 \chapter{Covariant and invariant systems}\label{ch:2}
 \markboth{\qquad 2. \quad Covariant and invariant systems \hfill}{}

 \section{Jacobian curves}\label{sec:2.1}\index{Jacobian curves}
 \markright{\hfill Jacobian curves \quad 2.1\quad}
 Let us consider on the surface $F$, which we may assume to be without any
singularity in a hyperspace, an irreducible linear system $|C|$ of dimension
2, namely a net. The locus of those points of the surface that are double
points for a curve of the net, is an algebraic curve
$C_j$, which we shall call the {\em Jacobian} of the net. The $C_j$ may be
also defined as the locus of those points in which two curves of the net
touch each other; in fact the pencil determined by these curves contains a
curve having a double point at the contact point. In the plane the Jacobian
of a net of curves of order $n$ $(>1)$ is a curve of order $3n-3$; and an
$i$-tuple base point of the net has in general the multiplicity $3i-1$ for
the Jacobian; thus a simple base point $O$ is, in general, a double point
for the Jacobian, but if the curves of the net have a fixed tangent line at
$O$ then the Jacobian has a triple point there, with that line as a fixed
tangent line\footnote{See Enriques--Chisini, Ibidem, Libro~III, Vol.~II,
Chapter~I, \S5.}. These properties also hold for any net on a surface $F$,
since the determination of the multiplicity of a base point for the Jacobian
of a net $|C|$ depends only on the differential behaviour of the curves $C$
under consideration and of the surface in which they are contained. This may
be approximated at the point $O$, to an arbitrarily high order, with an
osculating paraboloid, which, in turn, can be set into a one-to-one
correspondence with a plane. For the Jacobians of nets which are contained
in a larger linear system $|C|$ the following theorem holds:

{\em The Jacobians of the nets contained in one and the same irreducible
linear system $|C|$ of dimension $r>2$ are equivalent.}

Let us take any two nets in $|C|$ and let us compare their Jacobians.
First we assume that the two nets have a pencil in common, and
therefore both belong to the same $\infty^3$ linear system\footnote{This
system is determined by the common pencil and by two curves taken in
the two nets, but not lying in the pencil.}. A generic point $P$ of the
surface $F$ is a double point for some curve $\Cbar$ of the
aforementioned $\infty^3$ system, and $\Cbar$, together with the common
pencil of the two nets, determines a net of curves $C$ whose Jacobian
passes through $P$. This way we see that the Jacobians of the
$\infty^1$ nets of curves $C$ contained in the $\infty^3$ system and
having a common pencil, form a pencil (since there is only one of these
Jacobians passing through any point of $F$); but this is a linear
pencil since it is rational, being in one-to-one correspondence with
the pencil of the nets of an $\infty^3$ linear system which have a
common pencil\footnote{This set can be identified with a pencil of
planes of $S_3$.}. Now if we consider two nets of curves in $|C|$
(which we assume of dimension $r>3$) having a common curve $\Cbar$, then
we can construct an auxiliary net containing $\Cbar$ and having a common
pencil with each one of the two nets: the Jacobian of the auxiliary net
turns out to be equivalent to the Jacobians of the two nets; so, by the
transitive property of equivalence, these Jacobians will be equivalent.

Finally, if we have in $|C|$ two nets without a common curve (the
system $|C|$ has then dimension $r>4$), we may compare their Jacobians by
constructing an auxiliary net having a common curve with each one of
the two nets. The Jacobians of the two nets will therefore be
equivalent. \qed

 \section{The Jacobian system}\label{sec:2.2}
 \markright{\hfill The Jacobian system \quad 2.2\quad}
 {From} what we have just proved it follows that:

{\em The Jacobians of the nets contained in an irreducible linear system
$|C|$ on the surface $F$ belong to one and the same complete linear
system which will be called the Jacobian linear system $|C_j|$ of $|C|$.}
For the time being, we suppose the dimension of $|C|$ to be $\ge2$.

The above definition is certainly correct when the system $|C|$ has no
base points, if it is intended that the system of Jacobians $C_j$ has to
be completed without imposing any base point. By contrast, if $|C|$ has
an $i$-tuple base point, it must be our intention to assign this as a
point of multiplicity $3i-1$ to the complete system $|C_j|$, because,
as we said, $3i-1$ is in general the multiplicity of the Jacobian of a net
contained in $|C|$.

This also holds if the base points are infinitely near. So, if for
instance $|C|$ possesses a simple base point with a fixed tangent line,
that is, two simple infinitely near base points $O,O_1$, then the Jacobians
of the nets contained in the system will actually have a triple point
at $O$ with $OO_1$ as a fixed tangent line\footnote{See
Section~\ref{sec:2.1}.}. However the system of these curves will in general
be contained in a larger system with two double base points $O$ and $O_1$,
which will be our complete Jacobian system $|C_j|$. And even if the
enlargement of the system is not actually possible, the singularity formed
by a triple point with an infinitely near simple point has to be considered
as virtually consisting of two double points $O$ and $O_1$. We supposed so
far the system $|C|$ to be irreducible, but is clear how the definition must
be extended to the case in which $|C|$ contains a fixed part $K$ (which we
assume to be simple). Here any point of $K$ appears as a base point with
a fixed tangent line for any net contained in $|C|$, and so the curve $K$ will be a component of multiplicity three of the Jacobians of all the
nets contained in $|C|$. It has to be added three times to these
Jacobians and the reducible system thus defined will be contained in
general in the larger irreducible complete system $|C_j|$.

As a matter of fact one proves in this way that $K$ appears at least
with multiplicity three in the Jacobians of the nets of $|C|$. In order
to prove that it does not actually appear with multiplicity bigger than
three, one may reduce the problem from the surface to the plane, by
replacing $F$ with a suitable osculating paraboloid at a point of the
curve $K$, or alternatively with a rational surface having a very high
contact with $F$ along the whole curve $K$\footnote{See Enriques--Chisini,
Libro~XV, Chapter~IV, \S39, Vol.~II, page~634 and following.}. One
realizes in this way that a curve $K$ which is added as a fixed component
to a net of irreducible curves, appears three times in the Jacobian of
this net; it cannot appear with multiplicity bigger than 3, unless it
is a part of the Jacobian of $|C|$.

 \begin{NOTE} One may give a direct analytic verification of the
theorem we obtained. In order to do this one needs to know how a
complete linear system is cut out on a surface $F$ (with normal
singularities) of ordinary space, namely the result that ``the adjoint
surfaces $\fie$ to $f$, that is, the surfaces passing simply through the
double curve of $f$, cut out complete linear systems on $f$''\footnote{See
the footnote to Section~\ref{sec:2.4} and Chapter~III.}. For the sake of
simplicity we will suppose that $f$ is a surface of order $n$, without
singularities, so that the complete intersections with all surfaces of a
given order
$m$ form a complete system on it.
 \end{NOTE}

If we are given a net of irreducible curves $|C|$ on $f$, which are not
complete intersections, and we add to them a fixed component $K$, then we
may always suppose that the net $|C|$ is the residual of a curve $L$, which
does not have $K$ as a part of it, so that the curves $C+L$ are complete
intersections with surfaces $\fie_i$ of order $m$. Therefore the net
$|K+C|$ is partially contained in the net represented by an equation of
the form:
 \[
 \begin{array}{c}
 \fie(x_1,x_2,x_3,x_4)\times\hbox{} \qquad\hfill\\
 \qquad\hfill [\la_1\fie_1(x_1,x_2,x_3,x_4)+\la_2\fie_2(x_1,x_2,x_3,x_4)+
 \la_3\fie_3(x_1,x_2,x_3,x_4)]=0
 \end{array}
 \]
According to our assumptions, $\fie_1,\fie_2,\fie_3$ do not vanish
simultaneously on $K$, which is instead a part of $\fie=0$, whose
order we will denote by $s$. Now we may write down the equation of the
Jacobian of the net described above, and see that the curve $\fie=0$, and in
particular $K$, which is a part of it, appears in the net as a component of
multiplicity 3. The points of the Jacobian are the contact points with $f$
of a surface of the system which cuts out the net. If we write down the
conditions for contact, we find a linear system, whose compatibility
condition is expressed by the vanishing of the determinant
 \[
 D(x_1,x_2,x_3,x_4)=
 \renewcommand{\arraystretch}{1.8}
 \left|
 \begin{array}{cccc}
 \frac{\partial (\fie\fie_1)}{\partial x_1} & \frac{\partial (\fie\fie_2)}{\partial x_1} & \frac{\partial (\fie\fie_3)}{\partial x_1} & \frac{\partial f}{\partial x_1} \\
 \frac{\partial (\fie\fie_1)}{\partial x_2} & \frac{\partial (\fie\fie_2)}{\partial x_2} & \frac{\partial (\fie\fie_3)}{\partial x_2} & \frac{\partial f}{\partial x_2} \\
 \frac{\partial (\fie\fie_1)}{\partial x_3} & \frac{\partial (\fie\fie_2)}{\partial x_3} & \frac{\partial (\fie\fie_3)}{\partial x_3} & \frac{\partial f}{\partial x_3} \\
 \frac{\partial (\fie\fie_1)}{\partial x_4} & \frac{\partial (\fie\fie_2)}{\partial x_4} & \frac{\partial (\fie\fie_3)}{\partial x_4} & \frac{\partial f}{\partial x_4}
 \end{array}
 \right|=0
 \]


Now, once the derivatives of $\fie\fie_1$, $\fie\fie_2$, $\fie\fie_3$ have
been computed, we may decompose the determinant $D$ as a sum of
determinants: if we neglect those determinants which are identically zero,
then the others have the factor $\fie^2$ in common; which we factor out to
leave us with four determinants, one of which contains the factor $\fie$
again. We prove now that the sum of the other 3 determinants (simply)
vanishes at the points of the common curve $\fie=0$. This sum can be easily
transformed in a single determinant. Let us write:
 \begin{eqnarray*}
 \lefteqn{ \Phi=
 \fie_1
 \left| \renewcommand{\arraystretch}{1.8}
 \begin{array}{cccc}
 \frac{\partial\fie}{\partial x_1} & \frac{\partial\fie_2}{\partial x_1} & \frac{\partial\fie_3}{\partial x_1} & \frac{\partial f}{\partial x_1} \\
 \frac{\partial\fie}{\partial x_2} & \frac{\partial\fie_2}{\partial x_2} & \frac{\partial\fie_3}{\partial x_2} & \frac{\partial f}{\partial x_2} \\
 \frac{\partial\fie}{\partial x_3} & \frac{\partial\fie_2}{\partial x_3} & \frac{\partial\fie_3}{\partial x_3} & \frac{\partial f}{\partial x_3} \\
 \frac{\partial\fie}{\partial x_4} & \frac{\partial\fie_2}{\partial x_4} & \frac{\partial\fie_3}{\partial x_4} & \frac{\partial f}{\partial x_4} \\
 \end{array}
 \right| + \hbox{} \kern1cm
 } \\[8pt]
 &&
 \kern1cm \hbox{} +\fie_2
 \left| \renewcommand{\arraystretch}{1.8}
 \begin{array}{cccc}
 \frac{\partial\fie_1}{\partial x_1} & \frac{\partial\fie}{\partial x_1} & \frac{\partial\fie_3}{\partial x_1} & \frac{\partial f}{\partial x_1} \\
 \frac{\partial\fie_1}{\partial x_2} & \frac{\partial\fie}{\partial x_2} & \frac{\partial\fie_3}{\partial x_2} & \frac{\partial f}{\partial x_2} \\
 \frac{\partial\fie_1}{\partial x_3} & \frac{\partial\fie}{\partial x_3} & \frac{\partial\fie_3}{\partial x_3} & \frac{\partial f}{\partial x_3} \\
 \frac{\partial\fie_1}{\partial x_4} & \frac{\partial\fie}{\partial x_4} & \frac{\partial\fie_3}{\partial x_4} & \frac{\partial f}{\partial x_4} \\
 \end{array}
 \right| +\fie_3
 \left| \renewcommand{\arraystretch}{1.8}
 \begin{array}{cccc}
 \frac{\partial\fie_1}{\partial x_1} & \frac{\partial\fie_2}{\partial x_1} & \frac{\partial\fie}{\partial x_1} & \frac{\partial f}{\partial x_1} \\
 \frac{\partial\fie_1}{\partial x_2} & \frac{\partial\fie_2}{\partial x_2} & \frac{\partial\fie}{\partial x_2} & \frac{\partial f}{\partial x_2} \\
 \frac{\partial\fie_1}{\partial x_3} & \frac{\partial\fie_2}{\partial x_3} & \frac{\partial\fie}{\partial x_3} & \frac{\partial f}{\partial x_3} \\
 \frac{\partial\fie_1}{\partial x_4} & \frac{\partial\fie_2}{\partial x_4} & \frac{\partial\fie}{\partial x_4} & \frac{\partial f}{\partial x_4} \\
 \end{array}
 \right| ;
 \end{eqnarray*}
Exchanging in the first determinant the 1st column with the 3rd and in the
second determinant the 2nd column with the 3rd, and taking into account the
signs, one finds the expansion of a 5th order determinant:
 \[
 \Phi=
 \left| \renewcommand{\arraystretch}{1.8}
 \begin{array}{ccccc}
 \fie_1 & \fie_2 & \fie_3 & 0 & 0 \\
 \frac{\partial\fie}{\partial x_1} & \frac{\partial\fie_1}{\partial x_1} & \frac{\partial\fie_2}{\partial x_1} & \frac{\partial\fie_3}{\partial x_1} & \frac{\partial f}{\partial x_1} \\
 \frac{\partial\fie}{\partial x_2} & \frac{\partial\fie_1}{\partial x_2} & \frac{\partial\fie_2}{\partial x_2} & \frac{\partial\fie_3}{\partial x_2} & \frac{\partial f}{\partial x_2} \\
 \frac{\partial\fie}{\partial x_3} & \frac{\partial\fie_1}{\partial x_3} & \frac{\partial\fie_2}{\partial x_3} & \frac{\partial\fie_3}{\partial x_3} & \frac{\partial f}{\partial x_3} \\
 \frac{\partial\fie}{\partial x_4} & \frac{\partial\fie_1}{\partial x_4} & \frac{\partial\fie_2}{\partial x_4} & \frac{\partial\fie_3}{\partial x_4} & \frac{\partial f}{\partial x_4} \\
 \end{array}
 \right|.
 \]
Subtracting from the 1st row multiplied by $m$ the 2nd row multiplied by
$x_1$, the 3rd row multiplied by $x_2$, the 4th row multiplied by $x_3$, the
5th row multiplied by $x_4$ and taking into account Euler's theorem, we
may replace the 1st row with
 \[
 0, 0, 0,-s\fie,-nf
 \]
where $n,m,s$ denote the orders of $f$, the $\fie_i$ and $\fie$ respectively.

It is clear that the determinant in which this line appears vanishes
when $\fie=0$ and $f=0$ simultaneously occur, since, when these terms are
zero, the first row identically vanishes. \qed

Let us add that $K$ cannot appear with multiplicity bigger than three in
the Jacobian of the net; or at least, if this happens, then $K$ is a part
of the Jacobian of the net from which the fixed part $K$ has been removed.

In fact a higher multiplicity for $K$ should be a consequence of a
contact between the surfaces $f$ and $\Phi$ along the curve $K$. Now let us
suppose that the tangent planes to $f$ and $\Phi$ coincide at a point $P$ of
this curve. We should then have
 \[
 \rho \frac{\partial f}{\partial x_i}=
 \frac{\partial \Phi}{\partial x_i} \quad\mbox{for $i=1, 2, 3, 4.$}
 \]
Let us compute $\partial \Phi/\partial x_i$. We may differentiate by
columns; in this way the derivative will be expressed as a sum of
determinants obtained by replacing a column with its derivatives with
respect to $x_i$. It is clear that, since we are at a point where $\fie=0$,
$f=0$, the determinants that are obtained by differentiating the first three
columns are zero. Therefore we are left with:
 \begin{eqnarray*}
 \lefteqn{ m\left(\frac{\partial \Phi}{\partial x_i}\right)_P=
 } \\[8pt]
 &&
 \left| \renewcommand{\arraystretch}{1.8}
 \begin{array}{ccccc}
 0 & 0 & 0 & -s\frac{\partial \fie}{\partial x_i} & 0 \\
\frac{\partial\fie_1}{\partial x_1} & \frac{\partial\fie_2}{\partial x_1} & \frac{\partial\fie_3}{\partial x_1} & \frac{\partial^2\fie}{\partial x_1\partial x_i} & \frac{\partial f}{\partial x_1} \\
\frac{\partial\fie_1}{\partial x_2} & \frac{\partial\fie_2}{\partial x_2} & \frac{\partial\fie_3}{\partial x_2} & \frac{\partial^2\fie}{\partial x_2\partial x_i} & \frac{\partial f}{\partial x_2} \\
\frac{\partial\fie_1}{\partial x_3} & \frac{\partial\fie_2}{\partial x_3} & \frac{\partial\fie_3}{\partial x_3} & \frac{\partial^2\fie}{\partial x_3\partial x_i} & \frac{\partial f}{\partial x_3} \\
\frac{\partial\fie_1}{\partial x_4} & \frac{\partial\fie_2}{\partial x_4} & \frac{\partial\fie_3}{\partial x_4} & \frac{\partial^2\fie}{\partial x_4\partial x_i} & \frac{\partial f}{\partial x_4} \\
 \end{array}
 \right|
 + \left| \renewcommand{\arraystretch}{1.8}
 \begin{array}{ccccc}
 0 & 0 & 0 & -s \frac{\partial \fie}{\partial x_i} & 0 \\
\frac{\partial\fie_1}{\partial x_1} & \frac{\partial\fie_2}{\partial x_1} & \frac{\partial\fie_3}{\partial x_1} & \frac{\partial^2\fie}{\partial x_1\partial x_i} & \frac{\partial f}{\partial x_1} \\
\frac{\partial\fie_1}{\partial x_2} & \frac{\partial\fie_2}{\partial x_2} & \frac{\partial\fie_3}{\partial x_2} & \frac{\partial^2\fie}{\partial x_2\partial x_i} & \frac{\partial f}{\partial x_2} \\
\frac{\partial\fie_1}{\partial x_3} & \frac{\partial\fie_2}{\partial x_3} & \frac{\partial\fie_3}{\partial x_3} & \frac{\partial^2\fie}{\partial x_3\partial x_i} & \frac{\partial f}{\partial x_3} \\
\frac{\partial\fie_1}{\partial x_4} & \frac{\partial\fie_2}{\partial x_4} & \frac{\partial\fie_3}{\partial x_4} & \frac{\partial^2\fie}{\partial x_4\partial x_i} & \frac{\partial f}{\partial x_4} \\
 \end{array}
 \right|.
 \end{eqnarray*}
Expanding the two determinants with respect to the first row we have:
 \[
 m\left(\frac{\partial \Phi}{\partial x_i}\right)_P=\rho m \left(\frac{\partial f}{\partial x_i}\right)_P=
 s\frac{\partial \fie}{\partial x_i}
 \left| \renewcommand{\arraystretch}{1.8}
 \begin{array}{ccccc}
 \frac{\partial\fie_1}{\partial x_1} & \frac{\partial\fie_2}{\partial x_1} & \frac{\partial\fie_3}{\partial x_1} & \frac{\partial f}{\partial x_1} \\
 \frac{\partial\fie_1}{\partial x_2} & \frac{\partial\fie_2}{\partial x_2} & \frac{\partial\fie_3}{\partial x_2} & \frac{\partial f}{\partial x_2} \\
 \frac{\partial\fie_1}{\partial x_3} & \frac{\partial\fie_2}{\partial x_3} & \frac{\partial\fie_3}{\partial x_3} & \frac{\partial f}{\partial x_3} \\
 \frac{\partial\fie_1}{\partial x_4} & \frac{\partial\fie_2}{\partial x_4} & \frac{\partial\fie_3}{\partial x_4} & \frac{\partial f}{\partial x_4}
 \end{array}
 \right| + \hbox{}
 \]
 \[
-n\frac{\partial f}{\partial x_i}
 \left| \renewcommand{\arraystretch}{1.8}
 \begin{array}{ccccc}
 \frac{\partial\fie_1}{\partial x_1} & \frac{\partial\fie_2}{\partial x_1} & \frac{\partial\fie_3}{\partial x_1} & \frac{\partial \fie}{\partial x_1} \\
 \frac{\partial\fie_1}{\partial x_2} & \frac{\partial\fie_2}{\partial x_2} & \frac{\partial\fie_3}{\partial x_2} & \frac{\partial \fie}{\partial x_2} \\
 \frac{\partial\fie_1}{\partial x_3} & \frac{\partial\fie_2}{\partial x_3} & \frac{\partial\fie_3}{\partial x_3} & \frac{\partial \fie}{\partial x_3} \\
 \frac{\partial\fie_1}{\partial x_4} & \frac{\partial\fie_2}{\partial x_4} & \frac{\partial\fie_3}{\partial x_4} & \frac{\partial \fie}{\partial x_4}
 \end{array}
 \right|
 \]
for $i=1,2,3,4$. If, for brevity, we denote by $A$ the first determinant and
by $B$ the second, this equation can be written as
 \[
 \rho m \frac{\partial f}{\partial x_i}=s \frac{\partial \fie}{\partial x_i} A
 - n \frac{\partial f}{\partial x_i} B,
 \quad\hbox{for $i=1,2,3,4$,}
 \]
which a priori gives two possibilities:

 \begin{enumerate}
 \renewcommand{\labelenumi}{(\arabic{enumi})}
 \item either $\partial f/\partial x_i=\la\partial \fie/\partial x_i$, namely
the surfaces $f=0$ and $\fie=0$ are tangent, contrary to our assumptions; or
 \item $A=0$ and then, as one easily deduces from the form of the determinant
$A$, the point $P$ lies on the Jacobian of the net.
 \end{enumerate}

The algebraic procedure developed here can be explained with some simple
geometric argument which we only sketch, referring to the simplest case in
which the surface $f$ of order $n$ of ordinary space has no singularities,
$\fie_1,\fie_2,\fie_3$ are three linearly independent surfaces of order
$m>n$, and $\fie$ is a surface of order $s$.

Let us remark that if we take the surface $f$ together with a surface
$\Psi$ of order $m-n$, then it can be linearly combined with
$\fie_1,\fie_2,\fie_3$; one may then construct the Jacobian surface of the
$\infty^3$ linear system thus determined, which will be a surface $D$ of
order $(4m-4)$\footnote{This is an immediate extension of the notion of
the Jacobian of a net of plane curves. See Enriques--Chisini, Ibidem,
Libro~III, Chapter~I, \S5, Vol.~II, page~31.}. $D$ cuts out on $f$ a curve
of order $(4m-4)n$ which consists of the Jacobian of the net of curves
$|C|$ cut out by
 \[
 \la_1\fie_1+\la_2 \fie_2+\la_3 \fie_3=0
 \]
and of the intersection curve $L$ of $f$ and $\Psi$; since the latter has
order $n(m-n)$, then the Jacobian of the net $|C|$ turns out to be of order
$n(3m+n-4)$.

Let us add to the net
 \[
 \la_1\fie_1+\la_2 \fie_2+\la_3 \fie_3=0
 \]
a surface $\fie$ of order $s$, cutting out a curve $K$ of order $ns$ on $f$;
if we add $\fie$ also to $f$, then we have an $\infty^3$ linear system of
surfaces of order $m+s$ in which $\fie$ appears as a fixed part. Therefore
$\fie$ appears with multiplicity four in the Jacobian of the system; its
intersection with $f$ (of order $ns$), counted with multiplicity one is a
component of the intersection curve of $f$ with $\Psi+\fie$; so it appears
three times in the Jacobian curve of the net $|K+C|$, whose order is now
 \[
 n[3(m+s)+n-4].
 \]

 \section{The fundamental theorem}\label{sec:2.3}
 \markright{\hfill The fundamental theorem \quad 2.3\quad}
 The above remarks yield the following result.

 \begin{Theorem}[FUNDAMENTAL THEOREM] Let $|C|$ and $|L|$ be two irreducible
linear systems on the surface $F$, which we always assume to be virtually
free of base points and of dimension at least 2, then the Jacobian system
$|C+L|_j$ of their sum can be expressed as
 \[
 |C+L|_j=|C_j+3L|=|3C+L_j|.
 \]
In fact the systems $|C|+L$ with a fixed curve $L$, and $C+|L|$ with a fixed
curve $C$, are contained in the system $|C+L|$; their complete Jacobian
systems are
 \[
 |C_j+3L| \quad\hbox{and}\quad |L_j+3C|
 \]
respectively, and they both coincide with the system $|C+L|_j$.
 \end{Theorem}

According to the fundamental theorem one may now define the complete
Jacobian system of a linear system $|C|$ whose dimension is less than
two, namely a pencil or an isolated curve $C$. Indeed it suffices to add
to $C$ an auxiliary, at least $\infty^2$, system $|L|$, and set
 \[
 |C_j|=|3C+L_j-3L|.
 \]
The fundamental theorem can be thus extended to the Jacobian system of
any linear system, however reducible. Some caution is needed only in
the case when one considers linear systems $|C|$ and $|L|$ with assigned
base points on the surface $F$. If $|C|$ has an $i$-tuple base point $O$ which
is not a base point for $|L|$, and $|L|$ has a $s$-tuple base point $O'$ which
is not a base point for $|C|$, then the Jacobian system $|C+L|_j$ has
(virtual) multiplicity $3i-1$ at $O$, and $3s-1$ at $O'$; by contrast $O$ has
multiplicity $3i$ for $|3C+L_j|$ and $O'$ multiplicity $3s$ for $|3L+C_j|$.
Therefore the equivalence between the two systems $|C_j+3L|$ and $|3C+L_j|$
only holds if the neighbourhoods of the points $O$ and $O'$ are respectively
added to the two systems. Namely one has
 \[
 |3C+L_j+O|=|3L+C_j+O'|
 \]
According to our convention, the meaning of this equivalence is the
following: if one makes a transformation of the surface $F$, changing $O$ and $O'$ into exceptional curves, then these curves must be respectively
added to the transformed systems of $|C_j+3L|$ and $|L_j+3C|$.

Now we go back to the case we will generally consider in what follows,
that is, we deal with {\em complete systems, virtually free of base points}
on the surface $F$. The equivalence $|C_j+3L|=|L_j+3C|$ may be reduced to
 \[
 |C_j-3C|=|L_j-3L|=|K|,
 \]
which in other words means that: {\em if the Jacobian $|C_j|$ of a system
$|C|$ contains the triple of $|C|$, then the Jacobian $|L_j|$ of any other
system $|L|$ also contains the triple of $|L|$, and the system $|K|$,
obtained as the difference of $|C_j|$ and $|3C|$, is independent of the
system $|C|$ one starts from.} This system $|K|$ is defined to be the
{\em (impure) canonical system} of the surface $F$.

But, even if $|C_j|$ does not contain $|3C|$, that is, if the difference
system $|C_j-3C|$ is only virtual, the symbolic equivalence written above
is still meaningful. The meaning of the equivalence between virtual
systems is nothing but a different way of writing the relation
expressing the fundamental theorem on Jacobians.

It is also convenient to give a simpler form of the above-mentioned
equivalence of virtual systems. Therefore we define the adjoint linear
system $|C'|$ to $|C|$ by means of the relation
 \[
 |C'|=|C_j-2C|,
 \]
and we notice that, for a system $|C|$ on a surface $F$, $|C'|$ (unlike $|K|$)
does in general effectively exist: this is already the case for the
linear systems of curves of order $m$ and genus $\pi>0$ in the plane, since
then the $C'$ are the curves of order $m-3$ passing $i-1$ times through each
$i$-tuple base point of $|C|$; and, as we will see later on, the same holds
for all linear systems whose genus exceeds a certain bound.

Once the adjoint system has been introduced, the canonical system may
be defined as:
 \[
 |K|=|C'-C|=|L'-L|=\cdots,
 \]
and the meaning of this equivalence of effective or virtual systems is
that
 \[
 |C'+L|=|L'+C|
 \]
This is the {\em fundamental relation for the adjoint systems} (without base
points), which in turn is just a simple translation of the relation for
the Jacobian systems given above.

 \section{Canonical curves}\label{sec:2.4}
 \markright{\hfill Canonical curves \quad 2.4\quad}
 As we saw, starting from a given linear system $|C|$ on a surface $F$, one
constructs the adjoint linear system $|C'|$ and then the canonical system
$|K|=|C'-C|$, which may be either effective or virtual; we also realized
the latter system to be independent of the choice of the system $|C|$,
which is supposed to be base point free on $F$. Even when $|C|$ has an
$i$-tuple base point $O$, simple for $F$, the system $|C'-C|$ may still be
considered equal to $|K|$, except for the infinitesimal curve
corresponding to the neighbourhood of $O$, which must be added to the $K$,
since $O$ is an $(i-1)$-tuple point for $|C'|$. The independence of $|K|$ of
the choice of $|C|$ means that the canonical system $|K|$ is invariant for
birational transformations of the surface.

In order to explain this better, let us consider two surfaces $F$ and
$\Phi$, of orders $n$ and $m$ respectively, with normal singularities in
ordinary space; we suppose that there is a birational correspondence between
them, and, to begin with, we make the hypothesis that this correspondence
has no fundamental points on either of the two surfaces. The plane sections
of the surface $F$ form a linear system $|C|$ whose transformed system has
no base points on $\Phi$, and the plane sections $L$ of $\Phi$ likewise form
a linear system on $\Phi$ whose transformed system has no base points on
$F$. Now one may construct the canonical system $|K|$ on $F$ either starting
from the plane sections system $|C|$, or starting from the system $|L|$, and
one has
 \[
 |K|=|C'-C|=|L'-L|
 \]
The same system may be constructed on $\Phi$, either starting from the
plane sections system $|L|$, or starting from the system $|C|$, and one has
again
 \[
 |K|=|L'-L|=|C'-C|.
 \]
Therefore the curves of the canonical system $|K|$, which are constructed on
$F$ starting from the plane sections system, are transformed into the curves
of the canonical system of $\Phi$, which in turn is constructed starting
from the plane sections system $|L|$. We may now understand the meaning of
this construction.

In order to construct the canonical system on $F$, one has first to
construct the Jacobian system of $|C|$. To do so we notice that a star of
plane sections of $F$, with its centre at a point $O$ of the space, is a net
whose Jacobian is the intersection of $F$ with the polar surface of order
$n-1$ of $O$, away from the nodal curve.

Thus the Jacobians of the nets contained in the plane sections system $|C|$
form an $\infty^3$ linear system which is cut out on $F$ by the polars, away
from the nodal curve; but this system has the cuspidal points of the
aforementioned nodal curve as simple base points. It is easy to see that
these points are actually to be considered as ``simple points'' of the
surface, and, according to our conventions, they are to be removed in order
to obtain the complete Jacobian system $|C_j|$. Indeed if we consider a
transformation of $F$ into a surface on which the nodal curve is represented
by the curve of neutral pairs with respect to the system $|C|$, then the
cuspidal points give on such a curve some contact base points for a net of
$C$, and double base points for a pencil of $C$: so these points are simple
points for the Jacobians of all the nets contained in the $\infty^3$ system
of the $C$.

Hence the complete Jacobian system $|C_j|$, which is virtually free from
base points, contains the whole system of curves cut out on $F$ by the
(adjoint) surfaces of order $n-1$ passing through its double curve. If we
now subtract 3 planes from these surfaces, then we have that the
surfaces of order $n-4$ passing through the double curve of $F$ (if they
exist) cut out canonical curves on $F$.

In a similar way the adjoint surfaces of order $m-4$, passing through
the double curve of $\Phi$, cut out canonical curves on $\Phi$.

Therefore the birational invariance of the canonical system means that
``the section curves of $F$ with the adjoint surfaces of order $n-4$, are
sent, with a birational transformation, to curves which are equivalent
to the sections of $\Phi$ with its adjoint surfaces of order $m-4$''.

This theorem becomes even more significant if we now take for granted
a result which we will prove in the next chapter, namely that ``the
adjoint surfaces of any order, and in particular those of order $n-4$,
cut out a complete linear system on the surface $F$ of order $n$'', and
therefore, in this case, the complete canonical system $|K|$.

So we have: the (canonical) curves, cut out on the surface $F$ (away from the
double curve) by its adjoint surfaces of order $n-4$ are sent, in a
birational transformation, to the sections of $\Phi$ with its adjoint
surfaces of order $m-4$.

We actually supposed so far that the birational correspondence between $F$
and $\Phi$ has no exceptions, namely that the system $|L|$ has no base point
on $F$, and $|C|$ has no base point on $\Phi$. But if, for instance, $|L|$
has an $i$-tuple base point $O$ on $F$, then the theorem still holds with
this change: the exceptional curve $\om$ corresponding to the point $O$ of
$F$ has to be added to the transformed curves of $|K|$, since it appears as a
fixed part of the impure canonical system formed by the sections of $\Phi$
with its adjoint surfaces of order $m-4$.

 \begin{NOTE} We supposed $F$ and $\Phi$ to have only normal singularities
(that is, a double curve with triple points); but this restriction may be
removed, provided that one defines the adjoint surfaces to a given one,
according to the higher singularities it might possess. It suffices to point
out here that an ordinary $i$-tuple curve of $F$ has to be of multiplicity
$i-1$ for the adjoint surfaces, and a conic point of order $s$, away from the
multiple curve, has to be in general of multiplicity $s-2$ for
the adjoint surfaces. We shall come back later to this subject.
 \end{NOTE}

 \section{Properties of canonical curves}\label{sec:2.5}
 \markright{\hfill Properties of canonical curves \quad 2.5\quad}
 The canonical curves on a surface $F$ (more precisely the impure
canonical curves, cut out on a surface $F$ of order $n$, away from the double
curve, by the adjoint surfaces of order $n-4$) enjoy a property which we
now wish to point out explicitly. As we shall see in the next chapter,
this property, under mild assumptions, does in fact characterize the
canonical curves.

Let us consider on $F$ an irreducible, base point free, linear system $|C|$, of dimension
$r\ge2$, genus $\pi$ and degree $n$. The Jacobian of a net of curves $C$
cuts out on a curve $C$ the group consisting of the $2n+2\pi-2$ double
points of the characteristic $g_n^1$, and this belongs to the sum of the
canonical series with the double of the $g_n^1$. If one subtracts 2C from
$|C_j|$, then one has curves $C'$ adjoint to $|C|$, and they cut out on $C$
canonical groups. If
$C$ is subtracted once more, then one finds that the canonical curves cut
out on $C$ residual groups to the characteristic series.

Hence we may state:

{\em The adjoint curves cut out on the curves of the system $|C|$ canonical
divisors. The canonical curves $K$ cut out on the curves $|C|$ residual
groups to the characteristic series.}

Whence it follows that the adjoint curves $C'$ cut out $2\pi-2$ points on a
curve $C$, of virtual genus $\pi$, and the canonical curves $K$, either
effective or virtual, cut out
 \[
 i=2\pi-2-n
 \]
points on $C$.

This equality holds for any curve, whether irreducible, reducible, or
even isolated, provided one considers its virtual characters, which are
defined by assuming that $C$ has no assigned base points on $F$. In fact
let us add to $C$ a suitable curve $L$, of genus $\rho$ and degree $\nu$ which
meets $C$ at $s$ points, such that the complete linear system $|C+L|$ is
irreducible of dimension $>2$; it suffices to take as $|L|$ the system cut
out by those surfaces of large enough order which pass through $C$. Now
the curves $C'+L$ belonging to the system $|(C+L)'|$ cut out
 \[
 2(\pi+\rho +s-1)-2
 \]
points on a $C+L$; by removing their intersections with $L$, whose number
is $2\rho-2+s$, and then the $s$ intersections of $C$ with $L$, one obtains
 \[
 (CC')=2\pi-2.
 \]
Likewise the virtual number of intersections of $C$ and $K$ is always
$(CK)=2\pi-2-n$.

 \begin{NOTE} We add that the group cut out on an irreducible curve $C$ on
$F$, without multiple points, by an adjoint curve $C'$ is always a canonical
group, and the group cut out by a $K$ is a residual group to the
characteristic series (either effective or virtual). It suffices to prove
the assertion for a curve $C$ which is a part of a composite curve $C+L$
belonging to an irreducible net. Going back to the previous argument, one
finds that a curve of
 \[
 |C_j|=|(C+L)_j-3L|
 \]
cuts out on $C$ a group which belongs to the canonical series summed with
the double of the virtual characteristic series of $C$ itself.
 \end{NOTE}

 \section{Geometric genus and linear genus}\label{sec:2.6}
 \markright{\hfill Geometric genus and linear genus \quad 2.6\quad}
 Since the canonical system is invariant under birational
transformations of the surface, its characters likewise have an
invariant meaning with respect to these transformations. However we
need to distinguish between:
 \begin{enumerate}
 \renewcommand{\labelenumi}{(\arabic{enumi})}
 \item absolute invariants, which do not change for any transformation, and
 \item relative invariants which are preserved under birational
transformations without exceptions, but are not preserved under
transformations having fundamental points, that change a (simple) point
of one surface into an exceptional curve of the other.
 \end{enumerate}

Indeed the dimension of the canonical system, or rather the number of
linearly independent canonical curves, gives us an absolute invariant,
called the geometric genus ($p$ or $p_g$), or simply the ``genus'', of the
surface.

Since the complete canonical system is cut out on a surface $F$ of order
$n$ of ordinary space by the adjoint surfaces $\fie_{n-4}$ of order $n-4$
(see Chapter~III), then the genus of a general surface of order $n$ is
 \[
 p=\frac{(n-3)(n-2)(n-1)}{6}
 \]
which is the number of linearly independent $\fie_{n-4}$.

Therefore the genus of the plane, of the quadrics and of the cubic
surfaces is
 \[
 p=0.
 \]
This can be directly verified for the plane by observing that the
complete system $|C|$ of plane curves of order $n\ge3$ contains the
adjoint system $|C'|$ of the curves of order $n-3$; thus, since $|C|$ is
larger than $|C'|$, then $|C'|$ cannot contain $|C|$: so $C'-C$ is not
effective, which means $p=0$.

The vanishing of the genus of the plane yields that all rational
surfaces (which can be put in one-to-one birational correspondence with
the plane) likewise have genus zero. Among these surfaces we find the
above-mentioned quadrics and the cubics (at least those which are not
cones).

Another family of surfaces having (geometric) genus zero are the
scrolls, and therefore also all the surfaces birational to them. This
assertion can be justified by remarking that a generator of a scroll $F$
cannot have an adjoint system, because this system should cut out on
the generator the canonical series. In general the adjoint $|C'|$ of a
linear system $|C|$ on the scroll cannot contain $|C|$ because, if the
$C$ cut out $n$ points on the generators, then the $C'$ cut out $n-2$ points
on them (see Chapter~III, Section~\ref{sec:3.3}).

Going back to surfaces of any kind of genus $p\ge0$, the genus of the
canonical curves (either effective or virtual) gives a relative
invariant, called the linear genus $p^{(1)}$ or also the relative linear
genus, which drops by one for any point of the surface $F$ that is
changed into an exceptional curve, and conversely increases by one for
any exceptional curve that is sent to a point. In fact we saw that,
when a point $O$ of the surface is changed into an exceptional curve $\om$ of $\Phi$, this curve $\om$ has to be added to the
transformed $\Kbar$ of the canonical curves. Now such a curve is not connected
with the above mentioned $K$, that is, its virtual number of intersections with
them is zero, since the point $O$ is -- at least virtually -- not a base point of
$|K|$. In fact the system $|K|$ has no assigned base point on $F$. Therefore
by adding $\om$, of genus 0 and degree $-1$, to the $\Kbar$, the genus of these
drops by one, since $\om$ is disconnected from them.

It is apparent here that it is more appropriate, at least for surfaces
of genus $p>0$ where the canonical system is effective, to consider,
together with the relative linear genus $p^{(1)}$, the genus of the pure
canonical system, from which all the exceptional fixed components have
been removed. This is in turn the genus of the canonical curves on a
transformed surface without exceptional curves, as we shall see more
precisely in Section~\ref{sec:2.7}. We find in this way an absolute
invariant, called the absolute linear genus of the given surface. Its value
is
 \[
 \pbar^{(1)}=p^{(1)}+\si,
 \]
where $\si$ denotes the number of the exceptional curves belonging to the
surface.

More precisely, we intend to show in the next section that, at least
for $p>0$, these curves are always exceptional curves of the first kind,
disconnected one from another, and we will examine in particular the
case of the reducible exceptional curves.

As the genus of the canonical system, its degree also gives a relative
invariant of the surface, which we may denote by $p^{(2)}$. When, by a
transformation of the surface $F$, an exceptional curve $\om$ of degree $-1$
disconnected from the $K$ is added to the $\Kbar$, $p^{(2)}$ also drops by 1.
And here too, together with the degree of the impure canonical system,
we may define the degree of the pure canonical system; this is an
absolute invariant, which is defined, at least for surfaces with genus $p>0$. But $p^{(2)}$, compared with the linear genus, gives no new
invariant character. Indeed the following relation holds between the
genus and the degree of the impure canonical system
 \[
 p^{(2)}=p^{(1)}-1.
 \]
In order to prove this it suffices to remark that the adjoint system to
the canonical system $|K|$ is
 \[
 |K'|=|2K|.
 \]
Therefore the number of intersections of $K$ with $2K$ is $2p^{(1)}-2$, and
then
 \[
 (KK)=p^{(1)}-1.
 \]
In the case $p>0$ (and in any other case in which the absolute
characters we deal with may be defined, that is, when the surface can be
transformed into another one which has no exceptional curves) the above
relation between the genus $p^{(1)}+\si$ and the degree $p^{(2)}+\si$ of
the pure canonical system still holds.

 \begin{NOTE}
 We will see later on that the only surfaces for which the absolute linear
genus cannot be defined (by considering a transformed surface without
exceptional curves) are those belonging to the family of ruled surfaces
(rational or not). Nevertheless for these too one arrives at the
construction of an absolute invariant, which is given -- according to
Castelnuovo\footnote{Sul genere lineare di una superficie e sulla
classificazione a cui esso da' luogo, {\it Rendic.\ Acc.\ Lincei}, Vol.~VI,
1897. {\it Memorie scelte}; page~443.} -- by the maximum of the relative
linear genus with respect to all possible transformations of the given
surface.
 \end{NOTE}

It is convenient here to compute the linear genus of the plane and of
the irrational scrolls of genus $p$ $(>0)$, for which that maximum value is
achieved.

The linear genus of the plane may be computed by considering a linear
system $|C|$ of plane quartics, of genus $\pi=3$ and degree $n=16$: the genus
$\pi'=0$ of the adjoint system, which is the net of lines, is expressed
by the formula
 \[
 \pi'=\pi+p^{(1)}+2\pi-2-n-1.
 \]
Setting here $\pi=3$, $n=16$, $\pi'=0$, one finds that the linear genus of
the plane is
 \[
 p^{(1)}=10.
 \]
Let us consider now a scroll of order $m$ and genus $p>0$. The double of
the plane sections system (which is cut out by quadrics) has genus
 \[
 \pi=2p+m-1
 \]
and degree
 \[
 n=4m.
 \]
The system adjoint to it is formed by groups of $m+2p-2$
lines\footnote{This may be verified in various ways; see Chapter~III.},
and therefore has genus
 \[
 \pi'=-(m+2p-2)+1=-m-2p+3.
 \]
 Therefore from the formula
 \[
 \pi'=\pi+p^{(1)}+2\pi-2-n-1,
 \]
one deduces that: the linear genus of the scrolls of genus $p>0$ is
 \[
 p^{(1)}=-8(p-1)+1.
 \]

 \section{The exceptional curves as fixed parts of the canonical
system}\label{sec:2.7}
 \markright{\hfill The exceptional curves as fixed parts of the canonical
system \quad 2.7\quad}
 It is useful to make more precise what we said above
about the exceptional curves on a surface of genus $p>0$, as belonging, as
fixed parts, to the canonical system $|K|$.

The fact that an exceptional curve $\om$ of the first kind on a surface $F$
belongs, as a fixed component, to the system $|K|$ follows directly from the
fact that it has genus $\rho=0$ and degree $\nu=-1$. In fact $\om$ has to be
cut by the $K$ at
 \[
 2\rho -2-\nu=-1
 \]
points; since this (virtual) number is negative, $\om$ has to be a part of
each $K$. But we can say more: on the surface $F$ of genus $p>0$ we cannot
have two irreducible exceptional curves of the first kind connected one to
the other, namely with some point in common. In fact, assume such a pair of
curves $\om$ and $\om'$, of genus 0 and degree $-1$, with $i$ $(\ge1)$
points in common, exists; they appear in all canonical curves $K$ as fixed
parts to be counted a certain number of times. But, if the former has to be
counted $r$ times and the latter $s$ times, the number of intersections of
$\om$ and $\om'$ with the residual curves is respectively
 \begin{eqnarray*}
 (K\om)-r (\om\om)-s (\om\om') &=& -1+r-si\\
 (K\om')-s (\om'\om')-r (\om\om') &=& -1+s-ri
 \end{eqnarray*}
Now, since the sum of these two numbers is negative, at least one of
them is negative, thus the corresponding curve, either $\om$ or $\om'$,
should appear in the $K$ with higher multiplicity, contrary to the
assumption.

With the same reasoning one proves that a surface $F$ of genus $p>0$,
cannot contain an exceptional curve $\om$ of the second kind. Indeed let
us consider a transform $\Phi$ of $F$, on which $\om$ becomes a curve of
degree $\nu\ge0$, but again with genus $\rho=0$. On such a surface the
virtual number of points $\om$ has in common with a canonical curve is
 \[
 2\rho -2-\nu=-2-\nu
 \]
which is negative; $\om$ is therefore a fixed part of the $K$; but its
multiplicity should be infinite, which gives a contradiction. Suppose
indeed its multi\-plicity is $s$; then the number of intersections with the
residual curves is
 \[
 (K\om)-s (\om\om)=-2-(s+1)\nu,
 \]
which is always a negative number.

We may summarize the results we obtained by stating the theorem:

{\em On a surface of genus $p>0$ one may have at most a finite number of
irreducible exceptional curves of the first kind, each one disconnected
from the others; there can never be exceptional curves of the second
kind.}

We notice that if one has only irreducible exceptional curves of the
first kind, then each of them, $\om$, appears with multiplicity one as a
component of the canonical system and is a fundamental curve for the
residual system $|K-\om|$. Anyhow it cannot be a priori excluded that $\om$
is again a fixed component of $|K-\om|$, thus appearing twice as a fixed part
of $|K|$. This means that, by contracting $\om$ to a point, this point turns
out to be a base point for all curves of the transformed canonical system.
But since this is not an assigned base point, it has to be considered as
virtually nonexistent; so on the given surface, $\om$ has still to be
considered as a part of the pure canonical curves. Indeed these curves are
obtained by subtracting only once the exceptional curves which are parts of
the impure canonical system.

\paragraph{NOTE I} The case in which one has reducible exceptional curves,
which arise from infinitely near base points of a transformed surface,
deserves particular attention. Here the multiplicity of the exceptional
curves as fixed parts of the impure canonical system may be determined by
using a principle of continuity; we know in fact that this case arises as a
limit of the case when the base points of a transforming system become
indefinitely close to each other. Taking into account the remark which
concludes Section~\ref{sec:1.12} of the preceding Chapter~1, we point out
that it has the following meaning: Every component $\om_i$ of a reducible
exceptional curve
 \[
 \Om=\om+\om_1+h_2\om_2+\cdots+h_i\om_i+\cdots+h_s\om_s
 \]
appears as a component of the impure canonical system with multiplicity
equal to the sum of the multiplicities with which it appears in the
exceptional curves containing it\footnote{A. Franchetta, Sulle curve
eccezionali riducibili di prima specie. Boll.\ Unione Mat.\ It., Bologna,
1940.}.

Suppose we have, for instance, an exceptional curve consisting of
three components, each one connected with the successive, corresponding
to three infinitely near points on a linear branch; then these
components give rise to three reducible exceptional curves
 \[
 \om+\om_1+\om_2,\quad\om_1+\om_2,\quad\om_2.
 \]
Thus $\om$ appears as a simple fixed component of the canonical system $|K|$,
$\om_1$ appears twice, $\om_1$ three times.

If by contrast the three above-mentioned components arise from
proximate points on a cuspidal branch (so that $\om$ and $\om_1$ are
connected to $\om_2$, but are not connected one to the other), one has
three exceptional curves
 \[
 \om+\om_1+2\om_2,\quad\om_1+\om_2,\quad\om_2.
 \]
So $\om$ again appears as a simple fixed part of the canonical system $|K|$,
$\om_1$ appear twice, and $\om_2$ as a 4-tuple curve.

The theorem to the effect that a surface $F$ of genus $p>0$ in general
possesses only irreducible exceptional curves of the 1st kind disconnected
one from another, clearly yields that $F$ may be transformed into another
surface which has no exceptional curves at all, as was pointed out by
Enriques in his {\it Introduction} of 1896. But a rigorous proof of this
fact, under weaker assumptions which include (as we already pointed out
above) all surfaces which do not belong to the family of ruled surfaces, was
achieved in the simplest way by Castelnuovo--Enriques (1901), by using the
so-called Riemann--Roch theorem for surfaces. Thus we defer this development
to a later time. We only point out now that the case in which one has an
exceptional curve composed of $s$ parts, gives no problem. In fact one of its
irreducible components is an exceptional curve, which can be eliminated, so
that one may argue by induction.

\paragraph{NOTE II} Let us explicitly point a result which is implicit in the
characteristic property of the canonical curves of cutting out on a
curve $C$ of degree $n$ and genus $\pi$ a group of $2\pi-2-n$ points. In view
of this property the difference
 \[
 2\pi-2-n
 \]
(which in general is positive, but may also be, at least virtually,
negative) appears to be an additive character for the system $|C|$ of
curves on the surface; and this additive property might also be
directly deduced from the formulas which give the degree and the genus
of reducible curves. Now we may claim that: {\em On a surface of genus $p>0$
an irreducible curve $C$ of degree $n$ and genus $\pi$ such that
 \[
 n>2\pi-2
 \]
may only be an exceptional curve of the first kind:
 \[
 \pi=0,\quad n=-1.
 \]}
In fact, if $2\pi-2-n$ is smaller than zero, then $C$ has to be a component
of the canonical curves $K$; but, if we assume that its multiplicity as a
component is $s$, then the number of intersections of $C$ with the residual
curves is
 \[
 2\pi-2-(s+1)n
 \]
and therefore it is always negative, unless $n=-1$ and, consequently
$\pi=0$.

This theorem can be extended to the case of reducible curves, by remarking
that a sum of characters $n_i-(2\pi_i-2)$ which are negative or zero can
never give a positive number. But then a rather subtle analysis is needed in
the case of reducible exceptional curves. One may avoid it by considering
some transformed surface without exceptional curves. Here is the result:
{\em On a surface of genus $p>0$ there are no curves of genus $\pi$ and
degree $n>2\pi-2$ (however reducible), except for those which can be
obtained by adding to curves with $n\le2\pi-2$ some exceptional curves
disjoint from it} (that is, such that the virtual number of points in common
with them is zero).

In particular one deduces that an exceptional curve of the first kind
(however reducible), belonging to a surface of genus $p>0$, is
characterized as a ``connected curve of genus $\rho=0$ and degree
$\nu=-1$''\footnote{With a small restriction, that is, excluding the case
when the curve belongs to a linear pencil of rational curves (with some
fixed component), the theorem can be extended to all surfaces ($p\ge0$).
See A.~Franchetta, Sulla caratterizzazione delle curve eccezionali ecc.,
{\it Boll.\ Unione Mat.\ It.}, 1941.}.

 \begin{EXERCISE}
The reader is invited to solve the following paradox: let us
start with a surface of order 4 (of genus $p=1$) $F_4$ containing a line
$a$; consider a cubic $C$ which is cut out on $F_4$ by a plane through $a$,
and make a transformation of $F_4$ into a surface $\Phi$ by means of a
linear system with a base point $O$ on $C$. So one has on $\Phi$ an elliptic
curve $C$ with an exceptional curve $\om$ which meets it at one point. Let us
look now at the sum
 \[
 C+2\om;
 \]
this is a curve of virtual genus $\rho=0$ and degree $\nu=-1$, and yet it
is not an exceptional curve!
 \end{EXERCISE}

 \section{Bicanonical and pluricanonical curves}\label{sec:2.8}
 \markright{\hfill Bicanonical and pluricanonical curves \quad 2.8\quad}
 As we saw, the virtual existence of the canonical system
 \[
|K|=|C'-C|
 \]
constructed from a system $|C|$ without base points on $F$, yields the
existence of numerical characters like $p^{(1)}$ and $p^{(2)}$ which are
defined independently of the effective existence of $|K|$. But the
consideration of a virtual canonical system which is not effective has
an even more concrete meaning when it happens that, although the system
does not exist, nonetheless it gives rise to an effective double or
multiple, which can be directly defined by the relation
 \[
 |2K|=|2C'-2C|
 \]
or
 \[
 |iK|=|iC'-iC|,
 \]
In fact the fundamental theorem for the adjoint systems gives in any
case
 \[
 |iC'-iC|=|iL'-iL|.
 \]
We also notice that the bicanonical system may be defined by
subtracting $|C|$ from its second adjoint system
 \[
 |(C+C')'|=|C+C''|=|2C'|;\quad |2K|=|C''-C|
 \]
The simplest example, and also the first one that occurred, of a surface of
genus $p=0$, which has a bicanonical curve (of order zero), is given by the
surface $F_6$ of the 6th order which passes doubly through the edges of a
tetrahedron; we shall come back to this surface later. The plane sections of
$F_6$ are sextics $C$ of genus 4; their adjoint curves are twisted sextics
$C'$ which are cut out on the surface by the cubic surfaces passing simply
through the edges of the tetrahedron; the curves $C$ and $C'$ are not
equivalent, but their doubles are equivalent, namely
 \[
 |2C'|=|2C|;
 \]
in fact the complete system $|2C|=|2C'|$ is cut out on $F_6$ by the
surfaces of order 6, which also have the edges of the tetrahedron as
double curves.

The surface $F_6$ has no exceptional curves, but if it is transformed
into a surface $\Phi$ in such a way that one of its points $O$ is sent to a
curve $\om$, then the exceptional curve $\om$, counted with multiplicity
two, is the only bicanonical curve of $\Phi$.

The characters of the bicanonical system, as well as those of the
pluricanonical ones, give invariants of the surface $F$. First of all the
dimension of the bicanonical system, or rather this dimension increased
by one, that is, the number of linearly independent bicanonical curves, is
an absolute invariant which is called the bigenus $P=P_2$. The genus and
the degree of the bicanonical system give, by contrast, relative
invariants of the surface. But it is clear that these characters can be
expressed in terms of the (relative) linear genus $p^{(1)}$.

In fact, denoting by $|K|$ the virtual canonical system of genus $p^{(1)}$ and
degree $p^{(2)}=p^{(1)}-1$, the genus of the bicanonical system $|2K|$ is
given by
 \[
 2p^{(1)}+p^{(2)}-1=3p^{(1)}-2
 \]
and its degree by
 \[
 4p^{(2)}=4(p^{(1)}-1).
 \]
Similarly the $i$-canonical system of the surface $F$, no matter if the
pluricanonical systems of lower order are effective or not, gives us:
an absolute invariant which is the $i$-genus $P_i$, that is, the number of
linearly independent $i$-canonical curves, and the relative invariants
given by its genus and degree. But the latter depend on the linear
genus $p^{(1)}$ of the surface; precisely the genus of the $i$-canonical
system is
 \[
 \frac{i(i+1)}{2}(p^{(1)}-1)+1
 \]
and its degree is
 \[
 i^2(p^{(1)}-1).
 \]
What we said in the previous section about the exceptional curves on a
surface of genus $p>0$, can be extended now to the surfaces of genus $p=0$
having bigenus $P>0$, or, in general, having some $P_i>0$. In fact the
effective existence of the bicanonical or $i$-canonical curves $|iK|=|K_i|$
again implies that a curve of genus $\pi$ and degree $n$, with $n>2\pi-2$,
has to be a fixed part of the $K_i$. In particular the exceptional curves of
the first kind, each counted in general $i$ times, will appear as fixed
parts of the $K_i$. And, as in the case $p>0$, there can be on the surface
only exceptional curves of the first kind not connected one to another. We
shall not spend more time in discussing in detail properties of reducible
exceptional curves, which do not give rise to new observations.

We shall only point out that, as a consequence of what we said, the
absolute linear genus of a surface with a plurigenus $P_i>0$ (even if
$p=0$), is always defined; its value is $p^{(1)}+\si$, where $p^{(1)}$ denotes the relative linear genus
and $\si$ is the number of exceptional curves. However a surface with $p=0$ and
with some nonvanishing plurigenus can be transformed into another one which
has no exceptional curves, and the linear genus of the latter gives the
absolute linear genus.

 \section{A historical note}\label{sec:2.9}
 \markright{\hfill A historical note \quad 2.9\quad}
 The notion of the genus of an algebraic surface $F$ occurred for the first
time to A.~Clebsch\footnote{Comptes rendus de l'Acad\'emie des Sciences de
Paris, 1868.} as the ``number of double integrals of the first kind (that
is, everywhere defined) belonging to $F$''. If one assumes $F$ to be of order
$n$ in ordinary space, then the algebraic differentials of the first kind
can be formed by means of the adjoint surfaces $\fie_{n-4}$ of order $n-4$,
as well as the differentials of the first kind of a plane curve of order $n$
are formed by means of the adjoint curves of order $n-3$. It is therefore
apparent that these $\fie$, or rather the (canonical) curves which they cut
out, away from the double curve, have an invariant character with respect to
the birational transformations of the surface. The first algebraic proof of
their invariance was given by M.~Noether\footnote{Zur Theorie des
eindeutigen Entsprechens algebraischer Gebilde, {\it Math.\ Ann.}, Band~2
(1869), Band~8 (1874).}. After that, Noether again noticed that the
canonical system $|K|$ gives, besides the genus $p$ (its dimension increased
by one) also another invariant of the surface, namely the genus of the $K$,
that is, the linear genus $p^{(1)}$, defined at least for $p>0$, which he
called ``Kurvengeschlecht'' in contrast with the ``Fl\"achengeschlecht''. As
for the degree $p^{(2)}$ of the canonical system, he proved that it can be
expressed in terms of $p^{(1)}$ by means of the formula
 \[
 p^{(2)}=p^{(1)}-1,
 \]
which is a consequence of the fundamental property of the $K$ of cutting
out on the generic curve of a linear system $|C|$ residual groups to the
characteristic series. Indeed the above formula, considered by Noether
to be valid only for the effective characters, holds, with no
exceptions, for the virtual characters, as defined by Enriques.

F. Enriques, took up again after twenty years the general study of
algebraic surfaces from the point of view of birational
transformations\footnote{Ricerche di geometria sulle superficie
algebriche, Mem.\ R. Acc.\ delle Scienze di Torino, 1893.}.
His first aim was to understand whether the fundamental property of the
canonical curves $K$ characterizes them, if one knows that it holds for a
particular system $|C|$: this is actually the case (with one exception)
if $|C|$ is an irreducible linear system, at least $\infty^2$, without
proper fundamental curves (that is, curves such that the genus of the
residual curves drop). Whence the author derives a geometric proof of
the invariance of the $K$ with respect to birational transformations:
indeed a birational transformation of the surface may be decomposed as
a sequence of transformations, each preserving a two dimensional linear
system of plane sections. These and other subjects which are treated in
the above-quoted {\it Ricerche} (which we shall mention again later on),
are taken up again by F. Enriques, in the paper Introduzione alla
Geometria sopra le superficie algebriche of 1896\footnote{{\it Mem.\
della Societ\`a Italiana delle Scienze} (detta dei XL) s.~III, t.~X,
1896.}. Concerning the canonical system $|K|$, the most important issue is
the extension of the invariance theorem to the case $p_g=0$, when $|K|$ is
no longer effective. Already in {\it Ricerche} the author considered the
adjoint system to a system $|C|$, which -- for $p>0$ -- is the sum $|C'|=|C+K|$.
Now he goes deeper into the study of this system independently of the
existence of $|K|$ (namely for $p=0$). The adjoint curves $C'$ to $|C|$ on a
surface $F$ cut out on a generic $C$ canonical groups (as do the
adjoint curves of order $n-3$ on a plane curve of order $n$). But this
property alone does not suffice to characterize the adjoint curves if $|C|$ possesses some proper fundamental curves; however one may define
them in any case, in such a way that the following functional condition
 \[
 |C+D'|=|C'+D|
 \]
holds with respect to any two linear systems (free from base points).

This fundamental property of adjunction yields, for $p>0$, the
invariance of the canonical system:
 \[
 |K|=|C'-C|=|D'-D|,
 \]
and extends in fact the invariance theorem. Indeed this theorem is
still meaningful when $|K|$, defined as a difference (and therefore a
priori as a virtual system), is not effective. The meaning of the
theorem has been pointed out by Enriques by stressing these aspects:

 \begin{enumerate}
 \renewcommand{\labelenumi}{(\arabic{enumi})}
 \item The fundamental relation between adjoint systems leads to the
definition of the virtual characters $p^{(1)}$ and $p^{(2)}$ of the
canonical system, independently of its effective existence; as we said,
these characters are related by the formula
 \[
 p^{(2)}=p^{(1)}-1.
 \]
 \item It may happen that, although $|C|$ is not contained in $|C'|$, its
double or a multiple $|sC|$ of it is contained in $|sC'|$; this yields
that a virtual canonical curve, which is not effective, may give rise
to an effective multiple system.
 \end{enumerate}

We already pointed out examples in which this possibility occurs. We
shall see later on the importance, mainly in the classification
problems, of the bigenus and the plurigenera, that is, the new characters
depending on the dimensions of the multiples of the canonical system.

A remarkable methodological progress in the theory of invariants outlined by
Enriques was determined by the idea, due to the same author\footnote{It was
treated in a course of lectures at the University of Bologna, in the year
1897-98. See the Program, published in the Bollettino di Bibliografia e
Storia delle Matematiche, n.~13 (1899). For the surfaces see Intorno ai
fondamenti della geometria sopra le superficie algebriche, {\it Atti Acc.\
Torino}, 37, page~19 (1901).} some years later, of defining the adjoint
systems and the canonical system simply by means of the Jacobians. The idea
is so simple, so easy to be carried through, so general in view of
extensions to varieties of any dimension, that somebody (not only
mathematicians who belonged to Enriques' school) could have believed that it
was a common patrimony of the geometers, and that it should have been known,
at least for curves, since a long time. But those who read the treatments of
the geometry on a curve by C.~Segre and by E. Bertini of 1894\footnote{C.
Segre, Introduzione alla Geometria sopra un ente algebrico semplicemente
infinito, {\it Ann.\ di Mat.}, s.~II, t.~22. E. Bertini, La geometria delle
serie lineari sopra una curva piana secondo il metodo algebrico, {\it Ann.\
di Mat.}, s.~II, t.~22.} will not find there any proof of the invariance of
the canonical series on a curve other than the one by Brill and Noether
based on the theorem of Riemann--Roch, according to which the canonical
series is the only $g^{p-1}_{2p-2}$ belonging to a curve of genus
$p$\footnote{See C.~Segre, ibidem, n.~75; Bertini, ibidem, n.~21, b) and
g).}. Were this not true, that is, had the definition of the canonical
series and the proof of its invariance, based on the property of the
Jacobian series of the $g_n$, been known at that time, then it would be hard
to understand the more laborious development of the theory of canonical
curves on a surface given by Enriques in {\it Ricerche} of 1893 and in
{\it Introduzione} of 1896.

Let us add that the idea of the Jacobian systems may be exploited in
various ways, and this gives rise to some variants in the treatments by
other authors.

The form which is adopted here, not too far from the one appearing in
{\it Fondamenti}, corresponds to the development of the theory of curves as
treated in the book {\it Lezioni} by Enriques--Chisini\footnote{Vol.~III.
See also, for an analytic treatment: F. Enriques, Sul teorema di
invarianza della serie canonica $g^{p-1}_{2p-2}$, ecc., Mem.\ R. Acc.\
di Bologna, 1914.} and -- unless we are mistaken -- it is of the greatest
simplicity.
