 \chapter[Irregular surfaces and continuous systems of\\ inequivalent
curves]{Irregular surfaces and continuous systems of inequivalent
curves}\label{ch:9}
 \markboth{\qquad 9.\quad Irregular surfaces and continuous systems \hfill}{}

 \section{Introduction}\label{sec:9.1}
 \markright{\hfill Introduction \quad9.1\quad}
 We recall some remarks made in Chapter~IV, Section~\ref{sec:4.12}.

To start, with we point out that, when speaking of curves and of linear or
continuous systems of curves $C$ on a surface $F$, we usually mean that we
are dealing with {\em irreducible curves without multiple
points}\footnote{If one wants to consider also curves $C$ with multiple
points, these points should be assumed to be assigned base points for the
linear or continuous systems one is referring to.}.

We have seen that on a regular surface a complete linear system $|C|$
of curves has complete characteristic series. And a surface containing
a continuous system of curves $C$, formed by $\infty^d$ distinct linear
systems, that is, a system $\{C\}$ containing $\infty^d$ inequivalent curves,
with $d>0$, is irregular, with irregularity
 \[
 p_g-p_a\ge d.
 \]

This statement can be immediately justified for the $\infty^{d+r}$ continuous
systems $\{C\}$ of irreducible curves, formed by $\infty^d$ linear systems
having a certain (generic) dimension $r$: in fact the characteristic series
cut out on a generic $C$ by the infinitely near curves of $\{C\}$ has
dimension
 \[
 r+d-1,
 \]
thus the characteristic series of the linear system $|C|$ turns out to
have deficiency (at least) equal to $d$; it therefore suffices to invoke
the theorem of Castelnuovo (Chapter~IV, Section~\ref{sec:4.12}) which says that
 \[
 d\le p_g-p_a.
 \]

Let us add that this inequality holds also for the continuous systems $\{C\}$
which are formed by reducible curves; in fact first of all one may leave out
of consideration the fixed components which can possibly be parts of the $C$;
next, if the variable $C$ are reducible, then by subtracting them from a
large enough linear system $|L|$ one may obtain a continuous system $\{L-C\}$
which -- away from fixed components -- consists of irreducible curves and
contains $\infty^d$ inequivalent curves -- just as $\{C\}$ does.

The main purpose of this Chapter is to treat the arguments leading to the
converse of the previous theorem, in the sense that: on an irregular
surface, of genera $p_a$ and $p_g>p_a$, thus of irregularity $q=p_g-p_a$,
there always exist continuous systems of curves consisting of $\infty^q$
distinct linear systems.

Thus the presence of continuous systems of inequivalent curves is the {\em
characteristic property} of irregular surfaces. And this statement, or
rather the theorem which makes this assertion more precise by involving the
value of $p_g-p_a$, may be referred to as the {\em fundamental
theorem}\index{fundamental theorem for irregular surfaces} in the theory of
irregular surfaces.

But before approaching the proof of the fundamental theorem, and in
consideration of the difficulties of rather subtle kind which appear in it,
it is convenient to consider the most general assumptions; therefore we
shall examine what can be said about the surfaces containing $\infty^q$
continuous systems of inequivalent curves, where we only know that, as we
have seen before,
 \[
 0\le q\le p_g-p_a.
 \]
We will thus distinguish between two notions of irregularity, which a priori
might not always coincide: the {\em numerical
irregularity}\index{numerical!irregularity}\index{irregularity}
 \[
 p_g-p_a
 \]
and the {\em geometric irregularity}\index{geometric!irregularity} $q$,
which is the largest possible dimension of a continuous series of
inequivalent curves lying on the surface:
 \[
 q\le p_g-p_a.
 \]
In any case the number $q$ denotes an invariant character of the surface
itself: all the known examples of irregular surfaces (surfaces with an
irrational pencil of curves, surfaces of the pairs of points of a curve,
etc.) give $q>0$, or rather $q=p_g-p_a$.

 \section{The arithmetic condition for a curve on a surface with geometric
irregularity $q$ to belong to an $\infty^q$ continuous series of
inequivalent curves}\label{sec:9.2}
 \markright{\hfill The arithmetic condition for an $\infty^q$ continuous
series \quad 9.2\quad}
 Let $q>0$ be the geometric irregularity of a surface F: this means
that $F$ contains an $\infty^q$, and not larger, continuous series $\{D\}$ of
inequivalent curves.

We ask the following question: can one give an arithmetic criterion in
order that a curve $C$ on $F$ belong to an $\infty^q$ series of inequivalent
curves as above, that is, to a series formed by $\infty^q$ distinct complete
linear systems?

The answer is given by computing the virtual dimension of the system
determined by the curve $C$ (of genus $\pi$, degree $n$, and index of
speciality $i$):
 \[
r=p_a+n-\pi+1-i.
 \]

Indeed, let $|C|$ be a linear system of curves, either irreducible or
reducible, of degree $n$, of genus $\pi$, and of index of speciality $i$,
which is arithmetically effective, that is, such that its virtual dimension is
 \[
 p_a+n-\pi+1-i\ge0.
 \]
We associate with $|C|$ another regular (irreducible, etc.) large enough
linear system $|D|$ so that $|C+D|$ is also regular. We may assume that the
system $|D|$ is contained in a continuous system $\{D\}$ formed by
$\infty^q$ inequivalent linear systems, one of which -- to contrast with
$|D|$ -- we will denote by $|\Dbar|$.

Now the system $|C|$ can be constructed by subtracting a curve $d$ from
$|C+D|$, and its dimension turns out to be $\ge p_a+n-\pi+1-i$, since $|C+D|$
cuts out on the curve $d$ a (complete or not) series which has precisely
the index of speciality $i$ (see Chapter~IV, Section~\ref{sec:4.11}).

Let us consider the series cut out by the same system $|C+D|$ on a $\Dbar$.
It is not possible that, for a generic choice of $\Dbar$ in $\{D\}$, the
aforementioned series has an index of speciality greater than $i$,
since -- by varying $\Dbar$ continuously -- one may reduce it to $d$ and, in
this limiting process, the index of speciality of the series cannot
decrease. Therefore the dimension of the linear system $|\Cbar|=|C+D-\Dbar|$
turns out to be (according to the well known computation which gives the
theorem of Riemann--Roch) $\ge p_a+n-\pi+1-i$: so, as $\Dbar$ varies, one
obtains $\infty^q$ distinct complete linear systems, which form an
irreducible series $\{C\}$.

Let us state the theorem that we have just proved:

On a surface of geometric irregularity $q>0$, a linear system of
degree $n$, genus $\pi$ and index of speciality $i$, which is arithmetically
effective, that is, such that
 \[
 p_a+n-\pi+1-i\ge0,
 \]
always belongs to an $\infty^q$ series of inequivalent linear systems.

With this theorem we have a very simple understanding of the notion
of geometric irregularity (thus of the difference $p_g-p_a$, if we take
into account the fundamental theorem which gives the characteristic
property of the irregular surfaces).

 \section{The Picard variety corresponding to an irregular
surface}\label{sec:9.3}
 \markright{\hfill The Picard variety corresponding to an irregular
surface \quad 9.3\quad}
 In the previous section we have seen how, on a surface of geometric
irregularity $q$ ($>0$), the $\infty^q$ linear systems forming a certain
continuous system $\{|C|\}$ or $\{C\}$ are related to those of another
analogous continuous system $\{D\}$: there is a one-to-one correspondence
between the elements (linear systems) of $\{C\}$ and those of $\{D\}$, in
which two homologous linear systems are residual one of the other with
respect to a $|C+D|$. This remark (due to Castelnuovo) can be made more
precise by saying that: The elements (linear systems) of the $\infty^q$
continuous series belonging to a surface of geometric irregularity $q$ (>0),
can be regarded as the points of one and the same Abelian variety $V$, which
Castelnuovo wanted to dedicate to the French geometer, naming it the Picard
variety corresponding to the surface.

Indeed the operation $+D-\Dbar$ by means of which one passes from a linear
system $|C|$ to another linear system $|\Cbar|$ contained in $\{C\}$, can
be interpreted as a pointwise transformation of the variety $V$ into
itself; it is obvious that all the analogous transformations are
permutable and form an $\infty^q$ simply transitive group without
exceptions of the aforementioned variety $V=V_q$ (of dimension $q$):
precisely for this reason $V$ is characterized as a nondegenerate
Abelian variety.

The meaning of the theorem we have obtained is explained by the following
considerations.

Let us consider the question of determining all the curves of a given order
$n$ belonging to an algebraic surface $F$.

In order to belong to $F$, these curves have to satisfy some algebraic
conditions, thus they fall a priori into a certain number of families, hence
forming, in general, some continuous systems.

These systems are linear systems when $F$ is regular. If, by contrast, the
geometric irregularity of $F$ is $q$ ($=p_g>p_a$) then these systems --
which may be seen as varieties whose elements are distinct linear systems --
are not arbitrary algebraic varieties, but they are Abelian varieties $V_q$
or, at least, their multiples form Abelian varieties.

In order to understand better the implications of this fact we will just
recall a few facts about the Abelian varieties (see Section~\ref{sec:9.9}).

 \section{The characteristic property of irregular algebraic surfaces: the
fundamental theorem for $p_g=0$}\label{sec:9.4}
 \markright{\hfill The characteristic property of the irregular surfaces
\quad 9.4\quad}
 The proof of the fundamental theorem of the irregular surfaces, which we
have stated in Section~\ref{sec:9.2}, can be given in a very simple way for
surfaces with $p_g=0$, whereas it requires subtler considerations for
$p_g>0$. It is therefore convenient to distinguish between these two cases.

Let us first of all assume $p_g=0$, thus $p_a=-p<0$.

Let us consider, on our surface $F$, a complete linear system $|C|$, which
is regular, nonspecial, of genus $\pi$ and degree $n$, and therefore of
dimension
 \[
r=-p+n-\pi+1.
 \]

Let us associate to it another linear system $|D|$, also regular, of
genus $\rho$ and degree $\nu$, and therefore of dimension
 \[
r'=-p+\nu -\rho +1,
 \]
and such that the sum $|C+D|$ is also a regular system (see Chapter~IV,
Section~\ref{sec:4.8}).

Denoting by $m$ the number of intersections of $C$ and $D$, we write down
the genus and the degree of $|C+D|$=|E|:
 \[
 \Pi=\pi+\rho +m-1 \quad N=n+\nu +2m,
 \]
 and then its dimension:
 \[
 R=-p+N-\Pi+1=-p+(n-\pi)+(\nu -\rho)+m+2,
 \]
that is,
 \[
 R=r+r'+p+m.
 \]

Let us recall that, in view of the principle of degeneration or of
reducibility\footnote{See Enriques--Chisini, Libro~V, Chapter~III, 35 (Vol.~III,
page~405).}, an irreducible curve, with a certain number $m$ of double
points, cannot be continuously deformed to a reducible curve whose
components, if they have no multiple points, have less than $m+1$ common
points (that is, the limit curve must have a further double point, by means of
which the two components of the reducible curve are connected). {From} this
principle one deduces that, by imposing that the curves $E$ possess $m$ double
points, one cannot obtain a continuous system $\Ebar$ of dimension
$r+r'+p>r+r'$ formed by irreducible curves, strictly containing the system of
the curves of $|E|=|C+D|$ which are reducible in a curve of $|C|$ and in a
curve of $|D|$; on the contrary one defines in this way a system such that
one at least of its parts (of the same dimension) contains a continuous
system of curves which are reducible into a curve of the same order as $C$
and a curve of the same order as $D$; therefore these two components describe
two continuous nonlinear complementary systems, say $\{C\}$ and $\{D\}$.

To any curve $C$ of the larger continuous system $\{C\}$ there corresponds,
as residual with respect to $|E|$, a linear system of curves $|D|$, and
conversely; so it is easy to compute the dimension of the continuous
system $\{C\}$, which has to be (at least) $r+p$, and similarly the dimension
of $\{D\}$, which has to be $r'+p$. Furthermore we can say that $\{C\}$
consists of (at least) $\infty^p$ inequivalent linear systems, of dimension
equal to $r$, since the dimension of the complete linear systems forming
$\{C\}$, which a priori is $\ge r$, cannot be $>r$; in fact these linear
systems can be continuously deformed to the system $|C|$ we started with,
which has exactly dimension $r$.

But if the geometric genus of $F$ is $p_g=0$, we know that $F$ cannot
contain a continuous system of curves formed by $\infty^d$ distinct
linear systems, where $d>p_g-p_a=p$; thus we conclude that, on the
irregular surfaces with geometric genus $p_g=0$, there are continuous
systems of $\infty^p$ inequivalent curves, and no system formed by a
larger series of such curves.

We summarize the results obtained by stating the Restricted Fundamental
Theorem:
 \begin{proc}
 $A$ surface with negative arithmetic genus $p_a=-p$ certainly
contains continuous systems formed by $\infty^p$ inequivalent curves, and,
if the geometric genus is $p_g=0$, it does not possess any larger system of
inequivalent curves;
 \end{proc}
in other words, the geometric irregularity is
 \[
 q\ge-p
 \]
 and for $p_g=0$ it is equal to the numerical irregularity
 \[
 p_g-p_a=p.
 \]

 \section{The fundamental theorem for $p_g>0$}\label{sec:9.5}
 \markright{\hfill The fundamental theorem for $p_g>0$ \quad 9.5\quad}
 Now we try to extend the fundamental theorem to the surfaces of geometric
genus $p_g>0$ and of any arithmetic genus $p_a<p_g$. In order to do this, we
take into consideration again, on the surface $F$ (which we suppose to be of
genera $p_a$ and $p_g$) two regular, nonspecial, linear systems $|C|$ and
$|D|$ with characters $n$, $\pi$ and $\nu$, $\rho$ such that the sum
$|E|=|C+D|$ is also regular and cuts out on a $d$ a complete and nonspecial
series $g_m+\nu$.

If we denote by $N$ and $\Pi$ the characters of $|E|$, then we see (as
before in the case $p_g=0$) that the linear system $|E|$ has the dimension
 \[
 R=p_a+N-\Pi+1=p_a+(n-\pi)+(\nu -\rho)+m+2
 \quad\hbox{where $(m=CD)$},
 \]
that is,
 \[
R=r+r'+m-p_a,
 \]
where $r$ and $r'$ are the dimensions of $|C|$ and $|D|$. Now, if the linear
system $|C|$ has to be contained in a continuous system $\{C\}$ of dimension
$r+q=r+p_g-p_a$ (formed by $\infty^q$ distinct linear systems), then it is
necessary that imposing to the $E$ to have $m$ double points on
$F$, so that these curves break up into $C$ of $\{C\}$ and $d$ of the
residual system $\{D\}$, yields only $m-p_g$ independent conditions, and not
$m$ conditions (or $m-p_a$, as the existence of $|C|$ and $|D|$ seems to
imply). If this happens, then the continuous system $\{C\}$, of dimension
$p_g+n-\pi+1$, has, on a given $C=C_0$, a complete characteristic series,
and therefore it cannot be contained in a larger continuous system.

In order to show that
 \begin{proc}
 imposing to the $E$ to have $m$ double points on
$F$, so that these curves break up into curves of $\{C\}$ and $\{D\}$, yields
at most $m-p_g$ conditions,
 \end{proc}
and therefore that the geometric irregularity $q$ of $F$ and the numerical
irregularity ($q=p_g-p_a$) are equal, we are led to examine the
characteristic series of the continuous system $\{C\}$; it is apparent that
we will accomplish the proof if we see that such a series is complete.

In order to prove that, it suffices to consider the curves $E$ of $|E|=|C+D|$
which pass through the group $G$ of the $m$ points common of the two
components $C_0$ and $D_0$ of a reducible $E$, $E_0=C_0+D_0$; these curves
$E$ cut out on $C_0$ a complete characteristic series, and one of them,
together with $E_0$, determines a linear pencil of curves which have a
contact at the points of $G$: whence it follows that also the curve of the
pencil which is infinitely near to $E_0$ has $m$ double points near to those
of $G$\footnote{See e.g., Enriques--Chisini, Lezioni, Libro~II, \S5, Vol.~I,
page~183 and Libro~III, Chapter~II, \S20, Vol.~II, page~167.}. Therefore, if
the degeneration principle can be applied also to these infinitely near
curves, then this curve has to be reducible into two curves $C_1$ and $D_1$,
the first of which -- near to $C_0$ -- cuts out on $C$ a group of the
characteristic series (that can be arbitrarily given).

Have we thus obtained the proof (for any $p_g$) of the fundamental
theorem?

No, we must not be misled by the appearance of the proof: the theorem we
deal with concerns an integral property, and we have only established a
differential property that is a consequence of it, and which does not
necessarily imply it. Indeed, we have at most proved that
 \begin{proc}
 the characteristic series of $C_0$ is completely cut out by curves $C_1$,
which are infinitely near to it,
 \end{proc}
 but we have not proved that the $C_1$ actually belong, as curves which tend
to a limit, to a continuous system of proper curves $\{C\}$.

Here there is a critical point of the proof, whose historical significance
we will point out later (Section~\ref{sec:9.6}): the condition that imposing
$m-p_g$ double points on the $E$ yields as a consequence $p_g$ more double
points has not been verified ``globally'' but only ``locally'', for the
curves infinitely near to $E_0$.

The ``local'' differential property follows, as we have said, from the
integral, or ``global'', property that we want to establish, but the
latter is not in turn a consequence of the former.

In order to explain the matter better, let us consider a variety $V_r$
of dimension $r$, which is defined in a space $S_R$ (with $R>r$) as the
intersection of a certain number of hypersurfaces (or of some
differentially distinct linear branches of one and the same algebraic
hypersurface): in order to compute the dimension $r$, we are led to
consider the number $s$ of linearly independent points in the first order
neighbourhood of a generic point $P$ of $V_r$; but it is clear that the
varieties as the intersection of which $V_r$ is defined may have a certain
contact at each of its points; in that case the number of the points
infinitely near to $P$, which we are discussing, will be $s>r$.

For instance, if $r=1$ then $V_1$ may be the contact curve of two or more
surfaces of the space $S_3$, a curve to be counted twice; then the number
of its independent points which are infinitely near to a point $P$, is no
more $r=1$, but $s=2$.

This explanation also clarifies which is the gap that has to be
filled in our proof of the general fundamental theorem for $p_g>0$, in
passing from the local property we have just proved, to the global one
we have mentioned above.

Let us consider the $V_r$ of $S_R$, defined as the intersection of $m$
branches of hypersurfaces, which correspond to the curves of $|E|=|C+D|$
having $m$ double points (and forming the larger continuous system
consisting of the curves $E$ which are reducible into a curve $C$ of the
same order as $C_0$ and in a $d$ of the same order as $D_0$): we need that
these $m$ branches do not touch each other all along $V_r$ or at one of its
generic points $P$. This means that we cannot have a linear branch of a
curve with centre at $P$ having a contact of finite order $s$ ($\ge1$)
with the aforementioned branches but not with $V_r$; in other words: if
such a linear branch has $h<s+1$ ($>0$) proximate points in common with $V_r$,
then it must also have the proximate $(h+1)$-tuple point in common with this
variety.

Returning now from the $V_r$ to our surface $F$, we need to show that
for any infinitely near curve $C_1$ in the first order neighbourhood of C0,
there is a sequence $C_1,C_2,\dots$ of infinitely near curves in the
successive neighbourhoods of order $1,2,3$,\dots; and this series has to
be regular, (that is, it has to be similar to a sequence of points on a
linear branch). So $C_1$ is a part of a reducible curve $E_1=C_1+D_1$ near
to $C_0+D_0$; $C_2$ will be a part of a successive curve $C_2+D_2$, and so
on.

In order to do this and thus accomplish the general proof of the fundamental
theorem, an idea arises in a very natural way\footnote{The part which
follows was not completed by the Author and therefore the arguments which
are developed here contain many gaps; we thought it was better to present it
anyway, because it contains ideas which perhaps, if suitably completed, may
lead to a satisfactory conclusion of the theory. (Editors' note).}: since
two infinitely near inequivalent curves, $C_0$ and $C_1$, are given on $F$,
the operation $C_1-C_0$ can be successively applied to a curve, for instance
to $C_1$, defining in this way the sequence of successive curves
$C_2,C3,\dots$, or rather the sequence of the linear systems which they
determine. Here a new idea comes into the picture, namely one assumes that
``one may operate on the surface $F$ by summing and subtracting curves, even
when dealing with the infinitely near ones''. We want to justify this idea.

We start from a somewhat analogous question which concerns the curves. If,
on a curve $L$ of genus $p>0$, we consider two inequivalent groups of $p$
points, $G$ and $G'$, then the operation $+G'-G$ can be successively applied
starting with $G$ and it leads to a series, that is (in general) infinite,
of groups
 \[
 G,G',G'',G''',\dots,
 \]
which correspond to the homologous of $G$ in the powers of a
transformation $T$ of the first kind of the Jacobian variety $V$ associated
to $L$. Therefore the sequence of the successive groups $G'',G''',\dots$,
is still defined\footnote{Precisely in the sense of the differential
conditions that define the infinitely near points. See
Enriques--Chisini, Lezioni, Libro~IV, Vol.~II.} when $G'$ becomes infinitely
near to $G$: in this case the transformation $T$ becomes an infinitesimal
transformation that, according to S.~Lie, generates the continuous
group of $V$, and defines an analytic series $t$, which is an orbit of the
Jacobian group of $L$.

In order to pass from this case to the consideration of the infinitely near
curves on a surface, it is first of all convenient to justify the
degeneration principle for a curve infinitely near to $C_0+D_0$, having (as
in the above mentioned construction) $m=C_0D_0$ double points near to the
common points of $C_0$ and $D_0$; thus we need to show that such a curve is a
$C_1+D1$, formed by two parts which are respectively near to $C_0$ and $D_0$.

In order to do this, we may assume that $C_0$ and $D_0$ are normal,
nonspecial curves belonging to spaces of very high dimension compared with
their genus: we may indeed always reduce to this case by means of a
transformation of the given surface $F$. Then it is easy to verify that, in
the space of $F$, the irreducible curves $E$ of order $N$ and genus $\pi$,
describing the family defined by the reducible $C+D$, form a variety of a
certain dimension $R$; inside this, one has a variety of dimension $R-m$
consisting of the (reducible) curves $E=\Ebar$ with $m$ double points,
containing the family of the curves $C+D$, so that the independent
reducibility conditions for the $E$ (not on the surface but) in the space of
$F$ are just $m$, and not fewer.

As for the proof, let us project the reducible curve $C_0+D_0$ with $m$
double points (the intersections of $C_0$ and $D_0$) from a suitable linear
space on a multiple (reducible) $N$-tuple line, with $2N+2\Pi-2$ critical
points.

This $N$-tuple line may be thought of as of an $N$-sheeted Riemann surface,
which -- for a certain choice of the loops, taken in a given order, and for a
suitable ordering of the sheets -- can be defined by a system of
transpositions of the sheets, similar to those which we find in the
statement of the theorem of L\"uroth--Clebsch\footnote{See Enriques--Chisini,
Lezioni, Libro~V, Chapter~I, \S3, Vol.~III, page~26; Chapter~III, \S33,
Vol.~III, pages~359--376.}; thus (if $n$, $\pi$ and $\nu$, $\rho$ are the
characters of $C_0$ and $D_0$ respectively) we have:
 \begin{enumerate}
 \renewcommand{\labelenumi}{(\arabic{enumi})}
 \item first of all $2n+2\pi-2$ loops corresponding to an even number of
similar transpositions $(12)(12)\cdots((n-1)n), ((n-1)n)$;

 \item then $m$ loops (going to the $m$ apparent critical points, images of
the common points of $C_0$ and $D_0$) each of which corresponds to the
product of two transpositions
 \[
 (n(n+1)) \ (n(n+1))=1;
 \]

 \item and finally $2\nu+2\rho-2$ loops corresponding to an even number
of transpositions like
 \[
 ((n+1)(n+2)), ((n+1)(n+2)),\dots,\quad
 ((n+\nu -1)(n+\nu)), ((n+\nu -1)(n+\nu)).
 \]
 \end{enumerate}

Now, by the existence theorem\footnote{See Enriques--Chisini, {\it Lezioni},
Libro~V, Chapter~III, \S33, ibidem.}, we may let the multiple line defined
in this way vary continuously, in such a way to separate the two branch
points that come together at one of the apparent critical points. It is
therefore evident that the reducible curve $\Ebar=C_0+D_0$ in the space of
$F$ is susceptible of varying in a certain $\infty^R$ variety $V$ of
irreducible curves $E$, and inside $V$ there is an $\infty^{R-m}$ variety
$\Vbar$ of reducible curves with $m$ double points; in this way the
infinitely near curves to $C_0+D_0$ with $m$ double points precisely form
the linear neighbourhood of this curve-element on $\Vbar$.

Therefore we have finally proved the fundamental lemma:
 \begin{proc}
 every curve $E$ with $m$ double points, infinitely near to $C_0+D_0$ (in
the space of $F$), belongs to an $\infty^1$ (regular) series of reducible
curves containing $C_0+D_0$, and therefore it is actually reducible in two
improper curves $C_1$ and $D_1$ of the same order as $C_0$ and $D_0$
respectively, near to these curves.
 \end{proc}

The fundamental lemma that we have just established means that
 \begin{proc}
 the infinitely near curve to $C_0$, which is a part of a curve near to
$C_0+D_0$ with $m$ double points, constructed on $F$ as we have just
explained, depends only on $C_0$ and not on the choice of a $D_0$, which can
be associated to it in infinitely many different ways, in a regular system
$|D|$.
 \end{proc}
 On this ground, we want to recognize that after the mentioned curve
$C_1$, we have again a curve $C_2$, in the second order neighbourhood of
$C_0$, and then a curve $C_3$ and so on.

Let us assume that here we can speak about infinitely near curves,
defined by differentiable conditions in the same way as the points on a
curve, as about proper curves; we intend to justify the use we make of
this terminology later on.

Let us construct on the surface $F$, which contains the above
considered linear systems $|C|$ and $|D|$, a normally large linear system $|L|$ of a certain genus $P$; so $|C|$ and $|D|$ cut out on a generic $L$, and
even on every (irreducible) $L$ with a double point, complete series of
order smaller than $P$\footnote{See Chapter~IV, Section~\ref{sec:4.10}.}. Let
us consider a general pencil of curves $L$ in $|L|$, with a certain group
$\Ga$ of base points: first of all we claim that
 \begin{proc}
 the infinitely near inequivalent curves $C_0$ and $C_1$ cut out
inequivalent groups on the $L$ of the pencil.
 \end{proc}

This claim is a consequence of the fact that $|C_0|$ cuts out on each
$L$ a complete series: so -- since $M=CL>C_0^2$ -- the equivalent infinitely
near curves to $C_0$ are as many as the equivalent infinitely near groups to
$C_0L$ on $L$; therefore if $C_1$ would cut out on a curve $L$ a group
equivalent to $C_0L$, then $C_1$ should also be equivalent to $C_0$.

Now we can slightly modify and make more precise our assumptions, by
assuming that $C_0$ belongs, inside $\{C\}$, to a regular system of dimension
$r=p_a+n-\pi+1$, so that the curve itself is determined by $r$ simple base
points which form a subgroup $A$ of $\Ga$ (and similarly that $|D_0|$ too, if
needed, is of dimension $r'=p_a+\nu-\rho+1$, so that the curve $d$ can be
determined by imposing $r'$ simple base points forming again a subgroup of
$\Ga$). Choose then in the base group $\Ga$ of the pencil of the $L$, away
from $A$ and $B$, $P-M$ more points (which may also be a single point counted
$P-M$ times), forming a certain group $S$. By adding $S$ to the group of $M$
points cut out by $C_0$ on $L$, one has a nonspecial group $G$ of $P$
points on $L$, and by adding $S$ to the group cut out by $C_1$, one
similarly has a group $G_1$ of $P$ points, which is infinitely near to $G$
and inequivalent to it. The correspondence between $G$ and $G_1$ (an
infinitesimal transformation of the group of the Jacobi variety of $L$) gives
rise on the $L$ of the pencil to an analytic series $t$ of groups of $P$
points, containing a group $G_2$ infinitely near to $G$ in the second order
neighbourhood of $G$, that is, a point that follows $G_1$. As $L$ varies in
the pencil, the group $G_2$, which is algebraically and rationally
determined\footnote{The differential expression that defines $G_1$, from which
the one defining $G_2$ can be derived, has been computed by Chisini. See
Enriques--Chisini, Lezioni, Libro~VI, Chapter~II, \S28, Vol.~IV, page~143.},
describes on the surface $F$ a curve $C_2$ infinitely near to $C_0$, in its
second order neighbourhood, and successive to $C_1$; to this curve one has to
add the infinitesimal curves, not connected with it, corresponding to the
points of the group $A$.

In order to justify the claim to the effect that, as $L$ varies in the
pencil, $G_2$ traces out a curve $C_2$ near to $C_1$ and of the same order, we
need to prove that the locus of $G_2$, which is defined up to irreducible
curves $L$ of the pencil, does not contain any of the points of its base
group $\Ga$ away from $A$, and therefore that on any $L$ of our pencil (which
was chosen generically inside $|L|$) the group $G_2$ (near to $G$ in its
second order neighbourhood) cannot be special.

To see this we notice that:

 \begin{enumerate}
 \renewcommand{\labelenumi}{(\arabic{enumi})}
 \item on a curve $L$ of genus $P$, the group $G_2$ cannot be special unless the
groups $G$ and $G_1$ are also special; inasmuch as a group which is
infinitely near to a nonspecial one is certainly nonspecial;

 \item since the group of $M$ points $LC_0$ (away from $A$) does not belong
to a linear series of dimension $>0$, by adding to it a generic group of
$P-M$ points (the group $S$) one obtains a nonspecial group\footnote{This is
a simple consequence of the theorem of Riemann--Roch. See for instance
Enriques--Chisini, Lezioni, Libro~V, Chapter~I, Vol.~III.}: in order to have a
special group of points $P$ one has to impose one condition on $S$ and in
order that the group $G=S+LC$ be special on $L$ one has to impose at least
one further condition on $S$; thus the choice of a group $S$ of $P-M$ points
of $L$ that, when added to $LC_0$ and to $LC_1$, gives rise to two special
groups $G$ and $G_1$ yields at least two conditions which $S$ has to satisfy.
 \end{enumerate}

With this in mind we go back to the construction outlined above of a pencil
of curves $L$ in $|L|$. To start with we take a particular (irreducible)
curve $\Lbar$ of $|L|$ (of genus $P$) and $r$ of its points, that will be
imposed on $|C|$ as base points. Then we take a group $S$ of
$P-M$ points, that will be added to those of the group $LC_0$, (from which
$A$ has been subtracted). Finally, on the same $\Lbar$, we take a group
$\Ga$ of the characteristic series $g_N^{R-r}$ of $\Lbar$ containing $A$. In
order that the group of $P$ points $G=S+LC_0$ be special on $\Lbar$ and
have, on the same curve, an infinitely near group $G_1=S+LC_1$ which is also
special, the choice of the group $S$ is subjected, as we have said, to two
conditions at least; on the other hand, in order that the analogous
conditions be satisfied on a curve $L$ of the pencil which has $\Ga$ as base
group, the choice of $S$, and therefore the choice of $\Ga$, must satisfy at
least one condition on $L$, since we have here one more parameter; so we
would have a pencil of curves $L$ that is not generically chosen inside
$|L|$.

In this way the following assertion has been proved: the group $G_2$ which
we have constructed as above on the curve $L$ of a generic pencil of $|L|$
describes a curve near to $C_1$, of the same order (and not passing through
any of the base points of $\Ga$). And it is worth noticing that our argument
works even if in a pencil of curves $L$ there is, in general, some
(irreducible) curves of effective genus $P-1$ with a double point $X$;
inasmuch as one may regard these curves as having virtual genus $P$
(considering the double point $X$ as a virtually nonexistent double point).
This means that a group $S$ of $(P-1)+1$ points on such an $L$
has to be considered as nonspecial, unless it belongs to an $\infty^1$
linear series having a neutral pair $X_1X_2$ in correspondence with $X$. So,
also in this case, the speciality of $G$ and $G_1$ yields at least two
conditions to be satisfied by the choice of the group $S$.

In the above discussion we talked about the curves $C_1,C_2,\dots$, that are
infinitely near to $C_0$, as if they were proper curves. To justify the use
of this terminology, it is convenient to show that the above mentioned
curves may be actually defined through a limiting process of proper curves.
This happens first of all for the curve $C_1$ which is infinitely near to
$C_0$ in the first order neighbourhood: in fact we have seen that $C_1$ is a
part of the reducible curve near to $C_0$ inside a limit pencil for $\la=0$ of
$C_0+D_0+\la E$ ($|E|=|C+D|$) whose base points are the $m$ points of the
group $C_0D_0$. It is in fact true that the curve of this pencil is
irreducible, for $\la\ne0$, and tends to $C_0+D_0$; but, according to the
fundamental lemma proved above, as $\la$ approaches zero, this curve
acquires $m$ base points and breaks up into $C_1$+D1: this means that the
$\infty^1$ analytic series $t$ of groups of points determined by $G$ and
$G_1$ on the $L$ of the aforementioned pencil does not depend on the choice
of $D_0$ or of the system $|D|$ which we associated with $|C|$.

Now we can construct a proper curve whose variation defines $C_2$, by
considering on a generic curve $L$ of the pencil:

 \begin{enumerate}
 \renewcommand{\labelenumi}{(\arabic{enumi})}
 \item the group $G$ cut out by $C_0$ with the addition of the points of $A$;

 \item the group $G'$ formed (by $A$ and) by the points of $C_0+D_0+\la E$
near to those of $G$;

 \item the group of $P$ points $G''$, which corresponds to $G'$ in the
transformation $(GG')$ of the Jacobi variety of $L$.
 \end{enumerate}

The locus described by $G''$, as $L$ varies in the pencil, defines the
curve $C_2$ which is successive to $C$ and $C_1$.

To tell the truth, as long as $\la\ne0$, this locus is an irreducible curve
tending to a curve larger than $C_0$, that is, to $C_0+D_0$; in fact a point of
$C_0+D_0+\la E$, near to $C_0$, may be interchanged, as $L$ varies in the
pencil, with a point near to $D_0$, so that the curve which is defined by
the variable $G''$ is generated by the sum of $G''$ and of a $\Gbar''$
analogously defined with respect to $D_0$ and near to this $D_0$; and the sum
$G''+\Gbar''$ describes a curve E2, that cuts out on the curves $L$ groups of
the linear series cut out by $|E|=|C+D|$, or (a priori) a multiple of this
series, whose limit -- for $\la=0$ -- is $C_0+D_0$\footnote{The justification
of this assertion is the following: let us have on a curve $L$ of genus $P$
two groups $G$ and $\Gbar$ of $P$ points, and two more $G'$ and $\Gbar'$ such
that $G+\Gbar=G'+\Gbar'$ and let us construct the groups $G''=G'+|G'-G|$ and
$\Gbar''=\Gbar'+|\Gbar'-\Gbar|$, then $G''+\Gbar''=G'+\Gbar'=G+\Gbar$.}.

But the groups $G,G',G''$ belong on $L$ to an $\infty^1$ analytic series of
groups of $P$ points that -- as $\la\to 0$ -- that is, when they become
infinitely near, only depends on $C_0$ and $C_1$ and not on $D_0$ or on
$|D|$, and therefore it actually defines, in the second order neighbourhood
of $C_0$, the curve $C_2$, that is, the irreducible part of $E_2$ for
$\la\to0$ whose existence we wanted to prove.

We have thus constructed on $F$ a curve $C_2$ near to $C_1$ and successive to
it, in the second order neighbourhood of $C_0$, which is a part of a
reducible $E_2=C_2+D_2$ of the linear system $|E|=|C+D|$. Similarly, starting
with $C_2$, one constructs a successive curve $C_3$, near to $C_0$ in its
third order neighbourhood, etc. So we accomplish the proof of our theorem,
that is, we prove that near to the reducible $E_0=C_0+D_0$ there are, in all
the successive neighbourhoods of any finite order, reducible curves of
the system $|E|=|C+D|$, which form a regular series (like the linear
branch of a curve); therefore the fundamental theorem is proved in
general.

On a surface of geometric genus $p_g\ge0$ and of arithmetic genus
$p_a<p_g$, any linear regular nonspecial system $|C|$ of irreducible
curves is contained in a continuous system $\{C\}$ which cuts out on a $C$ a
complete characteristic series and is therefore formed by $\infty^q$
($q=p_g-p_a$) distinct linear systems. So: the geometric irregularity of a
surface is always equal to the numerical irregularity:
 \[
q=p_g-p_a.
 \]

And (Section~\ref{sec:9.2}) on a surface of irregularity $q$ any curve $C$,
with the characters $n$, $\pi$ and $i$, which is arithmetically effective,
that is, such that
 \[
 p_a+n-\pi+1-i\ge0,
 \]
belongs to an $\infty^q$ continuous series of inequivalent curves.
 \section{The history of the theory of continuous
systems}\label{sec:9.6}
 \markright{\hfill The history of the theory of continuous systems \quad
9.6\quad}
 The theory of continuous systems of curves lying on the irregular
surfaces has a fundamental significance, not only in the
algebro-geometric study of the surfaces, but also for the
(transcendental) theory of the simple integrals (of total
differentials) which belong to them, thus giving a way of relating
these two different aspects of the science of the surfaces. Therefore
we will take the general point of view consisting in understanding the
genesis of this theory. In particular we want to go more deeply into the
general theory of these continuous systems by putting them into a historical
perspective, so that one may also see the meaning of the errors through
which the theory itself had to go: this seems to us even more relevant since
we believe that the development of the theory has not yet attained its final
stage.

The memoir of Castelnuovo--Enriques of 1900 Sopra alcune questioni
fondamentali nella teoria delle superficie algebriche\footnote{{\it Ann.\ di
Mat.}, t.~VI, s.~II, page~165.} on the one hand fruitfully closes a series of
efforts, but on the other it raises new problems, such as the determination
of the surfaces with all plurigenera equal to zero (thus of the surfaces with
genus $p_g=0$ and negative arithmetic genus). In order to investigate these
problems and in general to make a progress in the knowledge of the surfaces,
also in connection with the transcendental theory of the French school, the
study of the continuous systems of inequivalent curves arose in a natural
way.

The general consideration of such systems goes back to a Note by G.~Humbert
of 1893\footnote{Sur une propri\'et\'e d'une classe de surfaces
alg\'ebriques, {\it Comptes Rendus}, 1893).}: the author proves that the
possess of such a system of curves yields the existence of simple integrals
of the first kind (Picard integrals of total differentials) attached to the
surface. C.~Segre called the attention of his colleagues to this Note and
after a while this subject became a field of research of the Italian school
of geometry. Already in 1896 Enriques gave an indirect
contribution\footnote{Un'osservazione relativa alla rappresentazione
parametrica delle curve algebriche. {\it Rendic.\ Circ.\ Mat.\ di Palermo},
1895.} proving that ``a rational series of curves on a surface is always
contained in a linear system''.

In 1899 Enriques starts the geometric study of the surfaces containing
continuous systems $\{C\}$ of inequivalent curves\footnote{Una propriet\`a
delle serie continue di curve appartenenti ad una superficie algebrica
regolare. {\it Rendic.\ Circ.\ Mat.\ di Palermo}, 1899.}, proving that they
are irregular. The author shows that the adjoint system of a linear system
$|C|$ contained in $\{C\}$ cannot cut out on a $C$ the complete canonical
series. When he communicated, before the publication, this result to
Castelnuovo, the latter replied by giving another proof of the theorem, in
which one shows that the characteristic series of $|C|$ is necessarily
deficient. And in the Note published by Enriques, he gives also an account
of the proof of his colleague, after having treated his own proof.

After this (in 1901) Enriques himself -- having in mind the objective of
understanding the connection between the geometrically defined irregular
surfaces ($p_g-p_a>0$) and those possessing simple integrals of the first
kind -- proved that ``the surfaces which possess $q$ simple integrals of the
first kind with $2q$ periods, contain continuous systems of inequivalent
curves\footnote{Sur les surfaces alg\'ebriques admettant des int\'egrales
de diff\'erentielles totales de premi\`ere esp\`ece, {\it Ann. de Toulouse},
(2), 3, page~77.}''.

On the other hand Castelnuovo succeeded in proving that ``a surface of
genus $p_g=0$ and $p_a<0$, containing a continuous system of inequivalent
curves, consequently possesses also an irrational pencil of curves''
(Section~\ref{sec:9.11}). This was a property whose meaning depended on
further possible developments of the theory, and Castelnuovo communicated it
to Enriques, who gave it publicity only later (1904), when he could deduce
from it that ``any surface with $p_g=0$ and $p_a<0$ always possesses such an
irrational pencil''.

Meanwhile, in 1902, F.~Severi, who had recently graduated at the University
of Torino, following advice of his teacher C.~Segre came as an assistant to
the chair of geometry at the University of Bologna, which at that time was
held by Enriques, in order to get in closer relationship to those who
studied the theory of surfaces. Enriques soon attracted the attention of the
young geometer to the subject of the continuous systems; he especially
suggested to him the study of a particular family of surfaces containing
such systems, that is, the surfaces that represent the pairs of points of a
curve; Severi in fact devoted a paper, published in 1903\footnote{Sulle
superficie che rappresentano le coppie di punti di una curva algebrica. {\it
Atti Acc.\ Torino}, Gennaio 1903. It is worth saying that, independently
from Severi, also M. De Franchis meanwhile published a paper Sulla variet\`a
$\infty^2$ delle coppie di punti di due curve o di una curva algebrica. {\it
Rendic.\ Circ.\ Mat.\ di Palermo}, 1903.}, to the study of these surfaces.
The study of the surfaces representing the pairs of points of a curve gave
the appropriate examples which shed light on some general questions: in
particular Severi encountered there the problem of the basis, which later on
has been the object of some of his most important studies. Meanwhile it gave
him the chance of introducing the more general concept of characteristic
series.

The idea was contained in germ in the above-mentioned proof that
Castelnuovo communicated to Enriques, about the deficiency of the
characteristic series of a $|C|$ contained in a $\{C\}$ formed by
inequivalent curves; here in fact one sees that the deficient series of
$|C|$ can be enlarged with the limit of the characteristic series of an
infinitely near linear system. More precisely Castelnuovo pointed out
in an explicit way this concept by referring to the (hyperelliptic)
surfaces which represent the pairs of points of a curve $L$ of genus 2;
on these surfaces one has a continuous system of curves $C$ of genus two,
which is of degree two and contains the images of the series of pairs
of points of $L$ with a fixed point: now a curve $C$ of such a system is
isolated, so that the linear system which should cut out on it the
characteristic series is missing; but this characteristic series, which
(since $p_g=1$ and the canonical curve is of order zero) is the canonical
$g^1_2$, is cut out on the aforementioned $C$ by the infinitely near curves
in $\{C\}$.

Castelnuovo communicated this idea to Severi, who was induced to
develop it in general; first of all he pointed out that the
characteristic series of a $C$ may be defined, independently on the
existence of a larger linear system containing $C$, as the difference
 \[
(C+D)C-DC,
 \]
 where $|D|$ is any auxiliary linear system (such that $|C+D|$ cuts out on $C$ a complete series). In fact this difference series does not depend on
the choice of the auxiliary system $|D|$\footnote{Severi, Osservazioni
sui sistemi continui di curve appartenenti ad una superficie algebrica.
{\it Atti Acc.\ Torino}, Febbraio, 1904.}. As we see, here we
can find the geometric or functional interpretation of the procedure
which was already used by Enriques to define the virtual degree of the
curve $C$, just by considering $|C|$ as a difference of two linear
systems of curves.

Once the characteristic series of a curve $C$ has been thus defined,
independently of the existence of a larger linear system $|C|$, there is
no difficulty in understanding that ``if $C$ belongs to a continuous system
$\{C\}$, then the curves of this system which are infinitely near to $C$ cut
out on it groups of the characteristic series''. Therefore Enriques' proof of
the theorems that we have recalled above was reduced to its simplest form.

After the result was proved to the effect that the possess of a
continuous system of inequivalent curves yields the irregularity of a
surface ($p_a<p_g$), the problem naturally arose of proving the converse
of the theorem, that is, of establishing that the possess of such
continuous systems is the characteristic property of the irregular
surfaces. But more than that, this problem took a more precise form. In
fact the geometers, in force of a number of examples, had departed from
the hypothesis implicitely assumed by Noether of the completeness of
the characteristic series of complete linear systems. Thus they
naturally wondered whether the characteristic series of a larger
continuous system were in fact complete. But the point was to find a
way of discussing the question; a criterion in fact was needed in order
to count in some way the conditions for the reducibility of a curve
contained in a linear system. Enriques (1904) found such a criterion in
the degeneration or reducibility criterion: if an irreducible curve $C$,
varying with continuity, becomes reducible in two components, these
must have at least a common point of connection, which is not the limit
of a variable double point of $C$.

{From} the topological point of view this is completely evident: in fact by
continuity the Riemann surface of $C$, which is connected, has to stay
connected in any passage to the limit which leads to a reducible
C\footnote{Only recently a remark by Noether was pointed out in which the
aforementioned principle was anticipated. In the memoir Ueber die
reductiblen algebraischen Curven (Acta Math., 1896), he develops the theory
of the series of groups of points (cut out by linear systems of curves or
surfaces) on a reducible, plane or twisted, curve, and points out that for
the twisted curves (!) the theory differs from the case of the irreducible
curves, unless the components have at least a common point.
 The reducibility principle, rather than from a topological point of view,
may be also established in an algebro-geometric way. See for instance
Enriques--Chisini, Libro~$V$, Chapter~III, \S36, Vol.~III, page~405.}.

By means of the reducibility principle Enriques believed he could
prove that ``the complete continuous systems have complete
characteristic series''\footnote{Sulla propriet\`a caratteristica delle
superficie algebriche irregolari. {\it Rendic.\ Acc.\ Bologna}, dicembre
1904.} so that a regular nonspecial linear system (with characters $n$ and
$\pi$), on a surface of irregularity $q=p_g-p_a$, belongs to a continuous
system of dimension $p_g+n-\pi+1$, thus composed of $\infty^q$ inequivalent
linear systems. In order to do this, he used a representation of the surface
on a multiple plane which -- for the proof itself -- is not at all essential;
in fact the argument may be repeated in a substantially identical way on the
surface itself, as was noticed by Severi\footnote{Intorno alla costruzione
dei sistemi completi non lineari, {\it Rendic.\ Circ.\ Mat.\ di Palermo},
1905.}.

Now it is worth pointing out that Enriques' argument contained a gap, as we
explained in Section~\ref{sec:9.5}: the author was not aware of the critical
point which one meets here and he passed, without realizing it, from a
differential property to an integral one. The difficulty, that is, the fact
that the reducibility principle is no longer sufficient for $p_g>0$, has been
discussed in Section~\ref{sec:9.5}, where we pointed out the real meaning of
this principle:
 \begin{proc}
 in order that a curve of a linear system $|C+D|$ becomes reducible into two
curves of the same order as $C$ and $d$ with $m$ common points, one has to
impose (not more than) $m$ conditions;
 \end{proc}
 by contrast, for $p_g>0$ one needs to see that these conditions are, in
general, only $m-p_g$. To show that this holds for the curves infinitely
near to a reducible curve $C+D$ of our linear system just means that one
proves a differential property, which one is not allowed to take in the
global sense. The argument only works for $p_g=0$ and ($p_a<0$), in which
case indeed there is no need to consider the characteristic series.

But the mistake or the gap in Enriques' argument was not noticed, for
fifteen years, by the geometers, and in fact we have to wait until 1921 to
find the criticism expressed by Severi; and at that time the fundamental
theorem, concerning the existence of $\infty^q$ series of inequivalent
curves, had been proved, in a different way, by means of transcendental
techniques, by Poincar\'e (1910)\footnote{H.~Poincar\'e, Sur les courbes
trac\'ees sur les surfaces alg\'ebriques, {\it Ann.\ Ecole Norm.\ Sup.},
page~3 (27).}. In 1904 Enriques' argument seemed to be confirmed also by the
treatment which Severi himself gave of it, which preserves -- as we said
above -- its essential structure\footnote{See the quoted memoir: Intorno
alla costruzione dei sistemi completi non lineari che appartengono ad una
superficie irregolare. {\it Rendic.\ Circ.\ Mat.\ di Palermo}, 1905.}.

Therefore the theorem, accepted in this way by the geometers, became
the starting point for important developments, which essentially
proceeded in two directions:

 \begin{enumerate}
 \renewcommand{\labelenumi}{(\arabic{enumi})}
 \item Enriques first of all attacks the classification of surfaces with $p_g=0$, and then the more general problems of the classification of
surfaces, which form the subject of the next two chapters X and XI.
Luckily enough these developments are mainly based, or can be based, on
the restricted fundamental theorem.

This is a priori clear for Chapter~X concerning the surfaces of genus $p_g=0$, but it holds also for the results about the surfaces of
arithmetic genus $p_a<0$, which we treat in Chapter~XI, where the proof
should anyhow be slightly modified in some points. Exceptions are given
by the surfaces with $p_a=0$ and $p_g=1$: these surfaces cannot have a
canonical curve of order zero and always possess a pencil of elliptic
curves.

 \item On the other hand, Castelnuovo and Severi turned to draw the
consequences of the fundamental theorem in view of the transcendental
theory: the question here was to clarify from a quantitative point of
view the connection between the irregular surfaces and those possessing
simple integrals of the first kind, upon which the mentioned theorem
shed some light. Concerning this we have already mentioned the first
attempt we made in order to prove the converse of Humbert's remark,
that is, the theorem that the surfaces possessing $q$ simple integrals of
the first kind with $2q$ periods contain continuous systems of inequivalent
curves and are therefore irregular. This theorem would have acquired a
larger significance if Picard's transcendental theory enabled us to know a
priori that the number of the aforementioned integrals of the first kind
attached to any surface is always equal to one half of the number of the
periods.
 \end{enumerate}

In the absence of this result, already before Enriques' note Sulla
propriet\`a caratteristica, etc., Severi investigated in a different way the
question about the surfaces possessing simple integrals of the first kind,
proving that ``the possess of integrals of the second kind (and therefore of
the first kind) yields that the characteristic series of a complete linear
system is deficient, and so also that the surface is
irregular''\footnote{Sulle superficie algebriche che posseggono integrali di
Picard della seconda specie, {\it Rendic.\ Acc.\ Lincei}, 1904. See
{\it Math.\ Ann.}, 1905.}. Although this proof establishes a fundamental fact
and contains an interesting analysis concerning the singularities on the
curve of poles of the integral, it does not seem possible to complete it in
order to find the number of the integrals of the first kind in terms of the
irregularity, without adding to it some essential argument\footnote{The
analysis of this question leads Severi to a result (Severi, Sulla differenza
fra i numeri degli integrali di Picard della prima e della seconda specie
appartenenti ad una superficie irregolare, {\it Atti Acc.\ Torino}, 1905).
The same results were obtained by Picard by his methods (Comptes rendus,
1905): see the Trait\'e by Picard and Simart, II, page~417.}.

On the other hand the consideration of the $\infty^q$ continuous systems of
inequivalent curves enabled Castelnuovo and Severi\footnote{G. Castelnuovo,
Sugli integrali semplici appartenenti ad una superficie irregolare, {\it
Rendic.\ Acc.\ Lincei}, Maggio-Giugno, 1905, F. Severi, Il teorema di Abel
sulle superficie algebriche, {\it Ann.\ di Mat.}, agosto, 1905.},
independently of one another, to refine Humbert's remark from a quantitative
point of view, proving that ``the irregular surfaces, with irregularity
$q=p_g-p_a$, possess exactly $q$ simple integrals of the first kind, with
$2q$ periods''. The proof of this essential theorem, for which the
consideration of the Fuchs' linear equation used by Picard is not
sufficient, is also achieved by Poincar\'e in a different way in the quoted
memoir; there he rediscovers, as we have already said, also the existence of
the $\infty^q$ continuous systems of inequivalent curves.

For the proof of the theorem on the integrals that we have mentioned above,
Castelnuovo introduces the important concept of the variety he called the
Picard variety attached to an irregular surface (see Section~\ref{sec:9.3}).
This notion enables one to construct in general the continuous system
defined by a curve $C$, starting from any other continuous system $\{D\}$
belonging to the surface. Enriques remarked that, in force of the theorem of
Riemann--Roch, this construction more precisely yields that ``a linear
system of curves with characters $n$, $\pi$ and $i$, which is arithmetically
effective, that is, such that $p_a+n-\pi+1-i\ge0$, always belongs to an
$\infty^q$ series of inequivalent linear systems'' (see
Section~\ref{sec:9.2}). This remark was communicated to Severi by the
author, and since the former had succeeded just at that time in remarkably
simplifying the proof of the theorem of Riemann--Roch, Enriques left it to
him to publish it\footnote{Severi, Sul teorema di Riemann--Roch e sulle
serie continue di curve appartenenti ad una superficie algebrica, {\it
Rendic.\ Acc.\ Torino}, 1905.}.

We come now to the criticism which revealed the gap in the proof of
Enriques' theorem about the continuous systems. As we have said, this was
made by Severi in 1921. It may be found in a series of Notes ``Sulla teoria
degli integrali semplici di prima specie appartenenti ad una superficie
algebrica''\footnote{{\it Rendic.\ Acc.\ Lincei}, 1921.}, in which one finds
the treatment, with some simplification at some points, of Poincar\'e's
proof of 1910, including the existence of the continuous systems and of the
integrals of the first kind\footnote{Poincar\'e's proof, with further
simplifications by Lefschetz, has been treated by O.~Zariski, Algebraic
Surfaces (Berlin, Springer, 1935). See VII, 4, 5 (page~128, 133).}. The
author remarks that (thanks to Poincar\'e) at least the essential part of
the theorem (the fundamental theorem as stated here in
Section~\ref{sec:9.5}) holds. Later on there will be other attempts to
extend this theorem by proving that the characteristic series of any
complete continuous system is always complete; will will discuss this later
(Section~\ref{sec:9.7}).

Enriques tried, in a few Notes in 1936-1937, Curve infinitamente vicine
sopra una superficie algebrica\footnote{{\it Rendic.\ Acc.\ Lincei}, aprile
1936 and {\it Rendic.\ Sem.\ Mat.\ R. Universit\`a di Roma}, 1936.} and
La propriet\`a caratteristica delle superficie algebriche irregolari e le
curve infinitamente vicine\footnote{{\it Rendic.\ Acc.\ Lincei}, ottobre
1937.}, to fill the gap which had been pointed out in the proof of the
fundamental theorem. He tried to prove that, given a curve $C$, (if the
characteristic series of $C$ has deficiency $q=p_g-p_a$) then not only there
are $\infty^q$ inequivalent curves infinitely near to it in the first order
neighbourhood, but also other successive curves in the second order
neighbourhood etc. The fundamental idea of this proof (which already appears
in the {\it Lezioni} by Enriques and Campedelli), relies on the fact that
``starting from two infinitely near points on an Abelian variety one
defines, as Lie does, an infinitesimal transformation which generates an
$\infty^1$ group.

Thus, in particular, in the Jacobi variety formed by all groups of $p$ points
of a curve of genus $p$, one can define, starting with two infinitely near
groups, an $\infty^1$ analytic series of elements (groups of points). This
construction can be extended to the continuous systems of inequivalent
curves on a surface, if one assumes the existence of a system whose elements
form the Picard (Abelian) variety. Enriques' considerations tend to justify
this construction, passing from infinitely near curves in the first order
neighbourhood to curves in the second order neighbourhood and so on; and this
independently of the assumption of the existence of the Picard variety, but
rather with the purpose of justifying its construction. The starting point is
therefore that, near to a reducible curve $C+D$, and in its first order
neighbourhood, there are curves $\Cbar+\Dbar$ which are reducible into two
components, one of the same order as $C$, the other one of the same order as
$D$, having the same number $m$ of common points. Here, as the author points
out, first of all one has to justify an implicit application of the
infinitesimal reducibility principle; but this is not a difficulty, inasmuch
as one may see that, by getting out of the surface, one may construct, in
the space, a series of reducible curves which contains
$C+D$ and $\Cbar+\Dbar$.

Once the effectiveness of the reduction process for the infinitely near
curves, and thus the effective existence of a inequivalent $\Cbar$
infinitely near to $C$ has been justified in this way, Enriques'
construction gives a series of operations on $C+D$ and $\Cbar+\Dbar$ which
turns out to depend on $C$ and $\Cbar$ only, and just leads to the extension
of the series of curves defined by $\Cbar$ in the successive neighbourhoods
of $C$.

On occasion of a Note in 1937, this proof was disputed by B.~Segre.

In any case Enriques (replying to Segre's objections) was willing to prefer
another proof, proposed by Segre himself, simply based on the continuity
principle, whose foundation anyhow was still to be discussed: we will
expressly discuss this interesting attempt and its developments for the
contribution of other geometers in the next Section~\ref{sec:9.7}.

Only after the failure of the aforementioned attempt we have taken up again,
as it was natural to do, the criticized proof, which -- with some explanation
and clarification -- we have treated in Section~\ref{sec:9.5} above.

 \section{On various attempts to prove and extend the fundamental
theorem}\label{sec:9.7}
 \markright{\hfill Attempts to prove and extend the fundamental theorem
\quad 9.7\quad}
 The proof of the fundamental theorem, which expresses the
characteristic property of the irregular surfaces, was obtained in
general ($p_g\ge0$) by means of some infinitesimal considerations which
take the place of a differential computation. Considerations of this
type, even if one remains in the field of the algebraic analysis, are
rather delicate and raise doubts in those mathematicians who are not
sufficiently familiar with them, or at least they give the impression
that, in contrast with the need of logic rigour, some intuitive element
has been introduced.

This impression was even better understood in view of some explanation a
little bit too fast and not so well justified in the quoted notes: whose
treatment\footnote{Especially following the way shown in the Notes of
1936.} we have taken up here again trying to explain and clarify the most
delicate points. For these reasons it appeared, and it may still appear,
desirable to give other proofs of the same theorem, which could be accepted
by everybody, without doubts and hesitations.

We have already said that such a proof, which makes use of the
integrals belonging to the surface, was given by Poincar\'e (1910), and
it can be seen today in the simplified treatment by Lefschetz,
reported on by Zariski. But, in the field of the algebro-geometric
theory, the attempt which, as we have said, was undertaken by B.~Segre\footnote{Un teorema fondamentale della geometria sulle superficie
algebriche e il principio di spezzamento, {\it Ann.\ di Mat.}, 1938.}
deserves a special quotation.

The leading idea of this author may be expressed as follows. One
considers the linear system $|E|=|C+D|$ which contains the reducible
curve $E_0=C_0+D_0$, having $m$ double points forming the group $C_0D_0$; and
one imposes on the curves $E$, near to $E_0$, $m-p_g$ double points (near to
$m-p_g$ points of this group); one wants to prove that the curves $\Ebar$ thus
defined are necessarily reducible into a curve of the same order as $C_0$
and a curve of the same order as $D_0$: and in order to do so it is
convenient to deduce a contradiction from the hypothesis that the
aforementioned curves $\Ebar$ are irreducible. Let us therefore construct on
a curve $\Ebar$ with $m-p_g$ assigned double points the complete
characteristic series $g$ of the continuous system $\{\Ebar\}$, and let us
pass to the limit for $\Ebar\to E_0$; the series $g$ has dimension
 \[
 2p_g+(n-\pi)+(\nu -\rho)+1,
 \]
but, in the limit, it apparently goes to the series $g_c+g_d$ formed by
the groups of the (completed) characteristic series of $C_0$ and $D_0$ with
the addition of the $p_g$ fixed points of the group $C_0D_0$ (the new double
points acquired by the reducible $E_0$); but now the dimension of the
limit series $g_c+g_d$ is
 \[
 2p_g+(n-\pi)+(\nu -\rho)
 \]
one less than the dimension of $g$!

The author claimed that this reduction of the dimension of a series is
absurd, being in contrast with the continuity principle. But Enriques, who
had made an analogous attempt (in the simplest case $p_g=1$) on his own, was
stopped by an embarassing paradox, and he treated the difficulty he had
met\footnote{Enriques, Sulla propriet\`a caratteristica delle superficie
irregolari, {\it Rendic.\ Acc.\ Lincei}, Giugno, 1983.}. Let us consider, in
the plane, the system of curves $C_8$ of order 8 with 15 double points; among
these $C_8$ there are the reducible curves formed by two double quartics
$C_4+\Cbar_4$, which have 15 common points $A_1,\dots,A_{15}$ limit of the
double points of $C_8$ and another sixteenth point $O$. Now on a variable
$C_8$ the canonical series $g=g_{10}^5$ is cut out by the quintics $C_5$
through the 15 double points; in the limit $C_8\to C_4+\Cbar_4$ the
aforementioned canonical series is nothing but the series which is composed
by the groups of the two canonical series $g^2_4$ on $C_4$ and $\Cbar_4$,
cut out by the curves $C_5$ through $A_1,\dots,A_{15}$, which consequently
have the common fixed point $O$; so it seems that in the limit the above
series $g$ goes to a series of lower dimension. The doubt arises whether, as
it seems to follow from the continuity principle, the dimension is preserved!

Thanks to Segre, who explained the paradox, this difficulty has been
clarified. The principle of conservation of the dimension of a linear series
actually follows from considerations of continuity and can be explained in
an explicit and irrefutable way; but when one looks for the limit of the
series $g$, as $C_8\to C_4+\Cbar_4$, one has to consider not only what it becomes of $g$ as an adjoint curve $C_5$ of $C_8$ tends to
a generic $C_5$ through $A_1,\dots,A_{15}$, but also the limit of $g$ in
correspondence with a variation in which $C_8$ and $C_5$ both tend to
reducible curves containing $C_4$ (or $\Cbar_4$) as a part, and where the
section of $C_8$ with $C_5$ becomes undetermined. In other words: the
paradoxical example brings to light the necessity of considering, when the
series $g$ goes to the limit, the whole limit!

B.~Segre well explains the importance of this necessity by means of the
example of a varying hyperbola which tends, in the limit, to the line pair of
its two asymptotes: the $g_2^1$ cut out on the hyperbola by the lines through
the centre, reduces to the fixed point which is the centre, but also to the
two symmetries with respect to the centre itself which one has on the
asymptotes. In a similar way, if one takes into account the fact that the
curves $C_8$ passing through $A_1,\dots,A_{15}$ on $C_4$ do not pass through
$O$ as a consequence, then the limit of the $g=g_{10}^5$ in Enriques'
example is given by two series of dimension 5 which are given by associating
groups of $C_4$ and of $\Cbar_4$. Just to describe one of these series, we
may say that it can be constructed by associating with the canonical groups
of $\Cbar_4$ the groups of a $g_6^3$ on $C_4$, that no longer has $O$ as a
fixed point, but is defined by setting $g_6^3=g_4^2+2O$.

After having explained in this way the embarassing paradox, B.~Segre saw a
gap in his own proof of the fundamental theorem, and thought about the
problem of filling it. He meant to do this by looking for a criterion for
the determination of the limit of the intersection of two curves which, as
they vary on a surface, come to acquire a common part. He gave an account of
this infinitesimal criterion in a Note, Sur un th\'eor\`eme fondamental de
g\'eom\'etrie sur les surfaces alg\'ebriques\footnote{{\it Comptes rendus}, 9
Janvier 1939.}, which was accompanied by another Note by
Enriques\footnote{Sur la propri\'et\'e caract\'eristique des surfaces
alg\'ebriques irr\'eguli\`eres. Comptes rendus, 3 Janvier 1939.} who was
glad to welcome the success of the Colleague.

Meanwhile B.~Segre left Italy and, as far as we know, he has not published
so far any more detailed paper on this subject.

Successively Severi made a critical revision\footnote{La teoria generale dei
sistemi continui di curve sopra una superficie algebrica. {\it Mem.\ Acc.\
d'Italia}, Vol.~XIX, 1941.} of the whole theory of the continuous systems of
curves on the algebraic surfaces. Unluckily this treatment is affected by
some mistakes, which were partly pointed out by Enriques in a Note, Sui
sistemi continui di curve appartenenti ad una superficie algebrica, and
which we need to consider here.

In order to complete the proof following Segre's argument, as we explained
above, Severi takes up again from Segre -- in a slightly modified version -- a
lemma of differential geometry, to the effect that, in passing to the limit
$\Ebar\to E_0=C_0+D_0$, the (complete) characteristic series $g$ of
the continuous system $\{\Ebar\}$ on $\Ebar$ necessarily has the $p_g$
points $P$ of the group $C_0D_0$ which are not in the limit of the $m-p_g$
double points of $\Ebar$ as fixed, and this no matter how the passage to the
limit is made, even if the curves which cut out the $g$ on $\Ebar$ tend to
curves having $C_0$ as a part, so that $\lim g$ is undetermined. The lemma of
Segre--Severi which would enable us to draw this conclusion expresses a
differential geometry property, that is, that the aforesaid points $P$ should
be the limits of points belonging to groups of the series $g$. This property
should in turn be a consequence of the fact, which was pointed out by
Enriques in the original proof of the fundamental theorem, that ``the
infinitely near curves to $C_0$ cut out on it the complete characteristic
series''; in this way the gap in the proof of that theorem would have been
filled.

Now Enriques, who had already found some difficulties in reconstructing
Segre's infinitesimal criterion on the basis of the indications which the
Author had given to him, was not able to understand this point and therefore
declared the lemma to be ``obscure and unconvincing''\footnote{{\it Comm.\
Math.\ Helvetici}: presented for publication on the 20th of March, 1942.}.
But he believed it might be possible to accomplish the proof by realizing the
series $\gbar$ on the variable $\Ebar$ as cut out by curves which stay
irreducible in the limiting process which one considers.

In this he was wrong and Severi was right in pointing out his
mistake\footnote{Intorno ai sistemi continui di curve sopra una superficie
algebrica. Comm.\ Math.\ Helvetici (30 Novembre 1942). Enriques was not
allowed by the direction of the Commentarii to reply to the latter article,
and therefore he could neither admit his mistake, nor point out the mistakes
which, in his opinion, Severi had made.}. But the understanding of this
mistake throws a shadow of doubt on the whole proof proposed by Segre and
Severi. The point is not only to correct the wrong lemma in Severi's Memoir,
as the Author himself\footnote{Note presented to the R. Acc.\ d'Italia, 16th
of January, 1942.} did after an example pointed out by BOMPIANI, but rather
to ask whether a differential property as it is expressed in the aforesaid
lemma actually suffices to exclude that the limit of $g$ can give rise on
$C_0$ to a fixed point free series. If one makes a careful analysis of the
question then one will be easily convinced that this doubt cannot be
removed, and, more than that, that the lemma itself appears here to be
completely inadequate, that is, that ``this lemma of Segre--Severi does not
answer the questions for which it has been introduced''. We do not insist on
this, since we believe that Severi himself has recognized the inadequacy of
his argument by now. In any case it is worth mentioning here an idea, which
-- if not today -- maybe tomorrow could have a fruitful application.

We have mentioned the fact that Severi (and B.~Segre before him) had
tried to extend the fundamental theorem on the irregular surfaces, by
proving that on any surface ``the characteristic series of an
irreducible continuous system without multiple points (however special
or superabundant) is always complete''.

Anyhow at the present time the Author recognized, with examples, that
certainly there are exceptions to more general result he had in
mind\footnote{And there are also exceptions to the uniqueness of the
continuous system to which a given irreducible curve belongs. The Author
believed he could prove this with the argument at the end of page~378
and at the beginning of page~379 of the memoir La teoria generale ecc.,
of 1941.}.

But we may restrict, as it is necessary, the choice of the continuous
systems of irreducible curves $\{C\}$ in such a way that the property holds
just by hypothesis: so let $\{C\}$ be an $\infty^{d+1}$ continuous system
having on $C$ a complete characteristic series of dimension $d$; it is
worth explaining the various cases which may occur.

The system $\{C\}$ certainly contains an infinite series of inequivalent
linear systems of curves, provided that $d+1$ is greater than the dimension
of the complete linear system $|C|$, that is, that this system of dimension
 \[
r=p_a+n-\pi+1-i+\om \quad \hbox{(with $\om\ge0$)}
 \]
 has a deficient characteristic series, with deficiency
 \[
q-\ep \quad (q=p_g-p_a).
 \]

Now we will distinguish between the elementary exceptional cases
which, together with all their combinations, give rise to all the
possible cases. It may a priori happen:

 \begin{enumerate}
 \renewcommand{\labelenumi}{(\arabic{enumi})}
 \item that the system $|C|$ is regular and special, with index of
speciality i:
 \[
0<i<p_g.
 \]
In the above notation:
 \[
 \om=\ep=0;
 \]

 \item that $|C|$ is nonspecial and superabundant, with superabundance
$\om>0$, whereas its characteristic series has the maximal deficiency
$q=p_g-p_a$: $i=0$, $\ep=0$.

 \item that $|C|$ is nonspecial and superabundant with superabundance
$\om=\ep$, and has the characteristic series with deficiency $q-\ep$:
$i=0$, $\ep>0$.
 \end{enumerate}

In case (1) (if it occurs) the dimension of $|C|$ is
 \[
r=p_a+n-\pi+1-i,
 \]
and that of $\{C\}$ is $r+q$, therefore the continuous system is formed by
$\infty^q$ linear systems with (in general) the same index of speciality.

In case 2), since the dimension of $|C|$ is $r=p_a+n-\pi+1+\om$, the
dimension of $\{C\}$ is again $r+q$, so $\{C\}$ is formed by $\infty^q$
linear systems with the same superabundance.

But in case 3), since the dimension of $|C|$ is $r=p_a+n-\pi+1+\ep$,
the dimension of $\{C\}$ turns out to be $r+q-\ep$, and we cannot say a
priori whether, for instance, $\{C\}$ is formed by $\infty^q$ regular linear
systems generically having dimension $r-\ep$, among which there is the
superabundant $|C|$ of dimension $r$, or by an $\infty^{q-\ep}$ series of
linear systems all with the same superabundance $\om=\ep$.

We have just mentioned only the extremal cases among all those which in the
hypothesis 3) appear to be a priori possible. The effective possibility that
these different assumptions occur, and in particular the first one (which by
a very hidden mistake was at a first moment excluded) was recognized by
Enriques in the quoted critical Note\footnote{Comm.\ Math.\ Helvetici, 20th
of March, 1942.}, where the example which we briefly mention here is
carefully studied.

Let us consider the scroll $F_5$ of the 5th order with a triple and a
double line: $F_5$ has geometric genus $p_g=0$, arithmetic genus $p_a=-p=-2$
and irregularity $q=2$.

On $F_5$ the plane section system $|C|$ is superabundant of
superabundance $\ep=1$, since it has degree $n=5$, genus $\pi=2$, and
dimension $r=p_a+n-\pi+1+\ep=3$, while the deficiency of the
characteristic series is $q-\ep=1$.

The linear system $|C|$ is contained in a continuous system $\{C\}$ of
dimension 4, and one can see that it is not formed by $\infty^1$ linear
$\infty^3$ system, but by $\infty^2$ linear regular $\infty^2$ systems,
among which there is the above mentioned $|C|$ of dimension 3. In order to
verify this assertion, one may add a pair of generators $a,b$ to the system
$|C|$ and, by means of the linear system thus obtained (which is
$\infty^6$), one may construct a surface $F_9$ of order 9, a normal scroll
in the space $S_6$, of which the $F_5$ appears to be the projection from a
pair of intersecting generators $a',b'$. $F_9$ has a double point at the
intersection point of $a'$ and $b'$, but it has no double curve; therefore
there are at most finitely many pairs of intersecting generators on it and
thus the $\infty^2$ linear systems forming $\{C\}$, that is, the systems cut
out on $F_9$ by the hyperplanes through a pair of generators, are, in
general, of dimension 2. \qed

On the other hand one may show the existence on $F_9$ of a continuous system
$\{L\}$ formed by $\infty^1$ superabundant linear systems, all of dimension
3 ($=2+1$): this is the continuous system cut out by the hyperplanes passing
through the double point $O$ and through a generator of the scroll. And it
should be pointed out that this example has to be considered as contained in
case 3), rather than 2), since the characteristic series of $\{L\}$ is a
nonspecial $g_5^3$.

 \begin{Remark} The $\infty^1$ linear systems $|L|$ forming $\{L\}$ are-on
the scroll $F_5$-residual of the generators with respect to the $\infty^5$
linear system
 \[
 |C+a+b-A-B|,
 \]
that has two base points, $A$ and B, on $a$ and $b$ respectively, forming a
neutral pair for the system
 \[
 |C+a+b|.
 \]

Now, if we replace $a$ and $b$ with two other generators $\abar$ and $\bbar$,
then the pair $AB$ will no longer be neutral for $|C+\abar+\bbar|$, thus by
subtracting a generator from these linear systems one is led to $\infty^2$
linear systems $|\Lbar|=|C+\abar+\bbar-A-B|$.

The two systems, that is, the system $\{L\}$ formed by $\infty^1$ linear
systems and the system $\{\Lbar\}$ formed by $\infty^2$ linear systems, are
distinct, namely $\{L\}+\{\Lbar\}$ is reducible, as a system of curves, in
two $\infty^4$ varieties; by contrast $|\Lbar|$, which contains the
$\infty^1$ variety of the systems $|L|$, has to be regarded as an
irreducible $\infty^2$ variety of linear systems. This circumstance is
expressed by saying that the system $\{L\}$ is exorbitant. The phenomenon of
the exorbitance (either of a linear system or of a continuous system of
linear systems) has been pointed out by Rosenblatt and Albanese, and later
by Severi\footnote{Nuovi contributi ecc., Lincei, 1916.}.
 \end{Remark}

Other examples of exorbitant systems are given by the system of the
plane sections of a completely general cone of order $n\ge3$.

For $n=3$ the $\infty^3$ linear system of the plane sections is contained in
the $\infty^1$ series of ($\infty^2$) linear systems formed by the triples
of generators of the cone, with the neighbourhood of the vertex added.

For $n=4$ the $\infty^3$ linear system of the plane sections of the
cone is contained in the $\infty^3$ series of ($\infty^1$) linear
systems formed by the 4-tuples of generators of the cone, with the
neighbourhood of the vertex added.

For $n\ge5$ the $\infty^3$ linear system of the plane sections of the
cone is contained in the $\infty^n$ series of ($\infty^0$) linear systems
formed by the groups of $n$ generators of the cone, with the neighbourhood
of the vertex added.

 \section{The paracanonical system}\label{sec:9.8}
 \markright{\hfill The paracanonical system \quad 9.8\quad}
 According to the fundamental theorem (Section~\ref{sec:9.4}) and to the
numerical condition of Section~\ref{sec:9.2}, the canonical system $|K|$ of
a surface $F$, of arithmetic genus $p_a\ge0$ and of geometric genus
$p_g>p_a$, is always contained in an $\infty^q$ ($q=p_g-p_a$) continuous
system of distinct linear systems, of dimension
 \[
r\ge p_a+p^{(1)}-1-p^{(1)}+1=p_a
 \]

It was believed that one could say more precisely what the dimension of this
paracanonical system is, by determining the deficiency of the characteristic
series of the canonical system $|K|$, at least under the hypothesis that the
pure canonical curves are irreducible, and with no multiple points, of genus
$p^{(1)}>1$\footnote{Enriques, Des courbes paracanoniques appartenant \'a
une surface alg\'ebrique irr\'eguli\`ere. Soc.\ Royale des Sciences de
Li\`ege (19th of October, 1939).}.

This deficiency should be $q=p_g-p_a$ and therefore the paracanonical
system (supposed to have a complete characteristic series on a given $K$)
would have dimension $2p_g-p_a-1$. But this argument is wrong\footnote{It
seems that essentially the same mistake is contained in the proof of
the same theorem which has been given by Severi in the quoted memoir on
La teoria generale dei sistemi continui ecc., of 1941.} as one may see
by revisiting the Author's argument and bringing into light a rather
delicate objection by which it is invalidated.

Let us indeed consider a regular, nonspecial linear system $|C|$, of
degree $n$ and genus $\pi$ on the surface $F$, whose adjoint system
$|C'|=|C+K|$ has dimension
 \[
 p_a+\pi-1
 \]
and cuts out on $C$ a deficient canonical series, of dimension
 \[
 \pi-1-q.
 \]

We denote by $G=CK$ the group of $2\pi-2-n$ common points of a $C$ and a $K$,
and we want to compute the number of conditions which $G$ imposes on the
curves of $|C'|$ that have to contain it. In order to do this, we assume
that the $C'$ which contain $G$ cut out on $C$ the complete characteristic
series of $|C|$, that is, the characteristic series $g$ of $\{C\}$, that is cut
out on the aforesaid curve $C$ by the (in general inequivalent) curves
$\Cbar$ infinitely near to it. We would like to justify this assumption by
saying that, among the residual curves of a $\Cbar$ with respect to $|C'|$,
there is at least a curve $\Kbar$ near to $K$, so that $C+K$ and
$\Cbar+\Kbar$ give rise to a pencil of curves $C'$ through $G$, passing
through an arbitrarily given group of $g$.

But this is in fact the mistake! If the paracanonical linear systems
have dimension lower than the dimension of the canonical system $|K|$,
that is, $p_g-1$, then a paracanonical system $|\Kbar|$ near to $K$ does not
contain a curve $\Kbar$ near to any given $K$: the limit of a linear system
$|\Kbar|$ is a smaller linear system contained in the canonical one.

So we have explained the mistake which invalidates the above argument, to
the effect that the deficiency of the characteristic series of $|K|$ is
$q=p_g-p_a$. Now we can actually see with examples that the theorem itself
is far from being true. Rather it seems likely that ``on a surface of
irregularity $q=p_g-p_a>1$, ($p_a\ge0$) with irreducible canonical curves,
without multiple points, the $\infty^q$ paracanonical linear systems are, in
general, regular and therefore the continuous system $\{K\}$ is
$\infty^{p_g}$: The characterstic series of $|K|$ (at least if this system
is not exorbitant) should have deficiency 1''.

Let us see that this is verified by the surface which represents the
pairs of points of a quartic curve $L$ of genus $q=3$.

This surface $F$ has the geometric genus $p_g=3$ and the arithmetic genus
$p_a=0$\footnote{See F.~Severi, Sulle superficie che rappresentano le
coppie di punti di una curva algebrica. {\it Atti Acc.\ Torino}, 25th of
January, 1903.}: the canonical curves of $F$ are given by the series of
pairs of points of $L$ taken from the 4-tuples of the special $g_4^1$, and
therefore there are $\infty^2$ of them; by contrast, the paracanonical
curves are given by the series of pairs belonging to 4-tuples of
nonspecial $g_4^1$, and therefore they form an $\infty^3$ continuous system,
consisting of $\infty^3$ inequivalent curves; inside this system one finds
the superabundant system $|K|$.

We can easily verify that the paracanonical systems $|\Kbar|$ have,
in general, dimension $p_a=0$ and not $>0$; otherwise the characteristic
series $g_6^x$ of $\{K\}$ on a $K$ (of genus $p^{(1)}=7$) would have
dimension
 \[
x\ge 3,
 \]
whereas a nonhyperelliptic curve of genus bigger than 4, like $K$,
cannot contain a $g_6^3$.

It seems that what we have proved for the surface representing the
pairs of points of a quartic curve $L$ of genus $q=3$, can be extended to
the surface $F$ which represents the pairs of points of a curve $L$ of
genus $q=4,5,\dots$

For $q=4$, $L$ can be realized as a sextic curve in $S_3$, and the canonical
curves $K$ of $F$ are given by the scrolls which are the intersections of
the congruence of the chords of $L$ with the line complexes of $S_3$, and
therefore one has
 \[
 p_g=6;
 \]
by contrast the paracanonical curves $\Kbar$ of $F$ will be obtained in
correspondence with the series of all the pairs of points of $L$ belonging
to its nonspecial $g_6^1$. If, as we believe, it happens that the $g_6^1$ of
a nonspecial $g_6^2$ correspond to the curves $\Kbar$ of a complete linear
system $|\Kbar|$, then the paracanonical system $\{\Kbar\}$ of $F$ turns out
to have dimension
 \[
 p_g=p_g-p_a+p_a=4+2=6
 \]
and contains $\infty^4$ regular linear systems of dimension
 \[
 p_a=2.
 \]

But the $\infty^5$ canonical system is not contained in this total
paracanonical system $\{\Kbar\}$, but it has to be exorbitant, since the
scroll of the chords of $L$ which belong to a special line complex cannot
be the limit of a scroll whose generators join the pairs of a (nonspecial)
$g_6^1$ of the curve $L$ itself.

Also for $q=5$, when the curve $L$ may be realized in $S_4$ as a curve of
order 8 (the base locus of a net of quadrics) it seems that the surface
$F$, which represents the pairs of points of $L$, possesses a
paracanonical system $\{K\}$ formed by $\infty^q=\infty^5$ regular linear
systems of dimension
 \[
 p_a=5 \quad (p_g=10);
 \]
one has to verify that the $g_8^1$ belonging to the nonspecial $g_8^3$ form
a (nonlinear) system contained in an $\infty^5$ complete linear system.

The examples that we have mentioned, the analysis of which has still
to be completed, suggest various problems about the irregular surfaces,
and in particular ``whether the paracanonical system $\{K\}$ of a surface
with $p_a\ge0$ and $p_g>p_a$, has dimension $p_g$ and consists, in general,
of $\infty^{p_g-p_a}$ regular linear $\infty^{p_a}$ systems''.

But at least the case in which the linear genus is $p^{(1)}=1$, and
therefore the canonical curves, for $p_g>2$, are reducible, and are
composed with the elliptic curves of a pencil, gives rise to an
exception.

In this case one deals with surfaces of irregularity $q=p_g-p_a$, with a
pencil of elliptic curves $k$ of genus $q$; in fact one can prove that that
the continuous system $\{C\}$, to which a curve $C$ belongs, cuts out on the
$k$ (the components of the canonical curves $K$) a series of equivalent
groups: if that were not the case, then one could find, together with the
pencil $|k|$, another elliptic pencil of curves and one could deduce from
this that the $k$ have a constant modulus and $p_a=-1$ (see Chapter~X).

Thus, if $p_a>0$, then the canonical curves of $F$ are composed of
$p_g-1>q-1$ curves $k$, forming a nonspecial series within the pencil:
therefore the paracanonical systems are equally of dimension $p_g-1$, and
they give rise to a superabundant continuous system of dimension
 \[
 2p_g-p_a-1.
 \]

By contrast, if $p_a=0$, then the canonical system $|K|$ is the
$\infty^p_g-1$ linear system formed by the $2q-2=2p_g-2$ curves of the
canonical groups in the pencil $|k|$, and the paracanonical systems have
dimension $p_g-2$, giving rise to a continuous system $\{K\}$ of dimension
 \[
 2p_g-p_a-2.
 \]

 \begin{Remark} It is convenient to add here that for $p^{(1)}=1$ and
$p_a=0$, the case of irregular surfaces (of genus $p_g=1$) with a pure
canonical curve of order zero cannot occur. In fact for a surface $F$ of
genus $p_g=1$ with a canonical curve of order zero, one cannot have
$p_a=0$\footnote{See Enriques, Intorno alle superficie algebriche di
genere lineare $p^{(1)}=1$, {\it Rendic.\ Acc.\ Bologna}, 9th of December,
1906 (n. 1).}. Indeed a linear system $|C|=|C'|$ of genus $\pi$ and degree
$n=2\pi-2$ on $F$ would give rise to an $\infty^1$ continuous series of
linear systems of dimension $\pi-1$, and one of these, different from
$|C|$ and not containing $|C|$, should cut out on a $C$ a nonspecial
series of degree $2\pi-2$ and dimension $\pi-1$.
 \end{Remark}

 \section{A digression on Abelian varieties}\label{sec:9.9}
 \markright{\hfill A digression on Abelian varieties \quad 9.9\quad}
 An Abelian variety $V_q$ (of rank 1) of dimension $q$ is defined as an
(algebraic) variety which admits a parametric representation by means of
(Abelian, that is,) $2q$-periodic functions of $q$ independent variables, a
representation which becomes one-to-one in the prismatoid of the periods: if
these functions are, as we assume, nondegenerate, then $V_q$ has an
$\infty^q$ simply transitive group of permutable transformations without
exceptions: this means that given any two (simple) points of $V_q$, there is
a birational transformation which sends the former point to the latter. The
simplest example of Abelian variety is the variety of the groups of $p$
($=q$) points $G_p$ of a curve $C$ of genus $p$, that is, the Jacobi variety
corresponding to the curve\footnote{See, for instance, Enriques--Chisini,
{\it Lezioni}, Libro~VI, Chapter~III, \S36, Vol.~IV, page~223.}.

This $V_p$ admits a mixed simply transitive group of birational
transformations into itself which is generated by the $\infty^p$ involutory
transformations of the second kind defined by the $g^p_{2p}$ which belong to
the curve; furthermore it has the $\infty^p$ permutable transformations of
the first kind, forming a continuous group $\Ga_p$, which are defined as the
product of an even number of transformations of the second kind. The Jacobi
variety $V_p$ contains an exceptional variety $V_{p-1}$ corresponding to the
series of the special groups $Gp$ on the curve; such a $V_{p-1}$ is
transformed into a $V_{p-2}$ of dimension $p-2$ by the transformations of
first or of the second kind, for instance by the involution between the
groups $G_p$ defined on $C$ by a $g^p_{2p}$\footnote{See Enriques--Chisini,
Lezioni, Libro~VI, Chapter~II, \S27 (vol.~IV, page~143).}.

When among the $p$ independent integrals of the first kind of the curve
of genus $p$ there are $q<p$ of them (linearly independent) reducible with
$2q$ periods, inside the $V_p$ of Jacobi corresponding to the curve one
finds an $\infty^{p-q}$ series of Abelian varieties $V_q$ which form an
imprimitive system for the $\Ga_p$; each of them has in fact an $\infty^q$
simply transitive algebraic group of birational transformations into itself;
moreover $V_p$ also has, by the theorem of Picard--Poincar\'e, an analogous
$\infty^q$ series of Abelian varieties $V_{p-q}$\footnote{See, for instance,
Enriques--Chisini, Lezioni, Libro~VI. Geometric interpretation by
Castelnuovo, Chapter~III, 4, 38, (Vol.~IV, page~234).}.

Abelian varieties $V_q$ of this type, imprimitively contained in a $V_p$ of
Jacobi, are the Picard varieties attached to irregular algebraic
surfaces\footnote{See R.~Torelli, Sulle serie algebriche semplicemente
infinite di gruppi di punti appartenenti a una curva algebrica, {\it
Rendic.\ Circ.\ Mat.\ di Palermo}, t.~XXXVII, 1914 (\S~II, n. 11).}: in
fact, on an irregular surface $F$, with irregularity $q$, the inequivalent
curves of an $\infty^q$ continuous system cut out on a curve of genus $p$
($>q$) a series formed by inequivalent groups of points or by $\infty^q$
linear series, which just form an Abelian variety $V_q$ in the $V_p$ of
Jacobi corresponding to the curve.

The previous remark has a general meaning since one can prove\footnote{See
G.~Castelnuovo and F.~Enriques, Sur les int\'egrales simples\dots, {\it
Ann.\ Ecole Norm.\ Sup.}, s.~3, t.~XXIII, 1906.}, with transcendental
techniques, that any Abelian variety (of rank 1) is a Picard variety
attached to some irregular algebraic surface; so we may state immediately
the theorem of Torelli: There are always Jacobian varieties imprimitively
containing a given Abelian variety.

If one does not like to use the result of Castelnuovo--Enriques, which
was obtained in a transcendental way, then one can make the following
considerations which lead to a theorem less precise than Torelli's
theorem, but sufficient for the further development of our treatment.

We take a more general geometric point of view and we define a $q$ dimensional Abelian variety $V_q$ to be a $V_q$ having an $\infty^q$ simply
transitive group $\Ga$ of birational transformations without exceptions
into itself, thus such that there is one nondegenerate transformation
of $\Ga$ in which any two (simple) points of $V_q$ correspond to each other.

The property of the group $\Ga$, of being simply transitive without
exceptions on the $V_q$, yields as a consequence that: there is no linear
system of dimension>0 of varieties $V_{q-1}$ on $V_q$ which is transitively
transformed into itself by $\infty^r$ ($r>0$) transformations of $\Ga$. In
other words: a variety $V_{q-1}$ in $V_q$, which is not transformed into
itself by a continuous subgroup of $\Ga$, cannot be transformed by $\Ga$ in
the varieties of a linear system $|V_{q-1}|$.

Let us consider, on the Abelian variety $V_q$, a linear system of dimension
$r>0$ of (irreducible) varieties $|V_{q-1}|$, which is transitively
transformed into itself by an $\infty^r$ (algebraic) subgroup of $\Ga$. We
may represent $|V_{q-1}|$ by means of a linear space $S_r$, in which a
transitive continuous group of linear transformations is given. Now we can
find, inside this group, a continuous subgroup $\Ga'$, of dimension $h\ge1$,
which fixes a given line $a$ of $S_r$ and permutes its points in $\infty^1$
ways. Therefore we find in the group $\Ga$ of $V_q$ a subgroup $\Ga_h$ of
dimension $h$, which permutes in $\infty^1$ ways the varieties $\Vbar_{q-1}$
of a linear pencil inside $|V_{q-1}|$ and carries a generic point of a
$\Vbar_{q-1}$ to the points of a $\Vbar_h$ formed by $\infty^1$ varieties
$\Vbar_{h-1}$ belonging to the $\Vbar_{q-1}$ of the pencil. But in this
pencil there is (at least) one $\Vbar_{q-1}$ which is invariant for $\Ga_h$
(corresponding to a fixed point on the fixed line of $S_r$) and therefore the
$\Vbar_h$ has to contain a $\Vbar_{h-1}$ which is invariant for $\Ga_h$: by
associating two points of this $\Vbar_{h-1}$ we have not one, but $\infty^1$
transformations of $\Ga_h$, thus contradicting the assumption that $\Ga$ is
simply transitive, without exceptions, on $V_q$.

Thus there is no linear system $|V_{q-1}|$ on $V_q$ which is invariant for a
continuous subgroup of transformations of $\Ga$, but there may be $|V_{q-1}|$
which are invariant for a finite group, which is a subgroup of $\Ga$ itself.

Now we may choose a $(q-1)$-dimensional variety $V_{q-1}$ in $V_q$ which is
not transformed into itself by any (non-identical) transformation of the
group $\Ga$. Indeed, since $V_q$ can be projectively realized in a suitable
space $S_r$, one of its generic hyperplane sections certainly is not
mapped to itself by any transformation of $\Ga$.

In this setting, the $\infty^q$ series of the transformed varieties of
the above $V_{q-1}$, which we denote by $\{V_{q-1}\}$, forms an algebraic
system which is identical with the group $\Ga$ and therefore also with the
variety $V_q$. In general we may find in the above series a finite number
$s$ of $V_{q-1}$ equivalent to the one we started from. So if we cut the
$V_{q-1}$ of the above series with a generic algebraic curve traced on $V_q$,
then we obtain a series consisting of $\infty^q$ inequivalent groups of
points, which can be represented as an Abelian variety $W_q$ (contained in
the Picard variety of the given $V_q$), that, for $s=1$, turns out to be
identical with the system ${V_{q-1}}$ and with the $V_q$ itself. If, by
contrast, $s>1$, then the mentioned system, and the $V_q$, are represented
on the multiple $W_q$ with multiplicity $s$, since $s$ equivalent $V_{q-1}$
correspond to a single point of $W_q$. If we consider the latter case,
$s>1$, then we have a $(1,s)$ correspondence between $W_q$ and $V_q$, by
means of which we may define an involution $I'$ of order $s$ on $V_q$; this
involution is invariant for the transformations of $\Ga$, and therefore it
has no double points, whose locus would give rise to exceptions for the
simple transitivity of $\Ga$.

In conclusion: an Abelian variety $V_q$ may always be represented
without branch points on another, simple or multiple, Abelian variety
$W_q$ which is imprimitively contained in the Jacobi variety corresponding
to a curve of genus $p$ ($\ge q$).

And conversely: any irreducible variety $V_q$, which can be represented
without branch points on a, simple or multiple, Abelian variety $W_q$
imprimitively contained in a variety of Jacobi, is itself an Abelian variety.

In order to prove this theorem, let us consider the correspondence of type
$(s,s)$, in which a given group $(A,B,\dots)$ of the involution $I'$
corresponds to another group of conjugate points $(A',B',\dots)$; this
correspondence is defined by a transformation of the Abelian group of
$W_q$ which sends its homologous points, $\Abar$ and $\Abar'$, one to
another. The problem is to prove that this correspondence of type $(s,s)$ is
the sum of $s$ birational correspondences, and so, for instance, that the
point $A'$ depends on $A$ not only algebraically, but also in a one-to-one
way, thus rationally. In order to do this, we let the point $A$ vary
continuously on $V_q$ describing a closed cycle $\ga$, and we see that $A'$
moves also continuously, equally describing a closed cycle, and therefore it
cannot be interchanged with, say, $B'$, but it certainly has to come back to
$A'$.

If the cycle described by $A$ is a zero cycle, then our assertion is
obvious, since it cannot enclose any branch point of the $(s,s)$
correspondence and the involution $I_s$ has no double points. But the same
can be proved for a nonzero cycle on the Riemannian space which
corresponds to the variety $V_q$. And the proof stems from a continuity
argument, by taking into account that, if we let $A'$ vary in such a way
that at the end it coincides with $A$, then the correspondence $(AA')$
between the cycle $\ga$ and the path $\ga'$ unambiguously described by the
point $A'$ homologous of $A$, is susceptible of varying continuously until
it becomes the identity; thus if the path $\ga'$ is assumed to be open,
it cannot become closed in the limit, since the conjugate points of the
involution Is are always distinct.

So it turns out that the variety $V_q$ possesses a simply transitive
group of birational transformations into itself, since there is just
one of its transformations carrying $A$ to another point $A'$. In
particular there will be a birational transformation which carries $A$ to
a point $B$ conjugate to it in the involution $I_s$, so the involution will
be generated by a finite group $G_s$ of $s$ transformations.

Now the birational transformations $(AA')$ which we have found on $V_q$ (and
that correspond to the permutable transformations of the $s$-tuple variety
$W_q$) cannot be said, a priori, to be permutable. Since they preserve the
involution generated by the group $G_s$, whose groups correspond to the points
of $W_q$, we may only say that, by transforming one of them, $\pi$, by means
of another one, $T$, one gets a transformation
 \[
 T\Pi T^{-1}=\Pi'
 \]
which differs from $\Pi$ by a cyclic transformation $\Om$ of $G_s$:
 \[
 \Pi'=\Pi\Om.
 \]

But $\Om$ has to remain unchanged as $T$ varies in the continuous group
which contains the identity, so
 \[
 \Om=1.
 \]

We thus deduce that $V_q$ possesses an $\infty^q$ simply transitive group
of permutable transformations into itself, that is, it is an Abelian
variety.

 \begin{NOTE} The transcendental theory of the Abelian varieties $V_q$ is
contained in the classical papers by Jacobi, Riemann and then by
Poincar\'e, Picard; a deeper study of the (hyperelliptic) surfaces -- the case
$q=2$ -- has been made by G.~Humbert and then also by the Italian school.
 \end{NOTE}

 \section{Continuation: the genus of the Abelian varieties}\label{sec:9.10}
 \markright{\hfill Continuation: the genus of the Abelian varieties \quad
9.10\quad}
 We want to complete what we have said about the Abelian varieties, by
computing their (principal, geometric) genus.

We start from the Jacobi varieties, proving that: The Jacobi variety,
$V_p=J_p$, corresponding to a curve of genus $p$, has genus one.

We shall prove that, on the $J_q$ of the $q$-tuples of points of a curve $C$
of genus $p$ ($q\le p$), the variety $K$ of the $q$-tuples which belong to
the groups of a $g_{2p-2}^{q-1}$ contained in the canonical series of $C$,
is a canonical variety. Let us first of all take $q=2$ ($p\ge2$). On the
surface $J_2$ of the (unordered) pairs of points of the curve $C$, one has
$\infty^1$ irreducible curves $M$, which are birationally equal to $C$, each
corresponding to the series of the pairs with a fixed point; thus the curves
$M$ form, on the surface, a system of index two.

Let us take a $g_{2p-2}^1$ contained in the canonical series of $C$ and let
$K$ be the image curve of the pairs which belong to this $\infty^1$ series;
let us determine the intersections of $K$ with a curve $M$. In order to do
this, let us consider the curve $M$ corresponding to the series of the pairs
of points with a (generic) fixed point $A$ of $C$ and let
 \[
 A, A_1, A_2,\dots, A_{2p-3}
 \]
be the group of the $g_{2p-2}^1$ which contains $A$. The points corresponding
to the pairs
 \[
(AA),\quad (AA_1),\dots, (AA_{2p-3})
 \]
belong to $M$, and form a canonical group on it; on the other hand, all
these points, except the first, also belong to $K$. Now the group of these
common points of $K$ and $M$ is residual to the characteristic series of $M$
with respect to the canonical series. Let us in fact consider, together with
$M$, the curve $M'$ which represents the pairs of points of $C$ having
another common point $B$: $M$ and $M'$ intersect at the point homologous to
the pair $(AB)$, and when $B$ indefinitely approaches $A$ this tends to the
point $(AA)$; thus $(AA)$ is a group of the characteristic series on the
curve $M$. Now, since $K$ cuts out on every curve of the system $M$ a
residual group of the characteristic series, $K$ itself will be a canonical
curve of $J_q$ by the third criterion of equivalence (see Chapter~III,
Section~\ref{sec:3.13}). \qed

After having thus proved our theorem for $q=2$, we shall prove it for $q=3$
and so on by induction. In fact, if we assume that the theorem is true for
the variety of the groups of $q-1$ points of $C$ ($q\le p$), we can prove
that the theorem is also true for the $J_q$ of the $q$-tuples.

To see this, let us consider on $J_q$ the $\infty^1$ series of irreducible
and birationally equivalent varieties $M_{q-1}$, each representing the
$q$-tuples with a fixed point: this series has index $q$. Let $K_{q-1}$ be
the variety of the $q$-tuples belonging to the groups of a $g_{2p-2}^{q-1}$
contained in the canonical series of $C$, and let us determine the
intersection of $K_{q-1}$ with $M_{q-1}$. Let $g_{2p-3}^{q-2}$ be the
residual series of the point $A$ with respect to the series that we are
considering. If we add $A$ to the groups of $q-1$ points belonging to the
groups of $g_{2p-3}^{q-2}$, then we obtain a variety $K_{q-2}$ which is
contained in both the $M_{q-1}$ corresponding to the point $A$ and the
$K_{q-1}$ which is the image of the $g_{2p-2}^{q-1}$. Now if we add to
$K_{q-2}$, on the $M_{q-1}$, the variety $M_{q-2}$ of the $(q-1)$-tuples
having $A$ as a fixed point (that is, the variety of the groups of $q$ points
in which $A$ appears twice) we obtain a (impure) canonical variety of the
$M_{q-1}$; this follows from the theorem which we assume to be true by
induction, since this variety corresponds to the series of the groups of
$q-1$ points belonging to the groups of the series
 \[
 g_{2p-3}^{q-2}+A,
 \]
a series having the fixed point $A$ and contained in the canonical
series of $C$.

So $K_{q-1}$ cuts out on $M_{q-1}$ a residual variety of $M_{q-2}$ with
respect to a canonical variety; therefore, by the third criterion of
equivalence (which can be easily extended from the case of the curves to the
case of varieties of any dimension) it suffices to prove that $M_{q-2}$ is a
variety of the characteristic system of $M_{q-1}$. Indeed, the $M_{q-1}$
corresponding to a fixed point $A$ and the $M'_{q-1}$ corresponding to a fixed
point $B$ have the variety $\Mbar_{q-2}$ of the $q$-tuples with the two fixed
points $A$ and $B$ as their common variety, and, when $B$ tends to $A$,
$M'_{q-1}$ becomes infinitely near to $M_{q-1}$ and $\Mbar_{q-2}$ just tends
to $M_{q-2}$.

So the theorem that we have stated is proved, and we may set $p=q$; we have:
on the Jacobi variety, which represents the groups of $p$ points of a curve
of genus $p$, the variety $K_{p-1}$ of the special $p$-tuples is a (a priori
impure) canonical variety. But this is just an exceptional variety, which can
be transformed in a variety of dimension $p-2$ on the variety, birationally
equivalent to $J_p$, which represents the groups of inequivalent $p$ points
on the curve $C$ of genus $p$. Indeed one obtains this transformation by
using the involutory transformation which interchanges on $C$ the groups of
$p$ points which are residual with respect to a $g_{2p}^p$.

Therefore as a consequence we have that the Jacobi variety $J_p$ has genus
one.

The theorem we have established for the Jacobi varieties $J_p=V_p$ can
now be easily extended to the more general Abelian varieties; we
briefly sketch its proof. First of all we refer to a simple $V_q$ imprimitively contained in a
$V_p$ ($q<p$); this $V_q$ belongs to a series of varieties of the same
dimension, a series of index one; thus the (impure) canonical variety of $V_p$
cuts out one canonical variety on each one of these $V_q$. Secondly the
theorem also extends to the case of a $V_q$ which is represented, without a
branch locus, on a multiple variety $W_q$ contained in a $V_p$ (see
Section~\ref{sec:9.9}). In fact the $(1,s)$ transformation of $W_q$ to $V_q$
maps the canonical variety of $W_q$ to a canonical variety of
$V_q$\footnote{Theorem of Painlev\'e--Castelnuovo, proved for $q=1$ in
Enriques--Chisini, Lezioni. Libro~$V$, Chapter~I, \S9 (Vol.~III, page~72).
For $q=2$ see Chapter~V, Section~\ref{sec:5.7} of this treatise. The theorem
may be extended without difficulties for all values of $q$.}.

In conclusion we can state: The Abelian varieties $V_q$ ($q\ge1$) always
have genus one, and their pure canonical variety is of order zero.

 \begin{NOTE} The computation of the genus of the Jacobi varieties $V_p$, and
in general of the Abelian varieties, is immediate from a transcendental
point of view, since one constructs at once a $p$-tuple integral of the
first kind attached to $V_p$. The geometric proof of the theorem that we
have treated above, was given by Severi in a note of 1911\footnote{Sulle
superficie e variet\`a algebriche irregolari di genere geometrico nullo,
{\it Rendic.\ Acc.\ Lincei}, 1911.}.

Another geometric proof of the same theorem was given by
B.~Segre\footnote{Determinazione di certi gruppi covarianti di due o piu'
serie lineari, {\it Rendic.\ Circ.\ Mat.\ di Palermo}, 1932.}; it consists in
showing that the images on the corresponding Jacobi variety of the linear
series $g_{2p}^{p-1}$ contained in a $g_{2p}^p$ on a curve of genus $p$ are
the varieties $V_{p-1}$ of a linear system whose characteristic system plus
the intersection of the image of the variety of the special groups, forms the
canonical system on a $V_{p-1}$. We limit ourselves to explain this fact in
the simplest case $p=2$, but we also point out that the extension of the
argument for $p>0$ is indeed less easy than one could wish.

Let us consider a curve $C$ of genus $p=2$, and a linear series $g_4^2$ on
it: the $g_4^1$ contained in this series give rise, on the Jacobi surface
$V_2$, to the curves of a net, without fundamental curves, which have a
variable intersection $A$ with the exceptional curve $V_1$, image of the
canonical $g_2^1$ of $C$, and a base point $B=V_0$ at the point which is its
transform by means of the involution defined by the $g_4^2$. The canonical
curves $K$ of $V_2$ are, a priori, defined on the surface by the property of
cutting out on a generic $L$ residual groups of the characteristic series
of $|L|$, to which has to be added the intersection $A$ of $L$ with the
exceptional curve $V_1$, or the base point $B$ (that is, the transform of $V_1$).

Therefore one has to prove that the series $A+B+g_6^2$ on a $L$ is
(totally) contained in the canonical series $g_8^4$. And the assertion
essentially derives from the geometric interpretation of the
Zeuthen's formula which gives the relation between the two canonical
series of the curves $C$ and $L$ in a $(2,3)$ correspondence\footnote{This
passage turns out to be very easy (for $p=2$) if one introduces the
auxiliary curve $M$ which represents the pairs of points of $L$ complementary
with respect to a $g_4^1$ contained in the $g_4^2$.}.

Thus the net of the curves $L$ on $V_2$, or rather on a transformed
surface without exceptional curves, turns out to be of genus $\pi=5$, of
effective degree 6 and virtual degree $n=8$, ($n=2\pi-2$), it has two base
points, and canonical characteristic series. The possess of such a net $|L|$, without proper fundamental curves, yields in fact that $V_2$ is a
surface of genus $p_g=1$ with a canonical curve of order zero. \qed
 \end{NOTE}

 \section{Irrational pencil on the surfaces with geometric genus
zero}\label{sec:9.11}
 \markright{\hfill Irrational pencil on the surfaces with geometric genus
zero \quad 9.11 \quad}
 {From} the fact that an Abelian variety $V_q$ of dimension $q\ge2$ has
geometric genus equal to one, one deduces that ``any non-exceptional surface
belonging to an Abelian variety $V_q$, has the geometric genus $p_g>0$''.

For $q=2$ this is evident, since $F$ coincides with $V_2$. To prove the
theorem in general one needs to remark that, for $q>2$, ``$F$ is contained
in a continuous system of analogous surfaces on $V_q$, of dimension at
least $q-2$''.

In fact, if one applies to $F$ the $\infty^q$ transformations of the group
$\Ga$ of $V_q$, one has, in general $\infty^q$ surfaces analogous to $F$;
the case in which $F$ admits a continuous series of transformations into
itself gives an exception; but in any case (by the simple transitivity of
the group $\Ga$) the transformations of $F$ will be at most $\infty^2$, and
therefore one has at least $\infty^{q-2}$ surfaces which are transformed of
$F$.

With this in mind, the above-stated property that $F$ is of positive
geometric genus is a consequence of the general property that: on a
variety $V_q$ of dimension $q$, which has geometric genus $P>0$, any surface
which belongs to a continuous system of dimension at least $q-2$, has the
genus $p_g>0$.

Let us prove this proposition referring first of all to the case $q=3$,
then to the case $q=4$, etc.

If we assume $q=3$, then the canonical surface which by hypothesis
exists on $V_q$ (even if it is just a single surface of order zero)
intersects $F$ along a curve which belongs to the residual system of the
characteristic system of $F$ with respect to the canonical
system\footnote{See Chapter~VIII, Section~\ref{sec:8.9}.}; this suffices to
prove the existence of the latter, that is, that the genus of $F$ is $p_g>0$.

If $q=4$, then the surfaces $F$, which are at least $\infty^2$, can be
arranged in at least $\infty^1$ simply infinite continuous systems, which
therefore give rise to an at least $\infty^1$ series of varieties $V_3$;
now, since the genus of $V_4$ is bigger than zero, a canonical variety of
$V_4$ cuts out on each one of the aforementioned $V_3$ some special surfaces,
which are residual of the characteristic system of $V_3$ with respect to the
canonical system, thus these $V_3$ have genus P>0. Whence it follows that the
$\infty^1$ surfaces describing them are also of genus $p_g>0$.

With an analogous argument one may pass from $q$ to $q+1$ and prove the
same assertion for a variety $V_q$ of any dimension.

The theorem that we have proved, to the effect that the surfaces belonging
to an Abelian variety have genus $p_g>0$, enables us to make more precise
the statement of the characteristic property of the irregular surfaces for
surfaces with genus $p_g=0$, by proving that: The irregular surfaces of
genus $p_g=0$ possess an irrational pencil of genus $-p_a=p$ of curves. And
conversely, any irregular surface of genus $p_g=0$ possessing an irrational
pencil of genus $p$ ($>0$) is an irregular surface of arithmetic genus $-p$.

Let $F$ be an irregular surface -- for the time being we do not make any
assumption about the values of its characters -- and let us consider a simply
infinite continuous system of curves $\{C\}$ on it, of index $i>2$, which we
may assume to be simple, that is, such that the $i$ curves $C$ passing through
a generic point of the surface do not have other common points as a
consequence.

Let us denote by
 \[
 C_i=\sum C
 \]
the sum of the $i$ curves of $\{C\}$ which go through a point of the surface
$F$.

Three cases may occur:

 \begin{enumerate}
 \renewcommand{\labelenumi}{(\alph{enumi})}
 \item the $\infty^2$ curves $C_i$ are all equivalent curves on the surface $F$;

 \item the aforementioned $C_i$ are arranged in $\infty^1$ simply infinite
systems of equivalent curves;

 \item the $C_i$ are pairwise inequivalent, or rather each of them is
equivalent to a finite number, $s\ge1$, of others.
 \end{enumerate}

Now we shall prove that in case a) all the curves $C$ are equivalent, that
is, the series $\{C\}$ is contained in a linear system.

In order to do this it suffices to prove that the $\infty^1$ curves $C$ cut
out a series of equivalent groups on any curve $M$ traced on $F$, and use,
for example, the first criterion of equivalence. Indeed the property we
claim for the series cut out on the curve $M$ is a consequence of a well
known equivalence criterion by Severi--Castelnuovo which concerns the
$\infty^1$ series of groups of points belonging to a curve\footnote{See III,
13. See also Enriques--Chisini, Lezioni. Libro~$V$, Chapter~IV, \S42
(Vol.~III, page~483)}.

Secondly we will prove that in case b) an irrational pencil of curves can be
constructed on the surface $F$.

In fact let a point $A$ move on $F$, so that the curves $C_i$ going through
it are equivalent to each other; since the condition that we have imposed is
algebraic, the point $A$ describes an algebraic curve $K$; and it is clear
that the curves $K$ defined in this way on $F$ cannot intersect, thus they
have to form a pencil (which has no base points, simple for the surface). We
assume, for simplicity, that the $K$ are irreducible (since otherwise we may
consider, instead of the $K$, their irreducible components), and we shall
prove now that the pencil $(K)$ is irrational.

Indeed the same consideration made before shows that the curves $C$ of
$\{C\}$ must cut out $\infty^1$ equivalent groups of points on any $K$.
So, by the second equivalence criterion\footnote{See III, 13.}, the $C$ are
equivalent up to curves of the pencil, that is, -- denoting by $C$ and $C_0$
any two curves of $\{C\}$ -- one has
 \[
 C+\sum K=C_0+\sum\nolimits'K
 \]
where the summations $\sum$ and $\sum'$ contain the same number of
summands. Consequently, if the curves $C$ are not all equivalent one to
another, then the pencil $(K)$ cannot be linear, since otherwise by
adding and subtracting the same number of curves $K$ one could pass from
a $C$ to an equivalent $C_0$.

An objection can be moved to this argument, based on the second
equivalence criterion, if the pencil $(K)$ contains some reducible
curves, like
 \[
 K=K_1+K_2 \quad\hbox{and}\quad \Kbar=\Kbar_1+\Kbar_2;
 \]
in fact then we could have, for instance,
 \[
 C_0=C+K_1-\Kbar_1+\cdots.
 \]

But the doubt raised here can be eliminated if one considers that the
curves $C_0$ and $C$ belong to one and the same continuous system $\{C\}$ and,
as $C_0$ varies in $\{C\}$, the components of the reducible $K$, whose number is
finite, stay fixed; whence, by continuously moving $C_0$ to $C$, it
necessarily follows that
 \[
 C_0\to C,\quad K_1=\Kbar 1.
 \]

Finally, in case c) one immediately proves that the genus of the surface
must be $p_g\ge1$. In fact if one represents the curves of $\{C\}$ with the
points of a curve $L$, then the $\infty^2$ curves $C_i$ define on the same $L$
a series $g_i$, which can be considered as a doubly infinite, simple or
multiple, variety, contained in the Jacobi variety corresponding to the
curve $L$, which represents the inequivalent linear series of $L$.

The above discussion makes clear that for an irregular surface $F$ of
genus $p_g=0$ any given series $\{C\}$ on it, of index $i$, formed by
inequivalent curves, gives rise to case b), so that one finds an irrational
pencil of curves $K$ on the surface.

Then we can see that the genus of $(K)$ is equal to the arithmetic
genus with the sign changed: $p=-p_a$. Because, if it were $p<-p_a$, in force
of the second equivalence criterion, one could construct a series of
inequivalent curves on $F$, that would cut out on any $K$ inequivalent groups
of points, and so (since case b) has to occur), another irrational pencil of
curves; but the existence of two irrational pencils, $(K)$ and (M), on $F$
would imply that this surface must have the geometric genus $p_g>0$: this is
evident if the two pencils are formed by unisecant curves\footnote{In fact a
canonical curve of $F$ is obtained by adding to the curves of (M) which pass
through the points of a canonical group of a $K$, the curves of $(K)$ which
pass through the points of a canonical group of a M.}; whereas the case in
which the curves of the two pencils cut each other at a certain number
$s>1$ of points leads to the representation of $F$ on a multiple ($s$-tuple)
surface, with two pencils of unisecant curves, whence it follows, a
fortiori\footnote{See $V$, 7.}, that its genus is $p_g>0$.

In conclusion the characteristic property of the irregular surfaces
of genus $p_g=0$, which consists, according to the above statement, in
possessing an irrational pencil of genus $p=-p_a$, has been proved.

\begin{NOTE} The theorem on the existence of an irrational pencil on the
irregular surfaces of genus $p_g=0$, has been stated by Enriques as a
consequence of the characteristic property of the irregular surfaces,
and precisely in his quoted note in {\it Rendic.\ Acc.\ Bologna} in
1904. Indeed, already in 1900 Castelnuovo remarked and communicated to the
colleague that ``from the existence of a continuous system of inequivalent
curves on a surface of genus $p_g=0$ the existence of an irrational pencil
can be deduced''. This property, recalled by Enriques in his Note, mentioning
its author, was justified with a transcendental argument as follows.

Let $\{C\}$ be an $\infty^1$ continuous system of inequivalent curves, on
the surface $F$ of genus $p_g=0$. Then, as G.~Humbert for the first time
remarked\footnote{Comptes rendus, 1893; J. de Math., s.~4, t.~X, 1894.}, one
can construct some simple integrals (of total differentials) of the first
kind on $F$, by summing the values taken by the Abelian integrals of the
first kind attached to the series $\{C\}$ in correspondence with the
$i$ ($>1$) curves $C$ passing through a given point $A$ of $F$. And these
simple integrals $I_1,I_2,\dots$ will necessarily be one a function of the
other, otherwise -- as was indicated by Noether\footnote{Ueber die totalen
algebraischen Differentialausdr\"ucke, {\it Math.\ Ann.}, Band~29, 1887. See
Picard et Simart, ibidem, Vol.~I, Chapter~V, \S15.} -- one could deduce from
them the existence of a double integral of the first kind, whence $p_g>0$.

After this Castelnuovo remarks that the curves $I=\hbox{const.}$ are composed
with the algebraic curves of an irrational pencil which are defined by
moving $A$ on the surface in such a way that the $i$ curves $C$ going through
a point describe equivalent groups of curves. This argument, especially in
the form that it acquires with the considerations of Castelnuovo's
Note\footnote{Sugli integrali semplici appartenenti ad una superficie
irregolare, {\it Rendic.\ Acc.\ Lincei}, 21st of May, 4th and 18th of June,
1905. See {\it Memorie scelte}, page~473 and following pages (see especially
n. 10, page~490).}, was translated by Severi\footnote{Sulle superficie e
variet\`a algebriche irregolari di genere geometrico nullo. {\it Rendic.\
Acc.\ Lincei}, s.~V, Vol.~XX, 1911.} into the geometric one we treated above.
\end{NOTE}

 \section{A note on the surfaces of irregularity one}\label{sec:9.12}
 \markright{\hfill A note on the surfaces of irregularity one \quad
9.12 \quad}
 In view of possible applications it is convenient to remark that the
arguments which we developed in the previous section also lead to the
characterization of the surfaces of irregularity $q=p_g-p_a=1$; for these
the following theorem holds: The surfaces $F$ of irregularity $q=1$ possess
an elliptic pencil of curves (of genus $\pi\ge0$). Any continuous system
of curves on them is obtained from a linear system by adding to it the
difference of two curves of the aforesaid pencil.

Let us remark that the Picard variety of $F$ is an elliptic curve. Let us
take now on $F$ an $\infty^1$ system of inequivalent curves, $\{C\}$, of
index $\nu>1$: There will be $\nu$ curves $C$ passing through any point $P$
of $F$, and, as $P$ varies, one obtains $\infty^2$ groups of $\nu$ curves;
therefore, starting with a particular one $P=\Pbar$ of these points, there
are $\infty^1$ points $P$ such that the group of $\nu$ curves $C$ passing
through them is equivalent to the group of the curves passing through
$\Pbar$. The locus of these points $P$ is a curve $K$ through $\Pbar$, and
all the analogous $K$ form an elliptic pencil of curves $(K)$ on $F$. Once
the first part of the theorem has been thus proved, in order to prove the
second part it suffices to point out that the curves $C$ of the continuous
system $\{C\}$ cut out on any $K$ equivalent groups, and accordingly (by the
second equivalence criterion) two inequivalent $C$ only differ by the
difference of two $K$.
