 \chapter{Surfaces with geometric genus zero}\label{ch:10}
 \markboth{\qquad 10. \quad Surfaces with geometric genus zero hfill}{}

 \section{Introduction}\label{sec:10.1}
 \markright{\hfill Introduction \quad 10.1\quad}
 The characteristic property of the irregular surfaces of genus $p_g=0$,
which we have established in Section~\ref{sec:9.4} and~\ref{sec:9.11} of
the previous chapter, enables us, as we shall see, to characterize -- in
a complete way -- the surfaces with genus $p_g=0$, and -- in particular
-- to assign the necessary and sufficient conditions defining the family
of ruled surfaces by the vanishing of the plurigenera.

The development of our deductions will proceed in the following order.

First of all we shall treat the case of surfaces which contain systems of
curves of genus $\pi$ and degree $n>2\pi-2$, showing that these can be
transformed into ruled surfaces, either rational or irrational; the
rational case corresponds (as we know) to the surface being regular.

After this, we shall treat a lemma concerning the reducible curves in an
irrational pencil; this will enable us to give to the second equivalence
criterion a more precise and simpler form, which we will need in what
follows.

Whence we are meanwhile going to deduce that the surfaces of genus $p_g=0$
and $p_a<-1$ are certainly birational to ruled surfaces of genus
$p=-p_a$. Then we will be left with the study of the surfaces with the
genera $p_g=0$ and $p_a=-1$, which give rise to:

 \begin{enumerate}
 \renewcommand{\labelenumi}{(\arabic{enumi})}
 \item surfaces birational to elliptic ruled surfaces, and

 \item surfaces which contain an elliptic pencil of curves of genus
$\pi\ge1$.
 \end{enumerate}

We shall deal with the two cases which occur, that is, $\pi>1$ and
$\pi=1$, separately, and we shall prove that, in both cases, the surface
contains another linear pencil of elliptic curves, which are birationally
identical.

Finally we shall go deeper into the analysis of the family of (elliptic)
surfaces defined by the above properties, we shall give their
construction, compute their plurigenera and see that they admit a simply
infinite, elliptic group of birational transformations into themselves,
etc. We will finish the chapter with the characterization of the surfaces
with all plurigenera equal to zero and of those with the plurigenera
equal to zero and one.

 \section{Surfaces containing a system of curves of genus $\pi$ and
degree $n>2\pi-2$}\label{sec:10.2}
 \markright{\hfill Surfaces with curves of genus $\pi$ and degree
$n>2\pi-2$ \quad 10.2\quad}
 Let $F$ be a surface which contains an irreducible linear system of
curves $|C|$, at least $\infty^2$, for which $n>2\pi-2$. The curves $C',C''$,
etc., the successive adjoint curves of $|C|$ cut out on $C$ a decreasing
number of points:
 \[
 2\pi-2,\quad 4\pi-4-n,\quad 6\pi-6-2n,\dots
 \]
and therefore there are a finite number of them; in other words, the
adjunction process starting with the linear system $|C|$ has to terminate
after a finite number of steps, giving rise to a last adjoint system
$|C^i|$, of dimension $\ge0$, which has no further adjoint curves of
order $>0$.

We already know of one case in which the adjunction terminates in this
way: that is, the case of the regular surfaces of genus $p_a=p_g=0$, with
bigenus $P=0$, in which the surface $F$ is known to be
rational\footnote{Castelnuovo's theorem, see Chapter~V,
Section~\ref{sec:5.4}.}. After this case, it is convenient then to
consider the hypothesis that the surface on which our system lies is
irregular, of genus $p_g=0$ and $p_a=-p\le1$, and therefore contains an
irrational pencil of genus $p$ of curves $K$. If the
$K$ have genus $\rho=0$, then $F$ can certainly be transformed into a
ruled surface of genus $p^{(2)}$. Then we need to examine the hypothesis
that the genus of the $K$ is $\rho\ge1$ and derive a contradiction from
this. It will be useful to treat the two cases separately:

 \begin{enumerate}
 \renewcommand{\labelenumi}{(\arabic{enumi})}
 \item $p>1$

 \item $p=1$.
 \end{enumerate}

The first case: $p>1$.

Let us denote by $s$ the number of intersections of the $C$ with the $K$,
and let us notice that the curves $C+pK-p\Kbar$ give rise to an $\infty^p$
series of distinct linear systems cutting out equivalent series on the $K$;
whence it follows that
 \[
s>1.
 \]

Now, since the (virtual) canonical curves of $F$ intersect the curves $K$,
of genus $\rho$, in $2\rho-2$ points, the curves $C'$ intersect the $K$ at
$s+2\rho-2$ points, the curves $C''$ at $s+2(2\rho-2)$ points, and so on.
Finally the curves of the last adjoint system $|C^i|$ cut out on the $K$ a
number of points which a fortiori is $m\ge2$. But this conclusion
immediately yields a contradiction, provided that the $C^i$ are
irreducible: in fact, since they possess an irrational involution
$\ga_m^1$ of genus $p$, which is cut out on them by the $K$, they must have
the (effective and virtual) genus greater than or equal to $m(p-1)+1>p$,
and therefore they must possess an adjoint system of dimension $>0$.

The conclusion (that contradicts the hypothesis we started with) can
be extended to the case in which $C^i$ is a sum of irreducible curves,
since in any case there is a component of $C^i$ which has to cut out at
least two points on $K$: this follows from the remark that the $C^i$ belong
to a continuous series of inequivalent curves, which can be obtained
by adding to them the difference of two curves of the pencil $(K)$:
$K-\Kbar$.

The second case: $p=1$.

We proceed as in the previous case: starting with the system $|C|$ with
characters $\pi$ and $n>2\pi-2$, we obtain a last adjoint system $|C^i|$,
consisting of curves meeting the $K$ at $m\ge2$ points; but it seems here
that the adjoint system of $C^i$ can actually be empty if the genus of
$C^i$ (effective and virtual) is equal to 1, as well as the genus of the
pencil of the $K$. But our hypothesis implies that the curves $C^i$, if
irreducible (or their components), belong to an $\infty^1$ elliptic series
of inequivalent elliptic curves; and since the curves of this series
cannot form another irrational pencil (distinct from $|K|$), we must admit
that the curves $C^i+K-\Kbar$ give rise to a continuous system $\{C^i\}$
of curves of degree $\nu\ge1$.

Whence we can deduce that the surface $F$ possesses $\infty^1$ rational
curves, and therefore it has to be birational to an elliptic ruled
surface, thus contradicting our assumptions. In fact, if $\nu=1$, then the
$\infty^1$ elliptic series of unisecant curves $C^i$ contains $\infty^1$
pairs of equivalent curves, and (by L\"uroth's theorem)\footnote{See
Enriques--Chisini, {\it Lezioni}, Libro~II, Chapter~I, \S3 (Vol.~I,
page~169) and Libro~V, Chapter~I, \S7 (Vol.~III, page~48)} their points of
intersections describe $\infty^1$ rational curves on $F$. If one has
instead $\nu>1$, then the (irreducible) curves $C^i$ with $\nu-1$ fixed
points give rise to an $\infty^1$ series of unisecant elliptic curves, so
that the previous argument may again be applied.

So we have proved the fundamental theorem:
 \begin{proc}
 The surfaces containing an irreducible linear system at least
$\infty^2$, of genus $\pi$ and degree $n\ge2\pi-2$, are birationally
equivalent to ruled surfaces, either rational or irrational.
 \end{proc}

And from this theorem, as we announced in Chapter~IV,
Section~\ref{sec:4.9}, one deduces that:

The surfaces belonging to the family of the ruled surfaces are the only
surfaces containing infinitely many exceptional curves; all the surfaces
outside of this family can be transformed into surfaces without
exceptional curves.

As for the fundamental theorem, it is convenient to add here that it holds
in the most general assumptions, regardless of the dimension of the system
$|C|$ whose characters verify the inequality $n>2\pi-2$, provided that the
curves $C$ do not contain exceptional curves as components.

In order to prove this, let us remark that the character $a(C)=n-(2\pi-2)$
is an additive character of the linear systems to which it refers, and
therefore it cannot be positive for the sum of two systems without being
positive for some of the summands. Thus it suffices to prove that:

Any surface $F$ on which one has an irreducible, non-exceptional curve $C$
of genus $\pi$ and degree $n>2\pi-2$ certainly belongs to the family of
the ruled surfaces.

In fact the $s$-multiple of the curve $C$, with $s$ sufficiently large, has
genus and degree
 \[
 \pi_s=s\pi+\frac{s(s-1)}{2}\,n-s+1, \quad n_s=s^2n;
 \]
 and therefore it has dimension
 \[
r\ge p_a+n_s-\pi_s+1>0,
 \]
 and it forms an irreducible system (for $n>0$) for which
 \[
n_s>2\pi_s-2.
 \]

And we also explicitly point out that what we have said still holds
for $\pi=0$, since, having excluded that $C$ be exceptional and so that
$n=-1$, we have
 \[
n\ge0;
 \]
and indeed, in the case $n=0$, the only one which has to be examined,
the multiple $|sC|$, for very large $s$, turns out to be always composed
with the rational curves of a pencil, whence it follows that $F$ can be
transformed into a ruled surface.

 \begin{NOTE}
 The theorem that the surfaces on which the adjunction terminates, and
therefore those containing linear systems with the characters $n$ and
$\pi$ for which $n>2\pi-2$, belong to the family of the ruled surfaces is
the fundamental result of the Memoir by Castelnuovo--Enriques, Sopra
alcune questioni fondamentali nella teoria delle superficie algebriche,
{\it Ann.\ di Mat.}, 1901.

The original proof of this theorem did not use the property of the
irregular surfaces of genus $p_g=0$ of containing an irrational pencil of
curves (this property was discovered, as we have said, in 1904). Instead
of this, it exploited an inequality coming from the comparison of the
Noether's expression of the Zeuthen--Segre invariant and of the theorem
of Riemann--Roch. We give a brief sketch of this argument.

If (on the surface $F$ that has to be of genus $p_g=0$) the adjunction
process starting with a linear system $|C|$ terminates, so that one
reaches a last adjoint system $|C^i|$, then the genus of the latter
system must be $\pi\le-p_a=p$.

Let us assume, for simplicity, that $|C^i|$ is irreducible, $\infty^1$ at
least; if $n$ denotes its degree, we have that the Zeuthen--Segre
invariant of the surface is
 \[
 \de-n-4\pi=-12p-p^{(1)}+9, \quad (p=-p_a)\footnote{Here $p^{(1)}$ denotes
the relative linear genus of the surface.}
 \]
whereas the theorem of Riemann--Roch gives its dimension
 \[
r\ge n-\pi+1-p.
 \]
Whence we deduce that
 \[
 5\pi+r\ge11p+p^{(1)}-8+\de\ge11p+p^{(1)}-8.
 \]

Then a suitable discussion, in which one has to distinguish between
the assumptions $p^{(1)}\le1$ and $p^{(1)}>1$, leads us to find on the
surface: either

 \begin{enumerate}
 \renewcommand{\labelenumi}{(\arabic{enumi})}
 \item a linear system of curves of genus $\pi$ and dimension
 \[
r\ge3\pi-5,
 \]
whence it follows that the surface is rational or birational to a ruled
surface of genus $\pi$\footnote{This is a consequence of the argument
which Enriques used in the Note Sulla massima dimensione dei sistemi
lineari di curve di dato genere appartenenti ad una superficie algebrica,
{\it Atti Acc.\ Torino}, 1894.}; or

 \item a linear system of curves that are reducible into rational curves,
whence it equally follows that the surface is birational to a ruled one.
 \end{enumerate}

This sketch just gives an idea of the way which was followed: a very
detailed analysis of various troublesome circumstances and a special
treatment of the cases in which $p<3$ are then needed.
 \end{NOTE}

The considerably simpler proof of our fundamental theorem, based on the
irrational pencil belonging to an irregular surface of genus $p_g=0$, has
been described in the {\it Lezioni} by Enriques--Campedelli\footnote{See
Chapter~III.} published in {\it Rendic.\ Sem.\ Mat.\ R. Universit\`a di
Roma} (1934). This proof has been developed here in a simpler form,
avoiding to invoke, as it was done there, the theorem by Castelnuovo and
Enriques on the double planes with all plurigenera equal to zero. Instead
of this, we have the elementary consideration we made above, that an
$\infty^1$ series of pairs of equivalent elements on an $\infty^1$
elliptic system is a $g_2^1$, thus it is certainly rational.

We finally say that the extension of the fundamental theorem to the case
of a single, non-exceptional curve, of genus $\pi$ ($\ge0$) and degree
$n>2\pi-2$, was brought into light by L.~Campedelli, Intorno alle
superficie algebriche su cui esistono curve di genere $\pi$ e di grado
$n>2\pi-2$, {\it Rendic.\ Acc.\ Lincei}, 19332) and the same Author
pointed out how an explicit use of this property leads to a
simplification of our theory in many points.

 \begin{NOTE} A special importance also has the proposition that we have
used in order to prove the fundamental theorem of this section. The
surfaces on which the adjunction terminates belong to the family of the
ruled surfaces.

Therefore, recalling the results we gave at the end of Chapter~VI,
Section~\ref{sec:7.1}, we may state that: The surfaces which do not
belong to the family of the ruled surfaces, that are therefore birational
to a model without exceptional curves, have the absolute linear genus
 \[
 p^{(1)}\ge1.
 \]
 \end{NOTE}

In Chapter~VI, Section~\ref{sec:6.1}, we only saw that for $p^{(1)}\le0$
all the plurigenera must be $P_i=0$, since the adjunction terminates;
here we see more, that is, that the surface must be birational to a ruled
surface (what contradicts the hypothesis that the exceptional curves can
be eliminated).

 \section{A lemma on the reducible curves in a pencil}\label{sec:10.3}
 \markright{\hfill A lemma on the reducible curves in a pencil \quad
10.3\quad}
 In view of the developments which follow, we need to analyze what is the
valency of a however reducible curve, belonging to a rational or
irrational pencil, in the count of the curves with a double point, whose
number appears in the expression of the Zeuthen--Segre invariant (Chapter~V,
Sections~2--3). For simplicity, we shall consider surfaces without
exceptional curves, thus not belonging to the family of the ruled
surfaces, and we will prove that: For the computation of the
Zeuthen--Segre invariant of a surface possessing a pencil $|C|$, rational
or irrational, of irreducible curves of genus $\pi>0$, without base
points, any reducible curve of the pencil has the valency $\ge0$.

Accordingly, the number $\de$ or $\De$ of the curves of the pencil with a
double point has always to be considered $\ge0$. If $\de=0$ or $\De=0$,
then the pencil cannot contain any reducible curve, except, when the curves
of the pencil are elliptic ($\pi=1$), for some multiple elliptic curves
(of degree zero).

To justify these assertions, it is convenient to recall Godeaux's formula
that, as we have said (Chapter~V, Section~\ref{sec:5.2}), expresses the
valency of a particular curve $\Cbar$ of the pencil, however reducible
into multiple components. Let us denote by $\debar$ the valency of a
particular $C=\Cbar$ that we assume to be decomposable into multiple
components of genus
$\rho_1$, $\rho_2$,\dots:
 \[
 \Cbar=t_1\theta_1+t_2\theta_2+\cdots
 \]
 meeting each other at
 \[
 s_{hk}=\theta_h\theta_k
 \]
 points. One has:
 \[
 \debar\ge\sum (t_i-1)(2\rho_i-2)+\sum s_{hk}(t_h+t_k-1),
 \]
where strict inequality $>$ occurs in the presence of possible
complications, for instance if some of the components $\theta_i$
possesses a double point, or if some of the points of connection of two
components $\theta_h$ and $\theta_k$ happen to coincide with a common
point of $r>2$ components\footnote{The inequality $\debar\ge0$ was proved
by Castelnuovo--Enriques ({\it Ann.\ di Mat.}, 1901) in the simplest case
in which $\Cbar$ breaks up into a multiple irreducible curve $\theta_1$
of genus $\rho_1$ and into another simple component $\theta_2$ which
intersects it at $s_{12}=\theta_1\theta_2$ ($>0$) points. The general
formula which is written above was given, by extending the argument
used in that case, by L.~Godeaux, Sur le calcul de l'invariant de
Zeuthen--Segre. (Mem.\ de la Soc.\ des Sciences de Hainart, 1920) and then
also by L.~Campedelli, Sul computo dell'invariante di Zeuthen--Segre;
Ancora sul computo ecc., Lincei, 1934. The latter author has proved that
$\debar\ge0$ and more precisely that one can have $\debar=0$ only when the
pencil $(C)$ is of genus $\pi=1$ and the $C$ is a multiple elliptic curve
of genus $\rho=1$:
 \[
 \debar=(t+1)(2\rho-2)=0.
 \]
This is in fact the property stated in our lemma, that also appears in
the Lezioni by Enriques--Campedelli (Sem.\ Mat.\ di Roma, 1934) and of
which A. Franchetta has recently given a much simpler new proof.}.

Let us remark that, since the surface has no exceptional curves, one
has, for every $i$,
 \[
s_{ii}=\theta_i\theta_i\le2\rho_i-2;
 \]
and therefore
 \[
 \debar\ge \sum s_{ii}(t_i-1)+\sum s_{hk}(t_h+t_k-1);
 \]
moreover, since one has $s_{hk}\ge0$ if $h\ne k$,
 \begin{eqnarray*}
 \sum s_{hk}(t_h+t_k-1)&=&\sum s_{hk}(t_h-1)+\sum s_{hk}(t_k-1)+\sum
s_{hk}\ge \\
 &\ge& \sum s_{hk}(t_h-1)+\sum s_{hk}(t_k-1).
 \end{eqnarray*}

Therefore one has
 \[
 \debar\ge \sum s_{ii}(t_i-1)+\sum s_{hk}(t_h-1)+\sum s_{hk}(t_k-1).
 \]

Let us introduce the curve
 \[
 C^*=\theta_1+\theta_2+\cdots
 \]
and denote by $\rho$ and $\nu$ its virtual characters. The expression on
the right hand side of the above inequality equals the number of the
virtual intersections $C^*(\Cbar-C^*)$; thus the theorem will be proved if
we show that $C^*(\Cbar-C^*)\ge0$, that is, if we deduce a contradiction from
the hypothesis
 \[
 C^*(\Cbar-C^*)=-i \quad (i>0).
 \]

Noting that $C^*\Cbar=C^*C=0$, this hypothesis implies that
 \[
 \nu=C^*C^*=i>0;
 \]
so the curve $rC^*$ would determine a linear system $|rC^*|$ of dimension
$d_r$ which, for sufficiently large $r$ would be
 \[
 d_r\ge p_a+r^2i-r\rho-\frac{r(r-1)}{2}\,i-r+1
 =p_a+\frac{r^2i}{2}-\frac{r}{2}(2\rho -2-i)+1
 \]
that is, an infinite of the second order as $r$ goes to infinity.

On the other hand the curve $r\Cbar$ determines a linear system
$|r\Cbar|$ composed with the curves of the pencil; its dimension is
given, for $r$ large enough, by $D_r=r-p$, where $p$ is the genus of the
pencil; that is, it is an infinite of the first order as $r$ goes to
infinity. But it is clear that the system $|rC^*|$ is partially contained
in the system $|rC|$, thus that $d_r\le D_r$. The hypothesis we started
with gives therefore a contradiction.

Moreover it turns out that we can have $\debar=0$ only if $\sum s_{hk}=0$,
and this is possible only if $C^*$ is formed by a single component
$\theta$, and if $\theta$ $\theta=2\rho-2=0$,
that is, if $\rho=1$.

 \section{Surfaces with genus $p_g=0$ and $p_a<-1$}\label{sec:10.4}
 \markright{\hfill Surfaces with genus $p_g=0$ and $p_a<-1$ \quad
10.4\quad}
 Let $F$ be a surface with geometric genus $p_g=0$ and negative arithmetic
genus $p_a=-p<-1$. We know (Chapter~IX, Section~\ref{sec:9.11}) that it
must contain a pencil of genus $p$ of curves $K$, with genus $\pi\ge0$.
Let us prove, by drawing a contradiction from the hypothesis $\pi>0$,
that we must have
$\pi=0$ and therefore $F$ must be birational to a ruled surface of genus
$p$.

So let us assume that $\pi>0$ and that $F$ (not birational to a ruled
surface) can be reduced to a surface without exceptional curves; its
absolute linear genus will be (Section~\ref{sec:10.2}, Note)
 \[
 p^{(1)}\ge1.
 \]
Let us now write the formula that gives the Zeuthen--Segre invariant of
$F$ in relation to the irrational pencil $K$. We have (Chapter~V,
Section~\ref{sec:5.2})
 \[
 \De-4+4(p-1)(\pi-1)=12p_a-p^{(1)}+9
 \]
and so, for $\pi>0$,
 \[
 \De\le 13-12p-p^{(1)}.
 \]
Whence it follows $\De<0$, since $p>1$, which contradicts the lemma of
Section~\ref{sec:10.3}.

Then we conclude that $\pi=0$. Thus
 \begin{proc}
 the surfaces with genus $p_g=0$ and $p_a=-p<-1$, are birational to ruled
surfaces of genus $p$.
 \end{proc}

 \section{Surfaces $F$ with $p_g=0$ and $p_a=-1$, possessing an elliptic
pencil of curves $K$ of genus $\pi>1$: Lemma I}\label{sec:10.5}
 \markright{\hfill Surfaces $F$ with $p_g=0$ and $p_a=-1$ \quad 10.5\quad}
 The analysis of the surfaces with genera $p_g=0$ and $p_a=-p=-1$ leads to
the study of the surfaces $F$ that possess an elliptic pencil of curves $K$ of
genus $\pi\ge1$; and it is convenient to distinguish between the two cases
that appear to be possible when $F$ does not belong to the family of the
ruled surfaces, namely:
 \[
 \pi>1 \quad\hbox{and}\quad \pi=1.
 \]

We discuss here the case $\pi>1$.

To start with, we notice that our surface $F$, that we may assume to
be without exceptional curves, has the absolute linear genus $p^{(1)}=1$.
In fact, by computing the Zeuthen--Segre invariant for the elliptic
pencil $(K)$\footnote{See V, 2 and 3.}, we find
 \[
 I=\De-4+4(p-1)(\pi-1)=\De-4=-12-p^{(1)}+9,
 \]
 so that, since $\De\ge0$, we deduce $p^{(1)}\le1$, and then
(Section~\ref{sec:10.2}, Note)
 \[
 p^{(1)}=1.
 \]

It is also useful to remark that for our pencil $(K)$ we have as a
consequence
 \[
 \De=0,
 \]
 and therefore (Section~\ref{sec:10.3}) the pencil itself (formed by
curves of genus greater than one) contains neither a curve $K$ with a
double point nor, in any case, a reducible one.

With this in mind, let us proceed by stating a series of lemmas which
will finally lead us to a theorem.

LEMMA I.-The surface $F$ has at least an elliptic parabicanonical
curve, cutting out on the curves $K$ of the pencil groups of the
bicanonical series $g_{4\pi-4}^{3\pi-4}$; and similarly an elliptic
para-$i$-canonical curve for $i>2$.

Let us in fact consider the linear system $|K''|$, the second adjoint
system of a curve $K$ of our pencil. Since $p^{(1)}=1$ and the virtual
canonical curves cut out groups of $2\pi-2$ points on the $K$, the genus
of the curves $K'$ is
 \[
 \pi'=\pi+1+2\pi-2-1=3\pi-2
 \]
and so the dimension of $|K''|$ is
 \[
r\ge-p+\pi'-1=3\pi-4,
 \]
thus at least equal to the dimension of the bicanonical series of $K$.
Now, if the series cut out by $|K''|$ on $K$ is not complete, then there is
at least a pencil of curves $K''$ passing through any of its groups,
whence we deduce the existence of a curve $K''$ which breaks up into $K$ and
a bicanonical curve, of genus
 \[
3p^{(1)}-2=1:
 \]
 this is either an irreducible elliptic curve or a curve which is
composed of elliptic curves without common points.

In the case in which $|K''|$ cuts out on $K$ the complete bicanonical
series, let us consider the (second paradjoint) continuous system
$\{K''\}$, formed by $\infty^1$ linear systems, that we obtain from a
particular $|K''|$ by adding the difference of two curves $K_1-K_2$ of the
pencil $K$.

Let us remark that all the linear systems forming $\{K''\}$ cut out the
same bicanonical series on $K$, and similarly the groups cut out by two
inequivalent curves of a paradjoint system (of order $1,2,3,\dots$)
always turn out to be equivalent.

Whence it follows that we have (at least) $\infty^1$ curves of $\{K''\}$
passing through a fixed bicanonical group of $K$, and therefore one has a
curve of this system which is reducible into a $K$ and a residual
parabicanonical curve: the latter, as the virtual bicanonical curve,
has genus
 \[
 3p^{(1)}-2=1
 \]
and therefore it is formed by one or more irreducible elliptic curves
without common points.

In a completely analogous way one proves that the paradjoint system
$\{K^i\}$ of order $i=3,4,\dots$, of dimension $(2i-1)\pi-2i$, contains
(at least) $\infty^1$ curves passing through a chosen group of the
$i$-canonical series on a $K$, and therefore it also contains a curve that
is reducible into the $K$ and a para-$i$-canonical curve, and this is
either elliptic or a sum of elliptic curves without common points.

But we cannot say that all these elliptic curves, for instance the
parabicanonical and the paratricanonical ones, are formed by distinct
irreducible components: in fact all of them might coincide with a
multiple paracanonical curve, namely an elliptic curve whose order (the
number of its intersections with the $K$) is $2\pi-2$.

 \section{Lemma II: a sketch of the proof}\label{sec:10.6}
 \markright{\hfill Lemma II: a sketch of the proof \quad 10.6\quad}
 On a surface $F$ (of genera $p_g=0$ and $p_a=-1$, with a pencil of curves of
genus $\pi>1$) there exist at least two elliptic curves, not formed by
the same components, whose multiples are two parapluricanonical curves,
which cut out equivalent groups of points on the $K$.

We have already constructed a parabicanonical curve (or a
parapluricanonical curve of higher order), say $C$, which is a curve of
genus one, formed by simple or multiple irreducible components without
common points. Let us try now to construct another curve $D$, of the same
order $4\pi-4$, which is not parabicanonical, and does not cut out on any
$K$ a bicanonical series, but a series whose $s$-multiple is the
$2s$-canonical series, cut out by the curve $sC$.

More precisely, in the case in which the genus $\pi$ of the $K$ is even,
and so we have the divisor $s=2$, we aim to construct an elliptic curve
$d$ which, although it has the same order $4\pi-4$ as $C$, does not cut
out on a $K$ a bicanonical group, but a group which belongs to a
semiquadricanonical series of this curve. And we will succeed in doing
this in the easiest case, in which the following two simplifying
assumptions, that we shall discuss in the next sections, are
satisfied:

  \begin{enumerate}
 \renewcommand{\labelenumi}{(\Roman{enumi})}
 \item on any curve $K$ a semibicanonical (but not canonical) series $\gbar$
can be rationally determined;

 \item a group can be rationally determined inside the aforementioned
$\gbar$.
 \end{enumerate}

In these assumptions we may first of all rationally determine on a $K$ a
semibicanonical series $\gbar=g_{2\pi-2}^{\pi-2}$ (different from the
canonical series $g=g_{2\pi-2}^{\pi-1}$) and then we may also determine a
group $G_{2\pi-2}$ in it; so we may construct a sufficiently ample linear
system $|L|$ of curves, which cuts out the corresponding series $\gbar$
on any $K$. Therefore the adjoint linear system of $|L|$, that is, the system
$|L'|$, cuts out on any $K$ the semiquadricanonical series $\gbar+g$. If
we compare now $|2L'|$ and $|2K''|$ ($|K''|$ is the second adjoint system of
a $K$) by using the second equivalence criterion (in its more precise form
that was given here with the lemma of Section~\ref{sec:10.3}), we see that
these only differ by a certain number $r$ of curves $K_1,\dots,K_r$ of
the pencil $(K)$, to be counted twice:
 \[
 |2L'|=|2K''+2(K_1+\cdots+K_r)|,
 \]
 whence we deduce that the system $|L'-(K_1+\cdots+K_r)|$ has the same
characters (degree, genus and thus virtual dimension $3\pi-4$) as $|K''|$.
Therefore it gives rise to a continuous system of dimension $3\pi-3$ (one
more than the dimension of the series cut out on a $K$), from which we
may subtract a $K$: the residual curve is the semiparaquadricanonical
curve that we were aiming to construct: $C$ and $d$ are certainly not
equivalent, and they are not even paraequivalent, whereas $2C$ and $2D$
are paraequivalent.

In case the assumptions II and I we mentioned above are not verified,
the argument can be repeated mutatis mutandis in the following sense:

 \begin{enumerate}
 \renewcommand{\labelenumi}{(\arabic{enumi})}
 \item If the hypothesis II is not verified any more (but I still is),
instead of a semibicanonical (noncanonical) series $\gbar$, one may
construct on a $K$ a series $\gbar'$ which is an $i$-multiple of the
mentioned $\gbar$, with $i$ an odd integer, for which the the hypothesis
is satisfied. So, instead of the system $|L|$, or rather of its adjoint
$|L'|$ which we have mentioned before, one may construct a linear system
$|M|$, cutting out on any $K$ the series $\gbar'$. Then the comparison of
$|2M|$ and $|2K^i|$ (where $K^i$ denotes the $i$th adjoint system of a
$K$) shows that the former system, from which a certain number of curves
$K_1,\dots,K_r$ of the pencil $(K)$, counted twice, have to be subtracted,
is paraequivalent to the latter. Whence we deduce that the system
$|M-(K_1+\cdots+K_r)|$ has the same characters as $|K^i|$; thus
(Section~\ref{sec:10.5}) it belongs to a continuous system in which there
is a curve reducible into a $K$ and a residual elliptic curve $D$. $d$
has the same order as $C=|K^i-\Kbar|$, the two curves are neither
equivalent nor paraequivalent, but their doubles are paraequivalent and
cut out on the $K$ equivalent groups.

 \item If the hypothesis I does not hold, then we may pass from the
surface $F$ to an $F^*$ with the same characters, which is represented on
the $m$-tuple $F$ ($m>1$) without branch curve, so that the simplifying
hypothesis (which is not verified by $F$) holds true for $F^*$. Then we
construct a semiparapluricanonical elliptic curve $d$ (of order $4\pi-4$
or $i(2\pi-2)$) on $F^*$, whose image on $F$ is still an elliptic curve
$d$ (of order $m(4\pi-4)$, or $im(2\pi-2)$). This curve $d$ is not
paraequivalent to the parapluricanonical curve $C$ of the same order
(that is, to the parabicanonical curve $C=|K''-\Kbar|$ that can be
constructed on $F$ according to Section~\ref{sec:10.5}); but the doubles
$2D$ and $2C$ are paraequivalent, and cut out equivalent groups on the
$K$.
 \end{enumerate}

In the case in which $\pi$ is an odd number, we denote by $s$ one of its
(odd) prime divisors, and we aim to construct on $F$ an elliptic curve of
order $4\pi-4$, cutting out on any $K$ a non-bicanonical group, belonging
to a series which is an $s$-submultiple of the $s$-bicanonical series. And
we will succeed in doing this in the simplest case in which the
assumptions I and II, analogous to the ones occurring in the case $\pi$
even and $s=2$, are satisfied.

Indeed in such assumptions a nonspecial series $\gbar=g_{2\pi-2}^{\pi-2}$,
$s$-submultiple of the $s$-canonical series, is given on any $K$, and a
group of points is also rationally given inside it; then we can construct
an ample enough linear system $|L|$, cutting out on any $K$ the
aforementioned series $\gbar$. Therefore the adjoint system $|L'|$ (of
order $4\pi-4$) cuts out on $K$ the series $\gbar+g$ which is an
$s$-submultiple of the $s$-bicanonical series. And we can also see that
$|sL'|$ differs from $|sK''|$ ($|K''|$ is the second adjoint system of a
$K$) by a certain number $r$ of curves $K_1,\dots,K_r$ of the pencil $(K)$
to be counted $s$ times:
 \[
 |sL'|=|sK''+s(K_1+\cdots+K_r)|.
 \]

As a consequence of this relation the system $|L'-(K_1+\cdots+K_r)|$
has the same characters (degree, genus and dimension $3\pi-4$) as $|K''|$
and it is contained in an $\infty^{3\pi-3}$ continuous system, containing
a curve reducible into a $K$ and a $D$: the latter is a curve of genus 1
and degree 0, the $s$-multiple of which is paraequivalent to the
$s$-multiple of the parabicanonical curve $C$.

But, as in the first case ($\pi$ even and $s=2$), it is convenient to
examine in which way the argument has to be modified if the aforesaid
assumptions I and II are not verified.

 \begin{enumerate}
 \renewcommand{\labelenumi}{(\arabic{enumi})}
 \item If the hypothesis II does not hold, then, instead of a $\gbar$, of
order $2\pi-2$, different from the canonical series, but an
$s$-submultiple of the $s$-canonical series, we may consider on $K$ a
non-pluricanonical series $\gbar'=(2\pi-2)\gbar -(\pi-2)g$, of order
$\pi(2\pi-2)$, rationally depending on $\gbar$; recall that according to
the hypothesis I, $\gbar$ is supposed to be rationally determined. In
this $\gbar'$ one group can be rationally determined. In this way we may
construct, instead of $|L|$, or rather of $|L'|$, a linear system $|M|$
cutting out on any
$K$ the corresponding series $\gbar'$. Then one proves that
 \[
 |sM|=|sK+s(K_1+\cdots+K_r)|
 \]
 and therefore
 \[
 |M-(K_1+\cdots+K_r)|
 \]
 has the same characters as $|K^\pi|$, and it contains a curve which is
reducible into a $K$ and a residual elliptic curve $D$: $d$ has the same
order $\pi(2\pi-2)$ as the para-$\pi$-canonical curve $C=|K\pi-\Kbar|$,
but the two curves are neither equivalent nor paraequivalent; only their
multiples, $sC$ and $sD$, give para-$s$-canonical curves, cutting out
equivalent groups on the $K$.

 \item Furthermore, if the hypothesis I does not hold, then one succeeds
in rationally constructing on a $K$ not one, but a group of $m>1$ series
$\gbar$ (or $\gbar'$) and one therefore obtains a surface $F^*$, with the
same characters, represented on the $m$-tuple $F$ without branch curve;
and on this $F^*$ each of the elliptic curves $K^*$ corresponding to the
$K$ contains a determined series $\gbar$ (or $\gbar'$). Therefore (as the
hypothesis I is verified for $F^*$) we can construct on $F^*$ an
elliptic curve $D^*$ which is a suitable submultiple of a
parapluricanonical curve, whose corresponding curve $d$ on $F$ is the
elliptic curve we were looking for.
 \end{enumerate}

In this way we have explained the general plan of the proof of our
lemma II; now we shall develop this proof in all its details,
separately discussing the two cases which occur: $\pi$ even and $s=2$,
$\pi$ odd and $s$ one of its prime divisors (different from 2).

 \section{The first case: the genus $\pi$ of the curves $K$ is
even}\label{sec:10.7}
 \markright{\hfill The first case: the genus $\pi$ of the curves $K$ is
even \quad 10.7\quad}
 To start with we recall that the bisection of a linear series
$g_{2n}^{2n-\pi}$ with $n>\pi$, on a curve of genus $\pi$, always leads to
$2^{2\pi}$ distinct series\footnote{See for instance Enriques--Chisini,
{\it Lezioni}, Libro~V, Chapter~III, \S35 (Vol.~III, page~394). See also
Libro~VI, Chapter~III, \S34 (Vol.~IV, page~199).}.

In particular the double of the canonical series on a $K$, that is, the
bicanonical series $g_{4\pi-_4}^{3\pi-4}$, gives rise to $2^{2\pi}$
halves; one of these is the canonical series $g=g_{2\pi-2}^{\pi-2}$ and
the others, which we will now consider, are $2^{2\pi}-1$ semibicanonical
series $\gbar=g_{2\pi-2}^{\pi-2}$, that are distinct from $g$ and distinct
one from the other. Since the curve $K$, as it varies in the pencil, stays
irreducible and never acquires double points, the aforementioned series
$\gbar$ never come together; still, they can be permuted by a circulation
of the curve $K$ in its own (irrational) pencil, and it is convenient to
suppose that in general one can rationally define -- not a single
series -- but a group of $m$ ($\ge1$) distinct series
$\gbar=g_{2\pi-2}^{\pi-2}$.

As we have announced, we shall introduce here two simplifying
assumptions, which will be justified later, in relation with the
consequences that we shall deduce from them.

Hypothesis I.-One has $m=1$, that is, for any $K$ it is possible to
determine uniquely, and therefore rationally, one semibicanonical (and
not canonical) series.

Hypothesis II.-In a series $\gbar$, which is assumed to be rationally
determined, it is possible to determine uniquely, and therefore
rationally, one group, which, as $K$ varies on $F$, describes a curve $L$
cutting out on the $K$ semibicanonical groups of $\gbar$. The latter
hypothesis may seem evident, since, as soon as we have the series $\gbar$
on $K$, we are led in general to transform the $K$ into a $K_{2\pi-2}$ of
the space $S_{\pi-2}$ defined up to projective transformations, on which
the hyperplanes cut out the series $\gbar$, thus this series is linearly
given, in such a way that a hyperplane of $S_{\pi-2}$ cuts out one of its
groups. But it is not a priori clear that a rationally given series
$\gbar=g^{\pi-2}$ on $K$ can be linearly defined in a unique way. In
fact, among the infinitely many projectively equivalent curves
$K_{2\pi-2}$ which are the images of the series $\gbar$, the choice of a
particular $K_{2\pi-2}$, that can be uniquely determined starting with
a group of $\gbar$, depends in general on some numerical irrationality
that one has to add to the field of rationality of $K$ and $\gbar$, and
such irrationalities give-as $K$ varies-some algebraic
irrationalities\footnote{In order to explain the difficulty, let us
consider a plane quartic $K_4$ defined as the curve enveloping an
$\infty^1$ series of quadritangent conics $K_2$ of index 2; in this way
the $g_4^1$ of the 4-tuples of the points of contact is rationally
determined on $K_4$; but this is not linearly defined by means of a well
determined $\infty^1$ linear system of conics with 4 base points. {From} an
abstract point of view the $g_4^1$ can be regarded as a rational curve
which can be represented on a line; but, in order to determine the
representation, a point of the rational curve is needed, and this yields
the addition of a quadratic irrationality. (Noether's theorem, see
Enriques--Chisini, {\it Lezioni}, Libro~V, Chapter~III,
\S32 (Vol.~III, page~341).}.

In force of the assumptions I and II, let us then construct a curve $L$ on
$F$ which cuts out on each $K$ of our elliptic pencil a group of $2\pi-2$
points of the semibicanonical series $\gbar$ (that we supposed to be
given). The adjoint curves $L'$ of $L$ cut out on the $K$ the groups of a
semiquadricanonical series $\gbar+g$, that is different from the
bicanonical series, and form a linear system $|L'|$; we may assume that
$|L'|$ is as ample as we like, since we may add to it -- if needed -- a
certain number of curves $K$. Let us now compare the linear system
$|2L'|$ with $|2K''|$ (the double of the second adjoint system to a K).
Since the two systems cut out equivalent groups on any $K$, by the second
equivalence criterion, made more precise by the lemma of
Section~\ref{sec:10.3}, they differ only by curves of the pencil $(K)$.
But on this elliptic pencil the difference of two groups of elements (the
first of which large enough) turns out to be equivalent to a positive
group of elements, thus we may write
 \[
 |2L'|=|2K''+K_1+K_2+\cdots|.
 \]

We distinguish now between two cases that a priori appear to be possible,
that is, that the number of the $K_1,K_2,\dots$ is an even number $2r$, or
that it is an odd number $2r+1$.

In the first case, if we let the group of the curves $K_1,K_2,\dots$
vary in the linear series $g_{2r}^{2r-1}$ that it determines in the
elliptic pencil, we can arrange the group to be equal to a group
counted twice:
 \[
 2K_1+2K_2+\cdots+2K_r
 \quad
(K_1=K_{r+1},K_2=K_{r+2},\dots).
 \]

In the latter case we can arrange the group to be equal to another
one formed by $r+1$ curves, $r$ of which are counted twice:
 \[
 2K_1+2K_2+\cdots+2K_r+K_0.
 \]

So we have
 \[
 |2L'-2K''-2(K+K_1+\cdots+K_r)|=|K_0|,
 \]
and therefore the characters ($\pi$ and $n=0$) of $K_0$ can be expressed
in terms of the characters (genus $\rho$ and degree $\nu$) of the virtual
curve
 \[
 L'-K''-(K+K_1+\cdots+K_r)
 \]
whose double is equivalent to $K_0$.

Then we have
 \[
n=4\nu=0
 \quad
\pi=2\rho +\nu -1=2\rho -1,
 \]
and consequently $\pi$ has to be odd. Since we assumed $\pi$ to be even,
the latter case is not possible and we must have
 \[
 |2L'-2(K_1+\cdots+K_r)|=|2K''|.
 \]

Now the linear system $|2L'-2(K_1+\cdots+K_r)|$ has two inequivalent
halves:
 \[
 |L'-(K_1+\cdots+K_r)|
 \]
and
 \[
 |K''|.
 \]

Therefore the first system has the same characters, genus and degree,
as the latter, that is,:
 \[
4\pi-4+\pi+1-1=5\pi-4
 \]
and
 \[
 2(4\pi-4)=8\pi-8.
 \]

Its dimension is therefore
 \[
3\pi-4,
 \]
which is the dimension of the complete linear series that it cuts out
on a $K$. Accordingly, a group of $3\pi-4$ points of this series belongs
to $\infty^1$ curves of the continuous system $\{L'-(K_1+\cdots+K_r)\}$
formed by $\infty^1$ inequivalent linear systems, and there is at least
one curve of this continuous system which breaks up into a $K$ and into a
residual curve $D$: the genus of $d$ turns out to be equal to the genus of
$C=|K''-\Kbar|$, that is, equal to one. And -- as required -- the elliptic
curve $D$, of the same order as the parabicanonical curve $C$, is not
equivalent (but semiequivalent) to it and therefore it is essentially
distinct from it, that is, it certainly contains some other (elliptic)
component.

It is now convenient to discuss the assumptions I and II which we
assumed in order to develop the previous argument. Let us start from the

 \subsection*{Discussion of the hypothesis II}

The hypothesis II, that is verified for $\pi=2$\footnote{In this case the
series $\gbar$ consists of a unique pair, which is therefore given
together with the $\gbar$.}, a priori is not true any more for higher
values of the even number $\pi$ ($\pi=4,6,\dots$). But we prove that, if
we cannot determine in general on $K$ one group $G$ of the $\gbar$ or of
the semibicanonical series series $\gbar+g$ (that is rationally obtained
from the given $\gbar$), we may in any case determine a group formed by
$i=\pi-1$ groups $G$ ($i$ odd), belonging to a non-pluricanonical
series -- say $\gbar'=i\gbar$ -- which is a half of the $2i$-canonical
series. Therefore we may construct on $F$ a curve M cutting out on any
$K$ a group of the corresponding $\gbar$, and by operating on $|M|$ as we
did on $|L|$, or rather on $|L'|$, that is, by subtracting a certain number
of curves $K$, we will obtain an elliptic, semipara-$2i$-canonical curve
$d$ (essentially distinct from $iC$), whose double is paraequivalent to
the bicanonical curve $C$ counted $i$ times.

Thus our argument based on the hypothesis II can be repeated with slight
modifications (by substituting the series $\gbar'$ for $\gbar$ or for
$\gbar+g$), if we prove what we have stated before: it is possible to
determine one group formed by $i=\pi-1$ (an odd number of) groups $G$ of
$\gbar$ (and therefore to determine $\gbar'$ linearly). In order to do
this, we consider in the space $S_{\pi-2}$ the family of all the
projectively equivalent curves $K_{2\pi-2}$, which are the images of the
semibicanonical series $\gbar=g_{2\pi-2}^{\pi-2}$ given on $K$. We try to
construct two quadrics $Q$ and Q', or rather a pencil of quadrics in this
space, with the property of being a (projective) covariant of
$K_{2\pi-2}$: then the self conjugate $i$-gon, with $i=\pi-1$, common to
$Q$ and $Q'$, is determined as a covariant of the pencil, and its faces
cut out $\pi-1$ groups of the $\gbar$ on $K_{2\pi-2}$.

The construction of the aforementioned pencil of $Q$ and $Q'$ can be
uniquely, and therefore rationally, performed by starting with the $K$ on
which the $\gbar$ is given. This is immediately seen if $K_{2\pi-2}$ does not
belong to any quadric of $S_{2\pi-2}$: the minimal double of the series
$\gbar$ is a (complete or not) bicanonical series cut out on $K_{2\pi-2}$
by the quadrics of its space and in that case it is linearly defined on
the corresponding bicanonical curve $K_{4\pi-4}$ of $S_{3\pi-4}$ (which is
also linearly given) by the hyperplanes of a linear system $\al$ that is
well determined by the curve $K_{4\pi-4}$. So a pencil of quadrics in
$S_{\pi-2}$ can be rationally determined as a function of $\gbar$ by
giving a pencil of hyperplanes of $S_{3\pi-4}$, chosen in $\al$ by
imposing that they pass through a suitable group of points of
$S_{3\pi-4}$, that can be fixed independently of the $K_{4\pi-4}$; this
is the same as giving a series $g_{4\pi-4}^1$ contained in the
bicanonical series
$g_{4\pi-4}^{3\pi-4}$ (and more precisely in the double of the series
$\gbar$).

But if the $K_{2\pi-2}$ belongs to $\infty^r$ quadrics of $S_{\pi-2}$
($r\ge1$), then the above argument fails, since a group of the minimal
double of the series $\gbar$, regarded as a hyperplane section of
$K_{4\pi-4}$, does not correspond any more to a single quadric of
$S_{\pi-2}$, but to an $\infty^{r+1}$ linear system of quadrics instead;
similarly an $\infty^t$ linear system of hyperplanes of $S_{3\pi-4}$
contained in $\al$ corresponds to an $\infty^{r+t+1}$ linear system of
quadrics, which cuts out on $K_{2\pi-2}$ a series $g_{4\pi-4}^t$ of lower
dimension. But, if we set
 \[
t=3\pi-4-2=3\pi-6,
 \]
then we have a linear system $|Q|$ of quadrics in $S_{\pi-2}$, whose
dimension is two less than the dimension of the total system: there is
now a harmonic system of dual quadrics associated\footnote{If in a linear
space of any dimension a quadric $a_x^2=0$ and a dual quadric
$a_u^2=0$ are given, then (according to Battaglini and Rosanes) there is
a simultaneous linear invariant of them, the harmonizer $a_a^2$, whose
vanishing defines the harmonic relation between a quadric and the
associated linear system of dual quadrics. See Enriques--Chisini,
{\it Lezioni}, Libro~III, Chapter~I, \S9 (Vol.~I, page~61).} with $|Q|$,
which defines a conjugate $i$-gon ($i=\pi-1$), and therefore a group of
$i$ groups of $\gbar$. \qed

 \subsection*{Discussion of the hypothesis I}

After having justified the hypothesis II (at least in the sense that,
if it is not verified for the series $\gbar$, then it is verified for
$\gbar'$), let us see how we can relatively justify also the hypothesis
I, in the sense that-if it does not hold for our surface $F$-it certainly
holds for an $F^*$ with the same characters as $F$ which is represented
on $F$ counted $m$ ($\ge1$) times, without branch curve; so it is possible
to construct an elliptic curve $D^*$ on $F^*$, whose image on $F$ is a
semiparapluricanonical curve $D$.

Let us assume that on each $K$ of $F$ we cannot uniquely determine a
single semibicanonical series $\gbar$, but we can determine a group of $m$
($m>1$) such series $\gbar$, that are permuted by a circulation of $K$ in
the pencil $(K)$. And, for simplicity, let us assume $\pi>2$ and therefore
$\pi\ge4$, and let us also assume that for each one of these series
$\gbar$ the hypothesis II holds; thus we can construct a linear image of
this series, as a rational function of it, that is, we can define a single
curve $K_{2\pi-2}$ of $S_{\pi-2}$, chosen among the infinitely many
corresponding projective curves.

The $\infty^1$ algebraic variety whose elements are our series $\gbar$ on
the $K$ (that is, the series associated to the K) can be abstractly
regarded as an $\infty^1$ series $\Si$ of spaces $S_{\pi-2}$, contained in a
space of higher dimension $S_R$. Since two of the $m$ series $\gbar$ on a
$K$ never come together, the series $\Si$ contains an elliptic involution
$\ga_m^1$ of spaces without double elements, and it is therefore also
elliptic.

Now in each one of the spaces $S_{\pi-2}$ of $\Si$ we can (by hypothesis)
construct a single curve $K^*=K_{2\pi-2}$, an image of the homologous
$\gbar$ on the corresponding K: the locus of the curves $K^*$ (which are
birationally equivalent to the K) is a surface $F^*$ that is represented
on the $m$-tuple $F$ (without branch curve) in such a way that any $K$ of
$F$ corresponds to a group of $m$ distinct curves $K^*$ on $F^*$. The
curves $K^*$ on $F^*$, birationally equivalent to the $K$, form an
elliptic pencil ($K^*$).

It is easy to see that the surface $F^*$, as well as $F$, and according
to the formula of the Zeuthen--Segre invariant of $F^*$ in relation to the
elliptic pencil of the $K^*$ without double points, has the arithmetic
genus $p_a=-1$ and the linear genus $p^{(1)}=1$.

In this setting we may say of $F^*$ what -- in the hypothesis I -- we have
said of $F$, since for any curve $K^*$ (of genus $\pi$) on $F^*$ a single
semibicanonical series $\gbar$, defined by the hyperplanes of $S_R$, is
rationally and linearly given. Whence we deduce that $F^*$ contains an
elliptic semiparabicanonical curve $D^*$, whose image on $F$ is again an
elliptic, semipara-$2m$-canonical curve $D$.

 \begin{Remark} In the above argument we have excluded the hypothesis
$\pi=2$, in which in any case the same discussion could be repeated with
slight modifications. But it suffices to notice that in this case, if the
$K$ are of genus $\pi=2$, on each of them the group of the 6 distinct
double points of the $g_2^1$ is determined, and -- as $K$ varies -- it
describes an elliptic semiparasexticanonical curve.
 \end{Remark}

 \section{The second case: the genus $\pi$ of the curves $K$ is
odd}\label{sec:10.8}
 \markright{\hfill The second case: the genus $\pi$ of the curves $K$ is
odd \quad 10.8\quad}
 We denote by $s$ an (odd) prime number which divides $\pi$, and we
consider on any $K$ one of the nonspecial series
$\gbar=g_{2\pi-2}^{\pi-2}$ obtained from the division by $s$ of the
$s$-canonical series. For the time being we introduce two simplifying
assumptions analogous to I and II of the previous section.

Hypothesis I: on the variable $K$ of the pencil $(K)$ a series $\gbar$ can be
uniquely, that is, rationally, determined.

Hypothesis II: if a $\gbar$ is rationally given on any $K$, we can
rationally determine a single one of its groups, which is uniquely
defined as $K$ varies in $(K)$; in other words the rationally determined
$\gbar$ can be linearly defined (giving rise to a single $K_{2\pi-2}$ in
$S_{\pi-2}$).

In force of these assumptions, we can construct a curve $L$ on $F$ cutting
out on any $K$ a group of $\gbar$; and $|L|$, defined up to summands
equal to $K$, can be assumed to be as ample as we like.

Let us denote by $|L'|$ the linear system, of order $4\pi-4$, adjoint to
$|L|$, and by $|K''|$ the second adjoint system to a $K$, which is of the
same order.

Let us compare, using the usual equivalence criterion, the two systems
$|sL'|$ and $|sK''|$; they must differ by a certain number of curves of
the pencil $(K)$ to be added to the less ample system $|sK''|$; moreover
we may assume that the curve to be added to this smaller system is formed
by a certain number of $s$-tuple curves
 \[
sK_1,sK_2,\dots,sK_r
 \]
and maybe also by a $t$-tuple curve $tK_0$ of the pencil, where $t<s$:
 \[
 |sL'|=|sK''+s(K_1+\cdots+K_r)+tK_0|.
 \]

But we prove that
 \[
 t=0,
 \]
since the genus of the virtual curve
 \[
((t/s)K_0)=|L'-K''-(K_1+\cdots+K_r)|
 \]
must be an integer $\rho$, whereas the genus of $tK_0$ (of degree 0) is
 \[
 t\pi-t+1=s\rho -s+1
 \]
so that
 \[
 t(\pi-1)=s(\rho -1):
 \]
 this relation cannot hold with $\rho$ an integer, since $s$ (a prime
divisor of $\pi$) is coprime with $\pi-1$ and $t$.

Therefore we have
 \[
 |sL'|=|sK''+s(K_1+\cdots+K_r)|
 \]
 and
 \[
 |sL'-s(K_1+\cdots+K_r)|=|sK''|.
 \]

So
 \[
 |L'-(K_1+\cdots+K_r)| \quad\hbox{and}\quad |K''|
 \]
have the same characters (degree $8\pi-8$ and genus $5\pi-4$) and
therefore the same virtual dimension $3\pi-4$, which is the dimension of
the series they cut out on a $K$. Whence we deduce that in the
$\infty^{3\pi-3}$ continuous system $\{L'-(K_1+\cdots+K_r)\}$ there is a
curve reducible into a $K$ and a residual curve $D$; $d$ (as well as the
parabicanonical $C=|K''-\Kbar|$) is elliptic, of order $4\pi-4$, and $sD$,
as well as $sC$, is a para-$2s$-canonical curve, and the two curves cut
out on the $K$ equivalent ($2s$-canonical) groups.

If the assumptions I and II do not hold, then the above argument can
still be repeated with some modification.

Assume that the hypothesis II does not hold, that is, it is not possible to
uniquely determine a group of the nonspecial series
$\gbar=g_{2\pi-2}^{\pi-2}$, which an $s$-submultiple of the $s$-canonical
series and, according to the hypothesis I, it is supposed to be
rationally given; then, instead of the nonspecial series $\gbar$,
$s$-submultiple of the $s$-canonical series, we may consider on $K$ the
series
 \[
 \gbar'=(2\pi-2)\gbar-(\pi-2)g;
 \]
we can rationally determine one of its groups by taking the sum of the
groups $\Gbar_{2\pi-2}$ of $\gbar$ having a point of a (canonical) group
$G_{2\pi-2}$ of $g$ as a $(\pi-2)$-tuple point and by subtracting then the
group $G_{2\pi-2}$ counted $\pi-2$ times: it is important to notice that
$\gbar'$, whose $s$-tuple is the $s\pi$-canonical series, is not the
$\pi$-canonical series, since otherwise from the relations
 \[
(2\pi-2)\gbar-(\pi-2)g=\pi g
 \]
and
 \[
(2\pi-2)\gbar =(2\pi-2)g
 \]
that is,
 \[
 s\gbar =sg,
 \]
we would deduce
 \[
 \gbar=g
\footnote{It suffices to solve the diophantine equation $(2\pi-2)x-sy=1$;
then we deduce that $[(2\pi-2)x-sy]\gbar=[(2\pi-2)x-sy]g$.}
 \]
since $2\pi-2$ and $s$ are coprime numbers.

Now, instead of $|L|$, or rather of $|L'|$, we may construct a
sufficiently ample linear system $|M|$ cutting out on each $K$ the series
$\gbar'$ (constructed as a function of $\gbar$, which in turn is
rationally given). We compare $|sM|$ with the linear system $|sK^\pi|$
multiple of $|K^\pi|$,the $\pi$th adjoint system of a $K$; if we denote by
$K_0,K_1,\dots,K_r$ suitable curves of the pencil $(K)$, then we have
 \[
 |sM|=|sK\pi+s(K_1+\cdots+K_r)+tK_0|,
 \]
 where $t=0$ and therefore
 \[
 |sM|=|sK^\pi+s(K_1+\cdots+K_r)|.
 \]

Whence we deduce that the linear system
 \[
 |M-(K_1+\cdots+K_r)|
 \]
has the same characters as $|K^\pi|$ -- degree, genus and virtual
dimension -- and therefore it belongs to a continuous system whose
dimension is one more than the dimension of the series $\gbar'$; therefore
there is inside this system a curve reducible into a $K$ and a residual
elliptic curve $D$, of order $\pi(2\pi-2)$, whose $s$-multiple is a
para-$s\pi$-canonical curve.

Let us finally discuss what can be done if the hypothesis I fails to
hold for the surface: this means that on a variable curve $K$ we cannot
rationally define a nonspecial series $\gbar$, among those whose
$s$-multiple is the canonical series, but we may only define a group of
$m$ ($>1$) series $\gbar$, permuted by a circulation of $K$ in the pencil $(K)$. As we have already seen in the first case ($\pi$ even and $s=2$), it
is convenient to pass from the surface $F$ to an $F^*$ with the same
characters, represented without branch curve on the $m$-tuple $F$ and such
that the hypothesis I holds for it. Then on $F^*$ an elliptic curve $D^*$
of a certain order $(2\pi-2)$, or more generally $\pi(2\pi-2)$, can be
constructed and the corresponding curve $d$ on $F$, of order
$m\pi(2\pi-2)$ is also elliptic: and $D$, which is an $s$-submultiple of a
para-$sm\pi$-canonical curve, certainly contains some component distinct
from the components of a para-$m\pi$-canonical curve $C$, constructed on
$F$ according to Section~\ref{sec:10.5}.

 \section{Lemma III}\label{sec:10.9}
 \markright{\hfill Lemma III \quad 10.9\quad}
 Let $F$ be a surface of genera $p_g=0$ and $p_a=-1$, possessing an elliptic
pencil of curves of genus $\pi>1$, and let $E$ be a curve on $F$ cutting out
on the curves $K$ of the pencil groups of points equivalent to those cut
out by a para-$i$-canonical curve $C$; if $E$ has no common points with $C$,
away from the possible common components, then it is a para-$i$-canonical
elliptic curve, like $C$.

Let us compare $E$ and $C$ with the second equivalence criterion (made
more precise with the lemma of Section~\ref{sec:10.3}): they are
equivalent up to the difference $K_1-K_2$ of two curves of the pencil
$(K)$:
 \[
 |E|=|C+K_1-K_2|.
 \]

Therefore $E$ has the same genus one and the same degree zero as $C$ and
it is, as well as $C$, a para-$i$-canonical curve.

 \section{Conclusion: the surfaces $F$ also possess a linear pencil of
elliptic curves}\label{sec:10.10}
 \markright{\hfill Conclusion: the surfaces $F$ also possess a linear pencil of
elliptic curves \quad 10.10\quad}
 The lemmas I, II and III that we established in the above sections
lead us now to a theorem which closes the analysis of our surfaces $F$.

The surfaces $F$, of genus $p_g=0$ and $p_a=-1$, with an elliptic pencil
of curves $K$ of genus $\pi>1$, also possess a base point free linear
pencil of elliptic curves.

Indeed let us construct on $F$ (Lemma~II) two elliptic para-$i$-canonical
curves $C$ and $D$, cutting out on the curves $K$ equivalent groups of
$n=i(2\pi-2)$ points. On each $K$ the above mentioned curves determine a
linear series $g_n^1$, possessing a certain number
 \[
 m\le2n+2\pi-2
 \]
of groups $G_n$ with a double point, away from those which can be absorbed by
the groups cut out by $C$ and $D$. The locus of these groups $G_n$ is a
curve $E$ that (with respect to the $K$) has the same order $mn$ as $mC$
and $mD$, and cuts out on the $K$ groups equivalent to sections with these
curves; moreover, it can never happen that, as $K$ varies in the pencil
$(K)$, a group $G_n$ coincides with the group cut out by $C$: in fact this
would imply the existence of double points for the elliptic involution cut
out by the $K$ over an elliptic component of $C$. Therefore the curve $E$
has zero intersections with $mC$, and it is consequently a
para-$mi$-canonical curve, as well as $mC$ and $mD$.

With this in mind, let us represent $F$ onto an $n$-tuple elliptic ruled
surface $\Phi$, in such a way that the $K$ correspond to the generators,
each of which represents the $g_n^1$ defined, as before, by the sections
of $C$ and $D$. We may realize $\Phi$ as a scroll in a certain space
$S_r$, and on this we have two minimal directrices corresponding to $C$
and $D$, belonging to two complementary spaces $\ga_1$ and $\ga_2$, of
dimensions $s$ and $r-s-1$; for instance, we may refer $\Phi$ to a cone
whose vertex corresponds to $C$ and such that $d$ is one of its
hyperplane sections. And on $\Phi$ we have an elliptic curve $E$ without
common points with $\ga_1$ and $\ga_2$. Now the $\infty^1$ linear
transformations of $S_r$ whose spaces of fixed points are $\ga_1$ and
$\ga_2$ act on the image of $E$ by transforming it in a rational
$\infty^1$ system of elliptic curves of degree 0, thus in a base point
free linear pencil of elliptic curves; accordingly, there is a base point
free linear pencil of elliptic curves $C$ on $F$ which corresponds to
this pencil of the $n$-tuple $\Phi$; so we have proved the existence of
our pencil. \qed

It cannot be excluded that the $C$ are reducible: in any case their
irreducible variable components form a linear pencil, which we again
denote by $|C|$.

Let us add that the computation of the Zeuthen--Segre invariant for
this irreducible linear pencil $|C|$ gives us
 \[
 \de-4=12p_a-p^{(1)}+9=-4,
 \]
thus $\de=0$: this means that the elliptic curves $C$, varying in $|C|$
and never acquiring a double point, all have the same modulus, that is, they
are birationally equivalent.

 \section{Surfaces with genera $p_g=0$ and $p_a=-1$ with an elliptic
pencil of curves of genus $\pi=1$}\label{sec:10.11}
 \markright{\hfill Surfaces with $p_g=0$ and $p_a=-1$ and an elliptic
pencil \quad 10.11\quad}
 The theorem which asserts the existence of a linear pencil of elliptic
curves on the surfaces with $p_g=0$ and $p_a=-1$ possessing an elliptic
pencil of curves $K$ of genus $\pi>1$, can now be extended to the surfaces
with the same characters ($p_g=0$, $p_a=-1$ and $p^{(1)}=1$) possessing an
elliptic pencil of curves $K$ of genus $\pi=1$.

It is convenient to start with the remark that, since the
Zeuthen--Segre invariant is
 \[
 \De-4=12p_a-p^{(1)}+9=-4,
 \]
 we must have
 \[
 \De=0;
 \]
 thus the elliptic $K$, varying in $(K)$ and never acquiring a double
point, are birationally equivalent one to the other.

Let us transform $F$, on which the $K$ are projectively realized as
curves of order $m$, into a surface, that we call $F$ again, of a suitable
space, in such a way that the $K$ become elliptic normal $K_m$, of the
same order $m$, each belonging to a space $S_{m-1}$: each $K$ of the
original $F$ is, in general, a projection of the corresponding $K_m$.
Now two elliptic normal $K_m$ with the same modulus are linearly
equivalent in a certain number $n$ of different ways: and precisely one
has $n=2m^2$ if the curves have general modulus, whereas $n=4m^2$ or
respectively $n=6m^2$, if they are harmonic or equianharmonic. Thus the
set $\Ga_n$ of $n$ linear transformations of the space referring two
$K_m$ one to another is rationally defined; by fixing one of the two $K_m$ and one of its points $P$, one defines a curve $C$ passing through
$P$ and intersecting the other variable $K_m$ at the $n$ points
corresponding to $P$ in the $n$ linear transformations of $\Ga_n$.

At a first glance one could believe that -- since the $K_m$ are
birationally identical -- the aforementioned linear transformation are
always distinct and therefore send a point $P$, fixed as above, to $n$
distinct points; thus the curve $C$ would contain an elliptic involution
$\ga_n^1$ without double points and so it would certainly be elliptic
and, as $P$ varies, it would give us a linear pencil $|C|$ of elliptic
curves, directrices of the elliptic pencil.

These curves can be reducible, in which case we denote again by $C$
their irreducible components, that are also elliptic without common
points, and we denote by $n$ again the number of their intersections
with the $K=K_m$. So we have, on the surface of irregularity $p_g-p_a=1$,
possessing an elliptic pencil of elliptic curves $(K)$, another pencil
$|C|$ of elliptic curves, directrices of the $K$, and this pencil is
necessarily rational\footnote{See Chapter~IX, Section~\ref{sec:9.11}.}.

But an objection can be moved to the argument used to prove that the
curves $C$ have genus one; this objection can only be overcome after a
deeper analysis, in which one has to take into account the fact that we
are actually dealing with surfaces of genus $p_a=-1$. In fact one cannot a
priori exclude that the linear transformations existing between two
variable $K_m$ become degenerate, and so -- supposing for instance that
$n=6$ -- that one has a pencil of normal sextics $K_6$ in $S_5$, inside
which there are degenerate sextics consisting of a double plane cubic
$K_3^2$: in that case there are not 36 any more, but 18 degenerate linear
transformations relating a generic $K_6$ and the $K_3$, and these induce
as many [1,2] correspondences between the two curves.

One also cannot a priori exclude that among the generically
projectively equivalent $K_m$ of an elliptic pencil there are cuspidal
curves or curves reducible in several simple or multiple parts; but
this case cannot occur on our surface of genus $p_a=-1$ (and $p^{(1)}=1$),
where, as we have seen, the number of $K_m$ with a double point has to be
 \[
 \De=13+12p_a-p^{(1)}=0
 \]

Therefore, since $p_a=-1$, we only have to remove the exception that the
pencil $(K_m)$ may contain some multiple elliptic curves, which have zero
valency in the computation of the Zeuthen--Segre invariant $\De$ (see
Section~\ref{sec:10.3}).

In order to do this, we notice that, by means of the two pencils $(K)$ and
$|C|$ our surface $F$ can be represented on a $n$-tuple cubic cylinder
$\Phi$, where the curves $C$ correspond to the normal plane sections $c$
of $\Phi$ and the $K$ correspond to its generators: on $\Phi$ we have a
branch curve formed by a certain number of plane sections $c$,
corresponding to multiple curves $C=sC_s$, and (possibly) also by a
certain number of generators $k$, corresponding to degenerate $K$
consisting of multiple elliptic curves $K=rK_r$. If $p_a=-1$, and so (for
the $K$) $\De=0$, then the multiple cylinder $\Phi$ gives the
representation of an $F$ that may be assumed without singularities, on
which a $C_s$ and a $K_r$ meet at $n/rs$ points (simple points for the
surface) corresponding to the point $A=ck$ of $\Phi$: indeed if one of
these intersection points of $C_s$ and $K_r$, say $A'$, were a double or
multiple point of $F$, then the limit degenerate curve of a $K$ would not
be $rK_r$ (as we have supposed) but $rK_r$ plus the neighbourhood of
$A'$, so that we would have (Section~\ref{sec:10.3}) $\De>0$. But now we
shall prove that if, by contrast, the point $A$ of the $n$-tuple cylinder
$\Phi$ comes from a point of the surface $F$, then this has to be a
singular (double or multiple) point of $F$, and thus the hypothesis of a
degenerate $K$ of the type $rK_r$ leads to a contradiction. Let us indeed
consider a plane $\albar$, which is transversal with respect to the
generators of $\Phi$ and passes through
$A$, and a variable plane $\al$ which does not pass through $A$ but tends
to $\albar$. The section of $\Phi$ with the plane $\al$ is a $n$-tuple
multiple curve $\ell$, on which there are two branch points $A_1$ and
$A_2$, the intersections with $c$ and $k$. Let us describe on the Riemann
surface of the multiple $\ell$ a closed path $\ga$ which goes around
$A_1$ and $A_2$; it can be considered as the sum of two elementary paths
which go around $A_1$ and $A_2$. The first path gives rise to a
substitution on the sheets equivalent to the product of
 \[
(s-1)\frac{n}{s}
 \]
 transpositions, and the latter to a substitution equivalent to
 \[
(r-1)\frac{n}{r}
 \]
transpositions. Therefore the closed path $\ga$, that in the limit -- as
$\al\to\albar$ -- becomes a path going around A, gives rise to a
substitution formed by at least $n$ transpositions: this means that the
curve $L$, homologous to $\ell$, cannot have any simple point
corresponding to A, since such a curve would give rise to a branch of
order $n$ at worst, which would correspond to a substitution product of
$n-1$ transpositions on the sheets of the Riemann surface of $\ell$!

We may therefore conclude that:

The surfaces $F$ of genera $p_g=0$ and $p_a=-1$, possessing a pencil of
elliptic curves $K$, also possess a linear pencil of elliptic curves $C$
which are directrices of the $K$.

 \begin{NOTE}
 In the above argument we have used the fruitful concept of the
singularity of a curve which appears in the Memoir of O.~Chisini, Le
singolarit\`a di un ramo superlineare di curva piana definite mediante un
prodotto di sostituzioni\footnote{{\it Atti Ist.\ Veneto}, t.~XXX,
parte 2a, 1921. This memoir, published after the second volume of the
Lezioni by Enriques--Chisini, gives a further important contribution to
the theory developed there.}.
 \end{NOTE}

It is worth pointing out explicitly that, if we remove the hypothesis
$p_a=-1$, then there are indeed surfaces possessing an elliptic pencil of
elliptic curves $K$ which contains reducible curves: say surfaces $F$ for
which $p_a\ge0$ and $p_g=p_a+1$, containing a linear pencil of curves $C$
of genus $\pi>1$ which are directrices of $(K)$. If we call elliptic
surfaces the surfaces with $p_g=0$ and $p_a=-1$ studied in the previous
sections (the motivation for this name will be explained later) then
these other $F$, presenting some remarkable analogies with the former,
can be called paraelliptic surfaces. The more relevant analogy is that on
the paraelliptic surfaces, as well as on the elliptic ones with $p_a=-1$,
there are two pencils of birationally equivalent curves: an elliptic
pencil of elliptic curves $(K)$ and a linear pencil of curves, which here
are not elliptic any more but of genus $\pi>1$. But, while the elliptic
surfaces have -- as we shall see -- an $\infty^1$ group of
birational transformations into themselves, this property is not shared
by the paraelliptic surfaces.

We limit ourselves to pointing out here two examples of paraelliptic
surfaces, constructed as multiple coverings of a cubic cylinder $f(x,y)=0$:

 \begin{equation}
 \begin{array}{c}
 u^2=(z-a)(z-b)(z-c)(z-d)\psi(x,y)\\
 f(x,y)=0 \quad (p_a=3, p_g=4);
 \end{array}
 \label{eq:10.11.1}
 \end{equation}
$\psi$ represents a quadric cylinder parallel to $f$ which cuts $f=0$ in
six branch generators, to be added to the 4 branch cubics
 \begin{equation}
 \begin{array}{c}
z=a,\quad z=b,\quad z=c,\quad z=d;\\
u^3=(z-a)(z-b)(z-c)(z-d)\psi(x,y)\\
f(x,y)=0 \quad (p_a=0, p_g=1);
 \end{array}
 \label{eq:10.11.2}
 \end{equation}
$\psi=0$ denotes here a cubic cylinder having a triple contact with $f$
along two generators, touching it along another generator $k$ and cutting it
further along a generator $k'$: the branch curve of the triple cylinder
$f$ consists therefore of the three cubics
 \[
 z=a,\quad z=b,\quad z=c,
 \]
 and of the two generators
 \[
 k\quad\hbox{and}\quad k'.
 \]
Finally we point out to the young researchers this interesting subject of
study: to determine and classify all the paraelliptic surfaces; we refer
to the Notes at page. 123 and 134 of the Enriques--Campedelli's
{\it Lezioni}, in {\it Rendic.\ Sem.\ Mat.\ R. Universit\`a di Roma}
(1934), where the reader can find some useful remarks and suggestions.

 \section{Elliptic surfaces}\label{sec:10.12}
 \markright{\hfill Elliptic surfaces \quad 10.12\quad}
 Let us summarize the results we have obtained in the previous sections:
Any irregular surface $F$ of genus $p_g=0$ which does not belong to the
family of ruled surfaces (rational or not) has the arithmetic genus
$p_a=-1$ and contains an elliptic pencil $(K)$ of curves of genus
$\pi\ge1$, and a linear pencil of elliptic curves $|C|$; therefore it can
be represented on a multiple cubic cylinder with multiplicity
$n=CK$, with a branch curve formed by normal plane sections of $\Phi$.

But conversely an $n$-tuple cubic cylinder $\Phi$, with the branch curve
formed by normal plane sections (which are fixed in such a way to
satisfy the existence conditions of an irreducible surface), does not
always represent a surface of genus $p_g=0$, since the images of the
plane sections of $\Phi$ can be reducible elliptic curves whose
components form a nonlinear pencil $(C)$ of genus $\rho\ge1$.

It is apparent here that our surfaces, with $p_g=0$ and $p_a=-1$, not
birational to ruled surfaces, are contained in a more general family of
surfaces $F$, with $p_a=-1$, possessing an elliptic pencil of curves $K$ of
genus $\pi\ge1$ and a pencil of genus $\rho\ge0$ of elliptic curves $C$;
for such $F$ the geometric genus is $p_g=\rho$.

Indeed, since $p_a=-1$, the existence of the two pencils $(C)$ and $(K)$
that characterize these more general surfaces $F$ yields that the
irregularity of $F$ is
 \[
q\ge \rho +1
 \]
 and therefore
 \[
 p_g\ge \rho ;
 \]
 but, on the other hand, the canonical curves of $F$, which are sums of
curves $C$, must cut out on each $K$ groups of the $g_{2\pi-2}^{\pi-1}$
composed with the double point free involution of genus
$\rho$\footnote{Painlev\'e-Castelnuovo's theorem. see Enriques--Chisini,
{\it Lezioni}, Libro~V, Chapter~I, \S9 (Vol.~III, page~72).} cut out by
$(C)$; therefore they have to belong to the canonical series
$g_{2\rho-2}^{\rho-1}$ in the pencil $(C)$. Then one has
 \[
 p_g\le \rho
 \]
and therefore
 \[
 p_g=\rho, \qed
 \]

These more general surfaces, characterized-as we have said-by the
arithmetic genus $p_a=-1$ and by the existence of two pencils $(K)$ and $(C)$
of irreducible curves on them, that is,

 \begin{enumerate}
 \renewcommand{\labelenumi}{(\arabic{enumi})}
 \item an elliptic pencil of curves $K$ of a certain genus $\pi\ge0$ (for
which, since $p_a=-1$, one has $\De=0$);

 \item and a pencil of genus $p_g$ of elliptic curves $C$, such that
$n=CK\ge1$, are called elliptic surfaces: and more precisely proper
elliptic surfaces those, with $\pi\ge1$, which are not birational to ruled
surfaces, and improper elliptic surfaces those, with $\pi=0$, which are
birational to ruled surfaces of genus 1.
 \end{enumerate}

The name ``elliptic'' comes from a parametric representation of the
aforementioned surfaces which is related to the $\infty^1$ group of their
transformations about which we will say later, and that has been
pointed out by P.~Painlev\'e\footnote{P.~Painlev\'e, Le\c{c}ons sur la
th\'eorie analytique des \'equations diff\'erentielles. (Paris, Hermann,
1897).}.

In relation to this parametric representation the number $n=CK$ has been
called determinant of the surface.

The parametric representation we have mentioned above translates into the
following geometric property: Any elliptic surface $F$ has an $\infty^1$
elliptic group of birational transformations into itself, whose
trajectories are the curves of the pencil $(C)$.

Let us prove this theorem, by showing that there exists a birational
transformation of $F$, for which all the curves $C$ are invariant and in
which two arbitrary points $P_0$ and $P_0'$ of one of these curves $C_0$
are sent one to the other.

Indeed let us choose another curve $C$ of the same pencil $(C)$ and a
point $P$ on it. If the $C$, which are birationally equivalent, are
elliptic curves with general modulus, there are-as it is well known-two
transformations of $C_0$ in $C$ mapping $P_0$ to P\footnote{See for
instance Enriques--Chisini, {\it Lezioni}, Libro~V, Chapter~III, \S27
(Vol.~III, page~262).}; and only one\footnote{See Enriques--Chisini,
Ibidem, Chapter~IV, \S39 (Vol.~III, page~446).} of these (rationally
determined) preserves the groups cut out by the curves $K$, forming on
each curve an elliptic involution $\ga_n^1$. Therefore, since the
transformation of the first kind $(P_0P_0')$ is given on $C_0$, a single
transformation of the first kind of $C$ is also determined together with
it and this sends a point
$P$ to a point $P'$, in such a way that $(PP')$ goes to $(P_0P'_0)$ as $C$
tends to $C_0$.

The proof can be easily extended to the cases in which the $C$ are
harmonic or equianharmonic elliptic curves.

It is convenient to remark that the property we have established for
the elliptic surfaces characterizes them:

Any surface $F$ which admits an $\infty^1$ elliptic group $\Ga$ of
birational transformations into itself is a, proper or improper,
elliptic surface of arithmetic genus $p_a=-1$ possessing an elliptic
pencil of curves $K$ of genus $\pi$ ($\ge0$) and a pencil of genus $p_g$
of elliptic curves $C$.

In fact in the first place $F$ must have a pencil $(C)$, of a certain
genus $\rho\ge0$, formed by the trajectories of the group $\Ga$, which are
elliptic curves birational to $\Ga$ itself. But secondly, we shall
show with the following construction the existence of an elliptic
pencil $(K)$ on F: let us consider an (irreducible) curve $L$ on $F$ and
the $\infty^1$ elliptic series $\{L\}$ formed by its images via the
transformations of the group $\Ga$; the curves $L$ of this series are
certainly not equivalent, since they have to cut out on a $C$ the
inequivalent groups of points corresponding to a given one in a
transformation of the first kind; but among the groups of curves $L$
passing through a variable point $P$ on $F$ we will find $\infty^1$
equivalent groups: this means that there is an elliptic pencil of curves
on $F$, on which the $L$ of the series $\{L\}$ cut out equivalent groups
(see Chapter~IX, Section~\ref{sec:9.11}).

After thus having constructed the two pencils $(C)$ and $(K)$ on $F$, we
easily determine the characters of the surface. Indeed the computation
of the Zeuthen--Segre invariant for $(K)$ gives us
 \[
 \De=13-12p_a-p^{(1)}\ge0
 \]
and so
 \[
 p_a\ge-1\quad (p^{(1)}\ge1).
 \]

On the other hand, the existence of an elliptic pencil $(K)$ and of a
pencil of genus $\rho$ yields that (there is an $\infty^{\rho+1}$ series of
inequivalent curves on $F$, and therefore that one has)
 \[
 p_g-p_a\ge\rho +1
 \quad p_g\ge\rho.
 \]

Finally an argument that we have already used before leads us to
establish that the canonical curves of $F$ -- which for $\rho>0$ are formed
by $2\rho-2$ curves $C$ -- must give a series contained in the canonical
series $g_{2\rho-2}^{\rho-1}$ in the pencil $(C)$; by comparing we get
 \[
 p_g=\rho, \quad p_a=-1. \qed
 \]

 \begin{Remark} {From} what we have said we deduce that on an elliptic
surface $F$ of determinant $n>1$ there is an involution $I_n$ formed by
the groups of $n$ points common to the $C$ and the $K$, and it is
generated by an Abelian group $\Ga_n$ of (permutable) birational
transformations, and precisely by the transformations of the group $\Ga$
of the surface which leave the $C$ and the $K$ fixed. If the groups of
points of $I_n$ are supposed to be elements (points) of a new surface
$f$, we obtain an elliptic surface of determinant 1, on which $F$ is
represented as a multiple, and precisely as an $n$-tuple, surface; if
$p_g=0$, then this improper elliptic surface is nothing but an elliptic
surface, or, if we like it better, a cubic cylinder $f(x,y)=0$.
 \end{Remark}

It is worth pointing out here that the elliptic surfaces $f$ of
determinant 1 are in general defined as the surfaces representing the $\infty^2$ variety of the pairs of points belonging to an elliptic curve $C$ and to a curve $K$ of genus $\pi$
($\ge0$).

Starting with these surfaces one can construct the elliptic surfaces
with determinant $n>1$, by introducing algebraic irrationalities by means
of which the group $\Ga_n$ can be solved. Concerning this, we can say
that, a priori, either only one radical or two non-superimposed
radicals are needed, according to whether the group $\Ga_n$ is cyclic or
Abelian with a basis of order two. In fact $\Ga_n$, that is a finite
group of transformations of an elliptic curve $C$ into itself, cannot
have a base of order $>2$\footnote{See Enriques--Chisini, {\it Lezioni},
Libro~V, Chapter~III, \S28 (Vol.~III, page~271) and Chapter~IV, \S39
(Vol.~III, page~446--451).}. In next section, where we shall limit
ourselves in particular to the hypothesis $p_g=0$, and again, more
precisely, in Section~\ref{sec:10.16}, we explain the meaning of this
construction, which yields the existence of the elliptic surfaces, and
the conditions under which it holds.

 \section{Construction of the elliptic surfaces of genus
$p_g=0$}\label{sec:10.13}
 \markright{\hfill The construction of the elliptic surfaces of genus
$p_g=0$ \quad 10.13\quad}
 In order to construct the (proper) elliptic surfaces $F$ of genus $p_g=0$,
with any determinant $n>1$, we start with a cubic cylinder $f(x,y)=0$,
that -- as we have said -- is a model of the (improper) elliptic surface
of genus $p_g=0$ and determinant 1.

The surface $F$ can be represented on an $n$-tuple cylinder $f$, by means
of the two pencils of irreducible curves $|C|$ and $(K)$ characterizing it;
the branch curve is formed by a certain number of normal plane
sections $c$, so that the images of the curves $K$ are the generators $k$
of the cylinder, and the (trajectories of the $\infty^1$ group of
transformations of $F$) $C$ correspond to the normal plane sections $c$
of the cylinder itself. The branch curve of the $n$-tuple cylinder
consists of a certain number of curves $c$, corresponding to multiple
curves of the type $C=sC_s$.

So the representation of the elliptic surface $F$ on the $n$-tuple
cylinder $f$ must satisfy the following conditions:

 \begin{enumerate}
 \renewcommand{\labelenumi}{(\arabic{enumi})}
 \item that the curve $K$ defined by a $n$-tuple $k$ is irreducible and
contains an involution $g_n$ either of cyclic type or of (properly)
Abelian type with base 2;

 \item that the curves $C$ corresponding to the normal plane sections $c$
of $f$, which are $n$-tuple curves without branch points, are generally
irreducible.
 \end{enumerate}

We notice indeed that, if condition 2) is not satisfied, then the
surface $F$, represented by the $n$-tuple cylinder, is not an (elliptic)
surface with determinant $n$ any more, but it has a determinant $<n$, and
it turns out to be, in general, of genus $p_g>0$. This happens, for
instance, for the surface
 \[
u^2=(z-a)(z-b)(z-e)(z-d)\psi^2(x,y)
 \quad
f(x,y)=0
 \]
which is an irreducible surface $F$ represented on the double cylinder,
with the branch curve formed by the plane sections $z=a$, $z=b$, $z=d$,
$z=e$.

In fact a generic normal section:
 \[
 z=c (c=const.)
 \]
 corresponds to a pair of distinct elliptic curves on $F$:
 \[
 u=\pm\sqrt{(c-a)(c-b)(c-d)(c-e)}\psi,
 \]
 so that the irreducible curves $C$ part of the images of the $c$ form a
pencil (no more linear, but elliptic) of unisecant curves of the K; and
therefore the geometric genus of $F$ is:
 \[
 p_g=1.
 \]

It is convenient to recall here the criteria in order that an $n$-tuple
elliptic curve without branch points is irreducible\footnote{See
Enriques--Chisini, Lezioni, Libro~V, Chapter~IV, \S39 (Vol.~III, page~446).}. There are two possible cases: the cyclic case and the Abelian
case (with basis two). In the first case the $n$-tuple curve can be
constructed, starting for instance with a cubic curve $f(x,y)=0$, by
extracting an $n$th root of a polynomial $\fie(x,y)=0$ on it,
 \[
 9 u^n=\fie(x,y)
 \quad
0 f(x,y)=0;
 \]
 in order that the multiple curve has no branch points, it is necessary
that the critical points
 \[
 f=0,\fie=0,
 \]
be apparent critical points, and therefore that the $\fie$, taken for
instance of order $n$, has three $n$-fold contact points with $f$.
Therefore the irreducibility of the mentioned multiple curve depends on
the fact that ``the series $g_3^2$ containing the triple of the apparent
critical points is different from the series cut out on $f$ by the lines
of the plane''.

In the Abelian case (in which $n=pq$) the construction of the $n$-tuple
curve requires the extraction of two not superimposed roots of index $p$
and $q$ of two polynomials $\fie(x,y)$ and $\psi(x,y)$, which we may
assume of order $p$ and $q$: in order that the branch curve has no branch
points, the curves $\fie$ and $\psi$ must have three points of contact
with $f$ of orders $p-1$ and $q-1$ respectively, that therefore form two
triples of apparent critical points; the irreducibility of the multiple
curve yields that these triples do not belong to the $g_3^2$ cut out on
$f$ by the lines of the plane, and moreover that they satisfy a certain
dissimilarity condition.

Let us develop this construction in the cyclic case, which one
encounters in particular when the determinant is a prime number. So we
have that the involution $I_n$ on the surface $F$, formed by the groups
$CK$, is generated by a cyclic group of transformations $\Ga_n$, of order
$n$.

Starting with the $n$-tuple cubic cylinder
 \[
 f(x,y)=0,
 \]
 we fix the normal plane branch sections
 \[
 z=a_1, z=a_2,\dots z=a_t;
 \]
 the $n$ points of the group $G_n$ corresponding to a generic point of $f$
can be given by the values of a radical
 \[
 u=\sqrt[n]{(z-a_1)^{r_1}(z-a_2)^{r_2}...(z-a_t)^{r_t}
\fie(x,y)}
 \]
 satisfying the following conditions:

 \begin{enumerate}
 \renewcommand{\labelenumi}{(\arabic{enumi})}
 \item for the exponents $r_i<n$ one has
 \[
r_1+r_2+\cdots+r_t\equiv 0 \mod n,
 \]
since otherwise there would be a branch point at infinity on the
generators of the cylinder $f$.

 \item If $h_i$ is the greatest common divisor of $n$ and $r_i$, so that
 \[
n=h_is_i,r_i=h_i\rho_i
 \]
then the following factor appears in the expression of $u$
 \[
 \sqrt[n]{(z-a_i)^{r_i}}=\sqrt[s_i]{\ep_i(z-a_i)^{r_i}},
 \]
where $\ep_i$ is a $h_i$th root of unity.

So on the $n$-tuple generator of the cylinder $f$, the point
 \[
 z=a_i
 \]
is a branch point corresponding on the homologous $K$ to a group $G_n$ (of
the $g_n$ cut out by the $C$) formed by $h_i$ groups of $s_i$ coinciding
points. In other words, the point $z=a_i$ is a branch point of order $s_i$
for the multiple $k$, with
 \[
 s_i=n/h_i,
 \]
and therefore we have
 \[
 \sum (n/s)(s-1)=2n+2\pi-2
 \]
(the summation is taken over all values of $s=s_1,s_2,\dots,s_t$), where
$\pi$ denotes the genus of the $K$.

 \item Each one of the numbers $s_i$ ($i=1,2,\dots,t$) is a divisor of the
lowest common multiple of the remaining ones.
 \end{enumerate}

This property is a general consequence of the Abelian character of $\Ga_n$. In fact since on the multiple line $k$ there are branch points at
infinity, the product of the cyclic substitutions on the sheets of the
algebraic function $u(xyz)$, in correspondence with the branch points
$a_i$, has to be equal to the identity:
 \[
 S_t\cdots S_2S_1=1
 \]
 so that, for instance
 \[
 S_1-1=S_t=\dots=S_3S_2,
 \]
that is,
 \[
 S_1{s_1}-1=S_t=\dots=S_3S_2.
 \]
But the substitution $S_1{s_1}$ is cyclic with period $s_1$, whereas the
product $S_t\cdots S_3S_2$ is cyclic with period a divisor of the lowest
common multiple of $s_2,s_3,\dots,s_t$\footnote{See for instance
L.~Bianchi, {\it Lezioni sulla teoria dei gruppi di sostituzioni $E$ delle
equazioni algebriche secondo Galois}, (Pisa, 1900), Chapter~III, \S30.},
since
 \[
 S_t^{s_t}\cdots S_3^{s_3}S_2^{s_2}= (S_t\cdots S_3S_2)^{s_2s_3
\cdots s_t}=1.
 \]

4) Since the $n$-tuple cylinder has no branch generators, the
polynomial $\fie(x,y)$ appearing in the expression of $u$ under the radical
must represent a cylinder having an $n$-fold contact with $f$ along any
common generator. And in order that the curves $C$ represented by
$z=\hbox{const.}$ be irreducible, the group of the apparent critical
generators must not be equivalent to a multiple of the triples of
generators cut out by the planes
 \[
 \al x+\be y+\ga=0.
 \]

Moreover we may take $\fie$ of order $n$, so that $\fie=0$ touches $f=0$
with an $n$-fold contact along three generators (not lying in a plane).
Indeed we can easily see that, if one lets $\fie$ vary continuously, then
the surface $F$ that one obtains is always birational to a given one,
since the curves $C$ and the $K$, represented by the plane sections
$z=\hbox{const.}$ and by the generators of $f$ respectively, are
preserved. So if we take, instead of the cylinder $\fie$, a cylinder
$\psi$ of order $nh$, with $h>1$, we may replace $\psi(x,y)$ with a
polynomial which is reducible into a simple factor $\fie$ of order $n$
and into a $n$-tuple factor $\fie_1^n$ with $\fie_1$ of order $h-1$:
 \[
 \psi=\fie_1^n\fie;
 \]
 thus we have
 \[
u=\fie_1\sqrt[n]{\fie(x,y)\cdot
(z-a_1)^{r_1}(z-a_2)^{r_2}\cdots(z-a_t)^{r_t}},
 \]
whence -- by changing $u$ into $u\fie_1$ -- we obtain precisely the
expression we wanted.

In conclusion: the typical elliptic surface with determinant $n$,
belonging to the cyclic case, is the surface
 \[
 u=\sqrt[n]{\fie(x,y)(z-a_1)^{r_1}(z-a_2)^{r_2}\cdots(z-a_t)^{r_t}\fie(x,y)}
 \quad
 f(x,y)=0
 \]
where the exponents $r_i$ satisfy the conditions mentioned above and
$\fie(x,y)=0$ is a cylinder of order $n$ touching $f$ with an $n$-fold
contact along three generators which do not lie in a plane.

The construction of the surfaces $F$ as $n$-tuple covers of a cubic
cylinder $f(x,y)=0$ can be repeated with few modifications in the case in
which the group is Abelian (with base 2). The only difference is that,
instead of having only one radical, we have now two non-superimposed
radicals in the expression of $u$. We do not dwell on this construction,
which we shall develop later on (Section~\ref{sec:10.16}) for the family
of surfaces $F$ for which the curves $K'$, as well as the $C$, are
elliptic. We only want to point out here that again, as in the cyclic
case, one has the equality
 \[
 \sum [n(s-1)]/s=2n-2\pi-2,
 \]
and each one of the numbers $s_i$ has always to divide the lowest common
multiple of the remaining ones.

 \section{Surfaces with all plurigenera equal to zero: characterization
of the ruled surfaces}\label{sec:10.14}
 \markright{\hfill Surfaces with all $P_i=0$: characterization of the
ruled surfaces \quad 10.14 \quad}
 We have proved that ``the irregular surfaces with geometric genus
$p_g=0$, which do not belong to the family of the ruled surfaces, are
elliptic surfaces with arithmetic genus $p_a=-1$''. Now we want to
determine the plurigenera of these elliptic surfaces and to verify that
these characters cannot be all zero, as it happens for the ruled surfaces
(rational or not). Thus, if we recall the rationality conditions of
Castelnuovo, given by the vanishing of the arithmetic genus and of the
bigenus (Chapter~VI, Section~\ref{sec:6.4}): $p_a=P_2=0$, then the
family of ruled surfaces is defined by the vanishing of the plurigenera,
and more precisely by the following theorem: The necessary and sufficient
condition in order that a surface is birational to a ruled surface is
that both its quadrigenus and its sextigenus vanish:
 \[
P_4=P_6=0.
 \]

Let us consider a (proper) elliptic surface $F$, of genus $p_g=0$ and
determinant $n\ge2$, represented on the $n$-tuple cubic cylinder $f(x,y)=0$
in the above described way, in relation to the two characteristic pencils
$(K)$ and $|C|$. We may determine the virtual canonical curve of $F$ by
adding to the transform of the canonical curve of the cylinder $f$ the
curve corresponding to the branch curve on this multiple cylinder (see
Chapter~V, Section~\ref{sec:5.7}).

Now the virtual canonical curve of $f$ can be obtained by subtracting the
linear system $2c$, formed by two normal plane sections of the cylinder,
from its adjoint system, consisting of the three lines cut out by the
plane at infinity; and since these lines are the images of exceptional
curves on $F$, we only have to add to the transform of the branch curve
the virtual curve $-2C$, corresponding to $-2c$. So, if we denote by
$z=a_i$ the components of the branch curve, each corresponding to a
multiple curve $C=s_iC_i$, then the pure canonical curve of $F$ is given
by
 \[
 D-2C=\sum (s-1)C_s-2C.
 \]

Since the curves $C$ homologous to the $c$ are irreducible, we
immediately see that the subtraction appearing in the above formula is
not possible, and so we prove once more that the irreducibility of the
mentioned curves $C$ implies that the genus of $F$ is $p_g=0$.

In fact the linear system $|D|$, defined by $D=\sum (s-1)C_s$, has
dimension $x=0$ and therefore it cannot contain the system $|C|$, let
alone the system $|2C|$. To justify this assertion, let us notice that,
since the curve $d$ is formed by elliptic components of degree 0, a
priori it has to define a linear system composed with the curves of the
pencil $|C|$ and with curves whose multiples lie in that pencil:
 \[
 |D|=|\sum (s-1)C_s|=|xC+\sum h_sC_s|.
 \]

Whence we deduce
 \[
 |xC|=|\sum k_sC_s|,
 \]
 with
 \[
 0\le k_s<s-1.
 \]
It is clear that such a relation cannot hold for $x>0$ (because the
generic curves $c$ are irreducible), since this would contradict the
reducibility principle, inasmuch as a variable (irreducible and
therefore connected) curve of $(C)$ would degenerate in parts $C_s$ which
are not connected to one another.

Now from the above formula we can derive the expression of the
pluricanonical curves and therefore the computation of the plurigenera
of $F$.

In general the $m$-canonical curves are given by
 \[
 |m\sum (s-1)C_s-2mC|=|m(t-2)C-m\sum C_s|,
 \]
 where the summation is taken over all the values of $s$ corresponding to
the $t$ multiple curves of the pencil $|C|$:
 \[
 s=s_1,s_2,\dots,s_t.
 \]

If we denote by $x=P_{m-1}$ the dimension of the complete $m$-canonical
system, then this system consists of $x$ variable (irreducible) curves $C$
and of some fixed components $C_s$, each to be counted a certain number
of times
 \[
 h<s.
 \]

So we have
 \[
 |m(t-2)C-m\sum C_s|=|xC+\sum hC_s|,
 \]
with $0<h<s$, the summation being extended to the values
 \[
 s=s_1,s_2,\dots,s_t
 \]
and to the corresponding values of
 \[
 h=h_1,h_2,\dots,h_t.
 \]

Now, since $sC_s=C$, from the above relation we deduce
 \[
 |(m(t-2)-x)C|=|\sum ((m+h)/s)C|:
 \]
 this means that the fractions are integers:
 \[
(m+h)/s=\rho,
 \]
and each of this integers ($\rho=\rho_1,\rho_2,\dots, \rho_t$) expresses
by excess the value of the fraction $m/s$, that is, it equals
 \[
[(m+s-1)/s],
 \]
the greatest integer smaller than $(m+s-1)/s$.

In conclusion we have
 \[
 x=m(t-2)-\sum \frac{m+s-1}{s},
 \]
and therefore the plurigenus of order $m$ ($\ge2$) is given by the
formula
 \[
P_m=1+m(t-2)-\sum\frac{m+s-1}{s},
 \]
where one has to take $P_m=0$ when the expression becomes negative.

Let us set here $m=2$; taking into account that -- for
$s=s_1,s_2,\dots,s_t$ -- we always have
 \[
 s\ge 2,
 \]
we have
 \[
 [(s+1)/s]=1, \sum [(s+1)/s]=t,
 \]
and therefore the bigenus is:
 \[
P_2=t-3.
 \]
Now, for $t>3$ we certainly have $P_2>0$ and therefore also $P_4>0$,
$P_6>0$, etc. If instead $P_2=0$, since the hypothesis $t=2$ has to be
excluded because it yields that $F$ is ruled, then we have
 \[
t=3.
 \]

But, if we denote by $n$ the determinant of $F$, then we have the
equality
 \[
 \sum n(s-1)=2n+2\pi-2,
 \]
whence (since $s_i\ge0$, $\pi\ge1$)
 \[
 \sum (1-(1/s))\ge1
 \]
 that is,
 \[
 t-\sum (1/s)\ge1
 \quad
 \sum (1/s)\le t-1.
 \]

So for $t=3$ we have
 \[
 \sum (1/s)=(1/s_1)+(1/s_2)+(1/s_3)\le 1.
 \]

Let us assume that the (integral and positive) numbers $s_1,s_2,s_3$
are in non-decreasing order:
 \[
 s_3\ge s_2\ge s_1;
 \]
 the above inequality gives us either
 \[
 s_1=2, s_2=3, s_3\ge 6
 \]
 or
 \[
 s_1=2, s_3\ge s_2\ge 4,
 \]
 or
 \[
 s_1\ge 3, s_3\ge s_2\ge 3,
 \]
 or finally
 \[
 s_3\ge s_2\ge s_1\ge 4.
 \]

In the first and the second case:
 \[
 [(s_1+2)/s_1]=2, [(s_2+2)/s_2]=[(s_3+2)/s_3]=1
 \]
 thus
 \[
P_3=0,
 \]
 whereas in the third and fourth case:
 \[
 [(s_1+2)/s_1]=[(s_2+2)/s_2]=[(s_3+2)/s_3]=1,
 \]
 and
 \[
P_3=1.
 \]

On the other hand, if we compute the quadrigenus
 \[
P_4=5-\sum [(s+3)/s],
 \]
 then we have:

in the first and third case
 \[
P_4=0,
 \]

in the second case
 \[
P_4=1
 \]

and in the fourth case
 \[
P_4=2.
 \]

And for the sextigenus
 \[
P_6=7-\sum [(s+5)/s]
 \]
 we find:

in the first and third case
 \[
P_6=1,
 \]

in the second case
 \[
P_6=0
 \]

in the fourth case
 \[
P_6=2.
 \]

For our purposes it is sufficient to notice that when $P_2=P_4=0$
(first and third case in the above discussion) then
 \[
P_6\ge1.
 \]

In conclusion for an elliptic surface, which is not birational to a
ruled surface, at least one of the two plurigenera $P_4$ or $P_6$ is
different from zero:
 \[
P_4+P_6\ge1, \qed
 \]

 \begin{Remark} The conditions for a surface to be ruled can also be
expressed by the vanishing of the genus of order 12:
 \[
P_{12}=0.
 \]

In fact for the elliptic surfaces with $P_4\ge1$ or $P_6\ge1$, one
certainly has
 \[
P_{12}\ge1.
 \]

It is also worth remarking that one has
 \[
P_{12}=1
 \]
for the elliptic surfaces (with a virtual canonical curve of order
zero) possessing an elliptic pencil of elliptic curves $K$, and, by
contrast
 \[
P_{12}>1
 \]
for the elliptic surfaces possessing an elliptic pencil of curves $K$ of
genus $\pi>1$, whose virtual canonical curves intersect the $K$ at
$2\pi-2$ points.

Indeed the difference between the two types of surfaces $F$ consists in
this:

in the case of the $K$ of genus $\pi=1$ one has the equality
 \[
 2n=\sum ((s-1)/s)n
 \]
that is,
 \begin{equation}
 t-2=\sum (1/s),
 \label{10.14.1}
 \end{equation}
whereas in the case in which the $K$ have genus $\pi>1$ the following
formula
 \[
 2n+2\pi-2=\sum \frac{s-1}{2}\,n
 \]
holds, which yields the inequality

 \begin{equation}
t-2>\sum \frac1{s}\,.
 \label{10.14.2}
 \end{equation}

It follows immediately that in case 1) one has
 \[
P_{12}=1,
 \]
and not $P_{12}>1$, not only by the formula for $P_{12}$, but also by
noticing that on the surface $F$, possessing two pencils of elliptic
curves, there cannot be (pure) pluricanonical curves of order $>0$.
 \end{Remark}

Now let us prove that, if the inequality 2) holds, then one
necessarily has
 \[
P_{12}>1
 \]
and not $P_{12}=1$. This proof cannot be simply deduced from the formula
for $P_{12}$, but one also has to take into account the existence
condition for the elliptic surfaces (related to the Abelian character of
the group $\Ga_n$) which we have already pointed out in
Section~\ref{sec:10.13}, that is, that ``each one of the numbers $s_i$ has
to divide the lowest common multiple of the remaining
ones''\footnote{This is in fact the way the inequality $P_{12}>1$ is
established in of the by Enriques--Campedelli, {\it Lezioni}, \S15 ({\it
Sem.\ Mat.\ di Roma}, 1934). But in the original Note by Enriques, Sulla
classificazione delle superficie algebriche ecc., {\it Rendic.\ Acc.\
Lincei}, febbraio 1914) the matter is not well explained, so that one
could be induced to believe that the result could be simply deduced from
the general formula that gives $P_{12}$. Such a deduction would be wrong,
as was pointed out by H. Geppert in his Bericht die Klassification der
algebraischen Fl\"achen ({\it Jahresbericht der Mathematische
Vereinigung}, 1931); indeed, by taking, for instance,
$s=2$, $s_2=s_3=5$, we find $P_{12}=1$.

Therefore Geppert proposes to consider $P_{24}=1$ instead of $P_{12}=1$.
But, as a matter of fact, for the elliptic surfaces with pluricanonical
curves of order $>0$ one already has $P_{12}>1$. The case of exception
which was pointed out does not correspond to an existing elliptic
surface, because 2 is not a divisor of 5~.}.

Let us write down the formula
 \[
P_{12}=12t-23-\sum \frac{s+11}{s}\,,
 \]
and let us discuss the cases that can occur for $t>4$, for $t=4$ and for
$t=3$.

For $t>4$, that is, $t\ge5$, one has
 \[
 \left[\frac{s_i+11}{s_i}\right]\le 6
 \]
thus we certainly have
 \[
 \sum \left[\frac{s_i+11}{s_i}\right]\le24,
 \]
 \[
P_{12}\ge12t-47\ge12(t-4)+1>1.
 \]

For $t=4$ the minimum value for $P_{12}$ is:
 \[
P_{12}=12(t-4)+1=1,
 \]
and it is attained only when the four numbers $s_i$ are all equal to 2:
 \[
 s_1=s_2=s_3=s_4=2,
 \]
so that the equality (1) holds, because this is the only case in which
for each of the $s_i$ we have
 \[
 \left[\frac{s_i+11}{s_i}\right]=6;
 \]
 by contrast, if one of these numbers is greater than 2, as the
inequality 2) requires, for instance if
 \[
 s_4\ge3,
 \]
 then we have
 \[
 \left[\frac{s_4+11}{s_4}\right]\le 4
 \]
 thus
 \[
P_{12}\ge3.
 \]

Finally, let $t=3$ and let us write all the integral solutions of the
inequality
 \[
(1/s_1)+(1/s_2)+(1/s_3)<1
 \quad
(s_1\le s_2\le s_3),
 \]
 that moreover satisfy the condition that each one of the $s_i$ divides
the lowest common multiple of the remaining ones. We have
 \[
 \left\{
 \begin{array}{rrr}
 s_1=2, s_2=4, s_3=8,\\
 s_1=2, s_2=5, s_3=10,\\
 s_1=2, s_2\ge 6, s_3\ge s_2,\
 s_1=3, s_2\ge 3, s_3>3,\\
 s_1>4, s_2\ge s_1, s_3\ge s_2.
 \end{array}
 \right.
 \]
In all these case we have
 \[
 \left[\frac{s+11}{s}\right]<12
 \quad
P_{12}>1. \qed
 \]

 \section{A historical note}\label{sec:10.15}
 \markright{\hfill A historical note \quad 10.15\quad}
 The classification of the irregular surfaces of genus $p_g=0$ ($p_a<0$),
as well as the proof that those which are not birational to ruled
surfaces are elliptic surfaces containing an elliptic pencil of curves $K$
of genus $\pi\ge1$ having an $\infty^1$ group of transformations into
themselves and another linear pencil of elliptic curves $C$, and the
characterization of the ruled surfaces by means of the vanishing of the
quadrigenus and the sextigenus, form the subject of F.~Enriques' memoir
Sulle superficie algebriche di genere geometrico zero, {\it Rendic.\
Circ.\ Mat.\ di Palermo}, 1905. In any case (apart from some minor lack
of accuracy in a few arguments) the proof of these results was affected
by a gap, concerning the lemma II of \S\S6--8; in fact the existence
of another elliptic curve essentially distinct from the parabicanonical
$C$ was simply deduced from the fact that the curves of the system
$\{K''\}$, the second adjoint system to a $K$, passing through a
bicanonical group of $K$, must form an $\infty^1$ system of index $>1$:
as a matter of fact one cannot exclude a priori that the reducible curves
of this system containing $K$ as a part consist of a single component to
be counted with a certain multiplicity.

In order to fill this gap, the author found the argument relying on
the construction of curves cutting out on the $K$ groups of the
semiquadricanonical, or, in general, a submultiple of a multiple of the
canonical series. This procedure was treated by him in a Note of the
Acc.\ dei Lincei, in 1930\footnote{Enriques, Sopra le superficie
algebriche trasformabili in rigate. (Luglio, 1930).} and then by
Enriques--Campedelli, in Lezioni sulla classificazione delle
superficie algebriche particolarmente di genere zero ({\it Rendic.\ Sem.\
Mat.\ R. Universit\`a di Roma}, 1934).

But in the new proof still remains a delicate weak point. In fact it
is not possible to assume a priori, as we temporarily did with the
hypothesis II of the \S\S6--8, that if a linear series is given
or added to the rationality field of a curve $K$, then a single group of
the series can be rationally determined. Luckily enough we were able to
overcome the difficulty, by proving that the hypothesis II is verified
for some multiple of the aforementioned series, so that we can justify
the construction procedure for the essentially distinct
parapluricanonical curves.

We still have to add a remark concerning the case in which our surfaces
possess an elliptic pencil of (birationally distinct) elliptic curves; in
order to make the construction of the trajectories $C$ rigorous, or
rather to prove that these are elliptic too, one has to exclude the
possibility that some of these curves $K$ can degenerate to multiple
curves (thus proving that for $p_a=-1$ there are no paraelliptic
surfaces): this point has been made precise by relying on the analysis of
the singularity arising from the crossing of two branch curves (Chisini)
and can be found in Enriques--Campedelli, {\it Lezioni} of 1934 (\S12).

Finally, as far as the effective construction of the elliptic surfaces is
concerned, in Enriques' original Memoir of 1905 only the cyclic case was
considered. The first examples concerning the Abelian case (with basis
two) have been brought into light by G.~Bagnera and M.~De Franchis in
their paper\footnote{Sopra le superficie algebriche che hanno le
coordinate del punto generico esprimibili con funzioni meromorfe
quadruplamente periodiche di due parametri, {\it Rendic.\ Acc.\ dei
Lincei}, 1907, 1.}.

The general construction of all the elliptic surfaces (which can be
reduced to the resolution of Abelian equations with a group of basis two)
was finally developed by O.~Chisini in the Notes of 1921\footnote{Le
superficie ellittiche il cui determinante e' un numero composto, {\it
Rendic.\ Acc.\ Lincei}, 1921, 2.}.

 \section{Surfaces with pluricanonical curves of order zero: the types
with elliptic curves $K$ with general modulus}\label{sec:10.16}
 \markright{\hfill Surfaces with pluricanonical curves of order zero
\quad 10.16\quad}
 In Section~\ref{sec:10.14}, and in particular in the Remark at the end,
we proved that:

The properly elliptic surfaces with genera $p_g=0$ and $p_a=-1$ can be
distinguished from the ruled surfaces by the 12-genus, since they have
 \[
P_{12}>0,
 \]
 instead of
 \[
P_{12}=0.
 \]

More precisely the elliptic surfaces with $p_g=0$ and $p_a=-1$ give rise
to two families:

 \begin{enumerate}
 \renewcommand{\labelenumi}{(\arabic{enumi})}
 \item surfaces with virtual canonical curve of order zero, characterized
by the property of possessing an elliptic pencil of curves $K$ of genus
$\pi=1$;

 \item surfaces with virtual canonical curve of order $2\pi-2$, containing
an elliptic pencil of curves $K$ of genus $\pi>1$.
 \end{enumerate}

The former (on which any pure linear system of genus $p$ has the degree
$n=2p-2$) have
 \[
P_{12}=1,
 \]
 whereas the latter (for which $n<2p-2)$ possess linear systems of
dimension $r>0$ of pluricanonical curves (of genus $p^{(1)}=1$) and have
 \[
P_{12}\ge 2.
 \]

Now we want to analyze more closely the classification of the surfaces of
class (1) for which $P_{12}=1$, showing that these give rise to few well
determined types, the equations of which can be written down.

Let us refer to the elliptic surfaces of the family 1); it is
convenient to distinguish between the cases in which:

Either the elliptic curves $K$ (which are birationally equivalent to
one another) have general modulus, or the curves $K$ are harmonic, or
they are equianharmonic.

Let us start here with the general case: we shall prove that the surface
$F$ has either determinant $n=2$ (the cyclic case) or $n=4$ (the Abelian
case).

In order to do this, we have to consider the involution $g_n^1$ cut out
on an elliptic curve $K$ by the trajectories $C$. This involution is
generated by an Abelian group and, since it is rational (and of genus
1), it has to contain some involutory transformation $I$ of the second
kind ($g_2^1$). Now, if the group $\Ga_n$ is not equal to the group
$\Ga_2$ formed by $I$ and by the identity, (since the products of two
transformations $I$ are transformations $\pi$ of the first kind) then $\Ga_n$
must contain as many transformations $\pi$ as I, say $m$ of each type: thus
$n=2m$. More precisely the transformations $\pi$ that are permutable with
a given $I$ are involutory, and so the Abelian group $\Ga_n$ generated by
the aforementioned $\pi$ and $I$ is the dihedral group $\Ga_4$ formed by
two involutions $I$, one involution $\pi$ and the identity.

Therefore our surface $F$ with a pencil of elliptic curves $K$ with
general modulus either has determinant $n=2$ or $n=4$, and the
corresponding $g_n^1$ defined on a $K$ belongs to one of the following two
types:

 \begin{enumerate}
 \renewcommand{\labelenumi}{(\alph{enumi})}
 \item $n=2$, $g_n^1=g_2^1$,

 \item $n=4$, $g_n^1=g_4^1$ is Abelian and dihedral, generated by two
permutable transformations of the second type, whose product is a
$\ga_2^1$ of the first series.
 \end{enumerate}

In the former case we may take as a model of the surface $F$ of
determinant $n=2$ the surface defined by the equations:
 \[
 {\rm I}_a: \quad
u=\sqrt{(z-a_1)(z-a_2)(z-a_3)(z-a_4)\fie(x,y)}
 \quad
f(x,y)=0,
 \]
where $f$ is a cubic cylinder and $\fie$ is a quadric cylinder having a
double contact with it along three generators not lying in a plane.

In the latter case, starting with the corresponding representation of
the quadruple line $k$ ($u=0$), we have to construct the elliptic curve
$K$ with a dihedral $g_4^1$ in the plane $(uz)$.

This construction can be explained either by using the general theory of
the Abelian equations\footnote{See for instance L.~Bianchi, Ibidem,
Chapter~IV, \S76.}, or, as we shall do here, specifically relying on the
properties of the elliptic curves\footnote{See Enriques--Chisini,
{\it Lezioni}, Libro~V, Chapter~IV, \S39 (Vol.~III).}.

Let us remark that our dihedral $g_4^1$ (on K) has 8 double points which
fall into two quadruples: a quadruple of double points of the former
$g_2^1$ formed by two pairs of conjugate points of the latter
 \[
 A_1A_2 \quad\hbox{and}\quad B_1B_2,
 \]
and a quadruple of double points of the latter $g_2^1$ formed by two
pairs of the former
 \[
 D_1D_2 \quad\hbox{and}\quad E_1E_2.
 \]

Therefore the 4-tuple line $k$ has four (double) branch points:
 \[
 abde,
 \]
and the rationality field of $K$ may be defined by setting
 \[
 u=\sqrt{\fie(z-a)(z-b)} \pm\sqrt{\psi(z-d)(z-e)},
 \]
(with $\fie$ and $\psi$ constant with respect to $z$ and $u$) so that the
two branch points $a$ and $b$ correspond to the pairs of double points
$A_1A_2$ and $B_1B_2$, and the points $d$ and $e$ correspond to the pairs
of double points $D_1D_2$ and $E_1E_2$.

The proof that the equation written above actually represents the
elliptic $K$ corresponding to our quadruple $k$ may be obtained by
remarking that the curve that we have represented possesses a $g_4^1$
with 8 double points; and anyway-by eliminating the radicals-we obtain an
equation of degree 4 in $z$ and $u$ and of degree 2 in $z$, that defines a
quartic with a double point at the point at infinity of the $z$ axis and
another double point infinitely near to it. Let us add that the two
transformations of the second type, that is, the series $g_2^1$ each having
4 double points, that compose the dihedral $g_4^1$, can be represented by
the equations
 \[
 \begin{array}{c}
u'=[\fie(z-a)(z-b)-\psi(z-e)(z-d)]/u
 \\
u''=[\psi(z-d)(z-e)-\fie(z-a)(z-b)]/u,
 \end{array}
 \]
as one may deduce from the identity:
 \begin{eqnarray*}
 \lefteqn{\frac{\fie(z-a)(z-b)-\psi(z-d)(z-e)}
 {\sqrt{\fie(z-a)(z-b)}+\sqrt{\fie(z-d)(z-e)}}=} \\
 && =\sqrt{\fie(z-a)(z-b)}-\sqrt{\fie(z-d)(z-e)}. \\
 \end{eqnarray*}

Therefore the equations of the elliptic surface $F$ with determinant
$n=4$, with $K$ of general modulus, can be reduced to the type:
 \[
 {\rm I}_b
 \quad
 u=\sqrt{(z-a)(z-b)\cdot\fie(x,y)}
 +\sqrt{(z-d)(z-e)\cdot\psi(x,y)}
 \quad
f(x,y)=0,
 \]
where $f$ is a cubic cylinder and $\fie$ and $\psi$ are two quadric
cylinders touching $f$ along two triples of generators, not lying in a
plane and not equivalent to one another.

 \section{Continuation: The harmonic case}\label{sec:10.17}
 \markright{\hfill Continuation: The harmonic case \quad 10.17\quad}
 In the harmonic case, besides the $g_2^1$ and the $g_4^1$ of the general
case, one can have on the elliptic curve $K$ a cyclic or Abelian $g_n^1$,
whose groups of $n$ points are transformed into themselves by a singular
transformation of order four. In fact the harmonic elliptic curves
possess, besides the ordinary transformations of the first and of the
second type, $\pi$ and $I$, two $\infty^1$ continuous series of singular
transformations of order 4, $\om$ and $\tau$; and the transformations of
the four types are related in the following way\footnote{See for instance
Enriques--Chisini, {\it Lezioni}, Libro~V, Chapter~III, \S27 (Vol.~III,
page~260).}:
 \[
 \begin{array}{cccc}
 \pi\om=\om_1 &\om_1\om=I &I\om=\tau &\tau\om=\pi \\
 \pi I=I_1 &I_1I=\pi &\om I=\tau &\tau I=\om \\
 \pi\tau=\tau_1 &\tau_1\tau=I &I\tau=\om &\om\tau=\pi \\
 \pi_1\pi=\pi_2 &I\pi=I_1 &\om\pi=\om_1 &\tau\pi=\tau_1.
 \end{array}
 \]

The various types of cyclic or Abelian $g_n^1$ belonging to an elliptic
harmonic curve $K$ can be classified by taking into account the following
remark: if the related group $\Ga_n$ is not generated only by ordinary
transformations (corresponding to $K$ with general modulus), then it must
contain an invariant subgroup $\Ga_4$ generated by a singular
transformation $\om$ or $\tau$. Now this $\Ga_4$ gives rise to a
$g_4^1$ with two quadruple points $A$ and $B$ and a pair of double points
$D_1$ and $D_2$ (forming a group of the $g_4^1$), that leads to the
representation of $K$ on a quadruple line $k$ ($u=0$) with two quadruple
branch points: $z=a$ and $z=b$, and a double branch point: $z=d$. If the
$g_n^1$ does not consist of this $g_4^1$ (that is, if $n>4$), then the group
$\Ga_n$ has to contain the transformations corresponding to the linear
transformations of the quadruple line fixing the point $d$ and permuting
the two points $A$ and B: these transformations generate an Abelian
$\Ga_8$, formed by 4 singular transformations of order 4, two $\om$ and
two $\tau$, by 2 involutions of the second type I, and by 2 transformations
$\pi$ of the first type: an involution $\ga_2^1$ and the identity.

In conclusion the $g_n^1$ on the elliptic harmonic curve $K$, provided
that it is not generated by ordinary transformations only, belongs to
one of the following two types:

 \begin{enumerate}
 \renewcommand{\labelenumi}{(\alph{enumi})}
 \item $n=4$, a cyclic $g_4^1$ generated by a singular transformation of
order 4: $\om$ or $\tau$; and

 \item $n=8$, a $g_8^1$ generated by an Abelian group $\Ga_8$ containing 4
singular cyclic transformations of order 4 (two $\om$ and two $\tau$)
and 4 ordinary transformations: two $I$ and two $\pi$ (that is, a $\ga_2^1$
and the identity).
 \end{enumerate}

In the first case the elliptic $K$ represented on the quadruple $k$, with
the branch points $a,b$ and $d$, can be realized on the $(zu)$ plane by
writing, for instance:
 \[
u=\sqrt[4]{\fie(z-a)(z-b)(z-d)^2}
 \]
with $\fie$ constant (with respect to $z$ and $u$).

The equation that we have just written corresponds to a certain choice of
the substitutions on the sheets of the algebraic function $u(z)$, in
relation to the branch curves $z=a,b,d$; if, by contrast, we take the
inverse substitutions then we find the similar type:
 \[
u=\sqrt[4]{\fie(z-a)^3(z-b)^3(z-d)^2}
 \]
which is birationally distinct from the first one.

So the elliptic harmonic surfaces with determinant $n=4$, cyclic case,
can be reduced to the type represented by the equations:
 \[
 II_a
 \quad
 u=\sqrt[4]{(z-a)(z-b)(z-d)^2\cdot\fie(x,y)}
 \quad
f(x,y)=0,
 \]
or to a similar type: where $f$ is a cubic cylinder and $\fie$ a cylinder
of order four touching $f$ with a fourfold contact along three generators
not lying in a plane. The similar types can be distinguished from one
another, as we have said, by the choice of the substitutions on the
sheets of $u$, and, although birationally distinct, they still have the
same characters.

In the properly Abelian case we can still represent in the $(zu)$ plane
a type of elliptic curve $K$ containing a $g_8^1$ with two groups, each
formed by two quadruple points, and a group formed by four double
points: in order to do this we have to construct an 8-tuple line with
three branch points
 \[
 z=a,z=b,z=d
 \]
such that the first two correspond to a cyclic substitution of order 4
on the sheets, whereas the third one corresponds to a simple
substitution of period 2.

This construction can be made according to the general principles of the
theory of the Abelian equations\footnote{See, for instance, L.~Bianchi,
{\it Lezioni sulla teoria dei gruppi di sostituzioni e delle equazioni
algebriche secondo Galois}, (Pisa, 1900) Chapter~III, \S30.}. On the other
hand the formula that one obtains in this way can be justified a
posteriori\footnote{See O.~Chisini, Le superficie ellittiche il cui
determinante e' un numero composto. {\it Rendic.\ Acc.\ Lincei}, s.~V,
Vol.~XXX, 1921, 2.} by looking at the substitutions on the sheets
corresponding to the branch points, as we mentioned above.

Let us say that the curve $K$ can be reduced to the type represented by
the equations
 \[
 u=\sqrt{(z-b)(z-d)\\fie}+\sqrt[4]{(z-a)(z-b)^3\\psi}
 \]
or to a similar type (which differs from this by a different choice of
the aforementioned substitutions): here $\fie$ and $\psi$ denote constants
(with respect to $u$ and $z$).

To justify this assertion, let us denote the 8 sheets of $u$ by means
of two indices:
 \[
u_{11} u_{12} u_{13} u_{14}
 \quad
u_{21} u_{22} u_{23} u_{24},
 \]
where the first index corresponds to the two values $+\sqrt{}$ and
$-\sqrt{}$ of the quadratic radical entering in the expression of $u$,
and the second to the four values $+\sqrt[4]{}$, $i\sqrt[4]{}$,
$-\sqrt[4]{}$, $-i\sqrt[4]{}$ of the radical of order 4.

The branch point $z=a$ corresponds the substitution on the sheets
 \[
S_a=(u_{11} u_{12} u_{13} u_{14})(u_{21} u_{22} u_{23} u_{24}).
 \]

And similarly the point $z=b$ corresponds to a cyclic substitution of
order 4, that is,
 \[
 S_b=(u_{11} u_{24} u_{13} u_{22})(u_{12} u_{21} u_{14} u_{23}).
 \]

The point $z=d$ corresponds, by contrast, to a substitution of period 2:
 \[
 S_d=(u_{11} u_{21}) (u_{12} u_{22}) (u_{13} u_{23}) (u_{14} u_{24}).
 \]

So we obtain three substitutions that, by multiplication, generate an
Abelian group of order 8, having the structure of the $\Ga_8$ we
mentioned above.

Let us add that the aforementioned substitutions satisfy the
existence condition\footnote{See, for instance, Enriques--Chisini,
{\it Lezioni}, Libro~V, Chapter~IV, \S38 (Vol.~III, page~429).} for the
algebraic function $u(z)$, (which is unbranched at the point $z=\infty$),
that is, that
 \[
 S_dS_bS_a=1;
 \]
and just in order to satisfy this condition we had to assign the
exponent 3 (instead of 1) to the term $z-b$ appearing under the fourth
order radical.

In this way the function $u(z)$ gives rise to the required elliptic
curve $K$ with an Abelian $g_8^1$, and moreover it gives the type of such
a curve, although we can also have another similar type, not birational
to the first one, given by
 \[
 u=\sqrt{(z-a)(z-b)^3\cdot\fie}+
 \sqrt[4]{(z-a)(z-d)\cdot\psi}
 \footnote{{From} the projective point of
vew, the curve $K$ representing the function u(z) has a tangent line
(parallel to the $u$ axis) with two fourfold contacts in correspondence
with the point $z=a$, a tacnode where two linear branches have a fourfold
contact (which therefore yields two quadruple points for the $g_8^1$) in
correspondence with $z=b$, and finally a quadruple tangent line with four
simple contacts in correspondence with $z=d$.};
 \]
other types which seem to occur can be reduced to the types mentioned
here with simple exchanges of the names of the branch curves.

It is worth adding that the two substitutions on the sheets of the
function $u$ in the neighbourhood of the branch points $z=a$ and $z=b$ can
be extended to birational transformations, of the same order 4, of the
curve $K$ whose equation
 \[
 \theta(zu)=0
 \]
can be obtained by eliminating the radicals appearing in the
expression of $u$; and the same can be said for the period two
substitution corresponding to the point $z=d$.

In fact one can prove, for instance, that $u_{12}$ is a rational
function of $u_{11}$ and of $z$; this follows from the consideration of
the two equations satisfied by a given value of $z$:
 \[
 \theta(uz)=0,\quad \theta(u'z)=0
 \]
and of their resultant
 \[
R(uu')=0:
 \]
this, together with $\theta(uz)=0$, defines $u'(uz)$, as a rational
function.

Now we can write the equations of an elliptic surface, harmonic type,
with determinant $n=8$:
 \[
 {\rm II}_b
 \quad
 u=\sqrt[2]{(z-a)(z-b)^3\cdot\fie(x,y)}
 +\sqrt[4]{(z-b)(z-d)\cdot\psi(x,y)},
 \quad
f(x,y)=0
 \]
where $f$ represents a cubic cylinder and $\fie$ and $\psi$ may be assumed
to be two cylinders, one of order 2 and the other of order 4, the
former tangent to $f$ along 3 generators and the latter tangent to it
with a fourfold contact along three other generators. In order that the
surface $F$ is actually of determinant $n=8$ (irreducibility of the
curves $C$ corresponding to $z=\hbox{const.}$ on it) one needs to suppose
that the two triples of generators of contact of $f$ and $\fie$ and of
$f$ and $\psi$ do not lie in planes, so that for $z=\hbox{const.}$ the
two curves
 \[
 u=\sqrt[2]{(z-a)(z-b)^3\cdot\fie(x,y)},\quad f(x,y)=0
 \]
 and
 \[
 u=\sqrt[4]{(z-b)(z-d)\cdot\psi(x,y)},\quad f(x,y)=0,
 \]
 are irreducible, and moreover the two curves
 \[
 u=\sqrt[2]{(z-a)(z-b)^3\cdot\fie(x,y)} \quad\hbox{and}\quad
 u=\sqrt[4]{(z-b)(z-d)\cdot\psi(x,y)}
 \]
are birationally distinct\footnote{See Enriques--Chisini, {\it Lezioni},
Libro~V, Chapter~IV, \S33 (Vol.~III, page~452).}: this yields that the
triple $g_1g_2g_3$ of the generators of fourfold contact of $\psi$ with
$f$ is not similar to the triple of generators $g'_1g'_2g'_3$ of contact
of $\fie$ with $f$, that is, the triple $g_1g_2g_3$, counted twice, forms a
sextuple which is not equivalent to the sextuple obtained by adding to
$g'_1g'_2g'_3$ three generators of $f$ lying in a plane.

 \section{The equianharmonic case}\label{sec:10.18}
 \markright{\hfill The equianharmonic case \quad 10.18\quad}
 In the equianharmonic case, if the elliptic curve $K$ contains a $g_n^1$
not generated by simple ordinary transformations ($n=2, 4$), then it has
to be generated by a cyclic or Abelian group $\Ga_n$ of transformations
(with basis 2) containing some singular transformation of order 3 or 6.
In fact the elliptic equianharmonic curves possess, besides the ordinary
transformations of first and second kind, $\pi$ and $I$, two $\infty^1$
continuous series of singular cyclic transformations of order 3, $\om$
and $\tau$, and two more series of cyclic singular transformations of
order 6, $\si$ and $\rho$\footnote{Enriques--Chisini, {\it Lezioni},
Libro~V, Chapter~III, \S27 (Vol.~III, page~260--263).}. And the
transformations of the various types are related in the following way:
 \[
 \left\{
 \begin{array}{cccccc}
\pi\si=\si_1 &\si_1\si=\tau &\tau\si=I &I\si=\om &\om\si=\rho &\rho\si=\pi
 \\
\pi\tau=\tau_1 &\tau_1\tau=\om &\om\tau=\pi &\si\tau=I &I\tau=\rho
&\rho\tau=\si
 \\
\pi I=I_1 &I_1I=\pi &\si I=\om &\om I=\si &\tau I=\rho &\rho I=\tau
 \\
\pi\om=\om_1 &\om_1\om=\tau &\pi\om=\pi &\si\om=\rho &\rho\om=I &I\om=\si
 \\
\pi\rho=\rho_1 &\rho_1\rho=\om &\om\rho=I &I\rho=\tau &\tau\rho=\si
&\si\rho=\pi.
 \end{array}\right.
 \]

Now for the Abelian $g_n$ not generated by ordinary transformations on
the equianharmonic curve $K$, three cases are possible:

 \begin{enumerate}
 \renewcommand{\labelenumi}{(\alph{enumi})}
 \item $n=3$, a cyclic $g_3^1$ generated by a singular transformation of
order 3, $\om$ or $\tau$;

 \item $n=6$, a cyclic $g_6^1$ generated by a singular transformation of
order 6, $\si$ and $\rho$ ;

 \item $n=9$, an Abelian $g_9^1$ (with basis two) whose group contains
three permutable singular cyclic transformations of order 3, say
$\om\om_1\om_2$, three more cyclic transformations as above
$\tau=\om^2,\tau_1=\om_1^2$, $\tau_2=\om_2^2$, and finally three ordinary
transformations of the first kind $\pi,\pi^2,\pi^3=1$.
 \end{enumerate}

This classification can be easily justified starting from the remark
that the group $\Ga_n$ has to contain an invariant cyclic subgroup $\Ga_3$,
generated by a singular transformation $\om$ or $\tau$. Since $\Ga_3$
generates a cyclic $g_3^1$ with three triple points, we are led to
examine the transformations of $K$ corresponding to the linear
transformations preserving a triple line of the type
 \[
 u^3=(z-a)(z-b)(z-d);
 \]
 and these are:

 \begin{enumerate}
 \renewcommand{\labelenumi}{(\arabic{enumi})}
 \item the involutions, for instance $(ab)(dd)$, having one of the points
of the triple $(abd)$ as a double point and permuting the other two
points; and

 \item the cyclic linear transformations $(abd)$ or $(adb)$.
 \end{enumerate}

Now by adding a transformation of type (1) to $\Ga_3$ one obtains a
cyclic $\Ga_6$ (with a sextuple point, two triple points and three double
points) that is not contained in a larger Abelian group; whereas the
addition of a transformation $(abd)$ of type (2) yields an Abelian
$\Ga_9$, containing three singular transformations of order three, which
in turn is not contained in a larger Abelian group.

We can easily construct the equianharmonic curves $K$ corresponding to
the aforementioned Abelian $g_n^1$, for
 \[
n=3, 6, 9,
 \]
 and accordingly the corresponding elliptic surfaces.

We have as a model of equianharmonic elliptic surfaces with
determinant $n=3$ (the cyclic case) the surfaces:
 \[
 {\rm III}_a
 \quad
 u=\sqrt[3]{(z-a)(z-b)(z-d)\cdot\fie(x,y)}
 \quad
 f(x,y)=0,
 \]
where $f$ is a cubic cylinder and $\fie$ is another cylinder of the same
order, touching $f$ with a triple contact along three generators not
lying in a plane.

And we have as the model of equianharmonic elliptic surfaces with
determinant $n=6$ (the cyclic case) the surfaces:
 \[
 {\rm III}_a
 \quad
 u=\sqrt[3]{(z-a)(z-b)^2(z-d)^3\cdot\fie(x,y)}
 \quad
f(x,y)=0,
 \]
where $f$ is a cubic cylinder and $\fie$ is another cylinder of order 6,
touching $f$ with a sixfold contact along three generators not lying in a
plane.

In the properly Abelian case, in which one deals with equianharmonic
elliptic surfaces $F$ of determinant $n=9$, a model of $F$ can be
constructed in a way similar to the one used in the harmonic case ($n=8$).

In order to do this, we need to construct the equianharmonic elliptic
curve $K$ of the plane $(uz)$ represented on a multiple line of order 9
$(u=0)$, the image of our Abelian $g_9^1$, generated by two singular
permutable transformations of order 3. As the type of this curve we can
take
 \[
 u=\sqrt[3]{(z-a)^2(z-d)\cdot\fie}+\sqrt[3]{(z-b)(z-d)^2\cdot\psi}
 \]
 with $\fie$ and $\psi$ constant.

To prove that the $u$($z$) thus defined actually corresponds to an
equianharmonic elliptic curve containing the aforementioned Abelian
$g_9^1$, it is convenient to compute the substitutions on the sheets of $u$ corresponding to the branch points
 \[
 z=a,z=b,z=d.
 \]

To this end, let us denote the sheets by means of two indices, the former
assuming the values $1,2,3$ in correspondence with the values of the
first cubic radical
 \[
 \sqrt[3]{},\quad \ep\sqrt[3]{},\quad \ep^2 \sqrt[3]{} \quad
(\hbox{where $\ep=e^{2\pi i/3}$})
 \]
 appearing in the expression of $u$($z$), while the latter index
accordingly assumes three values in correspondence to those of the
latter cubic radical. We have, in correspondence with the three branch
points, the following substitutions:

for $z=a$
 \[
 S_a=(u_{11}u_{31}u_{21})(u_{12}u_{32}u_{22})(u_{13}u_{33}u_{23});
 \]

for $z=b$
 \[
 S_b=(u_{11}u_{12}u_{13})(u_{31}u_{32}u_{33})(u_{21}u_{22}u_{23});
 \]

for $z=d$
 \[
 S_d=(u_{11}u_{23}u_{32})(u_{12}u_{21}u_{33})(u_{13}u_{22}u_{31}).
 \]

The third substitution is the inverse of the product of the other
two, so that the existence condition for the existence of the multiple
line $u=0$, not branched at the point at infinity $z=\infty$, is
satisfied\footnote{See Enriques--Chisini, {\it Lezioni}, Libro~V,
Chapter~IV, \S38 (Vol.~III, page~429).}:
 \[
S_dS_bS_a=1.
 \]

Thus one sees that the aforesaid substitutions generate an Abelian
group $\Ga_9$ giving rise to an, equally Abelian, $g_9^1$ on an
equianharmonic elliptic curve.

So one eventually proves that the equianharmonic elliptic surfaces
with determinant $n=9$ (the Abelian case) give rise to the type
 \[
 {\rm III}_c
 \quad
 u=\sqrt[3]{(z-a)^2(z-d)\cdot\fie(x,y)}
 +\sqrt{(z-b)(z-d)^2\\psi(x,y)}
 \quad f(x,y)=0,
 \]
 where $f$ is a cubic cylinder and $\fie$ and $\psi$ can be also assumed
to be cubic cylinders osculating $f$ along two triples of inequivalent
generators not lying in a plane.

 \section{Summary}\label{sec:10.19}
 \markright{\hfill Summary \quad 10.19\quad}
 We can summarize the analysis of the elliptic surfaces of genus $p_g=0$
with virtual canonical curve of order zero by stating the following
theorem:

The elliptic surfaces of genus $p_g=0$ and $p_a=-1$ with virtual
canonical curve of order zero, characterized by the value $P_{12}=1$,
that is, the surfaces with $p_a=-1$ possessing an elliptic pencil of
elliptic curves $K$ and another linear pencil of elliptic curves $C$, fall
into seven families with determinant $n=2,4$; $4,8$; $3,6,9$, and are
represented by the equations of one of the types I$_a$, I$_b$, II$_a$,
II$_b$, III$_a$, III$_b$, III$_c$, or by a similar type.

The types I$_a$ and I$_b$ (respectively the cyclic type and the Abelian
type) correspond to elliptic curves $C$ with general modulus; II$_a$ and
II$_b$ (the cyclic type and the Abelian type) to harmonic curves $C$, and
finally III$_a$ and III$_b$ to a cyclic type with equianharmonic $C$ and
III$_c$ to an Abelian equianharmonic type.

The aforesaid elliptic surfaces can also be distinguished, to a
certain extent, by means of the values of the plurigenera smaller than
$P_{12}$, that can be computed with the formula of
Section~\ref{sec:10.14}. The surfaces I$_a$ and I$_b$ (determinant 2 and
4) have the characters:
 \[
P_2=1, \quad P_4=1, \quad P_{12}=1;
 \]
the surfaces of the harmonic type II$_a$ and II$_b$ (determinant 4 and 8)
have:
 \[
(P_2=0), \quad P_4=1, \quad P_6=0, \quad P_{12}=1;
 \]
and finally the surfaces of the equianharmonic type III$_a$ (determinant
3) have
 \[
P_2=0, \quad P_3=1, \quad P_4=1, \quad P_{12}=1,
 \]
 while III$_b$ and III$_c$ (determinant 6 and 9) have
 \[
P_2=0, \quad P_3=0, \quad P_4=1, \quad P_{12}=1.
 \]

\begin{NOTE} The elliptic surfaces with virtual canonical curve of order
zero were already noticed by Enriques in his study of the surfaces of
genus zero (1905); they appear as particular hyperelliptic surfaces (more
precisely as involutions on the Jacobi surface corresponding to a
degenerate curve of genus two), and from this point of view they have
been considered by G.~Bagnera and M.~De Franchis in the memoir of 1907
Sopra le superficie algebriche che hanno le coordinate esprimibili con
funzioni meromorfe quadruplamente periodiche di due
parametri\footnote{{\it Rendic.\ Acc.\ Lincei}, s.~V, Vol.~XVI, 19071.}.

These authors discover that such surfaces give rise to a finite number of
types and succeed in classifying them by giving their hyperelliptic
parametric representations, corresponding to our 7 typical equations.

Independently of Bagnera and De Franchis, but after them, also Enriques
and Severi found this result again in a different way; their starting
point is still the parametric representation of our surfaces with
hyperelliptic functions\footnote{F.~Enriques and F.~Severi, Intorno alle
superficie iperellittiche irregolari, {\it Rendic.\ Acc.\ Lincei}, s.~V,
Vol.~XVII, 19081 (cf.\ by the same authors, ibidem, Vol.~XVI, 1907, 1).
M\'emoire sur les surfaces hyperelliptiques, {\it Acta Math.}, Vol.~32 and
33 (1909). See especially the \S\S56 and 57.}.

It seems to us particularly remarkable that in this treatment there is
the idea of constructing the hyperelliptic or properly Abelian surface as
a multiple elliptic surface without branch curves, thus computing the
characters defining it ($p_a=-1$, $p_g=1$, $P_4=1$ according to a theorem
of Picard--Enriques). For instance, if we refer to the simplest elliptic
surface, the one of type I$_a$, then one has a $F_m$ of a certain order
$m$ in $S_3$, possessing a biadjoint surface $\fie_{2n-8}$ and one
obtains a proper hyperelliptic surface ($p_a=-1$, $p_g=1$) by extracting
the $\sqrt[2]{\fie_{2n-8}}$ on $F_m$.

The present treatment of the question, that is, the algebraic classification
of our elliptic surfaces (which, in the constructive part, uses the paper
of O.~Chisini quoted above), was the object of a Note by Enriques, Sulle
superficie ellittiche di genere zero\footnote{{\it Rendic.\ Acc.\
Lincei}, s.~VI, Vol.~XIX, the 18th of February 1934.} of 1934, and then
it has been treated in the {\it Lezioni} by Enriques--Campedelli of the
Sem.\ Mat.\ di Roma (1934).
 \end{NOTE}
