 \chapter{Linear systems of curves}\label{ch:1}
 \markboth{\qquad 1. \quad Linear systems of curves \hfill}{}

 \section{Linear pencils}\label{sec:1.1}
 \markright{\hfill Linear pencils \quad 1.1\quad}
Let $F$ be a surface, for the sake of simplicity assumed to belong to
ordinary space and endowed with normal singularities\index{surface!with
normal singularities}, as defined in the preceding introduction.

Consider a rational function\index{rational!function} $t$ defined on $F$,
which is not reduced to a constant:
 \begin{equation}
 t=\frac{\fie(x,y,z)}{\psi(x,y,z)}.
 \label{eq1}
 \end{equation}
For any given value $t_0$ of $t$ there will be points on $F$ at which $t$
takes this value: the locus $C$ of these points is said to be a {\em level
curve} of the rational function $t$ on $F$. As $t$ varies, we shall have on
$F$ a simply infinite system of level curves which is said to form a {\em
linear pencil}\index{linear!pencil}. Among the curves $C$ of the pencil
there is the {\em curve of zeros}, namely the curve $\fie=0$ which
corresponds to $t=0$, and the {\em curve of poles}, namely that defined by
$\psi=0$, for which $t=\infty$.

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

The definitions given above are perfectly analogous to those concerning
curves and linear series $g^1_n$ on them, and give rise to similar remarks.
The pencil of curves $C$ defined by (\ref{eq1}) is equally well given by the
rational function
 \[
\tau=\frac{\al t+\be}{\ga t+\de}\,;
 \]
the only effect of the rational substitution is to change the value of $t$
corresponding to each individual curve $C$. Thus all the curves $C$ still
appear in the pencil, so that the curve of zeros and the curve of poles is
of no particular significance. The rational function corresponding to a
given pencil is determined by the arbitrary choice of the curves $\tau=0$
and $\tau=\infty$, together with that of the unit level curve $\tau=1$,
since these choices determine the preceding substitution. If two functions
$t$ and $\tau$ have the same curve of zeros and poles, they differ by a
multiplicative constant.

We remark moreover that the rational function (\ref{eq1}) is defined only
modulo $F$. Thus two different functions $\fie/\psi$ and $\fie_1/\psi_1$ are
to be identified on $F$ if we have
 \[
 \frac{\fie}{\psi}=\frac{\fie_1}{\psi_1}
 \]
for all points of $F$. That is,
 \begin{equation}
 \fie\psi_1-\psi\fie_1\equiv 0 \mod F,
 \label{eq2}
 \end{equation}
or
 \[
 \fie\psi_1-\psi\fie_1=AF,
 \]
where $A$ represents a polynomial.

All the curves $C$ of a linear pencil have the same order; if the order of
one of them apparently drops, it means that the curve has become reducible
and contains one part at infinity. To eliminate this apparent reduction of
the order, if we do not want to use a projective transformation of the
surface, we have only to adopt homogeneous coordinates $x_1,x_2,x_3,x_4$.
Therefore, if for instance the degree of the polynomial $\fie$ in
(\ref{eq1}) were less than that of $\psi$, then, on passing to homogeneous
coordinates, $\fie$ gets multiplied by a suitable power of $x_4$ and then
clearly, the curve of zeros contains also the section of $F$ with the plane
at infinity $(x_4=0)$ counted a certain number of times.

The definition of a linear pencil of curves $C$ has a simple projective
meaning: indeed, the equation
 \begin{equation}
 \fie(x,y,z)-t\psi(x,y,z)=0
 \label{eq3}
 \end{equation}
represents a pencil of surfaces which cut out the curves $C$ on $F$; and
(\ref{eq2}) says that
 \begin{proc}
 {the same pencil of curves can be cut out on $F$ by different pencils
of surfaces.}
 \end{proc}

The pencil of surfaces (\ref{eq3}) may have a base curve $\Ga$ which in
general does not belong to $F$ and meets $F$ in a finite number of points.
These {\em base points}\index{base points} of the linear pencil of curves
$C$ are points of essential indeterminacy of the rational function
(\ref{eq1}). In fact, a function of the form $0/0$ at a base point, has a
nonremovable singularity there: indeed, if one tries to define the value of
the function at such a point by a limiting argument, then one finds that the
values so obtained by continuity will be different when one approaches the
base point along different curves $C$.

A base point for the curves $C$ of a linear pencil may have a certain
multi\-plicity $i>1$ for the $C$, in which case it will be said to be an
{\em $i$-tuple base point of the pencil.}

Finally, it may happen that the base curve $\Ga$ of the pencil (\ref{eq3}),
or some part $K$ of $\Ga$ if it is reducible, lies on $F$;\index{fixed part}
then, as we have already said, $K$ may either be considered as a fixed
component of the curves $C$ of the pencil (which is thus {\em reducible}),
or one can forget about $K$ and keep only the pencil of curves $C$
consisting only of the variable intersection of $F$ with the surfaces
(\ref{eq3}).

 \section{Irrational pencils}\label{sec:1.2}\index{irrational pencil,
pencil!irrational}
 \markright{\hfill Irrational pencils \quad 1.2\quad}
 A linear pencil of curves $C$ on $F$ is characterized geometrically by the
following properties:

 \begin{enumerate}
 \renewcommand{\labelenumi}{\arabic{enumi})}
 \item it is an $\infty^1$ (algebraic) series of {\em index
$1$},\index{index} that is, only one $C$ passes through any generic point of
$F$;
 \item this series is {\em rational},\index{rational!series} that is, it can
be put into one-to-one algebraic correspondence with the points of a line or
with the set of values of a parameter $t$.
 \end{enumerate}

In fact the function $t$ defined in this was is single valued, and is
therefore a rational function of the points of $F$, and the curves $C$ are
its level curves. Thus if one calls a {\em pencil}\index{pencil} a series of
curves of index 1, namely a series enjoying property (1), we can say that
 \begin{proc}
 any rational pencil is linear.
 \end{proc}
We may also have {\em irrational pencils} on a surface, whose curves are
elements of an algebraic but not rational system, having a certain genus
$p>0$. Thus, for example, the generators of a scroll of genus $p$ (that is,
with plane sections of genus $p$) form an irrational pencil of lines.

We can construct an example of a surface with an irrational pencil of genus
$p$ of curves by intersecting any hypersurface of $S_4$ with a double cone,
formed by the planes of the same $S_4$ projecting the points of a curve of
genus $p$ from a line vertex.

We should point out that
\begin{proc}
a pencil of curves belonging to the surface $F$ is certainly rational if it
has a base point at a simple point of the surface.
\end{proc}

In fact, if the base point\index{base point!ofirrational pencil} $O$ is
simple for the curves $C$ of a pencil, then the series of these curves is in
one-to-one correspondence with the pencil of tangent lines at $O$. On the
other hand, if the point has multiplicity $i>1$ for the $C$, then they
correspond in general to the groups of an involution of order $i$ in the
pencil of tangent lines at $O$, and this involution is rational (L\"uroth's
theorem\footnote{See Enriques--Chisini, Ibidem, Libro~II, Chapter~I, \S3,
Vol.~I, page~168.}).

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

 \section{Linear systems}\label{sec:1.3}
 \markright{\hfill Linear systems \quad 1.3\quad}
The definition of a linear pencil of curves on $F$ can be generalized by
considering {\em linear systems}\index{linear!system} of dimension $r>1$. An
$\infty^r$ linear system $|C|$ of curves $C$ is cut out on $F$ by a linear
system of surfaces (or hypersurfaces)
 \begin{equation}
 \la_0\fie_0+\la_1\fie_1+\cdots+\la_R\fie_R=0;
 \label{eqa1}
 \end{equation}
the {\em dimension} $r$ turns out to be smaller than $R$ when the system
(\ref{eqa1}) contains $\infty^h$ surfaces that contain $F$ as a part, and
then $r=R-h-1$. One defines the curves which are {\em fixed
components}\index{fixed!components} of the $C$ and the {\em base
points}\index{base points} of $|C|$ for linear systems $|C|$ with several
degrees of freedom in the same way as for a linear pencil; a base point is a
common point of all the $C$, where all the $\fie$ vanish simultaneously.

A linear system $|C|$ can be defined in an intrinsic way, independently of
the projective realization of the surface, in the same way as we did for the
linear pencil. Indeed, consider $r$ rational functions of the variable point
of $F$, all having the same curve of poles, and represent them as
 \[
 t_1=\frac{\fie_1}{\fie_0}\,,\quad t_2=\frac{\fie_2}{\fie_0}\,,\dots,
 \quad t_r=\frac{\fie_r}{\fie_0}\,,
 \]
where we assume, for the sake of simplicity, that they are linearly
independent on $F$; in other words, it is not possible to find $r+1$
constants $a_0,a_1,\dots,a_r$, not all zero, such that
 \[
 a_0\fie_0+a_1\fie_1+\cdots+a_r\fie_r\equiv0 \mod F
 \]
holds identically. Then the level curves of the function
 \[
 t_0=\la_1t_1+\la_2t_2+\cdots+\la_rt_r
 \]
describe the linear system $|C|$ of dimension $r$.

It might seem that the curve of poles $\fie_0$ assumes a special position in
this system; but we may choose any curve of the system to be the curve of
poles, since one and the same system $|C|$ may correspond to infinitely many
linear systems of functions, derived from one other by linear substitutions.

The dimension $r$ of the linear system $|C|$ turns out to be defined
geometrically by the following property:
 \begin{proc}
there is one and only one curve of the system passing through $r$ generic
points of the surface.
 \end{proc}

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

The dimension $r$ and the degree $n$ of a linear system $|C|$ are related by
the inequality $r\le n+1$; indeed, the series cut out on any curve $C$ of
$|C|$ by the remaining curves of the system, away from the base points, is a
$g_n^{r-1}$, from which we get $r-1\le n$. The linear series $g_n^{r-1}$
defined on a generic curve $C$ is of fundamental importance in the study of
the linear system and is called the {\em characteristic
series}\index{characteristic series}\footnote{The characteristic series of a
linear system was introduced by C.~Segre and G.~Castelnuovo around 1890. The
term {\em characteristic series} was adopted for the first time by
G.~Castelnuovo, Ricerche generali sopra i sistemi lineari di curve piane,
{\it Mem.\ Acc.\ di Torino}, s.~II, t.~XLII, 1891, reproduced on page~137
ffoll.\ of Castelnuovo's {\it Memorie scelte}, Zanichelli, Bologna, 1937.}
of $|C|$.

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

 \section{The characteristic property of linear systems}\label{sec:1.4}
 \markright{\hfill The characteristic property of linear systems \quad
1.4\quad}
We observed in Section~\ref{sec:1.3} that if $|C|$ is a linear system of
dimension $r$ on a surface $F$, there is one and only one curve $C$ passing
through $r$ generic points of $F$. For $r>1$, this is a characteristic
property\footnote{F. Enriques, Una questione sulla linearit\`a ecc., {\it
Atti Acc.\ Lincei}, 1893. Cf.\ C.~Segre, Introduzione alla geometria sopra
un ente algebrico semplicemente infinito, {\it Ann.\ di Mat.}, s.~II, t.~22
(1894).}, whereas we have seen that for $r=1$ one may also have nonlinear
pencils.

The proof of this theorem is given as follows.

Assume for simplicity that the curves of the system $|C|$ are irreducible,
and consider first the case $r=2$. The $\infty^2$ curves intersect pairwise
in the groups of an involution $I_n$ of order $n$ equal to the degree of
$|C|$. Let us take these groups as elements (points) of a new surface $F'$;
then on $F'$ we have an $\infty^2$ system of curves $C'$, the transforms of
the curves of $|C|$, which intersect pairwise at one point. The curves $C'$
passing through a point $A'$ form a pencil with a base point, and is thus a
rational pencil (see Section~\ref{sec:1.2}). In addition to this pencil, let
us consider another with a base point $B'$. The two pencils of $C'$ with
centres at $A'$ and $B'$ can be set in one-to-one correspondence
respectively with two pencils of lines $A$ and $B$ in the plane, by means of
projective transformations that make the curve $C'$ joining $A'$ and $B'$
correspond to the line $AB$. This assignment gives a one-to-one
correspondence between the points of $F'$ and the points of the plane, in
which the curves $C'$ correspond to the lines of the plane and the pencils
of $C'$ correspond to the pencils of lines; this implies that the $C'$ on
$F'$, and so also the $C$ on $F$, form a linear system (a {\em
net}\index{net}).

Once the theorem is established for nets, it extends to the case of linear
systems that are $\infty^3$, and so in general, passing from dimension $r$
to dimension $r+1$. If $|C|$ is an $\infty^3$ system of curves such that
only one curve $C$ passes through three generic points, then the $C$ passing
through a fixed point $A$ will form a net. Now nets defined by two base
points $A$ and $B$ may be set in a projective one-to-one correspondence with
two nets of planes $A'$ and $B'$ of ordinary space in such a way that the
curves of the pencil common to the two nets correspond, in the same way for
the two correspondences, to the planes common to the two nets of planes.
{From} this one gets a bijective (homographic) correspondence between the
system of the $C$ and the planes of the space $S_3$, where the planes of a
pencil correspond to a pencil of the $C$, and the planes of a net correspond
to the curves $C$ of a net; this yields that the system $|C|$ is a linear
system. \qed

 \section{Extension of the theorems of Bertini}\label{sec:1.5}
 \markright{\hfill Extension of the theorems of Bertini \quad 1.5\quad}
Bertini\index{Bertini's theorems} has given two fundamental
theorems\footnote{E. Bertini, {\it Rendic.\ Ist.\ Lombardo}, 1880.} for
linear systems of plane curves: the first says that the curves of a linear
system cannot have variable double or multiple points not belonging to a
fixed component; the second one, which is a corollary of the first, states
that if the variable parts of a linear system are reducible, then they are
composed of the curves of a pencil. These theorems extend easily to linear
systems of curves belonging to any surface\footnote{See F.~Enriques,
Ricerche di geometria ecc., {\it Mem.\ Acc.\ di Torino}, 1893 and
Introduzione alla geometria sopra una superficie algebrica, {\it Mem.\ della
Societ\`a Italiana delle Scienze (detta dei XL)}, 1896. The reducibility
theorem was already observed in Noether, {\it Math.\ Ann.}, VII, 1874.}. The
extension turns out to be immediate, following the proof indicated in the
lectures of Enriques--Chisini\footnote{Ibidem, Libro~II, Chapter~I, \S5,
Vol.~I, page~180.}.

If the generic curve $C$ of the linear system $|C|$ possesses a variable
double or multiple point $O$, then the curve $C$ has at least two
intersections at $O$ with any infinitely near curve, and thus with the
curves of all the pencils to which it belongs. This means that $O$ is either
a base point or a point belonging to a curve $K$ which is a fixed component
of the $C$. We therefore state:
 \begin{proc}
the curves of a linear system on a surface cannot have double or multiple
variable points {\em (away from the base points)} not belonging to a fixed
component.
 \end{proc}
The second theorem of Bertini follows as a corollary.

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

We therefore state the following result:
 \begin{proc} {if the generic curve of a linear system $|C|$ on
$F$ is reducible then: either}
 \begin{enumerate}
 \renewcommand{\labelenumi}{{\rm\arabic{enumi})}}
 \item $|C|$ contains a fixed part, and, after we remove this, the remaining
system is irreducible, or
 \item the curves $C$ are composed with $r>1$ curves variable in a
pencil, rational or otherwise, (and in this pencil, regarded as an
$\infty^1$ system, they form a $g_r$), or finally
 \item the curves $C$ are composed of the curves of a pencil as in case
2) together with some fixed curves added to these.
 \end{enumerate}
 \end{proc}

Cases 2) and 3) include all the linear systems of dimension $r>1$, and of
degree $n=0$ (see Section~\ref{sec:1.3}).

 \section{The image surfaces of linear systems}\label{sec:1.6}
 \markright{\hfill Image surfaces of linear systems \quad 1.6\quad}
Consider a linear system $|C|$ that is irreducible of dimension $r\ge3$, and
therefore of degree $n>0$. Referring $|C|$ projectively to the system of the
hyperplanes of a space $S_r$ gives rise to a transformation of our surface
$F$ into a surface $F'$ of order $n$, on which the transforms of the $C$ are
cut out by the hyperplanes. In fact the $C$ may be viewed abstractly as the
elements (hyperplanes) of a space $S_r$; thus the projective transformation
we have indicated sends the $\infty^{r-1}$ linear system of the $C$ passing
through a point $P$ to the star of hyperplanes passing through a point $P'$
of $S_r$ and, as $P$ varies on $F$, the point $P'$ describes a surface $F'$.

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

Actually, one cannot exclude the possibility that the condition that the $C$
pass through a generic point of $F$ implies that they pass through other
points $P_1P_2\dots P_{m-1}$; in this case the groups analogous to
$PP_1P_2\dots P_{m-1}$ will form an {\em involution}\index{involution} on
$F$ of order $m$, and that the $C$ {\em belong to this involution}. The
resulting surface $F'$ of $S_r$ determined above will then be a surface of
order $n/m$, to be considered as a multiple surface of order $m$.

It is in fact a particular circumstance for a linear system $|C|$ of
dimension $r\ge3$ to belong to an involution, as follows from an easy count
of constants: there are at least three curves passing through $P$ that are
linearly independent, and in general there is no reason for one of these to
pass through the points of intersection of the other two. In contrast, an
$\infty^2$ linear system (that is, a net) always belongs to an involution of
order $n$, in which case it is of degree $n>1$; in fact, only two linearly
independent curves $C$ pass through a point $P$, and all the other $C$
through $P$ form a pencil, having their intersection as base points. Thus a
net $|C|$ of degree $n$ leads to a {\em multiple plane of
order}\index{multiple plane,order!of a multiple plane} $n$, with the $C$
having multiple lines of order $n$ as images; here the {\em branch
curve}\index{branch curve} is of special importance, that is, the locus of
points in the plane corresponding to groups of the involution endowed with a
double point.

We may give an example of a linear system $|C|$ of dimension $r\ge3$
belonging to an involution, by considering the system of the curves $C$
which are intersections of the surface $F$ with the cones of a certain order
$n$ having vertex at a fixed point of the space; the linear system $|C|$
will now belong to an involution of order $m$ and will be of degree $mn$,
where we denote the order of the surface $F$ by $m$.

When we want to exclude that a linear system $|C|$ belongs to an involution,
we say for short that it is a {\em simple system}\index{simple!linear
system}.

Going back in general to the transformation of the surface $F$ determined by
a linear system $|C|$ of dimension $r\ge3$, we note that the projective
correspondence discussed above can be translated into formulas as follows:
if the system $|C|$ is represented by
 \[
\la_0\fie_0+\la_1\fie_1+\cdots+\la_r\fie_r=0,
 \]
we should take the coordinates of the points of $S_r$ as
 \[
X_0\equiv\fie_0,\quad X_1\equiv \fie_1,\dots,\quad X_r\equiv \fie_r.
 \]

We conclude by stating a theorem which extends to surfaces a well-known
statement about curves and linear series on them:
 \begin{proc}
when we study the invariant relations of surfaces using a simple linear
system (at least $\infty^3$), the geometry of birational trans\-formations
on the surfaces is reflected in the projective geometry of the image surface
of the system.
 \end{proc}

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

Concerning base points, it will be useful to take into account the following
remarks. If $O$ is an $i$-tuple base point on the surface $F$ of a system
$|C|$ (that is simple for $F$), then it will be a fundamental point for the
transformation which sends $F$ into the image surface $\Phi$ whose
hyperplane sections are the $C$; and if the $C$ have no fixed tangent line
at $O$, then $\Phi$ will have an {\em exceptional curve}\index{exceptional
curve} of order $i$ corresponding to $O$. However, if $O$, as an $i$-tuple
base point of $|C|$ with $i\ge1$, is such that the $C$ have a fixed tangent
line (that is, a simple base point $O_1$ infinitely near to $O$) without a
higher contact at $O$, then we can still say that an exceptional curve of
order $i$ on $\Phi$ corresponds to $O$. But this curve will, in general,
consist of a curve of order $i-1$ (whose points correspond to the variable
points of the first order neighbourhood of $O$) and of a line which is the
image of the neighbourhood of $O_1$ (whose points belong to the second order
neighbourhood of $O$). In particular if $i=1$, then the whole neighbourhood
of the simple point $O$ is mapped to the neighbourhood of a conic double
point on $\Phi$, to which the exceptional line image of $O_1$ has to be
added (consider, for instance, the plane representation of the quadric
cone).

 \begin{NOTE}
In general, if the simple point $O$ of the surface $F$ is an arbitrarily
complicated singularity for the curves $C$ of $|C|$, one may resolve this
singularity, transforming $F$ into a surface $\Phi$ on which the
neighbourhood of the point $O$ (taken in its entirety) will be represented
by an exceptional curve consisting of as many parts as the branches (linear
or otherwise) that form the singularity of the curves $C$ at $O$. The
connections between these parts will reflect the way in which the stated
singularity is composed of multiple or simple infinitely near points, and
their proximity relations\footnote{Enriques--Chisini, Ibidem, Libro~IV,
Vol.~II, page~327 and following.}. This has been an object of analysis of
E.~Barber and O.~Zariski\footnote{Cf.\ E. Barber and O.~Zariski, Reducible
exceptional curves of the first kind, {\it Amer.\ J. Math.}, 1935
(page~119). P.~Du Val, {\it Amer.\ J. of Math.}, 1936.}. We will have
occasion to return to this argument later.
 \end{NOTE}

 \section[Equivalent curves and complete linear systems]{Equivalent curves
and complete linear\\ systems}\label{sec:1.7}\index{complete linear
system,linear system!complete}
 \markright{\hfill Equivalent curves and complete linear systems
\quad 1.7\quad}
The following considerations are a step-by-step extension of the well-known
results for linearly equivalent groups of points and complete linear series
on a curve\footnote{Enriques--Chisini, Ibidem, Libro~V, Vol.~III (1924).} to
linear systems of curves on a surface.

Two curves (of the same order) on a surface $F$ are said to be {\em
equivalent}\index{equivalent curves} if they both belong to a linear system
of dimension $r\ge0$: if they are distinct, this means that they belong to a
linear pencil; allowing systems of dimension 0 to be considered means that a
curve is to be taken as equivalent to itself.

The equivalence of two curves $C$ and $K$ is expressed by writing
 \[
C\equiv K,\qquad \hbox{or sometimes}\quad C=K.
 \]
Two equivalent curves $C$ and $K$ may always be considered as the curve of
zeros and the curve of poles of a rational function $t$ on F; the two curves
are interchanged if one substitutes $1/t$ for $t$. The relation of
equivalence between curves on a surface enjoys the three fundamental
properties of equality:

 \begin{enumerate}
 \renewcommand{\labelenumi}{\arabic{enumi})}
 \item the {\em reflexive} property, expressed by $C\equiv C$;
 \item the {\em symmetric} property: if $C\equiv K$ then also $K\equiv C$;
 \item the {\em transitive} property: if $C\equiv K$ and $K\equiv L$ then
also $C\equiv L$.
 \end{enumerate}

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

As a consequence of the above properties, one has the fundamental theorem:
 \begin{proc}
On a surface $F$ the set of all curves equivalent to a given curve $C$ forms
a linear system, which is said to be {\em complete}\index{complete!linear
system}.
 \end{proc}

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

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

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

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

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

This is therefore the projective meaning of the theorem on complete linear
systems. Consider, for example, a surface $F$ of a certain order $n$ in
ordinary space; it may happen that $F$ can be obtained as a projection of
two surfaces $F_1$ and $F'_1$ of the same order in $S_4$, the projections
being performed from two points $O$ and $O'$ respectively. Now if $F_1$ and
$F'_1$ are normal surfaces, then it follows that they will be projective to
one another, that is, projectively equivalent. If not, it means that $F_1$
can be thought of as a projection of a surface $F_2$ of $S_5$ and $F'_1$ as
a projection of a surface $F'_2$ of $S_5$, and so on. Proceeding in any
possible way from a surface to another of a higher dimensional space, we
eventually arrive at a surface $F_r$ of the space $S_{r+3}$ which will be
normal. And this $F_r$ will always be projectively well defined, whatever
sequences of surfaces $(F_1, F_2,\dots)$ or $(F'_1,F'_2,\dots)$, derived
from one another by projection, we used to arrive at it.

Examples of normal surfaces have already appeared in the theory of rational
surfaces\footnote{See Enriques--Conforto, {\it Le superficie razionali},
Bologna, 1939 (1945).}. Thus a surface $F_5$ of order 5 in ordinary space,
with elliptic plane sections and a double curve of order 5 belonging to a
quadric cone and having a triple point at the vertex of the cone that is
also a triple point for the surface can be considered as a projection of
several surfaces $F_5$ of the same order of the space $S_4$. These surfaces
are not projective to one another, but they are all projections of a normal
surface of $S_5$ that is uniquely defined up to projective equivalence.
These assertions can be justified by looking at the plane representation of
the $F_5$ by means of an $\infty^3$ linear system of cubics $C_3$ with four
simple base points $A_1A_2A_3A_4$.

Indeed such an $\infty^3$ system is contained in $\infty^1$ 4-dimensional
linear systems of curves $C_3$ with the same base points $A_1A_2A_3A_4$,
which are homographically (and also birationally) distinct in the plane, and
therefore represent surfaces $F_5$ of $S_4$ that are not projective to one
another. By contrast the normal surface of $S_5$ of which these $F_5$ are
projections (from an external point) is defined up to projective
transformations by the complete system of the $C_3$ through $A_1A_2A_3A_4$.

 \section{Sum and difference of linear systems: the residue
theorem}\label{sec:1.8}
 \markright{\hfill Sum and difference of linear systems: the residue
theorem \quad 1.8\quad}
 Curves composed of equivalent curves, namely sums of equivalent curves,
are equivalent. Indeed, let $C_1$ and $C_2$, $K_1$ and $K_2$ be two pairs of
equivalent curves. Take $t$ a rational function having $C_1$ as
curve of poles, $C_2$ as the curve of zeros, and $\tau$ a rational function
having $K_1$ as curve of poles and $K_2$ as curve of zeros. Then the
function $\tau t$ has $C_1+K_1$ as curve of poles and $C_2+K_2$ as curve of
zeros. Thus from
 \[
 C_1\equiv C_2\quad\mbox{and}\quad K_1\equiv K_2
 \]
one deduces
 \[
C_1+K_1\equiv C_2+K_2.
 \]
This yields that, given any two linear systems $|C|$ and $|K|$, there
exists a well determined complete system $|C+K|$, which contains all the
reducible curves $C+K$; this will be called the complete system sum of
$|C|$ and $|K|$.

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

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

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

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

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

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

In conclusion the complete linear system difference of $|C|$ and $|K|$:
 \[
 |L|=|C-K|
 \]
is defined by the relation
 \[
 |C|=|L+K|.
 \]

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

In particular, on a surface of ordinary space without singularities,
the surfaces $\fie$ of a given order cut out a complete system.
 \end{NOTE}

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

This is M.~Noether's Restsatz\footnote{{\it Math.\ Ann.}, Band~VIII. For the
case of curves see: Brill and Noether, {\it Math.\ Ann.}, Band~VII;
Enriques--Chisini, Ibidem, Libro~V.}, which, as in the case of the geometry
of curves (as remarked by G.~Castelnuovo, 1890), contains two distinct
statements: the former concerns the subtraction of complete systems, the
latter adds a projective complement to the first and concerns the
completeness of the system cut out on a surface $F$ of the space $S_3$ by
its adjoints $\fie$ of a given order.

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

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

 \begin{Remark} The definition of the system $|L|=|C-K|$ requires an
existence assumption, since the difference has an effective meaning
only if the system $|C|$ partially contains $|K|$. But it will also be
convenient in what follows to talk about virtual curves, defined by
subtraction, even if this is not actually possible. Thus we shall look
at the symbolic relation
 \[
 |C-K|=|L-M|
 \]
as equivalent to
 \[
 |C+M|=|K+L|.
 \]
The virtual curves are defined as the negative numbers in Peano's
theory of operators, namely as possible summands. To add the virtual
curve $C-K$ to a system $|D|$, means adding $C$ and then subtracting $K$ from it, and this will turn out to be possible for sufficiently ample
systems. In any case for the virtual curves $C-K$ and $L-M$ to be
equivalent means that the linear systems
 \[
 |D+C-K|=|D+L-M|
 \]
are equivalent, where
 \[
 |D+(C-K)|=|D+C-K|.
 \]
One can also give, as we shall see later on, a more restricted
definition of virtual curves as curves which do not effectively exist,
but such that the double or a multiple does exist. And this concept,
rather than to the notion of negative numbers, is related to the notion
of ideals, which are met in the theory of algebraic numbers fields.
 \end{Remark}

 \begin{Notice} The idea, if not the name, of ``virtual curves'', was
considered by Enriques in connection with the notion of canonical curves.
Enriques realized that the curve $C-K$, defined as the difference of two
linear systems, should correspond to a real mathematical object, even in
the case the subtraction does not lead to an effective curve. This is
indeed the main theme of the new development of the theory in passing from
the Ricerche of 1893 to the Introduzione of 1896, where one basically looks
for the invariant properties of the aforesaid canonical curves in the most
general setting, even when the genus $p$ is equal to 0. The said author
defines the virtual characters, degree and genus (see
Section~\ref{sec:1.10}), of the difference curve $C-K$, and notices that,
even if this curve does not exist, it may have an effective double. The
word ``virtual'', referred to the curve as well as to its characters (which
were so named by Enriques), can be found in Severi's note, Sulle curve
algebriche virtuali appartenenti ad una superficie algebrica, {\it Rendic.\
Ist.\ Lombardo}, 1905. The topological interpretation of the virtual curves
as two dimensional cycles has been given by S.~Lefschetz.
 \end{Notice}

 \section{The image surface of the sum system}\label{sec:1.9}
 \markright{\hfill The image surface of the sum system \quad 1.9\quad}
 Let us assume we have on the surface $F$ two irreducible linear
systems $|C|$ and $|K|$ of dimension $r$ and $s$ respectively, where
$r\ge1$ and $s\ge1$. One may construct in general a transformed surface
$\Phi$, the image of the sum system $|C+K|$, in a space of dimension R,
where $R>r+s$; the sum system which we are considering may be either
the minimal sum system, or, if one prefers, the complete system.

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

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

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

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

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

 \section{Virtual characters}\label{sec:1.10}
 \markright{\hfill Virtual characters \quad 1.10\quad}
 Let $|C_1|$ and $|C_2|$ be two irreducible linear systems on the
surface $F$, and let us denote by $n_1$ and $n_2$ their effective
degrees, by $\pi_1$ and $\pi_2$ their genera, and by $i$ the number of
variable intersections of a curve $C_1$ with a $C_2$. The degree of the
system $|C_1+C_2|$ will be the number of intersections
 \[
 (C_1+C_2)^2=C_1^2+C_2^2+2C_1C_2,
 \]
and therefore its value will be
 \[
 n=n_1+n_2+2i.
 \]
The genus of the sum system will be
 \[
 \pi=\pi_1+\pi_2+i-1,
 \]
since this is the expression of the genus of a curve which becomes
reducible in two parts; this expression has been determined by Noether
by an argument of continuity\footnote{See Enriques--Chisini, Ibidem,
Libro~V, Vol.~III, page~399.}.

These formulas lead to the definition of virtual degree and virtual
genus of a curve $C$ on $F$, independently of the possibility of extending
the $\infty^0$ linear system which they determine. In order to do this,
let us associate to $C$ an irreducible system $|K|$ such that the system
$|C+K|$ turns out to be irreducible; this can be easily achieved by
taking surfaces of sufficiently high order through $C$ and then the
residual intersections as the curves $K$. Now, denoting by $n$ the degree
of $C$ which we want to define, by $m$ the degree of $K$ and by $i$ the
intersection number of a $C$ with a $K$, the degree of $|C+K|$ will be
given by
 \[
 N=n+m+2i,
 \]
whence one deduces
 \[
 n=N-m-2i.
 \]
The virtual degree of $C$ thus defined is independent of the choice of
the auxiliary system $|K|$ which was associated to $C$. Indeed let
$|K_1|$ and $|K_2|$ be two auxiliary systems satisfying the required
conditions, of degrees $m_1$ and $m_2$, and let $i_1$ and $i_2$ be the
intersection numbers $CK_1$ and $CK_2$, and $j$ the number of variable
intersections of a curve $K_1$ and a $K_2$.

One can compare the degree $n_1$ defined with respect to $|K_1|$ and
the degree $n_2$ with respect to $|K_2|$, by computing the degree of
the system
 \[
 |(C+K_1)+K_2|=|(C+K_2)+K_1|=|C+(K_1+K_2)|.
 \]
Then one has
 \[
 N=(n_1+m_1+2i_1)+m_2+(2j+2i_2)
 =(n_2+m_2+2i_2)+m_1+(2j+2i_1),
 \]
which yields
 \[
 n_1=n_2.
 \]
In a completely similar manner one can define the virtual genus $\pi$ of an
isolated curve $C$, and this character is meaningful even if $C$ is a
multiple curve, or a reducible curve containing some multiple part.

Let us add that the definition of the virtual characters (degree and
genus) can be also extended to the case of a virtual curve $C-K$, defined
as a difference, independently of its effective existence (see
Section~\ref{sec:1.8}). It is sufficient indeed to add to $C-K$ a curve $L$ in
such a way that $|M|=|C-K+L|$ is an irreducible linear system, and then to
compute the characters of the difference $|M-L|$, which turn out to be
independent of $|L|$.

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

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

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

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

 \section{Exceptional curves}\label{sec:1.11}
 \markright{\hfill Exceptional curves \quad 1.11\quad}
 As we said above, the neighbourhood of a (simple) base point of the
surface $F$, which happens to be a base point for a linear system of
curves, may be regarded as an {\em infinitesimal curve} which, by means
of a transformation of the surface, can be changed into an actual curve
(an exceptional curve). Conversely, an exceptional curve is a curve on the
surface $F$ which can be transformed into the neighbourhood of a simple
point.

In order to take a closer look at the exceptional curves, let us start
by making this distinction:
 \begin{quote}
An {\em irreducible} curve $\om$ on the surface $F$ will be said to be an
{\em exceptional curve of the 1st kind}\index{exceptional curve!of the 1st
kind} if it can be transformed into a simple point by means of a linear
system $|C|$ with no base points on $\om$. In contrast, a curve $\om$ will
be called an {\em exceptional curve of the 2nd kind}\index{exceptional
curve!of the 2nd kind} if it can be transformed into a point by a linear
system $|C|$ having some base point on it.
 \end{quote}

An example of exceptional curves of the first kind is given by the 27
lines lying on a general cubic surface $F_3$. Indeed, $F_3$ can be
represented on the plane in such a way that an arbitrarily given one
among these lines corresponds to the neighbourhood of one of the 6
base points of the system of cubics representing $F_3$. By contrast,
the plane does not contain any exceptional curves of the first kind. But
lines, conics and more generally, plane curves that are fundamental
curves for a Cremona transformation are exceptional curves of the
second kind: one may transform a line into a point by means of a
homaloidal net (for instance a net of conics) having two base points on
the line; and one may transform a conic to a point by taking for example
a homaloidal net of cubics with a double base point and four simple base
points on the conic.

It follows already from these examples that the distinction between
exceptional curves of the first and of the second kind only has
projective meaning or is only {\em relatively invariant} under
transformations without base points: indeed {\em we can always replace
an exceptional curve of the second kind $\om$ belonging to a surface $F$
by a exceptional curve of the first kind} on a transformed surface $\Phi$.
For this, consider the transforming linear system $|C|$ taking $\om$ into
a point; it has, by hypothesis, a certain number $s$ of base points, with
certain multiplicities $i_1, i_2,\dots, i_s$ on $\om$, and the number of
intersections with $\om$ is precisely equal to the sum
$i_1+i_2+\cdots+i_s$. Now if we add to $|C|$ a linear system $|L|$
without base points, and for which $\om$ is not a fundamental curve,
then the system $|C+L|$ will change $\om$ into a fundamental curve for
the transformed system of $|C|$ (which will have no base points on it),
that will thus be an exceptional curve of the first kind.

Let us try to compute the genus and the degree of an irreducible exceptional
curve $\om$ of the first kind, by referring to the simple point $O$ whose
neighbourhood corresponds to $\om$ on a transformed surface. Consider on
such a surface any linear system $|C|$ for which $O$ is not a base point.
Imposing a base point $O$ on $|C|$ reduces its degree by one; moreover the
$C$ through $O$ have one point in common with the neighbourhood of $O$. Thus
writing $n$ and $\pi$ for the degree and genus of $|C|$, and $\nu$ and
$\rho$ for the degree and genus of the infinitesimal curve that corresponds
to the neighbourhood of $O$, that is, of the exceptional curve $\om$, we
will have
 \[
 n=n-1+\nu+2\cdot1 \quad\hbox{and}\quad \pi=\pi+\rho+1-1,
 \]
and thus
 \[
 \nu=-1 \quad\hbox{and}\quad \rho=0.
 \]
Actually, the formula $\rho=0$ follows directly from the rationality of the
neighbourhood of $O$.

Of course the same computation can be repeated on the transformed surface
$\Phi$ of $F$, containing $\om$. Here one has a system $|C|$ for which $\om$
is a fundamental curve (having no moving intersections with $C$); then the
curves $C$ passing through a point of $\om$ break up into $\om$ and a
residual curve $C_1$, which is unisecant to $\om$: writing the degree and
genus of $|C|=|\om+C_1|$, we recover the above formulas.

We therefore state that:
\begin{proc}
an irreducible exceptional curve of the first kind has virtual genus
$\rho=0$ and virtual degree $\nu=-1$.
\end{proc}

Turning now to exceptional curves of the 2nd kind, it suffices to recall
that such a curve $\om$ can be transformed into an exceptional curve of the
first kind, provided that one imposes certain base points to a suitable
transforming system $|C|$; therefore the virtual genus and degree of an
exceptional curve of the 2nd kind will be
 \[
 \rho=0 \quad\hbox{and}\quad \nu\ge 0.
 \]
For instance the degree of the generators of a scroll is
 \[
 \nu=0;
 \]
the degree of a line in the plane is
 \[
 \nu=1,
 \]
the degree of a conic is
 \[
 \nu=4, \quad\hbox{etc.}
 \]
Note that the constraint for the virtual genus of an exceptional curve to be
$\rho=0$ yields that an irreducible exceptional curve of the 1st kind $\om$
cannot have double or multiple points. Indeed the virtual genus $\rho$ is
computed by calculating the genus of the sum system $|C|=|C_1+\om|$.
Therefore a double point of the component $\om$, which is not a base point
of $|C|$, has to be considered as non-existing, hence, being already
$\rho=0$, the curve $\om$ would turn out to be reducible.

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

The transformation procedure which we used here leads to the following
result: if the surface $F$ contains a certain number $s$ of exceptional
curves which cut the $C$ in $i_1,i_2,\dots,i_s$ points respectively,
where one has
 \[
 i_1+i_2+\cdots+i_s> 2 \pi-2-n
 \]
then one may construct on $F$ an irreducible linear system of degree N
and genus $\Pi$ for which
 \[
 N>2 \Pi -2.
 \]
As we shall see later on, the existence of systems whose characters satisfy
such an inequality defines a particular family of surfaces. It
characterizes indeed the family of surfaces which are rational or
birational to scrolls; on a surface which does not belong to this
particular family, for any linear system of genus $\pi$ and degree $n$ the
fundamental inequality
 \[
 n \le 2 \pi-2
 \]
holds.

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

Likewise one sees that
 \begin{proc}
 a surface which contains exceptional curves of the second kind belongs to
the family of ruled surfaces.
 \end{proc}

Indeed, to start with, one may transform, as we said, an exceptional curve
of the second kind into a curve $\om$ of genus $\rho=0$ and degree
$\nu\ge0$. Here, since $\nu\ge0$, the procedure of adding $\om$ to a system
$|C|$ never ends because the intersection number of the curves of the
system with $\om$ increases, or at least does not decrease. Therefore one
will certainly arrive at a linear system of genus $\Pi$ and degree N, for
which
 \[
 N>2 \Pi -2.
 \]

 \section[A note on the reducible exceptional curves]{A note on the
reducible exceptional\\ curves}\label{sec:1.12}
 \markright{\hfill A note on the reducible exceptional curves \quad
1.12\quad}
 When one transforms a surface $F$ into another surface $\Phi$ by means of
a linear system $|C|$ having distinct base points $O,O_1,O_2,\dots$ on $F$,
the exceptional curves corresponding to these points, which are at the same
time fundamental curves for the system $|L|$ (transformed of the plane
sections of $F$), are irreducible curves, disconnected from one another.
But what happens when the points $O,O_1,O_2,\dots$ become infinitely near?

Suppose that the aforementioned points $O,O_1,O_2,\dots$ have
multiplicities $i,i_1,i_2,\dots$ for $|C|$, where $i>i_1>i_2,\dots$ and
the differences $i-i_1,i_1-i_2,\dots$ are as large as one needs. To
these points will in general correspond on $\Phi$ some exceptional
curves $\Om,\Om_1,\Om_2,\dots$ of orders $i,i_1,i_2,\dots$
respectively. But if the point $O_1$ becomes infinitely near to $O$, then
the curve $\Om$ of degree $i$ becomes reducible into two parts $\om$ and
$\om_1(=\Om_1)$: the former, of order $i-i_1$, corresponds to the points
of the neighbourhood of $O$ except $O_1$, the latter, of order $i_1$, is
the limit of the exceptional curve corresponding to $O_1$.

In general if several points $O_1O_2\dots$ become infinitely near to a
proper point $O$ on one or more branches, which may be either linear or
super\-linear, then the exceptional curve corresponding to the point $O$
will break up into several parts, connected between them, and
containing the exceptional curves which arise from the single points.
To be more precise we consider the characteristic case in which
$O,O_1,O_2,\dots,O_s$ form a sequence of infinitely near points on one
branch, and we start from the simplest examples. If
$O,O_1,O_2,\dots,O_s$ follow each other along a linear branch, then the
exceptional curve of order $i$ which corresponds to $O$ becomes a
composite curve of order $(i-i_1)+(i_1-i_2)+\cdots+(i_{s-1}-i_s)$:
 \[
 \Om=\om+\om_1+\om_2+\cdots+\om_s,
 \]
whereas the curves coming from $O_1, O_2,\dots$ become
 \begin{eqnarray*}
\Om_1&=&\om_1+\om_2+\cdots+\om_s\\
 \cdots &&\cdots \\
 \Om_{s-1}&=&\om_{s-1}+\om_s \\
 \Om_s&=&\om_s.
 \end{eqnarray*}
And, according to the principle of continuity, these reducible curves
have to be considered as exceptional curves, whereas their parts (e.g.
$\om_1,\om_2,\dots,\om_{s-1}$) do not form complete exceptional curves,
but only components of them. Indeed the component $\om$ will not be in
one-to-one correspondence with the (whole) neighbourhood of $O$, but with
this neighbourhood with the point $O_1$ deleted, and so $\om_1$ will be
in correspondence with the neighbourhood of $O_1$ with the point $O_2$
deleted, etc. We may pass from the surface $F$ to the $\Phi$ by means of
several subsequent transformations from $F$ to $F_1$, from $F_1$ to
$F_2$, \dots, from $F_{s-1}$ to $F_s=\Phi$; the first of these
transformations sends $O$ to a curve $\om$, the second sends a point
$O_1$ of $\om$ to a curve $\om_1$, the third sends a point $O_2$ of
$\om_1$ (different from the common point of $\om$ and $\om_1$) to a
curve $\om_2$, etc. Therefore each $\om_i$ (for $i<s$) will be connected
with the following $\om_{i+1}$ at one point and its degree (which is
$-1$ on $F_i$) will turn out to be $-1-1=-2$ on $F_{i+1}$ and on
$\Phi$. Accordingly, the composite exceptional curve
$\Om=\om+\om_1+\om_2+\cdots+\om_s$, corresponding to a sequence of
infinitely near points $O,O_1,\dots,O_s$, on a linear branch, will have
genus
 \[
 \rho=0+0+\cdots+0+(s-1)-(s-1)=0
 \]
and degree
 \[
 \nu=-2-2-\cdots-2-1+2(s-1)=-1.
 \]
At the end we find again the same characters of the irreducible
exceptional curves, of which the composite curves
$\Om,\Om_1,\Om_2,\dots,\Om_s$ appear as limits.

The same values of the characters must be found also when the points
$O,O_1, O_2,\dots,O_s$ form a sequence on a superlinear branch. However
in the definition of the limit curves we meet something surprising: the
components of the curves $\Om_i$ become in general multiple curves!

Let us consider, for the sake of simplicity, the case of three points
$O,O_1,O_2$ on a cuspidal branch: this case may be handled by looking
at the cuspidal branch as a limit of a linear branch, where $O_2$,
varying in the neighbourhood of $O_1$, goes to the position of its
satellite point. It is apparent here that the curve of order $i-i_1$,
corresponding to the neighbourhood of $O$ (without $O_1$), breaks up into
a curve $\om$ of order $i-i_1-i_2$ and a curve $\om_2$ of order $i_2$,
which corresponds to the neighbourhood of $O_2$; consequently the latter
curve turns out to appear twice in
 \[
 \om+\om_1+2\om_2.
 \]
In order to understand more clearly what is going on here, let us make
three successive transformations of the surface $F$, changing first of all
$O$ into $\om$, then a point $O_1$ of $\om$ into $\om_1$, and finally the
common point $O_2$ of $\om$ and $\om_1$ into $\om_2$: the position of the
point $O_2$ thus defined corresponds on the surface $F$ to the the
satellite point of $O_1$. But, according to what we said above, since $O_2$
is a double point for the curve $\om+\om_1$, then the transformed
exceptional curve $\om_2$ has to be counted twice. The multiplicities of
the components of the reducible exceptional curves corresponding to the
sequences of infinitely near points on a superlinear branch have been
determined by E.~Barber and O.~Zariski\footnote{Barber and Zariski, Amer.\
J. of Math., 1935.}, who also found out how these curves are connected
to one another. In this way another interpretation has been given, from a
new point of view, of Enriques' theory of singularities\footnote{See
Enriques--Chisini, Lezioni etc., Libro~I, Vol.~II.}. We shall restrict
ourselves here to state the following general rule: the reducible
exceptional curve $\Om$, which corresponds to a sequence of infinitely near
points $O_1,\dots,O_s$ which are proximate to a proper point $O$ on a
superlinear branch, is formed by $s+1$ components; the $i$th component
$\om_ih_i$ appears with the multiplicity $h_i$ which equals the minimum
order of a branch issuing from $O$\footnote{When we talk about a branch
issuing from $O_1$ (or from $O_2$, etc.) we mean we have previously made a
transformation which sends $O_1$ to a proper point.} and containing the
points $O_1,\dots,O_i$.

The same rule can be applied to the composite exceptional curves
$\Om_1,\Om_2,\linebreak[2]\dots,\linebreak[2]\Om_s$. Furthermore the
curve $\om_i$ is not necessarily connected with the following curve
$\om_{i+1}$, but -- in general -- to the curve $\om_{i+t}$ which follows
it after $t$ steps; this corresponds to the fact that the points
$O_{i+1},\dots,O_{i+t}$ are proximate points of $O_i$ (this is also
evident just by looking at the diagram of the branch
$O_{i+1},\dots,O_{i+t}$). Let us add that the curve $\om_i$ turns out
to be of degree $-1-t$, as follows from the fact that it represents
the neighbourhood of a point $O_i$ from which $t$ proximate points have
been removed. Taking into account the multiplicities of the curves
$\om_i$ in $\Om_i$, the composite exceptional curves
$\Om,\Om_1,\dots,\Om_s$ always turn out to be of degree $\nu=-1$ and
genus $\rho =0$, as the irreducible exceptional curve of which they are
the limit.

We finally point out that the set of irreducible exceptional curves
disconnected from one another, of which we consider the limit as
$O_1,O_2,\dots,O_s$ approach $O$, tends to a curve
 \[
 \Om+\Om_1+\cdots+\Om_s
 \]
which is composed of $\om,\om_1,\om_2,\dots,\om_s$, each counted with
multiplicity equal to the sum of the multiplicities which it has in
each $\Om_i$ containing it. We will see later on the meaning of this
remark.

 \section{The virtual characteristic series}\label{sec:1.13}
 \markright{\hfill The virtual characteristic series \quad 1.13\quad}
 We explained how one may determine the virtual characters, and in
particular the degree $n$, of a curve $C$ on a surface $F$, even if the
curve itself is an isolated curve, not contained in a system $|C|$ of
dimension $r>0$. The procedure for the computation of $n$\footnote{See
F.~Enriques, Introduzione etc., 1896: no.~15 and~16.}, leads also, for
$n>0$, to the definition of the characteristic series of $C$, independently
of whether it belongs to a larger linear system\footnote{F.~Severi,
Osservazioni sui sistemi continui di curve, {\it Atti Acc.\ Torino}, 1904.}.

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

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

 \begin{NOTE} In this form the characteristic series of a curve $C$
belonging to a continuous system implicitly occurred to G.~Castelnuovo, as
it appears from a letter to Enriques\footnote{See {\it Rendic.\ Circ.\ Mat.\
di Palermo}, dic.~25th, 1898 (n.4).}. Later on Castelnuovo himself
explicitly pointed out its meaning, in relation with the (hyperelliptic)
surface which represents the pairs of points of a curve of genus two, and
drew this example to the attention of Severi; who was therefore led to the
definition we have treated above, as he himself says in a footnote to the
first page of his paper.
 \end{NOTE}
