $a$ of space, not meeting $C'$, as its
axis. One will impose the following conditions on $\fie_{n-2}$:

 \begin{enumerate}
 \renewcommand{\labelenumi}{(\arabic{enumi})}
 \item that is passes through the $2\pi-2$ points of a canonical
group, section of $C'$;
 \item that is passes through $s-1$ points $A_1,A_2,\dots,A_{s-1}$
chosen among the $n$ common points of $C$ and of the line $a$, where
$s$ denotes the dimension of the characteristic series of $C$ when
completed, that is, the dimension of the complete series to which
the $g_n$ cut out on $C$ by the lines of the plane $a$ belongs;
 \item finally, touches a plane $\ga$, passing through a point $A_1$
of this group of $s-1$ points and not containing $a$.
 \end{enumerate}

Conditions~1--2 determine, on a generic plane through $a$, a
pencil of curves $\fie_{n-2}$, containing the curve consisting of the
line $a$ and of the $\fie_{n-3}$ passing through the canonical group
of $C$ section of $C'$. If one adds Condition~3, then a single curve
$\fie_{n-2}$ in the plane is determined; and, as this plane varies,
the $\fie_{n-2}$ thus constructed describes a surface
$\Phi_{n-2+h}$, adjoint to $F$, that a priori will be of order
$n-2+h>n-2$, passing $h$ ($>0$) times through the line $a$. For our
purposes, we need to show that $h=0$. Therefore we shall assume that
$h>0$ and we shall show that this assumption leads to a
contradiction.

In order to do that, let us remark that, since $h>0$, one has $h$
tangent planes to $\Phi_{n-2+h}$ at a point $A_s$ of $a$. Any one of
these planes $\abar$ cuts this surface along a curve consisting of
the line $a$ counted $h$ times and a $\fie_{n-2}$ passing through
the $s$ points $A_1,A_2,\dots,A_{s-1},A_s$. Now one can take as
$A_s$ one of the $n-(s-1)$ intersections with $a$ of the surface
$F$, or of the curve $\Cbar$ section of $\abar$, away from
$A_1,A_2,\dots,A_{s-1}$: since the $\fiebar_{n-3}$ section with
$\abar$ does not pass through $A_1$ (as $a$ does not intersect $C'$),
the group of points $A_1,A_2,\dots,A_s$ must belong, not to one
curve, but to a pencil of curves $\fie_{n-2}$ adjoint to the section
$\Cbar$ of $F$ with $\abar$, and thus this group has to belong to a
special $g^1_s$, of dimension (at least) equal to 1. But this cannot
happen on a particular plane $\abar$ containing $a$, unless it
happens for all the planes $a$. Therefore one deduces that $h=0$,
and so we succeed in constructing a surface $\Fie_{n-2}$ adjoint to
$F$ satisfying the conditions of:

 \begin{enumerate}
 \renewcommand{\labelenumi}{(\arabic{enumi})}
 \item passing through a curve $C'$ section of $F$ with an adjoint
$\Phi_{n-3}$;
 \item passing through $s-1$ among the $n$ points of intersection of
$F$ with a line a;
 \item touching a plane $g$ through one of these points.
 \end{enumerate}

By varying the points appearing in Condition~2, we can obtain
$\infty^{s+1}$ adjoints $\Fie_{n-2}$ through $C'$, which cut out
$\infty^s$ curves $\fie_{n-2}$ on a generic plane, that is, all the
adjoint curves $\fie_{n-2}$ passing through the canonical group
$G_{2\pi-2}$ section with $C'$. This means that the $\Fie_{n-2}$ through
$C'$ cut out on $F$ a linear system $|C|$ of dimension $s+1$, whose
characteristic series is complete, \qed

To establish Castelnuovo's theorem it suffices now to remark that:
``on a curve $C$ of genus $\pi$ ($>0$), the minimal sum of the
canonical series $g^{\pi-1}_{2\pi-2}$ and of another, base point
free, complete series $g^s_n$, not composed with an involution, is
always complete''.

This follows by remarking that a group $G_n$ of the latter series
imposes $s$ independent conditions on a group of the same series
$g^s_n$ that must contain it and $n-s$ conditions on a group of the
canonical series, therefore\footnote{See Castelnuovo, Sui multipli
di una serie lineare di gruppi di punti appartenente ad una curva
algebrica, {\it Rendic.\ Circ.\ Mat.\ di Palermo}, 1893, reprinted in
{\it Memorie scelte}, VIII.} at least
 \[
 s+n-s-1=n-1
 \]
conditions on a group of the minimal sum series
$g_{2\pi-2+n}=g^{\pi-1}_{2\pi-2}+g^s_n$; so the dimension of this series
exceeds the dimension of the canonical series by (at least) $n-1$,
hence it is equal to
 \[
 (n-1)+\pi-1=n+\pi-2.
 \]
This means that the minimal series $g^{\pi-1}_{2\pi-2}+g^s_n$ has the
same dimension as the complete series $g^{\pi-2+n}_{2\pi-2+n}$, thus
coincides with it.

 \begin{NOTE} Castelnuovo's theorem that expresses the regularity
conditions for a surface, with respect to the system $|C|$ of its
plane sections, extends easily also to the irreducible linear
systems $|C|$ that are not simple, as the author himself pointed
out. Thus we have:

{\em The necessary and sufficient condition in order that a surface
be regular ($p_a=p_g$) is that the adjoint system $|C'|$ of an
irreducible linear system $|C|$, of genus $\pi>0$, at least
$\infty^2$, cuts out the complete canonical series on the generic
curve $C$, that is, its dimension is $p_g+\pi-1$.

Every irreducible complete linear system belonging to a regular
surface has complete characteristic series.}

Later we shall again obtain this theorem as a particular case of a
property of the deficiency of the characteristic series of the
complete linear systems belonging to an irregular surface.
 \end{NOTE}

 \section{Examples of irregular surfaces}\label{sec:4.7}
 \markright{\hfill Examples of irregular surfaces \quad 4.7\quad}
 We now illustrate the previous results by some examples. For the
general surface of order $n$, without singularities, the adjoint
surfaces $\Phi_{n-4}$ are all the surfaces of order $n-4$; the actual
number of these, linearly independent, is equal to the virtual
number\footnote{If one wants to
apply the general postulation formula, one has to regard the double
curve as having order $n=0$ as genus $\rho=1$.}
 \[
 p_g=p_a=\binom{n-1}{3}=(n-1)(n-2)(n-3)/6
 \]
Thus, the general surfaces of any order are regular; in particular the
planes, the quadrics and the cubic surfaces that are not cones,
therefore also all the rational surfaces have superficial genus
 \[
 p_g=p_a=0.
 \]

Now if one imposes that a surface of order $n$ has a double curve of
relatively small order, one also obtains regular surfaces; if
instead one requires that the surface has a double curve of quite
large order, then a priori one finds a number of conditions that
exceeds the number of parameters on which the surface of given order
depends; therefore it is difficult to obtain examples of irregular
surfaces this way.

But let us consider a cone $F$ of order $n>2$, and of genus $p>0$,
and let $O$ be its vertex (a $n$-tuple point); here there are
certainly no adjoint surfaces of order $n-4$, that would have to pass
through the point $O$ with multiplicity $n-2>n-4$, so that it
follows that
 \[
 p_g=0
 \]
(as we already know). For the same reason there are no adjoint
surfaces $\Phi_{n-3}$ of $F$ of order $n-3$, that should cut out on
$F$ the $p_a+p$ linearly independent adjoint curves of the plane
sections, so that
 \[
 p_a+p \le 0;
 \]
that is,
 \[
 p_a\le-p.
 \]
But on the other hand there exist adjoint cones $\Fie_{n-2}$ (with $O$ as
an $(n-2)$-tuple point) that cut out on a plane the complete system of
adjoint curves of order $n-2$ to the section of $F$. Thus by our results we
can say that the numerical genus is
 \[
 p_a=-p.
 \]
In conclusion: {\em the ruled surfaces of genus $p$ have numerical
genus $-p$ and irregularity}
 \[
 p_g-p_a=p.
 \]
This theorem is proved by transforming the ruled surface into a cone;
but even without performing this transformation one can repeat the
previous argument, based on the following facts:

 \begin{enumerate}
 \item there do not exist any adjoint curves of the plane sections
$C$; and
 \item the curves of $|C+C'|$ (sections with the adjoints
$\Fie_{n-2}$) cut out the complete series on a $C$.
 \end{enumerate}

The example of the ruled surfaces has been known since the first
studies of A.~Cayley\footnote{On the deficiency of certain
surfaces\dots, {\it Math.\ Ann.}, Band~III.} (1871) and of
M.~Noether, who proved that the virtual expression for the genus
becomes negative in this case. Later on the first example of
irregular surface, for which $p_g>0$, was pointed out by
G.~Castelnuovo\footnote{See G.~Castelnuovo, Osservazioni intorno
alla geometria sopra una superficie, {\it Rendic.\ Ist. Lombardo},
1891. Note I. {\it Memorie scelte}, n.~XVII.}.

Let us impose on a surface $F_6$ of order 6 the condition of having 8
triple points $A_1,A_2,\dots,A_8$ with arbitrarily fixed position in the
space. The $F_6$ depend on 83 parameters, imposing a triple point yields
ten linear conditions, so the $F_6$ that pass triply through
$A_1,A_2,\dots,A_8$ are $\infty^3$, and split as triples of quadrics
(belonging to a pencil) that pass through those 8 points. But
instead of taking $A_1,A_2,\dots,A_8$ in arbitrary position, let us
assume that they are the base group of a net of quadrics $\Phi_2$.
In this case, by taking linear combinations of the triples of
$\Phi_2$, one obtains a linear system of irreducible
$F_6$, of dimension 9\footnote{One can show that the system of $F_6$
with the 8 triple points cannot have dimension $r>9$, by computing
the dimension of the system that the $F_6$ cut out on a quadric of
the net.}.

Now the $F_6$ thus constructed turn out to be of genus $p_g=3$, since
they have the quadrics $\Phi_2$ of the net that we are considering as
adjoints of order $6-4=2$. But the eight base points
$A_1,A_2,\dots,A_8$ impose on the $\Phi_2$ seven linearly independent
conditions instead of eight, which is the virtual number of these
conditions. So the virtual genus of $F_6$ is
 \[
p_a=p_g-1=2.
 \]

It is interesting to remark that we have thus constructed the first
instance of a series of analogous examples. In general it is possible
to construct irreducible surfaces of order $2n$ ($n\ge3$) that have 8
$n$-tuple points at the base points of a net of quadrics: these
$F_{2n}$ turn out to be irregular of genus
 \[
 p_g=\frac{n(n-1)}{2}
 \]
and
 \[
 p_a=n-1.
 \]

It is important to remark now that the quadrics of the net cut out on
one of the surfaces $F_{2n}$ a linear system composed with quartic
elliptic curves, and that these quartics form on $F_{2n}$ an
irrational pencil of genus
 \[
 p_g-p_a=\frac{(n-1)(n-2)}{2};
 \]
in fact the genus of the pencil equals the genus of the osculating
cone to $F_{2n}$ at one of the points $A$, which is a base point for
the pencil of these quartics.

Let us add that the above series of examples, pointed out by
Castelnuovo, can be generalized as L.~Campedelli (1936)\footnote{L.
Campedelli, in {\it Atti Acc.\ Torino}, 1936.} has shown. The
construction of an irreducible $F_{2n}$ with eight $n$-tuple points
actually depends on the construction of a linear pencil of elliptic
curves of some order $m$ dividing $n$ ($n=ms$), having eight
$s$-tuple base points on a quadric: such a group of base points
always belongs to an elliptic quartic, and it gives on it a group
resulting from the division by $n$ of the series cut out on the
curve by all the surfaces of order $2n$. Such a pencil of elliptic
curves can be projected into the plane from one of its base points
to a Halphen pencil of curves $C_{3m}$ with $9m$-tuple base points.

Starting with the pencil that we have described on a quadric, one
obtains irregular surfaces $F_{2n}$, with eight $n$-tuple points,
that have always an irrational pencil of genus $p_g-p_a$.

 \begin{Remark} Using the theorem of Chapter~VI to the effect that the
characteristic series of a complete linear system on a regular
surface is complete, one sees in general that {\em the existence of
an irrational pencil (of genus $\pi>0$) implies that the surface is
irregular.}

More precisely we shall find that the surface has irregularity
$p_g-p_a\ge\pi$. But more of this later.
 \end{Remark}

 \section{The theorem of Riemann--Roch for surfaces: systems larger than
the canonical}\label{sec:4.8}
 \markright{\hfill Riemann--Roch Theorem: systems larger than the
canonical \quad 4.8\quad}
 We have seen that a linear system of degree $n$ and genus $\pi$, which is
the adjoint of another irreducible system on a surface of numerical
genus $p_a$, has dimension
 \[
 r\ge p_a+n-\pi+1,
 \]
and this fundamental inequality, even under the restrictive
assumptions that we have made here, is an extension to surfaces of the
well-known theorem of Riemann--Roch for curves. It is convenient now to
understand the meaning of the restriction that $|C|$ is an irreducible
system. For the sake of simplicity we assume that $|C|$ is base point
free on the surface $F$, and that the same happens for the system $|L|$
to which it is adjoint. Denoting by $|K|$ the canonical system we have
 \[
 |C|=|L+K|.
 \]
So the condition imposed on $|C|$ of being the adjoint of another
system can be translated in the condition of containing the canonical
system, with a residual system $|L|$ of degree greater than zero, or, as
we shall say, of being more ample than the canonical system.

A system $|C|$ more ample than the canonical system $|K|$ can always
be regarded as the adjoint of the system
 \[
 |L|=|C-K|,
 \]
but it is not possible to say a priori that $|L|$ is irreducible. On the
contrary there are cases in which, even though one can increase the
dimension of $|C|$ by multiplying it by an arbitrarily large number,
the system $|L|$ always turns out to be reducible, containing for
instance some fixed components, not connected with the variable
components. Indeed this happens if $|C|$ is the system of plane
sections of a surface $F$ that has a multiple point that affects the
genus, for instance an isolated multiple point, of multiplicity $n\ge3$,
that corresponds to a fundamental curve $\theta$ of $|C|$. Nonetheless it is
easy to see that in this case the above inequality (dimension
$\dim|C|\ge p_a+n-\pi+1$) a fortiori holds true. In fact, denoting by
$\nbar$ and $\pibar$ the characters of the system $\Cbar=|C-q|$, we
shall have that $|\Cbar|$ is the adjoint of the irreducible system
$|\Lbar|=|L-q|$:
 \[
 |\Cbar|=| \Lbar '|,
 \]
and so the dimension of $|\Cbar|$ is
 \[
 \rbar\ge p_a+\nbar-\pibar+1;
 \]
now the degree of $\theta$ is $-n$ and its genus $r=(n-1)(n-2)/2\ge1$,
thus we have
 \[
 n=\nbar-n
 \pi=\pibar+r+n-1
 \]
and accordingly the dimension of $|C|$, $r>\rbar$, will be
 \[
 r>p_a+n-\pi+1.
 \]
But, in order to proceed rigorously with the greatest generality, we
shall prove that any linear system $|C|$ more ample than the canonical
system has dimension $r\ge p_a+n-\pi+1$, making use of another argument
that holds equally well, both for reducible and irreducible systems
$|C|$.

Given an irreducible or reducible curve $C$ on the surface $F$, we may
always assume that the adjoint surfaces $\Phi_{n-4+s}$ (of large enough
order $n-4+s$) of $F$ passing through $C$, do not pass as a consequence
through another curve or through points away from $C$ or the double
curve\footnote{This follows by remarking that in the postulation
formulas for a reducible curve (or for a curve and a group of points)
each component actually yields a certain number of conditions.}. In
this setting, the curve $C$ is contained in the system $|L_s'|$ cut out
on $F$ by the $\Phi_{n-4+s}$, and gives with respect to it an
irreducible (base point free) residual system
 \[
 |D|=|L_s'-C|.
 \]
Now, denoting by $r_s'$ the dimension, by $\pi_s'$ the genus and $n_s'$
the degree of $|L_s'|$, we have found that
 \[
 r _s'\ge p_a+n _s'-\pi_s'+1
 \]
(indeed the equality actually holds in this case); subtracting a curve
$D$, one finds that the residual system $|C|=|L_s'-D|$ has to have
dimension
 \[
 r\ge p_a+n-\pi+1,
 \]
provided that one knows that the series cut out by $|L_s'|$ on $D$ is
nonspecial; and this is the case when $|C|$ is more ample than the
canonical system $|K|$, the curves of which cut out on $D$ residual
groups of the characteristic series of $D$.

Let us carry out the computation introducing the characters, genus
$\rho$ and the degree $n$ of $D$ and the number $t$ of the
intersections of $C$ and $D$. The series cut out on $D$ by $|L_s'|$ has
degree $n+t$, and (if it is complete) it has dimension $n+t-r$, so that
if one subtracts $D$ then the dimension $r_s'$ of $|L_s'|$ decreases at
most by $n+t-r+1$; then we have
 \[
 r\ge r_s'-(n+t-r+1).
 \]
On the other hand
 \[
 n_s'=n+n+2t \quad
 \pi_s'=\pi+r+t-1
 \]
 hence
 \[
 n_s'-\pi_s'=n-\pi+n-r+t+1;
 \]
as a consequence we have
 \[
 r\ge p_a+n-\pi+1.
 \]
Thus we have proved that: for the irreducible or reducible linear
systems of degree $n$ and genus $\pi$ that are more ample than the
canonical system, the value of the dimension is always
 \[
 r\ge p_a+n-\pi+1.
 \]
This inequality is the so-called theorem of Riemann--Roch for surfaces,
and it will be extended later-as we shall see-to arbitrary linear
systems.

Meanwhile, referring to the systems more ample than the canonical
system, we may distinguish:

 \begin{enumerate}
 \renewcommand{\labelenumi}{(\arabic{enumi})}
 \item the regular systems, for which the equality holds
 \[
 r=p_a+n-\pi+1
 \]
 \item and the superabundant systems for which instead
 \[
 r>p_a+n-\pi+1.
 \]
 \end{enumerate}
We have already seen that some regular systems do exist, since the
adjoints of the multiples of quite large degree of an irreducible
system $|L|$ are such.

The result that we have obtained yields at once some remarkable
consequences.

Let $|C|$ be an irreducible linear system of genus $\pi$ and degree
$n$, more ample than the canonical system and regular on the surface
$F$, and let us add to it an irreducible curve $D$ of genus $r$ and
degree $n$, which intersects the curves $C$ at a certain number $m$
($>0$) of points.

The degree and the genus of $|D+C|$ are respectively given by
 \[
 n+n+2m \quad\hbox{and} \quad\pi+r+m-1,
 \]
hence their difference is
 \[
 (n-\pi)+(n-r)+m+1>n-\pi
 \]
provided that it is
 \[
 m\ge r-n.
 \]
One deduces that under this assumption the system $|C+D|$ has dimension
greater than
 \[
 p_a+n-\pi+1
 \]
which is the dimension of $|C|$, and therefore it is certainly
irreducible.

Moreover the system $|C+D|$ will be regular and it will have to cut out
a (nonspecial) complete series on $D$, otherwise subtracting $D$ one
would obtain a superabundant system $|C|$.

We notice that this extension of $|C|$ would be impossible for $m=0$,
that is, if the curve $D$ were fundamental for $|C|$; in this case
therefore one has $n-r<0$, as it follows a priori by remarking that the
fundamental curve $D$ of genus $r\ge0$ must have negative degree:
$n<0$, where $-n$ is the number of intersections of $D$ with its
residual curves (of $|C-D|$).

If one interchanges the roles of $|C|$ and $|D|$ in the previous
remark, one is led to state that: given a linear system $|C|$
(virtually free from base point), however reducible on the surface $F$,
it is always possible to add to it an irreducible regular linear system
$|D|$ so that the sum is an irreducible linear system $|C+D|$ which is
again regular.

It suffices to take as the system $|D|$ the multiple $|L_s|=|sL|$ of
a regular system $|L|$ with no fundamental curves, whose number of
intersections with the $C$ (or with their components) thus grows
with $s$. In the case in which the curves $C$ are reducible, e.g.,
$C=C_1+C_2+\cdots$, one firstly adds to $C_1$ a multiple of $|L|$,
then adds to $C_2$ the irreducible system (as ample as one wishes)
thus obtained and so on, until one obtains an irreducible system
$|sL+C|$.

 \section{Elimination of the exceptional curves}\label{sec:4.9}
 \markright{\hfill Elimination of the exceptional curves \quad
4.9\quad}
 We continue the analysis of the previous section, about the addition
of a curve to a regular linear system $|C|$ on a surface $F$. And we
take as the curve to be added to $|C|$ an exceptional curve $\om$ of
genus $r=0$ and degree $n>-1$. If this is not fundamental for $|C|$,
and so it meets the $C$ at $m>0$ points, adding it to $|C|$ one
always obtains a linear system of greater dimension, irreducible and
regular as $|C|$. And the curves of this sum system have $m+n$
intersections with $\om$. Therefore $\om$ can be successively added
to $|C|$, thus obtaining an infinite sequence of systems of
increasing dimension, provided that it is $n>0$, that is, in the
case in which $\om$ is an exceptional curve of the second kind.
Conversely, if $\om$ is of the first kind ($n=-1$), then the
sequence of systems thus obtained stops at the system $|C+m\om|$ for
which $\om$ becomes fundamental.

Now let us consider, for every linear system $|C|$ of degree $n$ and
genus $\pi$, belonging to the surface $F$, the difference $2\pi-2-n$
(number of intersections of its curves with the virtual canonical
curves) that yields an additive character $a(C)$ of the systems on
$F$. This difference decreases at least by one every time that one
adds $\om$ to $|C|$ or to one of the systems thus obtained, hence,
if the family of these is infinite, then it will certainly end up to
be negative, so that one will have systems of genus $\pi$ and degree
$n>2\pi-2$ on $F$.

Now, we wish to assume that the surface $F$ does not contain any
system for which the difference $-a(C)=n-(2\pi-2)>0$. Under this
assumption it does not contain any exceptional curve of the second kind
and it contains at most a finite number $s$ of exceptional curves of
the first kind.

For the sake of simplicity we shall assume that these are
irreducible exceptional curves, but we shall explain later how the
argument works in the other case (which corresponds to exceptional
curves that are the images of infinitely near points).

Let us assume that the (irreducible, regular) system $|C|$ is, not only
virtually, but also actually free from base point on $F$. If, as we
have said earlier, the $C$ intersect the exceptional curve $\om$ at $m$
points, then $\om$ is a fundamental curve (of degree-1, thus decreasing by
one the degree of the residual curves) for the system $|C+m\om|$. And
transforming the surface $F$ into another surface $F'$ via this system, $\om$ is transformed into a simple point $O$ of $F'$.

Now each of the other $s-1$ exceptional curves is not connected with
$\om$, and therefore it still gives an exceptional curve of the first
kind (or a simple point of the surface) on $F'$, otherwise one would
have on $F'$ exceptional curves of degree $\ge0$ (that is, of the second
kind), hence one would be able to construct on the surface $F'$, and
also on $F$, systems $|C|$ for which the character
 \[
-a(C)>0.
 \]
So by transforming $F$ into $F'$ one succeeds in decreasing the number
of exceptional curves belonging to the surface, so that after $s$
transformations we shall obtain a surface $F$ completely free of
exceptional curves.

Actually we have simplified our discussion by assuming that the
exceptional curves of $F$ are all irreducible. But if there on $F$
is an exceptional curve of the first kind consisting of $h$ parts
(given in a certain order $\om,\om',\dots, \om^{(h-1)}$), the last
one of these components is always an exceptional curve of degree
$n=-1$, and when one reduces this curve to a point with the
transformation that takes $F$ into $F'$, the preceding component
becomes an exceptional curve (of the first kind) of degree $n=-1$.
So the exceptional curve consisting of $h$ parts (which form $h$
exceptional curves of $F$, one irreducible and $h-1$ reducible)
becomes an exceptional curve on $F'$ consisting of $h-1$ parts. And
eventually it disappears after $h$ transformations, since it is
taken (on $F$) to a group of $h$ infinitely near simple points.

Therefore we conclude: In general a surface $F$ can be transformed into
another one free of exceptional curves. This transformation fails to
exist only for the surfaces (of genus $p_g=0$, bigenus $P=0$, and so
on) on which there exist linear systems $|C|$ for which the character
$-a(C)=n-(2\pi-2)>0$, that is, as we shall see, for the surfaces that
belong to the family of ruled surfaces.

 \section{Completeness of the series cut out by a small linear system
on the curves of a large system}\label{sec:4.10}
 \markright{\hfill Completeness of the series cut out by a small linear
system \quad 4.10\quad}
 The extension of the theorem of Riemann--Roch to the case of
arbitrary linear systems (even to the ``special'' ones, that are less
ample than the canonical system) will be given in the simplest way
by exploiting the completeness of the series that a ``small'' linear
system cuts out on the generic curve of a ``normally large'' system.
``Small'' and ``large'' are relative terms, that are defined as
follows: one regards as ``large'' with respect to a given linear
system $|C|$ (virtually free from base point on the surface $F$) a
linear system $|D|$ that contains $|C|$ in such a way that the
difference $|D-C|$ equals the multiple $|s L|$, with
$s$ an arbitrarily large integer, of an irreducible base point free
linear system $|L|$, that is, it is a system ``more ample'' than $|s
L|$. And one says that $|D|$ is normally large, with respect to
$|C|$, if the previous condition is satisfied with a system $|L|$
free of fundamental curves, so that picking $s$ large enough one can
achieve that the system $|B|=|D-C|$ is
irreducible\footnote{This is no longer true if $|L|$ and
so also $|L_1|=|D|$ has a fundamental curve $\theta$ that is not
fundamental for for $|C|$, since in that case $\theta$ is a fixed part
for $|D-C|$. So also the main lemma does not hold any more for $|C|$
with respect to this (not normally) large system $|D|$: for instance
the canonical system $|K|$ does not cut out any more on $D$, in
general, the complete series (cut out by $|K+\theta|$).}.

 We wish to prove the Main Lemma.

A complete, reducible or irreducible, linear system $|C|$ cuts out a
complete series on the irreducible curve of a linear system $|D|$, which
is normally large with respect to it.

To prove this lemma, let us take on a curve $D$ an arbitrary group $G$ of
the complete series $CD$, cut out by $|C|$, and associate with it a group
$G'$, whose existence will be shown right away, satisfying the two following
properties:

 \begin{enumerate}
 \renewcommand{\labelenumi}{(\arabic{enumi})}
 \item G+G' is a group of the (usually incomplete) characteristic
series cut out on $D$ by the other curves of $|D|$;
 \item $G'$ is a group of the series cut out on $D$ by an irreducible
curve $B$ of $|D-C|$.
 \end{enumerate}

Then $G+G'$ is the base group of a pencil of curves $D$ and there is
an irreducible fundamental curve $B$ with respect to it passing
through $G'$; accordingly the aforesaid pencil contains a curve $D$
consisting of $B$ and of a $C$ through G; therefore every group $G$
of the complete series $CD$ on $D$ is cut out by a $C$.

It is only necessary to see that, when $|D|$ is quite large with
respect to $|C|$, then for every group $G$ of the complete series
$CD$ actually there is a group $G'$ satisfying properties (1) and (2)
that can be associated to it. In order to do this, we consider the
two, a priori incomplete, linear series $DD-G$ and $BD$, cut out
on $D$ by the $D$ through $G$ and by the $B$ respectively; they are
contained in the same complete series, and we should prove that they
certainly have an effective linear series in common, which is
described by the groups G'. Thus it suffices to prove that the
deficiencies of these two series are smaller than their dimensions,
because then, as one easily sees, the sum of the dimensions of the
two series exceeds the dimension of the complete series.

Now the last condition is certainly satisfied because one sees that,
for large $s$ going to infinity, the dimension of the characteristic
series of $|D|$ on $D$, hence also the dimension of the series
$DD-G$, becomes goes to infinity to the second order as
$\frac{s^2}{2}n$ (where $n$ is the degree of $|L|$), and the same
happens for the dimension of the series $BD$; on the other hand the
deficiency of the characteristic series, and thus also of the series
$DD-G$, may very well grow indefinitely with $s$, but we know a
priori it cannot become infinity of order greater than one.

We shall limit ourselves to carry out this simple computation in the
case in which
 \[
 |D|=|L_s| \quad |C|=|L|.
 \]
Denoting by $n$ and $\pi$ the degree and the genus of $|C|$ and by
$n_s$ and $\pi_s$ the degree and the genus of $|L_s|$, the dimension
$r_s$ of $|L_s|$ is given by the theorem of Riemann--Roch:
 \[
 r_s\ge p_a+n_s-\pi_s+1,
 \]
where
 \[
 n_s=s^2n, \quad \pi_s=s \pi+\frac{s (s-1)}{2}n-s+1,
 \]
and so
 \[
 r_s\ge \frac{s^2}{2}n-s \pi-\frac{s}{2}n+s-1;
 \]
thus the actual dimension of the characteristic series
$|DD|=(L_sL_s)$ is equal to
 \[
 r_s-1\ge \frac{s^2}{2}n+\cdots
 \]
disregarding the terms of order $s$. And, since the group $G=CL_s$
is made up of $sn$ points, the dimension of the effective series
$DD-G$ will be
 \[
 \frac{s^2}{2}n+\cdots,
 \]
disregarding the terms of order smaller than two.

On the other hand the dimension of the series cut out on $D=L_s$ by
the curves $B$ of $|L_{s-1}|=|L_s-C|$ is also
 \[
 \frac{s^2}{2}n+\cdots,
 \]
It is now possible to compute the deficiency of these series, or at
least an upper bound for it, depending on the deficiency of the
characteristic series $DD$ and thus on its index of speciality. If this
series is complete (of dimension $r_s-1=p_a+n_s-\pi_s$) then its
index of speciality is $p_a$, if instead it has deficiency $d$, then
its index of speciality is
 \[
 i=d+p_a.
 \]
In any case this index $i$ is smaller than the degree of the residual
series, that is, than the number of intersections of $D=L_s$ with a
canonical curve $K$:
 \[
 DK=s (2\pi-2-n),
 \]
and accordingly the deficiency of the aforementioned series $DD$, and
thus also the deficiencies of $DD-G$ and of $BD$, cannot become
infinite with $s$ of order greater than one. \qed

The same computation can be extended, with slight modifications, to
the case in which one has again $|D|=|L_s|=|sL|$ and $|L|=|B+C|$, where
$|L|$ is a system free of fundamental curves, that is, to the case in
which one has
 \[
 |D|=|L_s+E|,
 \]
and $|C|$ coincides with $|L|$ or it is partially contained in it:
 \[
 |L|=|B+C|.
 \]
We arrive again at the conclusion that the complete system $|C|$
cuts out a complete series on the curve $D$ of a normally large
system.

 \section{The theorem of Riemann--Roch for arbitrary linear
systems}\label{sec:4.11}
 \markright{\hfill The theorem of Riemann--Roch for arbitrary linear
systems \quad 4.11\quad}
 Let us take now, instead of the linear system $|C|$ considered in the
previous section, the canonical system $|K|$, thus we are assuming $p_g>0$,
and let us construct an irreducible linear system $|D|$ that is normally
large with respect to $|K|$. We may say that $|K|$ cuts out the complete
residual series of the characteristic series on a generic curve $D$ of $|D|$.

This statement holds also for $p_g=0$, meaning with it that ``on a
surface $F$ of genus $p_g=0$, the characteristic series of a normally large
system $|D|$ is a nonspecial series''. To prove this, take an
auxiliary linear system $|C|$ and consider, together with it, its
adjoint system $|C'|$. If $|D|$ is large enough with respect to
$|C|$ and to $|C'|$, then both these systems cut out complete series
on a generic $D$: say $g$ and $g'$. If the characteristic series $DD$
is special and thus contained in the canonical series $DD'$, where
$|D'|=|D+C'-C|$, then the series $g$ has to be contained in $g'$, and
precisely a group $G$ of $g$ will be part of a group $G'=G+G_1$ of
$g'$. We can see that, under this assumption, $|C'|$ contains $|C|$,
too, and so $p_g>0$, contrary to the assumption.

In order to do this it suffices to remark that the curve $C$ passing
through $G$ (and, in the reducible case, each of its components) has
a number of intersections greater than $CC'$ with the (normally
large) curve $D$, and so it is part of the $C'$ through $G+G_1$.

We have thus established that {\em on a surface of genus $p_g>0$ the
characteristic series of a normally large linear system {\em (with
respect to the canonical system)} has index of speciality $p_g$.}

Let us now consider on the surface $F$ a system $|C|$, however special,
of index $i$ ($\ge0$), and a system $|D|$ normally large with respect to
it. We want to see that {\em the index of speciality of the series
cut out on a $D$ by the system $|C+D|$ is equal to the index of
speciality of $|C|$, that is, to the number $i$ of linearly
independent canonical curves which contain a given $C$ as a part.}

Indeed, since the series cut out by $|K|$ on $D$ is complete, the
linearly independent canonical curves $K$ passing through a group $G$ of
the series $CD$ on a $D$ are as many as the index of speciality of
the series cut out by $|D+C|$ on $D$; each one of these $K$ has a
number $CD>CK$ of intersections with $C$ (or, in the reducible case,
with any component of $C$) and therefore it contains $C$ entirely.

Now we can deduce the dimension $r$ of the system $|C|$ of degree
$n$, genus $\pi$ and index of speciality $i$, that one obtains by
subtracting a $D$ from the system $|D+C|$; in order to do this, we
have to subtract from the dimension of the latter system, which is
regular, the dimension of the series cut out on $D$ by $|D+C|$, of
which we know the index of speciality, but it may be not complete.
In this way we find a lower bound for $r$: precisely we find, as for
the systems more ample than the canonical system: $r\ge p_a+n-\pi+1$
if $i=0$, that is, for all the {\em nonspecial} systems $|C|$, and
in general
 \[
 r\ge p_a+n-\pi+1-i
 \]
{\em for the special systems of index $i$.

This inequality is the most general form of the theorem of
Riemann--Roch for surfaces.}

Since it has been proved for the irreducible or reducible base point
free linear systems $|C|$ on the surface $F$, one deduces that it holds
true for any linear system, with arbitrarily assigned base points: in
fact an $i$-tuple base point imposed on the $C$ lowers their degree by
$i^2$, their genus by $i(i-1)/2$, and their dimension by at most
 \[
 \frac{i(i+1)}{2}=i^2-\frac{i(i-1)}{2}.
 \]
The linear systems $|C|$ for which the dimension $r$ is equal to
$r=p_a+n-\pi+1-i$ are said to be {\em regular}; they are said to be
{\em superabundant} if instead
 \[
 r>p_a+n-\pi+1-i.
 \]
The theorem of Riemann--Roch for curves of genus $p$ gives us the
dimension of the complete special series $g^r_n$ expressed in terms of the
dimension of its residual series with respect to the canonical
$g^{p-1}_{2p-2}$. The obvious complement of that theorem is the remark
that the residual series of the $g^r_n$ has degree $2p-2-n$. It is also
possible to complete the theorem of Riemann--Roch for special linear
systems on a surface, by expressing characters as genus and degree of
the residual system by means of those of the given system. Let in fact
$|C_1|$ and $|C_2|$ be two linear systems, each residual to the other with
respect to the canonical system $|K|$. Let $n_1$ and $\pi_1$ be the
characters, genus and degree, of the former system; $n_2$ and $\pi_2$
the characters of the latter, and let us denote by $m$ the number of
intersections of $|C_1|$ and $|C_2|$. Introducing the linear genus
$p^{(1)}$ of the surface, we may express the genus and the degree of
$|K|$ by means of the formulas
 \[
 p^{(1)}=\pi_1+\pi_2+m-1,\quad
 p^{(1)}-1=n_1+n_2+2m.
 \]
 But one can compute $m$ by taking into account that $|K|$ cuts out on
$C_1$ a group of
 \[
 2\pi_1-2-n_1=n_1+m
 \]
points, so that
 \[
 m=2\pi_1-2-2n_1.
 \]
Therefore we have
 \begin{eqnarray*}
 p^{(1)}&=&\pi_1+\pi_2+2\pi_1-2-2n_1-1
 =3\pi_1+\pi_2-2n_1-3p^{(1)}-1 \\
 &=& n_1+n_2+4\pi_1-4-4n_1=4\pi_1-3n_1+n_2-4, \nonumber
 \end{eqnarray*}
whence we deduce
 \begin{eqnarray*}
\pi_2&=p^{(1)}-3\pi_1+2n_1+3 \\
n_2&=p^{(1)}+3n_1-4\pi_1+3.
 \end{eqnarray*}
{\em These formulas express the characters of the system $|C_1|$
residual of the system $|C_2|$ with respect to the canonical system
by means of the characters of $|C_1|$.}

We notice that the dimensions of the two systems $|C_1|$ and $|C_2|$,
each residual to the other with respect to the canonical system, are
given by
 \begin{eqnarray*}
 r_1&=&p_a+n_1-\pi_1-r_2+\om_1 \\
 r_2&=&p_a+n_2-\pi_2-r_1+\om_2,
 \end{eqnarray*}
where $\om_1$ and $\om_2$ denote their respective superabundances.
So by the previous formulas one obtains
 \[
 \om_1=\om_2
 \]
that is: {\em two special systems one residual to the other with
respect to the canonical system have the same superabundance.}

 \section{The deficiency of the characteristic
series}\label{sec:4.12}
 \markright{\hfill The deficiency of the characteristic
series \quad 4.12 \quad}
 Consider on the surface $F$ of numerical
genus $p_a$ and geometric genus $p_g\ge p_a$ an irreducible linear
system of genus $\pi$, of degree $n$ and dimension $r$, that for the
sake of simplicity we assume to be nonspecial. On the generic curve
$C$ we have two linear series, one residual to the other with
respect to the canonical series: that is, the characteristic series
$g^{r-1}_n$, cut out by the other curves $C$, and the series of
degree $2\pi-2-n$ and dimension $p_g-1$ cut out by the canonical
curves $K$. If these two series have deficiency $d$ and $d'$
respectively, then they are contained in two complete series of
dimension
 \[
 r-1+d and p_g-1+d'.
 \]
 Applying the theorem of Riemann--Roch on the curve, one has
 \[
 r-1+d=n-\pi+p_g-1+d'+1
 \]
 that is,
 \[
 r=n-\pi+p_g+d'-d+1.
 \]
 So if $d'=0$ we have
 \[
 r=p_g-d+n-\pi+1.
 \]
Now, if $|C|$ is a regular linear system, normally large with respect
to the canonical system, then its dimension is
 \[
 r=p_a+n-\pi+1
 \]
and on the other hand, we know that the canonical curves $K$ cut out the
complete series on $C$, so that one has $d'=0$.

Whence we deduce that
 \[
 d=p_g-p_a.
 \]
In words: the characteristic series of a complete, normally large,
irreducible linear system, which is regular\footnote{Later we shall
see that ``every normally large linear system is regular''.}, has
deficiency
 \[
 d=p_g-p_a.
 \]
With this in mind we wish to prove that the characteristic series of an
arbitrary irreducible linear system $|C|$ belonging to the surface $F$
has always deficiency
 \[
d\le p_g-p_a.
 \]
In order to obtain this result we shall establish the following:

Lemma: If a linear system
 \[
 |C|=|C_1+C_2|,
 \]
sum of two irreducible linear systems, cuts out the complete series on
a curve of the system $|C_1|$, then the deficiency $d_1$ of the
characteristic series of $|C_1|$ is smaller than or equal to the
deficiency $d_2$ of $|C_2|$:
 \[
 d_1\le d_2.
 \]
Let us denote by $n_1,\pi_1,n_2,\pi_2$, the characters of $|C_1|$ and
$|C_2|$, by $r_1$ and $r_2$ their respective dimensions, and by $s$ the
number of intersections of a $C_1$ with a $C_2$; finally let $r$ be the
dimension of $|C|$. The series cut out by $|C|$ on a $C_1$ is complete
by assumption. Therefore the $C$ passing through the group $G$ of the
$s$ common points of a $C_1$ and a $C_2$ also cut out a complete series
of dimension $r_1-1+d_1$ on $C_1$.

On the other hand the dimension of the complete series cut out on $C_1$
by $C$ is
 \[
 r-r_2-1
 \]
 hence the dimension of the series cut out on $C_1$ by the $C$ through $G$ is
 \[
 r-r_2-1-(s-e),
 \]
 where $s-e$ denotes the number of linearly independent conditions that
the group $G$ of $s$ points imposes on the $C$ that have to contain it. Thus
we have
 \[
 r_1-1+d_1=r-r_2-1-(s-e),
 \]
 that is,
 \[
 d_1=r-(r_1+r_2)-(s-e).
 \]
This equality holds also independently of the assumption that $|C|$
cuts out the complete series on $C_1$, provided that one replaces $d_1$
with the deficiency $d'_1$ of the characteristic series of $|C_1|$ with
respect to the series cut out by the $C$ through $G$ ($d'_1\le d_1$); by
applying the same argument to the system $|C_2|$ one finds:
 \[
 d_2\ge d'_2=r-(r_1+r_2)-(s-e).
 \]
Comparing the expressions of $d_1$ and $d_2$ it follows in any case
 \[
 d_1 \le d_2 \quad \qed
 \]
Now, given an arbitrary irreducible, at least $\infty^1$, linear system
$|C|$ of degree $n$ and genus $\pi$ on the surface $F$, let us
construct on $F$ another irreducible linear system $|D|$, which is
regular and large enough so that deficiency of its characteristic
series is $p_g-p_a$.

We know (Section~\ref{sec:4.8}) that (since the number of intersections
$(CD)$ is greater than $n-\pi$) by adding $|C|$ to $|D|$ one obtains
again a regular linear system $|C+D|$ cutting out the complete series on
$C$. Therefore the deficiency $d$ of the characteristic series of $|C|$
has to be
 \[
 d\le p_g-p_a.
 \]
In words: the characteristic series of an irreducible system on a
surface of genera $p_a$ and $p_g\ge p_a$ has deficiency
 \[
 d\le p_g-p_a.
 \]
So the irregularity $p_g-p_a$ of a surface is also defined as the
maximum value of the deficiency of the characteristic series for the
irreducible linear systems that belong to it.

On a regular surface ($p_g=p_a$) every irreducible complete linear
system has complete characteristic series. Or, from the projective
point of view: a regular normal surface, in some space, has normal
curves as its hyperplane sections. If the sections of a normal surface
are not normal, then one deduces that
 \[
 p_a<p_g.
 \]
We may elucidate what we have said by means of the example of the
ruled surfaces.

We know that the ruled surfaces (with sections) of genus $p$ `have
geometric genus $p_g=0$ and numerical genus $p_a=-p$. Therefore in
general the normal ruled surfaces of order $n$ belong to spaces of
dimension $r=n-2p+1$, and actually C.~Segre\footnote{C.~Segre,
Courbes et surfaces regl\'ees, {\it Math.\ Ann.}, 1892.}, already in
1890, has proved that a nonspecial ruled surface of order $n$ and
genus $p$ that is not a cone, belongs precisely to a space
$S_{n-2p+1}$; by contrast a special ruled surface, whose plane or
hyperplane sections are special curves of index $i$, is normal in a
space of dimension $n-2p+1+i$ (we always exclude the case of a cone
in which the system of sections turns out to be superabundant); so
the characteristic series of the system of hyperplane sections has
deficiency $p$ in any case.

 \begin{Remark} The theorem on the deficiency of the characteristic
series of the complete linear systems has led G.~Castelnuovo to
recognize that: every surface containing an irrational pencil of genus
$p$ ($>0$) is irregular with irregularity
 \[
 p_g-p_a\ge p.
 \]
This theorem is a consequence of a more general one proved by
F.~Enriques\footnote{Una propriet\`a delle serie continue di curve
appartenenti ad una superficie algebrica regolare, {\it Rendic.\
Circ.\ Mat.\ di Palermo}, t.~XIII, 1899.}: a surface that contains
an irreducible algebraic system $\{C\}$ of inequivalent curves is
irregular; more precisely if $\{C\}$ contains $\infty^d$
inequivalent curves then the irregularity of the surface is
 \[
 p_g-p_a\ge d.
 \]
 \end{Remark}

Enriques computed indeed the deficiency of the (canonical) series cut
out on $C$ by the adjoint system $|C'|$. On that occasion
Castelnuovo -- whom he had informed of the result -- pointed out the
deficiency of the characteristic system of the complete system to which
a curve $C$ belongs. The proof takes a simpler form in the treatment
of F.~Severi\footnote{Osservazioni sui sistemi continui di curve
appartenenti ad una superficie algebrica regolare. {\it Atti Acc.\
Torino}, 1904.}. Indeed it suffices to remark that the complete
continuous system $\{C\}$ consists of $\infty^d$ linear systems
$|C|$ having, in general, some dimension $r$: so the characteristic
series of $|C|$ on a generic curve $C$ has dimension $r-1$ but it is
contained in the characteristic series of the continuous system of
dimension $r-1+d$, that is, in the series cut out on $C$ by the
curves of $\{C\}$ infinitely near to $C$.

We explicitly notice that the theorem about the irrational pencil is
a consequence of the theorem thus stated: it suffices to remark that,
by subtracting the groups of $p$ curves of a pencil of genus $p$ from a
large enough system, one obtains a continuous system consisting of
$\infty^p$ inequivalent linear systems.

 \section{Historical note}\label{sec:4.13}
 \markright{\hfill Historical note \quad 4.13\quad}
 The numerical genus of a surface has been considered since the first
studies of A. Cayley, H.G. Zeuthen and M.~Noether. After that
Noether had defined the (geometric) genus $p$ of a surface $F_n$ of
order $n$ as the number of linearly independent adjoint surfaces
$\Phi_{n-4}$, it was natural to look for an expression of $p$ in
terms of the characters of the double curve of $F_n$; and Cayley's
postulation formulas were useful to this purpose. Cayley
himself\footnote{A. Cayley, On the deficiency of certain surfaces.
{\it Math.\ Ann.}, Band~III (1871).} remarked that, in the case of
the ruled surfaces of genus $p$, the aforementioned computation
leads to the negative value $-p$, whereas the geometrically defined
genus cannot, by definition, become negative and, in this case, it is
precisely zero.

Meanwhile Zeuthen was led, by the study of correspondences between
algebraic surfaces, to prove (although with some restriction on the
singularities) that ``the arithmetic (virtual) expression of the
genus, according to Cayley, is invariant under birational
transformations of the surface, and this invariance is independent of
its (supposed) geometric meaning and of the proof of the invariance
of the canonical system provided by Noether''.

Noether's interpretation of these results\footnote{{\it Math.\
Ann.}, Band~VIII (1875).} was that the genus of a surface $F_n$ of
order $n$ can be defined in two ways: geometrically as the actual
number of linearly independent adjoint surfaces $\Phi_{n-4}$, and
numerically by means of the postulation formulas; so one introduces
the geometric genus and the numerical genus of the surface, both a
priori invariant for birational transformations; they are supposed
to coincide in general, except that the numerical genus can become
negative (for the ruled surfaces) and therefore it seems to include
the other, as a more meaningful character.

But Noether's conjecture, that one should have $p_a=p_g$ for
nonnegative $p_a$, was disproved by the new examples of irregular
surfaces that were added to the example of the ruled surfaces. As we
have said before the first such example was provided by
G.~Castelnuovo in a Note of the Ist.\  Lombardo in 1891. Other
examples came from the investigations of E.~Picard and G.~Humbert on
the surfaces having integrals of total differentials of the first
kind, and in particular on the hyperelliptic surfaces, for which
$p_g=1$ and $p_a=-1$.

Subsequent studies have brought to light large classes of irregular
surfaces, as we showed in the Remark of Section~\ref{sec:4.7}.

In the meantime the investigations begun in 1893 and continued in
1896 (in the {\it Introduzione} quoted earlier) have led F.~Enriques
to recognize the geometric meaning of the difference $p_g-p_a$ with
respect to the deficiency of the (canonical) series that the system $|C'|$ adjoint to an irreducible linear system $|C|$ cuts out on the
generic curve of $|C|$; whence it follows the completely general
proof of the invariance of $p_a$, while the invariance of $p_g$ had
been already established earlier. According to Enriques the irregularity
$p_g-p_a$ is the maximum value of the aforesaid deficiency, but a priori
it is still possible that there exist systems $|C|$ for which the
aforementioned deficiency is $<p_g-p_a$. At this point E. Picard
completed the theory, by proving (by transcendental methods) that the
adjoint system $|C'|$ of a simple irreducible system $|C|$ (at least
$\infty^3$) is always regular, and therefore $p_g-p_a$ can be simply
defined as the deficiency of the series cut out by the adjoint surfaces
$\Phi_{n-3}$ on the plane section of a surface $F_n$\footnote{E. Picard,
Sur quelques questions se rattachant \'a la connexion lin\'eaire dans la
th\'eorie des fonctions alg\'ebriques de deux variables ind\'ependantes.
v. Crelle, Band~129, 1905. See Picard and Simart, Trait\'e t.~II,
page~437. An algebro-geometric proof (under milder assumptions) of the
theorem, has been given by Severi, in the Note Sulla regolarit\`a del
sistema aggiunto (Lincei, 8 nov. 1908); however this proof uses the
theorem of completeness of the characteristic series of a continuous
system.}. This theorem includes for $p_g=p_a$ Castelnuovo's theorem
stated in Section~\ref{sec:4.6}; more about it will be said in next
section.

The extension to surfaces of the theorem of Riemann--Roch was sketched
for the first time by M.~Noether in 1886\footnote{Extension du
th\'eor\`eme de Riemann--Roch aux surfaces alg\'ebriques, {\it Comptes
Rendus de l'Acad\'emie des Sciences}, Paris, t.~103.}. The author
considers a complete irreducible linear system of dimension $r$, of
degree $n$ and genus $\pi$, having a certain index of speciality
$i\ge0$, on a surface of (geometric) genus $p$. The characteristic
series of $|C|$, cut out by $|C|$ on a generic $C$, is a $g^{r-1}_n$
residual of the series $g^{p-i}{2\pi-2-n}$ cut out by the canonical
system $|K|$; and Noether, without stating it explicitly, assumes that
the two series, cut out by complete systems, are complete; under this
assumption the theorem of Riemann--Roch applied to the curve $C$ yields
the equality
 \[
 r=p+n-\pi+1.
 \]
The matter is reconsidered by Enriques since the 1893 in the paper {\it
Ricerche}, where he succeeds in proving that for the regular surfaces of
genus $p>1$ the characteristic series of a complete linear system is
complete, provided that the same happens for the characteristic series
of the canonical system; under this assumption (that later will be seen
to be superfluous) the following inequality is thus proved
 \[
 r\ge p+n-\pi+1,
 \]
and the author points out that this cannot be changed into an equality,
that is, that superabundant systems do exist, since it suffices for the
existence of such systems the presence of fundamental curves of genus
$r>0$.

In the {\it Introduzione} of 1896, with the new definition of the
numerical genus, Enriques gave the extension of the theorem of
Riemann--Roch for all surfaces, regular and irregular, to the adjoint
systems, which means-- substantially --to the systems more ample than the
canonical system.

In order to extend the theorem to arbitrary linear systems, it was
necessary to find the maximum value of the deficiency of the
characteristic series proving that it is given by $p_g-p_a$. This was
precisely the result obtained by G.~Castelnuovo in two
memoirs\footnote{Alcuni risultati sui sistemi lineari di curve
appartenenti ad una superficie algebrica. Mem.\ della Societ\`a
Italiana delle Scienze, detta dei XL (1896, Vol.~X). Alcune
propriet\`a fondamentali dei sistemi lineari tracciati sopra una
superficie algebrica, {\it Ann.\ di Mat.}, s.~II, Vol.~XXV. Reprinted
in {\it Memorie scelte}, XXII and XIII.} in the years 1896 and 1897.

The method used by Castelnuovo is based on a suitable extension of
the idea that leads to find the simplest conditions for the regularity
of a surface, as we have explained in Section~\ref{sec:4.6}. These are
methods that still seem to be usefully applicable, but that turn out to
be quite cumbersome.

A remarkable simplification in this type of questions was given by
F.~Severi, in two Notes of 1903 and 1905\footnote{Sulla serie
caratteristica di un sistema lineare di curve appartenente ad una
superficie algebrica, {\it Rendic.\ Acc.\ Lincei}, s.~V,
Vol.~XII (1903). Sul teorema di Riemann--Roch e sulle serie continue
di curve appartenenti ad una superficie algebrica. {\it Atti Acc.\
Torino, Vol.~XL (1905)}.}; in particular in the latter Note the
author is able to reverse the order of the deductions, proving first
of all the theorem of Riemann--Roch and then deriving from it the
maximum of the deficiency of the characteristic series of the
irreducible systems.

The starting point of these developments is the well known Noether's
theorem of $Af+Bf$ extended to surfaces in the following form:

Every surface $f_m(xyz)=0$, of order $m$, not containing the plane
$z=0$, passing through the points of the group $G$ common to the plane
$z=0$ and to the curve $D_s$ of order $ns$, complete intersection of
the surface $f_n$ of order $n$, without proper multiple points, with
the surface $f_s$ of order $s$, has an equation of the form
 \[
 f_m=Af_n+Bf s+z f_{m-1}=0,
 \]
where $f_{m-1}$ is a polynomial of degree $m-1$ and $A$ and $B$ are
polynomials in $x,y$ of degree $m-n$ and $m-s$ respectively.

Assuming directly that the surface $f_n$ have normal singularities,
from this theorem it follows that the surfaces $f_m$ of any order,
passing through the double points of a $D_s$, cut out a complete series
on $D_s$: actually one proves that if the proposition is true for $m$
then it is true for $m-1$.

Whence Severi derives the:

Lemma.-For $s>n-4$ the surfaces $\Phi_{n-4}$ adjoint to $f_n$ cut out a
complete series on the curve $D_s$ (complete intersection of $f_n$ and
$f_s$).

Indeed the surfaces of order $n-4$ passing through the double points of
$D_s$ have to contain the double curve of $f_n$ entirely and therefore
they are adjoint to it.

As a consequence of the previous lemma one has that ``the canonical
curves $K$, and then also the curves of a (special) complete system $|C|$
contained in $|K|$, cut out the complete series on the curves
$D_s$''. Thus, subtracting $D_s$ from the complete system $|Ds+s C|$,
one obtains the inequality for the dimension of $|C|$ that expresses
the theorem of Riemann--Roch.

On the other hand, once one has verified that the complete system
$|D_s|=|sD|$ (free of proper fundamental curves) is regular (see
Section~\ref{sec:4.15}), the above lemma again implies that the
characteristic series of a $D_s$ has deficiency $p_g-p_a$.

Whence one deduces (as we did in the previous section) that the
characteristic series of an arbitrary irreducible linear system $|C|$, at
least $\infty^1$, has deficiency $\le p_g-p_a$.

The argument that leads to such a result is the same used by Enriques
in the {\it Ricerche}, in order to pass from the characteristic
series of the canonical system to the characteristic series of
another arbitrary linear system, on a regular surface. The same
argument was taken up again in the most general form, but as a quite
secondary tool, by Castelnuovo to prove the theorem on the
deficiency of the characteristic series in the special cases that
escaped its direct proof (nonsimple systems).

The new proof of the theorem of Riemann--Roch for arbitrary linear
systems, as given in this treatise\footnote{See F.~Enriques, Sur
l'extension du th\'eor\`eme de Riemann--Roch aux syst\`emes
lin\'eaires de courbes appartenant \`a une surface alg\'ebrique.
Bull.\ Sci.\ Math., 1940 (see in particular n.~5).} is conceptually
at least as simple as Severi's. In our opinion it has the advantage
of showing the deep reason of the fact that the canonical curves $K$
(or the curves of a special system) cut out a complete series on the
$D_s$: this is not because the $D_s$ are complete intersections of
the given surface $F_n$ with a surface $F_s$, but because $|K|$ is
quite small in comparison with the system $|D_s|$. Moreover our
method yields the most direct proof of the theorem in consideration,
by keeping the same point of view from which one treats the
preliminary case of the adjoint systems or of the systems more ample
than the canonical system, that is, one does not have to appeal to
some new argument, as it appears to be necessary following the ways
of Castelnuovo and Severi.

And moreover, since the theorem of Riemann--Roch is obtained by
considering the series that a system $|L|=|C+D|$ cuts out on a curve
$D$, that has to be subtracted from $|L|$, then it is proved for the
system $|C|=|L+D|$, however reducible this is.

 \footnote{The following paragraph appears in the main text of
the Italian edition, but from the context, is presumably written by
Castelnuovo or a subsequent editor.}
Now the argument involving subtracting a curve $D$ from $|L|$ already
appears in Enriques' {\it Ricerche} (of 1893)\footnote{Ibidem, IV, 4.}.
Therefore one might think that Enriques himself, and Castelnuovo along
with him, knew at a very early stage, at least implicitly, about the
possibility of extending the theorem of Riemann--Roch to reducible
systems. According to their inductive attitude of mind the result
occurred to them by starting from concrete examples. The extension to
reducible systems is formulated in their joint memoir of
1900\footnote{Castelnuovo--Enriques, Sopra alcune questioni
fondamentali nella teoria delle superficie algebriche, {\it Ann.\ di
Mat.}, t.~VI, s.~III, n.~4.}.

The theorem of Riemann--Roch also yields some notable criteria for
virtual curves to exist as effective curves, as we shall see in the
next section.

 \section{Virtual curves}\label{sec:4.14}
 \markright{\hfill Virtual curves \quad 4.14\quad}
 Let us bear in mind that the computation of the dimension of a linear
system $|C|=|A-B|$, residual of a curve $B$ with respect to a regular and
nonspecial system $|A|$, has been obtained by computing the dimension of
the series cut out by $|A|$ on the curve $B$; from this it would seem that
the theorem of Riemann--Roch, expressing the result of this computation,
should also yield a criterion for the effective existence of the virtual
curves of $|A-B|$. First of all this is evident in the case in which $|A|$
cuts out a nonspecial series on $B$.

The formula that gives the dimension of the adjoint system
 \[
 r\ge p_a+\pi-1
 \]
to another system of genus $\pi$ is a first application of this criterion,
which has already been encountered in Enriques' {\it Introduzione} of
1896. The formula that gives the dimension of the $i$-canonical system on
a surface of linear genus $p^{(1)}$ is another application:
 \[
 P_i-1\ge p_a+\frac{i(i-1)}2(p^{(1)}-1) \qquad (i>1).
 \]
This appears in a note by Castelnuovo of 1897\footnote{Sul genere lineare
di una superficie ecc., {\it Rendic.\ Acc.\ Lincei}, 1897: reprinted in
{\it Memorie scelte}, XXIV.} and in the paper by Castelnuovo--Enriques of
1900\footnote{Sopra alcune questioni fondamentali della teoria delle
superficie, {\it Ann.\ di Mat.}, 1900.}: actually the $i$-canonical system
$|iK|$ can be obtained from a regular $i$-tuple adjoint system $|A|=|B+iK|$
of a regular system $|B|$, by subtracting $|B|$ from it, and $B+iK$ cuts out
precisely a nonspecial series on $B$.

A third application of the theorem of Riemann--Roch as a criterion for
the effective existence of virtual curves is given by Severi (1905) and
concerns the general case in which $|A|$ cuts out an arbitrarily special
series on the curve $B$: we shall say more about this later on.

If one wants to examine the question of the effective existence of the
curves of a difference system $|C|=|A-B|$, one may assume that the two
systems $|A|$ and $|B|$ defining the virtual curves $C$ are both
irreducible, regular and free of fundamental curves, and also free of
base points on a nonsingular surface $F$; this is because one can always
add to the given systems $|A|$ and $|B|$ another regular system free of
fundamental curves.

Now we remark that:

 \begin{enumerate}
 \renewcommand{\labelenumi}{(\arabic{enumi})}
 \item The effective existence of the virtual curves of $|C|=|A-B|$,
requires first of all that the degree of the system from which one
subtracts is greater than or equal to the degree of the system which is
subtracted
 \[
 A_2\ge B_2.
 \]
Indeed if $A_2<B_2$, the curves $C$ have a negative intersection number
with the curves of the system $|A+B|$, since $(A+B)(A-B)=A_2-B_2$, and
therefore they should be fixed components of this system, which is
irreducible by the assumptions we have made.

 \item Secondly, in order that there exist curves of $|A-B|$, it is
necessary that the intersection number of the curves $A$ and $B$ be in
between the degrees of the two systems $|A|$ and $|B|$:
 \[
 B_2\le AB\le A_2.
 \]
Otherwise the curves $C$ would have negative intersection number with
the curves $B$ or with the curves $A$, since one has respectively: for
$AB<B_2$
 \[
 B(A+B) < 0
 \]
and for $AB>A_2$
 \[
 A(A-B) < 0.
 \]
 \end{enumerate}

The conditions (1) and (2) are necessary both for the effective
existence of $|C|=|A-B|$ and for the existence of a multiple
$|i(A-B)|$. Indeed if
 \[
 A_2>B_2,
 \]
we must have
 \[
 B_2<AB<A_2,
 \]
otherwise the $C$ (or their $C$) -- that are not actually curves of
order zero -- would have zero intersections with the $B$ or with the $A$
and therefore they would be fundamental curves for the corresponding
system (that we have assumed to possess no such curves).

To the two necessary conditions stated above there corresponds the
sufficient condition for the existence of a multiple of $|C|$:

If the virtual degree of $|C|=|A-B|$ is positive and $A_2>AB>B_2$,
then the curves of the multiple system $|iC|$ is always effective for
large enough $i$.

Indeed let us consider the series cut out by the system $|iA-(i-1)B|$
on a curve $B$ (assuming that this system actually exists); this series
is of degree
 \[
 AB+(i-1)(AB-B_2),
 \]
 that for $i$ large enough becomes greater than the order of the
canonical groups of $B$; therefore the aforesaid series is
nonspecial, and it is possible to compute the dimension of the
residual system
 \[
 |i A-(i-1)B-B|=|i A-i B|
 \]
 with the formula
 \[
 r_i\ge p_a+n_i-\pi_i+1,
 \]
 where
 \[
 n i=i 2n \quad
 \pi_i=i \pi+n i(i-1)/2-i+1
 \]
($n=(A-B)^2=$ virtual degree of $|C|$: $\pi=$virtual genus of $|C|$).

{From} the previous formula it follows $r_i>0$ for large enough values
of $i$, hence the effective existence of the curves of $|iC|$ also
follows.

Nevertheless in the previous argument we have assumed a priori that
the system $|iA-(i-1)B|$ existed; this assumption can be justified
for the values of $i$ that make $r_i>0$, since under such conditions
it is possible to compute the virtual dimensions of $|iA-B|$,
$|iA-B-B|=|iA-2B|,\dots$, up to $|iA-(i-1)B|$, that a fortiori all
turn out to be greater than $r_i$: indeed for the aforementioned
values of $i$, the dimension of the series cut out on $B$ by
$|iA-(i-2)B|$ is a fortiori greater than the dimension of the
canonical series, and therefore it is nonspecial, and so on.

The criterion for the existence of virtual curves that we have thus
proved can be applied to the pluricanonical system of a surface for
which the linear genus is $p^{(1)}>1$, and so the virtual degree of the
canonical system is
 \[
 p^{(2)}=p^{(1)}-1>0.
 \]
 Thus one finds the formula already written above
 \[
 Pi\ge p_a+(p^{(1)}-1)i (i-1)/2+1.
 \]
 Let us notice that the criterion does not lead to any conclusive
answer about the existence of the multiple $|iC|$ of a virtual system
of degree $n=0$, hence it does not enable us to claim a priori the
existence of pluricanonical curves for $p^{(1)}=1$; although this
existence is then proved for surfaces free of exceptional curves
(not belonging to the family of ruled surfaces) by means of other
considerations.

The case in which the degree of $|C|$ is $n$ <0 remains a fortiori
unsettled; one cannot say whether or not the two systems $|A|$ and $|B|$,
or their multiples $|iA|$ and $|iB|$, for $i$ large enough, give rise to
an effective difference.

In order to complete this discussion we still want to consider the
case in which the systems $|A|$ and $|B|$ have the same degree
 \[
 A_2=B_2.
 \]
 In this case the difference $|A-B|$ or one of its multiples cannot be
effective if the intersection number of $A$ and $B$ is different from
$A_2$; then one has the necessary condition for the existence
 \[
 AB=A_2=B_2,
 \]
 and the virtual degree of $|C|=|A-B|$ is
 \[
 n=(A-B)^2=0.
 \]
Now the curve $C$ cannot be effective unless is a curve of order zero,
that is, when $A\equiv B$. Analogously the curve of $|iC|$ will be
effective, again as a curve of order zero, when
 \[
 iA\equiv iB.
 \]
We notice here that if one does not have $iA\equiv iB$ for some value
of $i$, then one may construct on the surface $F$ (that we may
assume to have the curves $B$ as plane or hyperplane sections) an
infinite family of systems $|iA-(i-1)B|$, all consisting of curves
of the same order and therefore contained in one or several
continuous series of inequivalent curves. This implies that the
surface is irregular:
 \[
 p_g>p_a.
 \]
Thus we can say that: {\em On a regular surface two linear systems of
curves $|A|$ and $|B|$ of the same degree, that meet at
 \[
 AB=A_2=B_2
 \]
points, are parts of a same multiple system:
 \[
 |iA|=|iB|,
 \]
so that the multiple according to $i$ of their difference is the
effective curve of order zero: $iA-iA$.}

Leaving the regularity of the surface aside, one can say in general
that: ``if two curves $A$ and $B$ of the same degree $n$ meet at $n$
points, then two of their multiples belong to the same continuous
series of curves, or as one also says, they are algebraically
equivalent''\footnote{See F.~Severi, {\it Math.\ Ann.}, 62 (1906),
page~194. A. Todd, {\it Proc.\ Camb.\ Phil.\ Soc.}, march 23,
1939. About the relations with a theorem on the base of W. V. Hodge,
{\it J. London Math.\ Soc.}, 12, 1937, see B. Segre, {\it Ann.\ di
Mat.},1937 and J. Bronowski, {\it J. London Math.\ Soc.}, 1938.}.

The criterion for the effective existence of $|C|=|A-B|$, that we
mentioned above and attributed to Severi\footnote{Sulle curve
algebriche virtuali appartenenti ad una superficie algebrica.
{\it Rendic.\ Ist. Lombardo} II, 38 (1905).}, just consists in the
formal extension to virtual curves of the theorem of Riemann--Roch,
which one obtains by defining the index of speciality $i$ of $|C|$ as
``the number of linearly independent curves of the residual system
of $|C|$ itself with respect to the (impure) canonical system
$|K|$'', that is, as the dimension of the system
 \[
 |K-C|=|B+K-A|
 \]
 plus one.
 Denoting by $n$ and $\pi$ the virtual characters of $|C|$, the inequality
 \[
 r\ge p_a+n-\pi+1-i,
 \]
 yields a lower bound for the dimension $r$ of $|C|$ and thus ensures the
effective existence of the system whenever one has
 \[
 p_a+n-\pi+1-i\ge0.
 \]
In order to justify in our circle of ideas what we have just said
(and restricting ourselves, for the sake of simplicity, to the
assumption $i>0$) it suffices to keep in mind the lemma that has led
us to the theorem of Riemann--Roch for special systems, and to
remark that the system $|B+K-A|$ can be regarded as small with
respect to the normally large system $|sB|$ for quite large $s$.
Therefore $|B+K-A|$ cuts out a complete series of dimension $i-1$ on
a curve of the aforesaid system and so the index of speciality of
the series cut out by $|A+(s-1)B|$ on the curve itself has precisely
index of speciality $i$: whence one deduces the above-written
formula, that gives a lower bound for the dimension $r$ of
 \[
 |A+(s-1)B-sB|=|A-B|.
 \]
 At this point it is convenient to add the remark that the index of
speciality $i$ of a virtual curve may turn out to be arbitrarily large,
unlike the index of an effective curve which is always
 \[
 i \le p_g.
 \]
 Indeed the residual system of $|C|$ with respect to $|K|$, that is,
 \[
 |B+K-A|,
 \]
is not a priori contained in $|K|$: actually the fact that it is
contained in $|K|$ expresses in a different way the necessary and
sufficient condition for a virtual curve to have effective existence.

 \section{Regular and superabundant systems}\label{sec:4.15}
 \markright{\hfill Regular and superabundant systems \quad 4.15\quad}
 Let us consider an irreducible complete linear system of genus $\pi$,
degree $n$ and index of speciality $i$ on a surface of arithmetic genus $p_a$:
the dimension of this system satisfies the inequality expressing the
theorem of Riemann--Roch
 \[
 r\ge p_a+n-\pi+1-i,
 \]
 that is,
 \[
 r=p_a+n-\pi+1-i+w \quad\hbox{where}\quad w\ge0.
 \]
The system is said to be regular if $w=0$, and superabundant if $w>0$;
in the latter case $\om$ denotes the superabundance of the system.

It is convenient to see whether there is any criterion that enables
one to compute the superabundance of a system $|C|$, or at least to
decide whether or not $|C|$ is regular.

The most important result, in this line of thought, concerns the
regularity of the adjoint system of an irreducible linear system $|C|$.
We have proved this for a large enough multiple of $|C|$ (at least
$\infty^3$ or $\infty^2$) and we have said that it has been firstly
recognized by Picard (Section~\ref{sec:4.13}) for the system adjoint to
$|C|$ itself.

Here we can establish in a very simple way the theorem of Picard
provided that we add a unnecessary restriction:

The system $|C'|$, adjoint to an irreducible linear system $|C|$ of
dimension $r\ge p_g-p_a$, is certainly regular (of dimension
$p_a+\pi-1$).\footnote{See Enriques, Bull.\ de Sci.\ Math., Ibidem
1940.}

The proof is based on a:

Lemma.-Let $|C|$ and $|D|$ be two irreducible linear systems, that for
the sake of simplicity we shall assume of dimension greater than one.
One can prove that ``if the dimension of $|C+D|$ is greater than
$p_g-p_a$, then $|C|$ cuts out on a curve of the sum system $|C+D|$ a
complete series''. Indeed consider on the curve $L$ of $|L|=|C+D|$ a
group $G$ of the complete series containing the groups of the
intersections with the $C$. Since the characteristic series of $|L|$ has
deficiency $d\le p_g-p_a$, among the $\infty^r$ groups LD there are
$\infty^{r-(p_g-p_a)}$ groups that added to $G$ form groups of the
characteristic series of $|L|$: let $G'$ be one of these groups. Now
there is a pencil of curves $L$ passing through $G+G'$, and the curve
$D$ through $G'$ is fundamental with respect to this pencil:
subtracting it, one obtains a curve of $|C|$ passing through $G$. Thus
the complete series of the groups $G$ is cut out on $L$ by the system
$|C|$.

Now, let us consider on a surface an irreducible linear system $|C|$,
of a certain genus $\pi$ and of dimension greater than $p_g-p_a$; we
wish to prove that the adjoint $|C+K|$ of this system (where $K$ denotes
the canonical system) is regular, of dimension $p_a+\pi-1$.

In order to do this, associate with $|C|$ an irreducible linear
system $|D|$ of dimension greater than $p_g-p_a$ and such that the adjoint
of this system is regular; and construct the system
 \[
 |L|=|C+D+K|.
 \]
Both the system $|C+K|$ and the system $|D+K|$, cut out on a curve $L$
complete series, which must be one residual to the other with respect
to the canonical series. Since one knows the dimension of the series
cut out by $|D+K|$, which is equal to the dimension of the system $|D+K|$,
one deduces the dimension of the series cut out by $|C+K|$, which equals
the dimension of this system.

Indeed denote by $r$ and $m$ the genus and the degree of $|D+K|$, and by
$s$ the intersection number of a $C$ and a $D+K$: the dimension of
the series cut out on $L$ by $|D+K|$ is
 \[
 p_a+m-r+1
 \]
and its degree is $m+s$, while the genus of $L$ is
 \[
 \pi'=\pi+r+s-1.
 \]
So the theorem of Riemann--Roch tells us that the dimension of the
series cut out on $L$ by $|C+K|$ is
 \[
 p_a+m-r+1-(m+s-\pi')-1=p_a+\pi-1.
 \]
Another regularity criterion (that is useful in finding the
postulation of a surface with respect to the varieties $V_{n-1}$ of a
space $S_n$ of dimension $n>3$) concerns the multiples of a linear
system: the multiples of very large order of an irreducible, at least
$\infty^3$, linear system $|C|$ without fundamental curves are regular.

Consider some integer $h$ such that the system $|h C|$ contains the
canonical system $|K|$ as a part, so that one has
 \[
 |h C-K|=|D|.
 \]
 We know that for $s$ large enough the adjoint system of $|s C|$,
namely $|s C+K|$, is regular. Now we have
 \[
 |(s+h)C|=|s C+K+D|,
 \]
where $D$, in general, is a curve consisting of several simple or
multiple components, of either positive or negative degree. In this
setting one has to show that when one successively adds the
irreducible components of $D$ to the regular system $|s C+K|$ one
obtains larger and larger regular systems (according to the
criterion of Section~\ref{sec:4.8}).

Indeed let us denote by $L$ an irreducible component of $D$, with
characters $r$ and $n$; since one has always $r\ge0$, then one sees
that the intersection number of $L$ with the curves of $|s C+K|$ that
have to be added to it (with $s$ arbitrarily large) is
 \[
 m=st+2r-2-r>r-n,
 \]
where $t>0$, since $L$ is not a fundamental curve for $|C|$. The
analogous inequality always holds for the intersection number of the
curves of $|sC+K+L|$ with a component of $D$, and so successively
for the sum of $|sC+K|$ with all the irreducible components of $|D|$.

Finally the system $|sC+K+D|$, that is a multiple of $|(s+h)C|$,
turns out to be regular, for $s+h$ large enough, \qed

It suffices to explain the argument by considering to the first
relevant case, where one has to add $|sC+K+L|$ to $L$ itself. Here we
assume that $L$ is a double component of a curve $D$ and is of genus
$r\ge0$ and of negative degree $n=-n'$. Then we claim that in this
case the intersection number of $L$ with the curves of $|sC+K+L|$ is
 \[
 st+2r-2+n'-n'=st+2r-2>r-n',
 \]
for $s$ large enough compared to $n'=-n$.

We may state the theorem that we have just proved by saying that:
{\em Any linear system which is normally large (with respect to the
canonical system) is regular.}

It is useful to remark explicitly that the property does not hold
for large systems that are not normally large.  The simplest example
is provided by the linear systems that have proper fundamental
curves: {\em If the linear system $|C|$ has a proper fundamental
curve $\theta$ of genus $r>0$, the arbitrarily large multiples of
$|C|$ have superabundance}\footnote{See Enriques, Ricerche\dots,
1893.}
 \[
 \om\ge r
 \]
Indeed the subtraction of a curve $q$, of degree $-n$ from $|sC|$,
decreases by one the dimension of the system, by $n$ its degree and by
$n+r-1$ its genus, and thus by $r-1$ its virtual dimension; this
means that the superabundance $\om$ of $|sC|$ exceeds by $r$ the
superabundance $w'$ of $|sC-\theta|$:
 \[
 \om=\om'+r.
 \]
We notice that, in general, (for $r>1$) one has also $\om'>0$, and
thus the genus $r$ of the fundamental curve is only a lower bound
for the superabundance of $|sC|$. It is instructive to recall here
the simple projective example of a surface $F$ of order $n$ (with
plane sections $C$), having an isolated multiple point $O$ of order
$i>3$, on which the surfaces $\fie_s$ (of large enough order $s$) cut
out the linear system $|sC|$.

One sees at once that not only the general $\fie_s$ cut out on $F$ a
superabundant system (for which the curve $\theta$ represented by the
neighbourhood of $O$ is a fundamental curve), but also that the systems
cut out on $F$ by the $\fie_s$ passing through $O$ with multiplicities
$1,2,\dots,i-3$ are superabundant, whereas the system cut out by the
adjoints $\fie_s$ that pass $i-2$ times through $O$ is regular.
Whence one easily deduces the computation of the superabundance
$\om$ of
$|sC|$.

 \section{More on the regularity of the adjoint system of a curve on a
regular surface}\label{sec:4.16}
 \markright{\hfill More on the regularity of the adjoint on a
regular surface \quad 4.16\quad}
 We wish to go deeper into the question of the regularity or
superabundance of the linear systems more ample than the canonical
system, referring to the simple case of the regular surfaces, with
$p_g=p_a=p>0$.

If $|C|$ is a linear system more ample than the canonical system $|K|$,
then one has
 \[
 |C-K|=|L|,
 \]
 where $|L|$ denotes an effective system, and so
 \[
 |C|=|L+K|=|L'|.
 \]
Thus $|C|$ can be considered as the adjoint system of an effective
curve $L$. Now if $L$ is irreducible, of genus $\pi$>0, then its adjoint
system is certainly regular, of dimension $p+\pi-1$, since it cuts out
on $L$ a (complete) series that cannot be larger than the canonical
series $g^{\pi-1}_{2\pi-2}$.

The same conclusion -- regularity of the adjoint system of $L$ -- can be
extended to the case in which the curve $L$ is an arbitrarily reducible,
connected curve, consisting of simple parts.

Connected curve consisting of simple components means ``curve such
that whenever it is divided in two parts in any possible way, always
gives rise to curves having some (connection) point in common'', that
is, topologically, a ``curve or Riemann surface on which it is always
possible to give a continuous path joining two arbitrary points''.

For connected curves, consisting of simple components, one has the:

Lemma of geometric composition of connected curves. {\em A connected
curve $L$ consisting of s simple components can always be obtained by
adding to a connected part consisting of $s-1$ components
$L_1+L_2+\cdots+L_{s-1}$ a further $s$th component $L_s$ with some
common (connection) point with it.}

This is proved by induction: assuming that the given $L$ contains a
connected part, consisting of $h$ simple irreducible components with
$h<s$, this part must be connected with the rest of the curve and
therefore with at least one of its components; adding such a
component to it, one obtains a connected component of $L$,
consisting of $h+1$ components.

As a Corollary:: given a connected curve $L$ consisting of $s$ simple
components, it is always possible to take these in some order
$L_1,L_2,\dots,L_s$ so that in the curve
 \[
 L=L_1+L_2+\cdots+L_s
 \]
 each part $L_1+L_2+\cdots+Lh$ ($h=1, 2,\dots,s$) is connected.

Using this property it is easy to recognize that ``the adjoint system
$|L'|$ is regular'', as we have stated above. Indeed denote by
$\pi_h$ the genus of $L_1+L_2+\cdots+L_h$, by $\pi_h'$ the genus of
$L_h$, and by $i_h$ the intersection number
$(L_1+L_2+\cdots+L_h)L_h$. We have that the genus of
 \[
 L=L_1+L_2+\cdots+L_s=(L_1+L_2+\cdots+L_{s-1})+L_s
 \]
 is
 \[
 \pi_s=\pi_s'+\pi_s-1+i s-1,
 \]
 and the series cut out on $L_s$ by $|L'|$ has degree
 \[
 n_s=2\pi_s'-2+i s
 \]
and dimension
 \[
 r_s \le \pi_s'-2+i s;
 \]
thus the subtraction of $L_s$ from $|L'|$ yields
 \[
 r_s+1 \le \pi_s'+i s-1
 \]
conditions.

Thus: if the system $|(L_1+L_2+\cdots+L_{s-1})'|$ adjoint to
$|L_1+L_2+\cdots+L_{s-1}|$ is regular then one has precisely
 \[
 r_s+1=\pi_s'+i s-1
 \]
and the system $|L'|$ adjoint to $|L|=|L_1+L_2+\cdots+L_s|$ is also
regular. The theorem thus stated is verified for a curve $L$ consisting
of $s$ components (that are simple and connected between them) if it is
verified for a curve consisting of $s-1$ components. Since $|L_1'|$ is
regular, the theorem is proved in general.

Contrary to what happens for connected curves, the adjoint systems of
disconnected reducible curves are not regular, but superabundant. More
precisely:

The system $|L'|$ adjoint to a curve $L$ consisting of $s$ simple
irreducible curves without common points has superabundance equal to
$s-1$.

This is proved with the same inductive argument used for connected
curves. Consider for instance a curve $L$ (of genus $\pi$) made of two
parts $L_1$ and $L_2$ of genera $\pi_1$ and $\pi_2$, without common
points, ($\pi=\pi_1+\pi_2-1$). Then the adjoint system
$|L'|=|(L_1+L_2)'|$ cuts out the complete canonical series
$g^{\pi_1-1}_{2\pi_1-2}$ on $L_1$, and the residual system of $L_1$ with
respect to it -- that is, $|L_2'|$ -- cuts out the canonical series
$g^{\pi_2-1}_{2\pi_2-2}$ on
$L_2$; the latter system $|L_2'|$ is regular, thus $|L'|$ is
superabundant, of dimension
 \[
 p+\pi=p+(\pi_1+\pi_2-1)-1+w
 \]
 where
 \[
 w=1.
 \]
Now it is easy to compute the superabundance of the system $|(L+\sum
s_i q_i)'|$ adjoint to a disconnected curve $L+\sum s_i q_i$, consisting
of an irreducible curve $L$ of degree $>0$, that, for the sake of
simplicity, we may assume to vary in an $\infty^2$ linear system, and of
several irreducible curves $q_i$ of degree $n_i<0$, fundamental for
this system $|L|$, each counted a certain number $s_i$ of times.

 \section{The adjoint system of a connected curve consisting of
multiple components on a regular surface (continued)}\label{sec:4.17}
 \markright{\hfill The adjoint system of a connected curve
(continued) \quad 4.17\quad}
 The above results do not settle the question of the regularity or
superabundance of the adjoint system of a curve $L$ on a regular surface,
when $L$ is however formed by simple and multiple irreducible components.
The difficulty that one has to face here is actually related to the
concept of ``connected curve'', that has to be precisely extended to
an arbitrary $L$, of the form described above.

We define a curve $L$, arbitrarily made of simple or multiple
irreducible components, to be (arithmetically) connected when any two
parts, $L_1$ and $L_2$, in which it is decomposed, always have a (virtual)
number of common points
 \[
 L_1L2>0:
 \]
it is not excluded that some part of $L_1$ has a negative number $-n$
of intersections with $L_2$ (thus containing at least one multiple
component of $L_2$, of negative degree), provided that then the remaining
part has more then $n$ points in common with $L_2$.

The consideration of connected curves, however formed of multiple
parts, gives rise to difficulties that have been explained by
A.~Franchetta\footnote{Sulla caratterizzazione delle curve
eccezionali riducibili di prima specie, in Boll.\ dell'Unione Mat.,
Bologna, 1941.}, and that are worth to be pointed out here.

Contrary to what intuition suggests in the case of simple parts
(having a nonnegative number of common points with other components)
one cannot say that in general, by adding to a connected curve $L$ a
component $C$ that meets it at $n>0$ points, one always obtains a connected
curve L+C.

Indeed Franchetta points out examples of disconnected curves
consisting of $s$ components, that can be split into a connected curve
made of $s-1$ components and one component that meets it at a positive
number of points.

As a consequence of this, also the lemma (of geometric composition)
that we have proved in the previous section for connected curves made
of simple components, does not hold any more: in general one cannot say
that an arithmetically connected curve made of $s-1$ components can
always be obtained by adding an irreducible component to a connected
curve made of $s-1$ components.

Nevertheless some properties still hold for arithmetically connected
curves; they have been pointed out by Franchetta\footnote{Ibidem.} and
take the place of the analogous properties in the case of simple
components. We state them as the following theorems:

Theorem I.-A curve $L$ (however formed by simple or multiple
irreducible components) is arithmetically connected if each of its
connected parts is connected with the remaining part (that is, it has a
positive virtual number of points in common with it).


Indeed if $L=A+B$ is not connected, then we have $AB\le0$. So either $A$
is connected and then there is a connected part of $L$ that intersects
the rest in a nonpositive number of points, or else it is possible to
decompose $A$ as the sum of two parts $A_1$ and $A_2$ disconnected one from the
other:
 \[
 A=A_1+A_2, \quad A_1A_2\le0;
 \]
from the fact that $AB$ is nonpositive it follows that at least one
of the two curves, say $A_1$, has a nonpositive number of
intersections with $B$; but in this case one has
 \[
 L=A_1+(A_2+B)
 \]
 with
 \[
 A_1(A_2+B)=A_1A_2+A_1B \le 0,
 \]
that is, there is a part $A_1$ of $A$, contained in $L$, disconnected from
the remaining part. Now either $A_1$ is connected or it contains a part
$A_{11}$ disconnected from the remaining part $(L-A_{11})$ and so on.
But the sequence of parts $A,A_1,A_{11},\dots$, whose number of
components decreases, ends with a connected curve, consisting of one or
several irreducible components: the last part thus obtained would be a
connected curve disconnected from the rest.

Theorem II.-Given an (arithmetically) connected curve $L$, it is
possible to remove from it a connected part $B$ (consisting of one or
several components) in such a way that the remaining part $L-B$ is again
connected.

Among all the possible connected parts of $L$, choose one, call it $B$,
having the smallest (positive) number of common points with the
remaining part $A=L-B$; we claim that $A$ is also connected. If this
were not the case, one would have a decomposition into disconnected
parts
 \[
 A=A_1+A_2;
 \]
 where one may assume that $A_1$ itself is connected (Theorem~I). But,
in this case, $B$ must have a positive number of intersections with each
of the parts $A_1$ and $A_2$ of $A$, since otherwise one would have
either $(B+A_1)A_2\le0$ or $(B+A_2)A_1\le0$ and thus $L$ would be
disconnected; so we have
 \[
 BA_1>0 \quad\hbox{and}\quad BA_2>0,
 \]
 thus
 \[
 BA_1 < BA=B(A_1+A_2),
 \]
 and a fortiori
 \[
 (B+A_2)A_1 < BA;
 \]
so $A_1$, a connected part of $L$, would have with the rest a number of
intersections smaller than $B$, contradicting our assumption.

Franchetta's theorems replace, to a certain extent, the intuitive
properties that characterize connected curves made of simple
components, as we shall see in what follows.

However it does not seem easy to deduce from these properties,
without exceptions, the regularity of the system $|L'|$ adjoint to an
arbitrary connected curve $L$ (with multiple components of negative
degree). One may try to follow a quite indirect way, strictly related
to our approach to the Riemann--Roch theorem, but subject to a working
hypothesis that will be clarified later on.

Consider on the surface $F$ a regular linear system $|C|$, which is a
multiple of large enough order of an irreducible system,, at least
$\infty^2$, free of fundamental curves. Hence, to start with,
$|D|=|C+L|$ is irreducible. One may assume that both $|L|$ and its
adjoint system $|L'|=|L+K|$ cut out complete series on a curve
$\Dbar$ of the normally large system $|C+L|$. Now one can prove that
also $|C|$ cuts out a complete series on this curve $\Dbar$. Indeed
let $G$ be a group of the complete series $CD'$ and let $G$ be a
residual group of $G$ with respect to the characteristic series of
$|D|$, section of $D'$ with $L$; $G+G$ is the base group of a pencil
of curves $D$, that have no intersection with the components of $L$
away from $\Dbar$.

Now, since $L$ is connected, if one imposes that a $D$ of this pencil
contains a point of $L$ away from $\Dbar$ then the whole connected
curve will be contained in $D$ and the remaining part of $D$ will be
a $C$ through $G$. \qed

The above argument is based on the assumption that, since $L$ is
arithmetically connected, when one imposes that a curve $D$ of the
above-said pencil contains a point of $L$, away from $G$, then the
whole of $L$ becomes a component of $D$, that is, each of its
components is contained in $D$ with the same multiplicity with which
it appears in $L$.

We believe this property to be true, but we have no satisfactory
proof of it. We shall introduce it as a working hypothesis, thus it has
to be clear that the above result and the consequences that we shall
deduce from it are subject to the validity of that assumption.

Now, we have seen that the regular system $|C|$ cuts out on the curve
$D$ of $|D|$ a complete series $g$, which is residual of the series cut
out by $|L'|=|L+K|$. Hence we are able to compute the dimension of the
latter series (and so of $|L'|$) that is equal to the index of
speciality of $g$ minus one. Denoting by $n$ the degree of $|C|$, by
$\rho$ its genus, and by $m$ the intersection number $CL$, the degree of
the series $g$ is
 \[
 n+m
 \]
 and its dimension is
 \[
 r=p+n-\rho+1,
 \]
 whereas the genus of $D$ is
 \[
 \pi+\rho+m-1.
 \]
 Thus the index of speciality $i$ of $g$ is given by:
 \[
 n+m-(\pi+\rho+m-1)+i=p+n-\rho+1,
 \]
whence it follows that the dimension of the series $(CL')$ residual of
$CD$ is
 \[
 i-1=p+\pi-1.
 \]
So $|L'|$ has exactly this dimension, and it is regular.

In conclusion: {\em On a regular surface of genus $p>0$, the adjoint
system of an arithmetically connected curve is always regular.}

 \section{On the reducibility of the canonical system}\label{sec:4.18}
 \markright{\hfill On the reducibility of the canonical system \quad
4.18\quad}
 The theorem proved in the previous section leads to remarkable
consequences concerning the bicanonical system and the pluricanonical
systems. In order to understand the meaning of these, it is convenient
to go back to the question of the reducibility of the canonical system.

Noether believed he was able to prove that the canonical system of a
surface of genus $p>0$ can be reducible only in two cases:
 \begin{enumerate}
 \renewcommand{\labelenumi}{(\arabic{enumi})}
 \item because of the existence of fundamental curves (of genus
$\rho=0$ and degree $n=-1$) that are fixed parts of the impure
canonical system;
 \item because the (absolute) linear genus is $p^{(1)}=1$, in which
case the pure canonical curves are in general composed with the
elliptic curves of a pencil.
 \end{enumerate}

So for $p^{(1)}>1$ the pure canonical system should always be
irreducible. But Noether's argument is wrong, and there are actually
examples of (pure) reducible canonical curves $K$, belonging to a
surface $F$ of genus $p>0$ and linear genus $p^{(1)}>1$:

 \begin{enumerate}
 \renewcommand{\labelenumi}{(\arabic{enumi})}
 \item the first example is provided by the double plane with branch
curve $C_{2n}$ of order $2n$, having a $(2n-6)$-tuple point $O$ and two
$[3,3]$ points $A_1$ and $A_2$, lying together with $O$ on a line $a$
 \[
 (p=2n-7,\quad p^{(1)}=4n-15).
 \]
The (double) images of the canonical curves $K$ are the curves
$C_{n-3}$ that have $O$ as a $(n-4)$-tuple point and pass simply
through $A_1$ and $A_2$, and therefore they are the sum of a $C_{n-4}$
with $O$ as a $(n-5)$-tuple point and of the fixed part $a$. Now this
line represents an (irreducible) curve $\theta$, a fixed component of the $K$,
which is not exceptional, since it has genus $\rho=0$ and degree $n=-2$:
as it appears also from the fact that it has two intersections with the
residual components of the $K$, since the number of these intersections
is:
 \[
 2\rho-2-2n=2,\quad
 (r=0,n=-2).
 \]

 \item A second example is provided by the double plane with branch
curve $C_{10}$, of order 10, with three $[3,3]$ points on a line
$a$, that is not a component of $C_{10}$. A simple count of
dimensions tells us that the linear system of $C_{10}$ with three
assigned triple points on $a$ and three triple points infinitely
near to these in assigned directions has dimension 29, and therefore
it is larger than the $\infty^{26}$ system consisting of $a$ counted
twice and of the curve $C_8$ with three tacnodes and given tacnodal
tangents. Now, on the double plane that we have constructed the
images of the canonical curves are the conics that split as the sum
of the fixed line $a$ and of a variable line. The corresponding
fixed part $\theta$ of the canonical system $|K|$ is a curve of
genus $r=1$ and degree $n=-1$; the genera of the surface $F$ are
 \[
 p=3,\quad p^{(1)}=6.
 \]
 One may compute the degree $n=-1$ of $\theta$ by remarking that
$\theta$ -- which is a fixed part of $|K|$ -- has two intersections with
the residual curves, so that
 \[
 2r-2-2n=-2n=2,\quad n=-1.
 \]
 On the other hand the complete curve corresponding to a on $F$ is a
curve of genus four consisting of three elliptic curves of degree $-1$,
corresponding to the $[3,3]$ points, and of our curve $\theta$, which has a
point in common with each one of them: the total degree of this
complete curve is
 \[
 -3-1+2.3=2.
 \]

 \item A third example is provided by the double plane with branch
curve $C_{12}$, of order 12, with four $[3,3]$ points belonging to a
line $a$ (which is not a component of $C_{12}$); counting dimensions
one shows the existence of irreducible $C_{12}$, that do not contain
$a$. Now the canonical curves $K$ are given on the double plane by
the cubics that split as the sum of the line $a$ and of a variable
conic. The line $a$ corresponds to a fixed component of the $K$ that
has genus $r=1$ and degree $n=-2$; indeed $\theta$ has 4
intersections with the residual curves, so that\footnote{We also
give the example of an irregular surface with $p_a=2$, $p_g=3$,
$p^{(1)}=11$, represented on the double plane with a branch curve
$C_{12}$ consisting of 3 quartics $C_4$ (belonging to a pencil) that
touch each other at 8 points lying on a conic $C_2$; the images of
the canonical curves split as the sum of the conic $C_2$, which
corresponds to a fixed curve $\theta$ of genus 3, and of a variable
line. The surface contains an irrational (elliptic) pencil of
curves, represented by the pencil of the $C_4$, according to a
theorem of M.~De Franchis' on the irregular double planes.}
 \[
 2r-2-2n=-2n=4,\quad n=-2.
 \]

The complete curve corresponding to $a$ on $F$ is a curve of genus
five consisting of four elliptic curves of degree $-1$,
corresponding to the
$[3,3]$ points, and of our curve $\theta$, that has a point in common
with each one of them: the total degree of this curve is
 \[
 -4-2+2\cdot4=2.
 \]
 \end{enumerate}

\begin{NOTE} It is not known at the present moment whether there exist
surfaces with $p^{(1)}>1$ whose canonical system is reducible in such a
way that its variable parts are composed with the curves $k$ of a
pencil\footnote{Examples have actually been constructed recently by G.
Pompilj: Alcuni esempi di superficie algebriche a sistema canonico puro
degenere, {\it Rendic.\ Acc.\ Lincei}, s.~8, v.~IV, 1948.}. About
this possibility we can only prove the following proposition:

If the pure canonical curves of a surface of genus $p>1$ (apart from
the fixed components) are composed with the curves $k$ of a pencil of
genus $\pi$ and degree $n$, then one has in any case
 \[
 \pi\le p^{(1)},\quad n\le p^{(2)}=p^{(1)}-1.
 \]
\end{NOTE}

Denote by $\theta$ the fixed components, if any, of the $K$:
 \[
 |K|=\sum\theta+|sk|,
 \]
and let $\rho$ and $n$ be the characters of one of the $\theta$. We
express the degree of $|K|$ as the sum of the intersection numbers of
its components (irreducible or not) $k$ and of $\theta$; we have
 \[
 s(2\pi-2-n)+\sum(2\rho-2-n)=p^{(1)}-1.
 \]
 Now, on a transformed surface without exceptional curves, we have for
each $\theta$
 \[
 2\rho-2-n\ge0
 \]
 therefore
 \[
 2\pi-2-n \le p^{(1)}-1.
 \]
 On the other hand a curve $k$ intersects the residual curve
$(s-1)k+\sum\theta$ in the virtual number of points
 \[
 2\pi-2-2n\ge0.
 \]
 Whence one deduces
 \[
 n\le\pi-1,
 \]
 and
 \[
\pi\le p^{(1)},n\le p^{(1)}-1. \quad\hbox{q.e.d.}
 \]

 \section{On the regularity of the bicanonical
system}\label{sec:4.19}
 \markright{\hfill On the regularity of the bicanonical
system \quad 4.19\quad}
 The possibility that the pure canonical system $|K|$ of a surface
with $p>0$ and $p^{(1)}>1$ may be reducible forces us to examine
more closely the question of ``whether the bicanonical system and
the pluricanonical systems are regular''. In order to do this, one
needs to prove that, although the $K$ may be arbitrarily reducible
into simple or multiple components, they are always connected (for
$p^{(1)}>1$); so that the adjoint system
 \[
 |2K|=|K'|
 \]
is regular, and similarly, since the curves of $|2K|$ are also
connected, the tricanonical system
 \[
 |3K|=|(2K)'|
 \]
is also regular, and so on.

In words: {\em on a regular surface of genus $p>0$ and of linear
genus $p^{(1)}>1$, the bigenus is exactly
 \[
 P=p+p^{(1)}
 \]
 and the $i$th genus is
 \[
 P_i=p+\frac{i(i-1)}{2}p^{(1)}.
 \]}

In order to prove this important theorem it is convenient to show
that, for $p>0$ and $p^{(1)}>1$:

 \begin{enumerate}
 \renewcommand{\labelenumi}{(\arabic{enumi})}
 \item the variable parts of the canonical system are irreducible;
 \item the canonical curves $K$, however reducible, are always
connected, with {\em order of connection} $\ge2$ (that is, any part
of $K$ has at least two intersections with the residual part);
 \item as a consequence the pure bicanonical system is regular of
dimension $p+p^{(1)}-1$ and not greater.
 \end{enumerate}

These results can be extended a fortiori to the pluricanonical
systems of order $>2$.

\paragraph{(1)} The first proposition is easily proved. First of all
if $p\ge2$ and $p^{(1)}>1$, then the variable parts of the canonical
curves are either irreducible or composed with the irreducible
curves $C$, of genus $\pi>1$, of a pencil; therefore the curves $C'$
adjoint to a curve $C$ form a linear system of dimension (at least)
$p+\pi-1$ that cuts out the complete canonical series
$g^{\pi-1}_{2\pi-2}$ on $C$; two of the $C'$ have at least $2\pi-2$
variable intersections and thus the $C'$ are not composed with the
curves of a pencil; a fortiori the bicanonical system
$|2K|$, that contains $|C'|$ as a part, does not have reducible
variable part composed with the curves of a pencil.

The theorem extends to the case $p=1$. In this case, if the variable
parts of the bicanonical curves are composed with the irreducible
curves $C$ of a pencil, first of all -- since $p^{(1)}>1$ -- the genus of
these curves has to be $\pi>1$\footnote{Actually we shall see later that
if a surface has a pencil of elliptic curves and its geometric genus is
positive, then its linear genus has to be $p^{(1)}=1$.}. Now, if the
number of the components $C$ that are parts of the curves of $|2K|$ is
equal to $s$ ($P=s+1$ is the bigenus of the surface), since $p=1$, there
will be $s$ curves $C_{1/2}$ such that their doubles belong to the
pencil $|C|$:
 \[
 K=\theta+s C_{1/2}.
 \]
Now the adjoint system $|C_{12}'|$ of a curve $C_{1/2}$, which is
contained in $|2K|$, should have the curves of $|C|$ as variable parts;
but this is impossible because the dimension of this system is not one,
but (at least) $p+\pi-1=\pi>1$\footnote{Proposition~1 can be also
proved in another way, first showing that $\pi=2$ and therefore
necessarily $p>1$. If the variable parts of the bicanonical curves
$2K$ are composed with $s$ ($>1$) curves of a complete linear
pencil, then, denoting by
$\theta$ the fixed parts of $|2K|$ we shall write
 \[
 |2K|=\sum\theta+|sC|.
 \]
Now denote by $n$ and $k$ the degree and the genus of $|C|$, and by
$\nu$ and $\rho$ the degree and the genus of a $\theta$, and let us
compute the additive character of the $2K$ expressing their
intersection number with a $K$; we have
 \[
 2K\cdot K=2p^{(1)}-2=\sum(2\rho-2-\nu)+s(2\pi-2-n).
 \]
But, referring, as we may, to a surface without exceptional curves:
\[
2\rho-2-\nu\ge0,
 \]
and therefore
\[
s(2\pi-2-n)\le 2p^{(1)}-2.
\]
Thus, in order that the pencil $|C|$ be complete, its dimension has to
be $r=1$; therefore, if $|C|$ is not contained in the canonical system
$|K|$, we have:
 \[
p+n-\pi+1\le1 \quad \hbox{and}\quad n<\pi-1;
 \]
and this inequality holds a fortiori if $|C|$ is contained in $|K|$,
because in that case $|2C|$ is contained in the adjoint system
 \[
 |C'|=|C + K|.
 \]

In this set up, if $\pi>2$, one has
 \[
 s(\pi-1)\le2p^{(1)}-2,\quad
 \pi-1\ge2,\quad
 p^{(1)}-1\ge s,
 \]
and as a consequence the dimension of $|2K|$ is
 \[
 \ge p+p^{(1)}-1>s,
 \]
contradicting the hypothesis that $|2K|$ is formed by the groups of $s$
curves of the pencil $|C|$ and has therefore dimension $s$. \endgraf

After having proved in this way our proposition in the case $\pi>2$,
one has to discuss the case $\pi=2$. \endgraf

In this case the presence of the pencil of curves $C$ of genus $\pi=2$
makes it possible to represent the surface on a double plane with
branch curve $f_{2m}$ of some even order $2m$, with a $(2m-6)$-tuple or
$(2m-5)$-tuple point $O$ and other elementary singularities: 4-tuple
points and $[3,3]$ points. \endgraf

The bicanonical curves have as their images in this plane curves
$\Phi_{2m-6}$ adjoint to $f_{2m}$, that are made of a fixed
$\Phi_{2m-6-s}$ and of $s$ lines through $O$. Since among the
bicanonical curves there is at least one canonical curve counted twice,
the $\Phi_{2m-6-s}$ must be the double of a curve $\Psi_{m-3-h}$
($s=2h$) that, together with $h$ lines through $O$, is the image of a
canonical curve. Since $h\ge1$, one deduces that $p>1$, and this case
is already settled by the first remarks in the text.}.

\paragraph{(2)} To prove that the canonical curves $K$ are connected,
it suffices to show that the bicanonical curves of $|2K|$ are
connected, and therefore one has to contradict the assumption that
they are disconnected. Now the bicanonical system $|2K|$ consists of
a fixed part $L$ (arbitrarily reducible) and of irreducible variable
parts $C$:
 \[
 |2K|=L+|C|.
 \]
If the curves of this system are disconnected, then they can be
decomposed in two parts having a number of common points which is
nonpositive: one of these parts will contain a $C$ and a (fixed)
part $E$ of $L$, whereas the other will be a fixed $D$:
 \[
 L=D+E,\quad (C+E)D\le0.
 \]
Now we may assume that $D$ is connected; if this were not the case
then $D$ could be decomposed into two parts $D_1$ and $D_2$ with a
nonpositive number of common points, and one of these parts -- say
$D_1$ -- would have
 \[
 D_1(C+E+D_2) \le 0
 \]
intersections with the residual one; continuing in this way, in any
case one would eventually find a connected part of $L$, and thus of
$C+L$, having a nonpositive number of intersections with the residual
curve.

Therefore we assume $D$ to be connected and denote by $r$ its genus and
by $n$ its degree: in order that $D$ be connected we should
have\footnote{This proposition is proved by induction, for the
curves $D$ consisting of $s$ irreducible components, assuming that
the proposition holds for the curves consisting of less than $s$
components. Indeed $D$ can be decomposed into two parts, $D_1$ and
$D_2$, each one connected and connected to the other, with $n$
common points; thus, denoting by $\rho_1$ and $\rho_2$ the genera of
these curves, one has
 \[
 r=\rho_1+\rho_2+n-1,\quad \rho_1\ge0,\rho_2\ge0,r\ge0.\qed
 \]}
 \[
 \rho\ge0.
 \]

Let us write the intersection number of $D$ with the residual curve
with respect to the bicanonical system $|2K|$:
 \[
 D(2K-D)=D(C+E)=4\rho-4-3\nu\le0.
 \]
Whence we deduce
 \[
 \nu\ge \frac43(\rho-1).
 \]
Now, if $\rho=0$, since $n$ is an integer, one has
 \[
 \nu\ge -1:
 \]
so that either $\nu\ge0$, hence $p=0$, or $D$ is an exceptional curve of
the surface $(\rho=0,n=-1)$. One may assume that the surface is
transformed in such a way that it does not contain any such curve, and
therefore one may exclude the case $\rho=0$, and take
 \[
 \rho\ge1.
 \]

But if $\rho=1$, since it is not possible to have $\nu>0$ (otherwise
$p=0$), then one has $n=0$, and so the bicanonical and pluricanonical
curves of $|iK|$ have zero intersections with $D$, whereas $D$ has
always zero intersections with the residual curves of $|iK-D|$,
$|iK-2D|$ and so on; whence it follows that the curves of $|iK|$ are
composed with the elliptic curves of a pencil and $p^{(1)}=1$,
contradicting our assumption ($p^{(1)}>1$).

In conclusion, if the bicanonical curves are disconnected, then there
is one of their parts $D$, which is connected and has a nonpositive
number of intersections with the residual curves, of genus $\rho>1$, and
degree
 \[
 \nu\ge\frac43(\rho-1)>\rho-1.
 \]

But this is impossible, since it contradicts the hypothesis that $D$ is
contained in the fixed part $L$ of the curves of $|2K|$; indeed the
dimension of the complete system $|D|$ of index of speciality $p$ is,
according to Riemann--Roch:
 \[
 p+\nu-(\rho-1)-p>0.
 \]
Thus the curves of the bicanonical system $|2K|$, and thus also the
curves $K$ of the canonical system, are connected: any part of a $K$, of
genus $\rho$ and degree $\nu$, has to have a positive even number of
common points with the residual curve:
 \[
 2\rho-2-2\nu\ge2,
 \]
 and so the order of connection of $K$ is $\ge2$.

\paragraph{(3)} {From} the fact that the pure canonical curves are
connected (for $p>0$, $p^{(1)}>1$) one deduces that the bicanonical
system $|2K|=|K'|$ is regular. In a similar way, all pluricanonical
curves turn out to be connected, and so the corresponding systems
 \[
 |K_i=|\{(i-1)K\}'|
 \]
are also regular.

 \section{Remark on the irreducibility of the bicanonical
system}\label{sec:4.20}
 \markright{\hfill Remark on the irreducibility of the bicanonical
system \quad 4.20\quad}
 After having proved that on a regular surface with $p>0$ and
$p^{(1)}>1$ {\em the pure bicanonical system is regular}, we may wonder
whether it is actually always irreducible. Since we have excluded that
its variable parts may be reducible, one has to show that {\em it has no
fixed part} (except for the exceptional curves).

But the argument fails because of a rather subtle problem arising in
case the canonical curves are however reducible into simple and
multiple components. We shall restrict ourselves to prove the theorem
in the case in which the (pure) {\em canonical curves $K$ consist of
simple components.}

In that case we know that a curve $K$ is still connected when one
subtracts from it an irreducible component $K_1$ having the least
possible number $n$ ($\ge2$) of common points with the residual curve
$K_2$.

Now, denoting by $\pi_1$ and $\pi_2$ the genera of $K_1$ and $K_2$
respectively, we have the genus of K given by:
 \[
 \pi=\pi_1+\pi_2+n-1.
 \]
So when one subtracts $K_1$ from $|K|$, the dimension of the regular
system adjoint to $|K|$ drops by $r_1+1=\pi_1+n-1>0$, where $r_1$
denotes the dimension of the series cut out by $|K'|$ on $K_1$. This
means that $K_1$ cannot be a fixed component of $|K'|=|K_1+K_2'|$, since
otherwise subtracting that part one would obtain the system $|K_2'|$
having the same dimension.

Now -- once the component $K_1$ has been subtracted -- the curves of
$|K_2|$ are still connected, but in general no longer has order of
connection $\ge2$: they may have {\em handles}, that is, some components
having {\em a single} point in common with the residual curve; these
components -- when they are of genus zero -- will be fixed parts of
$|K_2'|$. Nevertheless these handles are not -- in any case -- fixed
parts of $|K'|$; this is because otherwise the series $g_{2\pi_1-2+n}$
cut out by $|K'|$ on $K_1$ would have fixed points and would not be
complete, whereas it must have dimension
 \[
 r_1=\pi_1-2+n.
 \]
Thus we have seen that, since the $K$ are connected curves with order
of connection $\ge2$, the adjoint system $|K'|$ cannot contain either
$K_1$ or the handles of the residual curve $K_2$ as fixed parts. Now,
removing these handles from $K_2$, one has a connected curve $\Kbar_2$,
with order of connection $\ge2$, and one proves in a similar way that
neither a component that can be subtracted without disconnecting the
curve, nor the handles of the residual curve, are fixed parts of
$|K_2'|$. Proceeding in this way, one sees that no component of $K$ can
be a fixed part of $|2K|=|K'|$. \qed

 \section{Genera of a reducible surface}\label{sec:4.21}
 \markright{\hfill Genera of a reducible surface \quad 4.21\quad}
 We want to add to the topics treated in this chapter some information
about reducible surfaces. Suppose that surface $F$ of some order $m$,
that we wish to assume with normal singularities, splits into two
surfaces $F_1$ and $F_2$ of orders $m_1$ and $m_2$ respectively. Then
we want to determine the numerical characters of $F$ in terms of those
of $F_1$ and $F_2$ and of the curve in which they intersect.

We should point out that the limit surface $F_1+F_2$ has a double curve
that is the limit of the double curve of $F$ and that consists of three
parts: a part of certain order $d_1$, double for $F_1$ and not contained
in $F_2$, a second part, the order of which we shall denote by $d_2$,
double for $F_2$ and not contained in $F_1$, and finally a third
part -- of order $d$, say -- contained in $F_1$ and $F_2$. Besides
this curve one has a curve $C$ of a certain genus $\pi$,
intersection of $F_1$ and $F_2$, that connects the two surfaces
$F_1$ and $F_2$, the sum of which is regarded as the limit of $F$.

Now we have to determine the numerical characters of the reducible
surface $F_1+F_2$ in terms of the analogous characters of the two
components and of the characters of the common curve $C$ connecting
them\footnote{{From} the degeneration principle applied to the plane
curves sections of $F$ it follows that there must be a curve of
connection of order greater than zero. See Enriques--Chisini, ibidem.}.

Let us start by discussing the arithmetic genus. We denote simply by
$p$ (equal to $p_a$) the arithmetic genus of $F$, by $p_1$ and $p_2$ the
arithmetic genera of $F_1$ and $F_2$ respectively, while we have
denoted by $\pi$ the genus of the curve $C$.

It is clear that one can compute $p$ by means of the postulation
formulas for the double curves of $F_1$ and $F_2$, or rather for the
curves of order $d_1,d_2$ and $d$ that are the limit of the double
curve of $F$ when $F$ degenerates as the sum of $F_1$ and $F_2$. In
this way one can find the formula
 \[
 p=p_1+p_2+\pi,
 \]
which is the extension to surfaces of Noether's formula that gives the
genus of a reducible curve\footnote{In general, for varieties of any
dimension: see F.~Severi, Fondamenti per la geometria sulle
variet\`a algebriche, {\it Rendic.\ Circ.\ Mat.\ di Palermo}, 1909.}.

But we can justify the formula without performing the computation
mentioned above.

Actually the surfaces $F_{m_1+m_2-4}$ of order $m_1+m_2-4$ adjoint to
$F_1+F_2$ (for which $C$ must not be regarded as a double curve) cut
out on $F_1$ the adjoint system of $|C|$ that has virtual dimension:
 \[
 p_1+\pi-1,
 \]
and for each one of these curves there are in general infinitely many
surfaces $F_{m_1+m_2-4}$; the virtual dimension of this system of
$F_{m_1+m_2-4}$ will be $p_2$, since there are $\infty^{p_2-1}$
surfaces $F_{m_1+m_2-4}$ that split into $F_1$ and a surface
$F_{m_2-4}$ adjoint to $F_2$. In this way one finds the dimension of
the system of $F_{m_1+m_2-4}$ adjoint to $F_1+F_2$:
 \[
 p-1=p_1+p_2+\pi-1.
 \]
One may object to this argument that since the surfaces $F$ adjoint
to $F_1+F_2$ are subject to pass through a curve of order $d_2$ not
belonging to $F_1$, they might cut out on $F_1$ a system of
dimension smaller than the dimension of the system $|C'|$ adjoint to
$|C|$. But, since we deal here with virtual dimensions, this
circumstance does not affect our computation, in which the given
adjoint surfaces might be replaced by adjoint surfaces of higher
order.

Therefore we conclude that the numerical genus of a reducible
surfaces whose components of genera $p_1$ and $p_2$ are connected by a
curve of genus $\pi$ is
 \[
 p_1+p_2+\pi.
 \]
Now we want to compute the linear genus $p^{(1)}$ of $F_1+F_2$.

In order to do this one needs to know the linear genera $p_1^{(1)}$ and
$p_2^{(1)}$ of $F_1$ and $F_2$, the genus of the curve of connection
$C$, that we have already denoted by $\pi$, and the degrees $n_1$ and
$n_2$ that this curve has on $F_1$ and $F_2$ respectively.

The linear genus of the reducible surface is
 \[
 p_1^{(1)}+p_2^{(1)}(1)+8\pi-n_1-n_2-9.
 \]
Indeed, as we have remarked, the surfaces of order $m_1+m_2-4$ adjoint
to $F_1+F_2$ cut out on $F_1$ curves $C'$ adjoint to $C$ that have genus
 \[
 p_1^{(1)}+3\pi-3-n_1;
 \]
similarly the curves that the same $F$ cut out on $F_2$ have genus
 \[
 p_2^{(1)}+3\pi-3-n_2.
 \]
Now a curve of the first system on $F_1$ and a curve of the second
system on $F_2$ taken together form the intersection of $F_1+F_2$ with a
$F_{m_1+m_2-4}$ whenever they have in common a canonical group of
$2\pi-2$ points, which are their intersections with $C$. So the genus of
the curves sum of these two is
 \[
 p^{(1)}=p_1^{(1)}+p_2^{(1)}+8\pi-n_1-n_2-9. \qed
 \]
