 \chapter{The general classification of surfaces}\label{ch:11}
 \markboth{\qquad 11. \quad The general classification of surfaces
\hfill}{}

 \section{Introduction}\label{sec:11.1}
 \markright{\hfill Introduction \quad 11.1\quad}
 The reader who has followed the development of this treatise, if -- as
it often happens to the mathematician -- he likes the general truths
best, may have formed the impression that the author has given too much
room to examples and particular cases, being somehow guided by the
feeling of curiosity of the naturalist, who gathers in a museum the
most different types of animals, or plants or minerals.

But, as the museum can give an idea of the richness of forms of life
and therefore suggests the consideration of general problems in
biology, also the collection of examples in this field of mathematics
assumes an essential significance from the heuristic or
historical-constructive point of view of the science. We can clarify
its value by repeating the words with which G.~Castelnuovo gave an
account of the efforts, that we made together, to dissipate the
darkness that we had met at the beginnings of the theory of
surfaces\footnote{La geometria algebrica e la scuola italiana, Lecture
at the International Congress of the Mathematicians, Bologna, september
1928.}. ``We had constructed, in an abstract sense I mean, a large
number of models of surfaces in our space or in higher dimensional
spaces; and we had distributed those models, so to speak, in two
show-cases. One contained the regular surfaces, for which everything
proceeded as in the best of all possible worlds; analogy made it
possible to extend to these the most important properties of the plane
curves. But when we tried to verify these properties for the surfaces
of the other show-case, the irregular ones, then the troubles started,
and we found exceptions of every kind. Finally, an assiduous study of
our models led us to predict some properties that had to be true, with
suitable modifications, for the surfaces of both show-cases; then we
tested these properties with the construction of new models. If they
stood the test then, as the last step, we tried to find their logic
justification''.

But now we can say more: the determination of particular families or
classes of surfaces, over which we lingered, does not simply give mere
examples, whose meaning is then surpassed by the inductive development
of the science; in fact these are brought together in a synthesis that
can be said to be the main purpose of this algebro-geometric theory;
namely the general classification of the algebraic surfaces.

To this aim, it is convenient to notice that algebraic geometry is, on
the whole, a higher form of arithmetic, where the criterion itself of
generality has only a relative meaning. Here there is a very deep
remark, which goes back to Poncelet and Noether: from a purely logical
point of view we cannot say, for instance, that the family of surfaces
which are not birational to a ruled surface is larger than the one of
the ruled surfaces. If one proceeds to the classification of surfaces
(either in the ordinary space or in a hyperspace) according to their
order $n$, the case of surfaces which are not birational to ruled
surface, whose hyperplane sections have genus $\pi$ with
 \[
n\le2\pi-2
 \]
appears to be more general; but if, by contrast, we proceed to the
classification of the surfaces according to the genus $\pi$ of their
hyperplane sections, then the more general case appears to be the one
in which the order is
 \[
n>2\pi-2,
 \]
that is, the case corresponding to surfaces birational to ruled surfaces.

We do not want to draw paradoxical consequences from what we have said.
Thus we will go on considering the study of surfaces not belonging to
the family of the ruled surfaces as more general than the one of the
ruled surfaces. But this is a {\em value judgement}, not an absolute
fact. In giving it we weigh in our hand -- so to speak -- the problems
concerning the two types of surfaces, the richness of the cases to
which they give rise, the possibility of relating the two families from
more meaningful points of view. {From} this we can draw this lesson: that
every family or class of surfaces possesses its own characters that
distinguish it from the others and it is an object of study equally
worth to be pursued; and the definition of such a family by the
integral invariant characters has often a general interest. For example
the definition of the ruled surfaces by the vanishing of the
quadrigenus and the sextigenus gives, rather than a particular result
concerning this family, a significant theorem for the general theory;
telling us of the existence of canonical or pluricanonical curves (of
order $\ge0$) on any surface which is not birational to a ruled surface.

Our classification makes a synthesis of the results of previous
analyses and succeeds in distinguishing 4 families of surfaces, which
have essentially different properties and are defined by the value of
$P_{12}$ and -- subordinately -- of $p^{(1)}$. These are:

 \begin{enumerate}
 \renewcommand{\labelenumi}{(\Alph{enumi})}
 \item the ruled surfaces (existence of linear systems of genus $\pi$
and degree $n>2\pi-2$, infinitely many exceptional curves that cannot be
eliminated, and also -- as we shall see -- continuous series of
birational transformations which do not form a group of finite
dimension);

 \item the surfaces (without exceptional curves) with pluricanonical
curves of order zero (pure linear systems of degree $n=2\pi-2$);

 \item the surfaces ($n<2\pi-2$ with $\pi>1$) having canonical or
pluricanonical curves composed with the elliptic curves of a pencil
($p^{(1)}=1$);

 \item the surfaces ($n<2\pi-2$) with $p^{(1)}=1$, that give rise to
canonical or pluricanonical surfaces offering a model whose projective
properties reflect the invariant properties of the class of surfaces.
 \end{enumerate}

Each one of these families is an object of study interesting in
itself, but the relevance of the characteristic properties
distinguishing each of them from the others has a significance for the
general theory, as we pointed out above.

Now, to let the classification acquire its proper meaning, it is
convenient to extend some of the remarks that we made in the case $p_g=0$
to the surfaces with $p_g>0$; and to this aim we need first of all to
make a deeper study of the surfaces admitting a continuous series of
birational transformations into themselves.

 \section{Surfaces admitting a continuous series of birational
transformations into themselves: the cases leading to ruled
surfaces}\label{sec:11.2}
 \markright{\hfill Surfaces with a continuous series of transformations
\quad 11.2\quad}
 We know that the surfaces birational to ruled surfaces possess
continuous series of birational transformations into themselves; in
particular on a rational surface, or rather on the plane, one
knows\footnote{Enriques, 1893. See Enriques--Chisini, Lezioni. Libro~V,
Chapter~II, \S28 (Vol.~III, page~200).} the types of continuous groups of
Cremona transformations; and on the other hand one also has continuous
series of transformations, like the quadratic transformations, which do
not generate continuous groups of finite dimension. And such series can
also be found on a ruled surface of genus $p>0$; where however one has
series of transformations acting on the surface in a intransitive way,
if $p>1$.

With some additional restriction, one can prove the converse of the
above remarks, that is, the possess of a continuous system of birational
transformations enables us to characterize the family of ruled
surfaces. This can be done from four points of view:

 \begin{enumerate}
 \renewcommand{\labelenumi}{(\arabic{enumi})}
 \item any surface {\em possessing an $\infty^1$ rational group of
transformations into itself} belongs to the {\em family of ruled
surfaces;}\index{ruled surfaces}

 \item any surface possessing a (transcendental or algebraic) {\em
continuous group of transformations into itself}, fixing a linear
system $|L|$ of curves, at least $\infty^1$, and acting transitively on
it\footnote{We mean that it permutes the elements (curves) of $|L|$ in
infinitely many ways.}, belongs to the {\em family of ruled surfaces;}

 \item any surface $F$ possessing a {\em continuous group} of dimension
$r\ge2$ of transformations into itself, acting intransitively on the
points of $F$, or also a continuous group of dimension $r\ge3$ acting
transitively on it, belongs to the family of ruled surfaces;

 \item any surface possessing a continuous series of transformations
into itself, that {\em does not generate a group} of finite dimension,
belongs to the {\em family of ruled surfaces}.
 \end{enumerate}

The theorem (1) follows directly from the theorem of Noether--Enriques
(1898)\footnote{See for instance Conforto, {\it Le superficie
razionali},  Libro~II, Chapter~I, \S\S4, 5, 6 (pages~245, 260, 262).}
concerning the surfaces with a pencil of genus $p\ge0$ of rational
curves, because the trajectories of the $\infty^1$ rational group form
in fact a pencil of rational curves. But it should be pointed out that
the result we are mentioning was obtained by Painlev\'e before
Enriques' extension to the case $p>0$ of Noether's theorem for $p=0$:
indeed the fact that the rational curves $C$ of a pencil are
trajectories of an $\infty^1$ group makes it easy to construct a
unisecant curve to the $C$, by means of which the $C$ can be rationally
transformed into lines.

The theorem 2) states, on the whole, that ``a surface transformed into
itself by a continuous group of linear transformations is either rational
or ruled''; the theorem was given in this projective form, with some
restriction, by Enriques (1893), and it was proved in general.by Fano
(1895, 96, 97)\footnote{See Conforto, {\it Le superficie razionali},
Libro~II, Chapter~II, \S37 (page~496).}.

It is however convenient to give a new proof of this theorem, which
also takes care of the case of a group $\Ga$ fixing a linear system $|L|$,
at least $\infty^2$, belonging to an involution, and the case of a $\Ga$
fixing a linear pencil $|L|$; and more than that, the proof that we
propose here just reduces the general theorem to the latter case.

We start from (Picard's) remark that ``a continuous group of
birational transformations of an algebraic surface into itself is
always contained in an algebraic continuous group''\footnote{Indeed the
conditions that a surface $F$ is transformed into itself by birational
transformations of a given order are algebraic conditions, and
therefore they define a mixed or continuous algebraic group. If the
group is mixed it contains a continuous group.}.

If the surface $F$ admits an algebraic continuous group of
transformations into itself, fixing a linear pencil $|L|$ of curves and
acting transitively on it, we may assume that the curves $L$ are permuted
by $\Ga$ in the same way as by an $\infty^1$ group, since otherwise we may
replace $\Ga$ with the subgroup of its transformations fixing one or two
particular curves $L$.

In this setting we need to distinguish between the following
assumptions:

 \begin{enumerate}
 \renewcommand{\labelenumi}{(\alph{enumi})}
 \item the group $\Ga$ is $\infty^1$; then its trajectories, which admit
$\infty^1$ transformations into themselves, are curves $K$ (a priori
rational or elliptic)\footnote{SCHWARTZ's theorem. See
Enriques--Chisini, {\it Lezioni}, Libro~V, Chapter~III, \S31 (Vol.~III,
page~309).} on which the curves $L$ cut out an invariant rational
involution, therefore they are rational\footnote{The curves admitting
infinitely many transformations into themselves fixing a $g^1$ can be
transformed, with a multiple of such a series, into curves (of a
certain space) with infinitely many projective transformations, and are
therefore rational. See, for instance, Enriques--Chisini, Ibidem,
Libro~V, Chapter~III.}: it follows that $F$ is birational to a ruled
surface.

 \item The group $\Ga$ is $\infty^1$ at least, and it acts transitively
on the points of the surface. Then the curves of the pencil $|L|$,
which are transformed into themselves by $\infty^1$ transformations of
$\Ga$, are either rational or elliptic.
 \end{enumerate}

If they are rational, then $F$ is birational to a ruled surface. If
instead they are elliptic, then let us consider the $\infty^1$ elliptic
series $\{D\}$ formed by the images of a curve $D$, chosen in a completely
general way on $F$: through any point $P$ of $F$ there is a certain
number $i$ of curves $d$ of $\{D\}$, and $P$ may vary on a curve $K$ in
such a way that the groups of $i$ curves $d$ passing through it are
equivalent; hence the curves $K$ thus defined must form a base point
free elliptic pencil $(K)$\footnote{See Chapter~IX,
Section~\ref{sec:9.12}.}.

Therefore, among the transformations of our group $\Ga$ that permute
the elements (curves) of $(K)$, there are $\infty^1$ transformations
forming a subgroup $\Ga'$ and having the $K$ as their trajectories. By
SCWARTZ's theorem, these $K$ are a priori either rational or elliptic,
but the latter case can be excluded; let us in fact remark that the
transformations of $\Ga'$ have to fix the linear series $g_n^1$ cut out on
a $K$ by the curves $L$ of $|L|$. Thus we deduce that $F$ is birational to a
ruled surface. \qed

After having established theorem 2) in the case in which the linear
system $|L|$ fixed by the group $\Ga$ is a pencil, it is easy to extend
the proof to the case in which $|L|$ is an $\infty^r$ linear system,
with $r>1$. In fact the transformations of $\Ga$ (that can be assumed to
be a continuous algebraic group) act on the elements (curves) of $|L|$
as linear transformations of an $S_r$, also forming an algebraic group
$\Ga'$, and the subgroups of dimension $\ge1$ formed by all the
transformations of $\Ga'$ permutable with a given $\om$ are algebraic,
too. Now such a subgroup, let us call it $\Gabar'$, fixes the points
fixed by $\om$, and therefore it has also a fixed line not formed by
fixed points: this line corresponds to a linear pencil of curves $L$
which is transitively invariant for $\Gabar'$. Therefore we fall again
in the case we have treated before.

Theorem 3)\footnote{See Castelnuovo--Enriques, Sur les surfaces
alg\'ebriques admettant un groupe continu de transformations
birationelles en elles-m\^emes. (Comptes rendus, Paris, the 29th of
July, 1895).} can be proved as follows.

Let us start by assuming that the $\infty^r$ group $\Ga$ of
transformations of $F$ acts intransitively on the surface, that is, that it
fixes the curves $C$ of a pencil on it. If $\Ga$ is (as we may always
assume) an $\infty^r$ algebraic group with $r\ge2$, then we can get an
$\infty^1$ algebraic subgroup $\Ga'$ of $\Ga$, by imposing on the
transformations of $\Ga$ that they fix $r-1$ generic points of $F$: say
$O$, $O'$,\dots

Now the group $\Ga'$ -- which is an $\infty^1$ algebraic series of
elements with $\infty^1$ transformations into itself -- is a priori
either rational or elliptic. But it cannot be elliptic, because it
contains an invariant series $g_n^1$, of a certain order $n>1$,
corresponding to the series of the points of $F$ infinitely near to
$O$. Thus it is rational and (according to Theorem~(1) $F$ is birational
to a ruled surface.

To tell the truth, the above argument fails if the transformations of
$\Ga'$ fix all the points of $F$ infinitely near to $O$; but in that
case we can repeat the same argument for the successive neighbourhood of
a point $O_1$ near to $O$ in the first order neighbourhood, and
similarly -- if needed -- for the first neighbourhood of a point $O_s$ near
to $O$ in the $s$th order neighbourhood, not entirely consisting of fixed
points of $\Ga'$.

In an analogous way one proves the second case of the theorem, where
one has an algebraic group $\Ga$ at least $\infty^3$, acting
transitively on the points of $F$. If we refer to the simplest
hypothesis, in which $\Ga$ has dimension 3, then we obtain an $\infty^1$
algebraic subgroup $\Ga'$ by imposing on the transformations of $\Ga$
that they fix a point $O$, and -- as in the previous case -- we see that
$\Ga'$ is a rational group.

If by contrast $\Ga$ has dimension $r>3$, then we similarly obtain an
$\infty^1$ rational subgroup $\Ga'$ of it by imposing on the
transformations of $\Ga$ that they fix a point $O$ of $F$, and then
another point $O_1$ infinitely near to $O$, etc.

Finally we prove theorem 4), which answers a subtle question raised by
E.~Picard\footnote{Castelnuovo--Enriques, Sopra alcune questioni
fondamentali nella teoria delle superficie algebriche, {\it Ann.\ di
Mat.}, 1901 (n. 19).}.

If the surface $F$ admits a continuous series $\Ga$ -- say $\infty^1$-of
transformations into itself, which does not generate a group (of finite
dimension), then this means that the products of the transformations of
$\Ga$, taken $r$ by $r$, form in general an $\infty^r$ series of
transformations $\Ga_r$, where $r$ is susceptible of becoming
arbitrarily large. Therefore the transformations of $\Ga_r$ transform a
curve $C$, chosen on $F$ in the most general way, in the curves $C_r$
of a continuous system $\{C_r\}$ of dimension $r$. If the curve $C$
belongs to a base point free linear system, of genus $\pi$ and degree
$n$, then the system $\{C_r\}$ is formed by linear systems $|C_r|$ of
the same genus and degree, with a certain number of base points of
multiplicity $i$ (for $i=1, 2,\dots$); and the degree and genus of a
$|C_r|$ are respectively given by
 \[
N_r=n+\sum h_i^2
 \]
and
 \[
 \Pi_r=\pi+\sum \frac{h_i(h_i-1)}{2}
 \]

Therefore we have
 \[
N_r-2\Pi_r=n-2\pi+\sum h_i.
 \]

But, since the degree of $\{C_r\}$ (which is a priori $\ge r-1$) grows
with $r$ as much as we wish, we can choose $r$ in such a way that
 \[
 \sum h_i^2>(2\pi-n)^2
 \]
 thus
 \[
 \sum h_i\ge \sqrt{\sum h_i^2}>2\pi-n.
 \]

So it is apparent that the surface $F$ possesses curves of genus $\Pi_r$
and degree $N_r>2\Pi_{r-2}$ and therefore\footnote{See Chapter~X,
Section~\ref{sec:10.2}.} it is birational to a ruled surface. \qed

 \section{Elliptic and hyperelliptic surfaces}\label{sec:11.3}
 \markright{\hfill Elliptic and hyperelliptic surfaces \quad 11.3\quad}
 The analysis of the previous section allows us to ``determine all
the surfaces $F$, not birational to a ruled surface, admitting a
continuous group of transformations into themselves''.

First, we may assume this group $\Ga$ (not contained in a larger group)
to be algebraic; therefore the group gives rise to one of the following
cases:

 \begin{enumerate}
 \renewcommand{\labelenumi}{(\arabic{enumi})}
 \item either $\Ga$ acts intransitively on the points of the surface $F$,
and then (Section~\ref{sec:11.2}) it is an $\infty^1$ elliptic group
fixing the elliptic curves $C$ of a pencil, which has a certain genus
$\rho\ge0$;

 \item or $\Ga$ is an $\infty^2$ group of transformations, acting
transitively on the points of $F$.
 \end{enumerate}

\paragraph{Case~1} In the first case the surface $F$ admitting an
$\infty^1$ algebraic group of birational transformations is an elliptic
surface, with the arithmetic genus $p_a=-1$, the geometric genus
$p_g=\rho$, and the linear genus $p^{(1)}=1$.

In fact the $\infty^1$ transformations of $\Ga$ map a curve $L$,
generically chosen on $F$, to the $\infty^1$ curves of an elliptic series
$\{L\}$; there is a certain number $i$ of curves $L$ going through a
given point $P$, and $P$ can be allowed to move along a curve $K$ in such
a way that the group of the aforementioned $i$ curves $L$ always stays in
the same equivalence class; therefore the curves $K$ form an elliptic
pencil $(K)$ on $F$, and the curves $L$ cut out equivalent groups on the
curves of this pencil: for this reason the curves $K$ are essentially
distinct from the trajectories $C$ of $\Ga$.

Now the surface $F$, containing a pencil of genus $\rho$ of elliptic
curves $C$ and an elliptic pencil of curves $K$ (of genus $>0$), is
elliptic (Chapter~X, Section~\ref{sec:10.12}) and it has the aforesaid
characters, which can be easily computed.

In order to do this, we compute the number $\De$ of the curves of $(K)$
with a double point, expressing it by means of the Zeuthen--Segre
invariant. Since $(K)$ is an elliptic pencil, we find
 \[
 \De-4=12p_a-p^{(1)}+9.
 \]

And since the curves of $(K)$ are interchanged by the transformations
of $\Ga$, that do not fix any of them, we find:
 \[
 \De=0;
 \]
 thus
 \[
 12p_a+p^{(1)}+13=0, \quad p_a=-1, \quad p^{(1)}=1.
 \]

Finally, if $\rho=0$ then we have $p_g=0$, and if $\rho>0$ then the
groups of the $g_{2\rho-2}^{\rho-1}$ in the pencil $(C)$ give canonical
curves of $F$ (Chapter~X, Section~\ref{sec:10.12}); therefore in any
case one has
 \[
 p_g=\rho.
 \]

 \begin{NOTE} The simplest example of elliptic surface of genus $p_g>0$ is
given by the surfaces $F$ of determinant 1 representing the pairs of
point of an elliptic curve and of a curve of genus $p_g$, which on the
other hand can be represented on a multiple cubic cylinder (see Chapter~X,
Section~\ref{sec:10.12}). The elliptic surfaces of genus
$p_g>0$ and of determinant $n>1$ can be constructed by starting with a
multiple $F$ of determinant 1, or also with a multiple cubic cylinder. On
the whole one can repeat the arguments developed in the case $p_g=0$ with
a few modifications.
 \end{NOTE}

It is therefore apparent that {\em the elliptic surface with $p_g>0$
give rise to infinitely many families corresponding to the following
integral characters}, which are, to a certain extent, independent:

 \begin{enumerate}
 \renewcommand{\labelenumi}{(\alph{enumi})}
 \item the geometric genus $p_g$, genus of the pencil of elliptic curves
$(K)$;

 \item the determinant $n$, number of the intersections of a curve $K$
with a curve of the elliptic pencil $(C)$ (formed by the unisecant
curves of the K);

 \item the number of the multiple elliptic curves ($K=sK$) belonging to
the pencil $(K)$ and the corresponding multiplicities ($s_i$): as for
$p_g$ one can relate these numbers to the plurigenera of the
surface\footnote{}.

 \item the genus $\pi$ of the curves of the elliptic pencil
(C):\footnote{Here there are interesting problems concerning the
classification: for instance ``to determine the elliptic surfaces
corresponding to the first values of $\pi$''. For $\pi=2$, $p_g=0$, we
quote the Memoir by L.~Campedelli, Intorno alle superficie ellittiche
con un fascio di curve di genere due. {\it Rendic.\ Sem.\ Mat.\ di
Padova}, 1935.}
 \[
 \pi=1+n(p_g-1)+\sum [(s_i-1)n]/2s_i.
 \]
 \end{enumerate}

\paragraph{Case~2} After having completed the discussion of Case~1, we
now examine case 2), in which we deal with surfaces admitting an
$\infty^2$ continuous group $\Ga$ of transformations into itself.

To start with we notice that, if the transformations of $\Ga$ are not
permutable, then the transformations which are permutable with a given
one form an $\infty^1$ algebraic subgroup of $\Ga$, and therefore we
obtain a particular case of 1).

Therefore we may assume $\Ga$ to be an $\infty^2$ group, transitive on
$F$, formed by permutable transformations. And moreover we may assume
that it is transitive without exceptions on the surface $F$ without
exceptional curves: otherwise one has on $F$ at least a fixed point $O$
or a fixed curve $\chi$ and, by fixing a point near to $O$, or a point
of the curve $\chi$, we would find an $\infty^1$ algebraic subgroup of
$\Ga$ with a fixed point, thus $F$ would be birational to a ruled
surface (Section~\ref{sec:11.2}).

Finally it is easy to see that the group $\Ga$ acts on the points of $F$ in
a simply transitive manner, that is, we cannot have a finite number $n>1$ of
transformations of $\Ga$ in which a point $A$ of the surface corresponds to
one and the same point $A'$. Indeed a contradiction can be drawn from
this hypothesis in the following way.

Assuming that the hypothesis is verified, let us consider, together with
$F$, the surface $F'$ (which gives the parametric representation of
the group $\Ga$ according to S.~Lie) whose points correspond, without
exceptions, to the elements (transformations) of $\Ga$; since the
transformations of $\Ga$ act on one another by multiplication, $\Ga$
acts on $F'$ in a simply transitive way: $F'$ is therefore a
hyperelliptic surface, that is, an Abelian surface (Chapter~IX,
Section~\ref{sec:9.9}). On the other hand there is a $(1,n)$
correspondence between $F$ and
$F'$, in which the points of $F$ correspond to the groups of an
involution $I_n$ on $F'$; thus $F'$ is represented on the $n$-tuple $F$
without branch points. Whence it follows that the involution $I_n$ on
$F'$ is generated by a finite Abelian group $G_n$ of transformations of
$\Ga$, which is invariant for the $\infty^2$ transformations of $\Ga$
itself; and therefore the $n$ transformations of $\Ga$ relating two
groups of the involution, correspond to only one transformation of $F$
relating the two homologous points $A$ and $A'$, against the hypothesis
we started from, that there were $n$ transformations of $\Ga$ sending
$A$ to $A'$.

In conclusion we may say that the surfaces $F$, not birational to
ruled surfaces, admitting an $\infty^2$ group of birational
transformations (case 2) are hyperelliptic surfaces, that is, two
dimensional Abelian varieties, according to the definition of Chapter~IX,
Section~\ref{sec:9.9}.

We have already computed the genus of these surfaces (Chapter~IX,
Section~\ref{sec:9.10}), that is
 \[
 p_g=1
 \]
 and it is easy to find the other characters.

First of all the hyperelliptic surface $F$ (without exceptional curves)
cannot have any canonical curve $K$ of order $>0$.

In fact the $\infty^2$ group of transformations of $F$, fixing $K$, would
not carry its points to other points on the surface, against the
transitivity of $\Ga$. It is convenient to notice here that $F$ cannot a
priori have a canonical system of dimension $p_g-1>0$; in fact, this
linear system would be fixed by the transformations of $\Ga$ and thus
$F$ itself would be birational to a ruled surface
(Section~\ref{sec:11.2}).

Now, since the surface $F$ has a canonical curve of order zero, its
(absolute) linear genus is
 \[
 p^{(1)}=1.
 \]

We are left with the computation of the arithmetic genus $p_a$. To do
this, it is convenient to prove that the irregularity of $F$ is
 \[
q=p_g-p_a=2,
 \]
 so that
 \[
 p_a=-1.
 \]

We first prove that ``the Jacobi variety $V_p$, corresponding to a curve
$C$ of genus $p$, which clearly has irregularity $\ge0$, has exactly the
irregularity $p$ and not greater, that is, we cannot have on $V_p$ an
$\infty^r$ series of inequivalent $V_{p-1}$, with $r>p$''.

This proposition follows from the third equivalence criterion
(Chapter~III, Section~\ref{sec:3.13}) which can be easily extended from
the surfaces to the higher dimensional varieties. According to this
criterion we may say that two inequivalent varieties $V_{p-1}$
belonging to $V_p$ must cut out inequivalent varieties $V_{p-2}$ on the
$V_{p-1}$ corresponding to the series of the groups of $p$ points of
$C$ with a fixed point $A_1$, and accordingly inequivalent varieties
$V_{p-3}$ on the $V_{p-2}$ corresponding to the groups of $p$ points of
$C$ with two fixed points
$A_1$ and $A_2$, etc.

Therefore the property that we have stated can be proved by showing
that the $V_2$, corresponding to the series of groups of $p$ points of
$C$ with $p-2$ fixed points $A_1A_2\cdots A_{p-2}$, that is, the surface
representing the pairs of points of the curve $C$ of genus $p$, has exactly
the irregularity $p$ and not $>p$.

In order to prove this, it suffices to remark that an $\infty^r$ series
of inequivalent curves on such a surface gives rise to an $\infty^r$
series of inequivalent groups of points on $C$, thus we certainly have
 \[
r\le p. \qed
 \]

Once we have established the proposition on the irregularity of a
Jacobi variety $V_p$, we deduce that ``a $q$-dimensional Abelian variety
$V_q$ imprimitively contained in the Jacobi variety $V_p$ (with $q<p$),
that is, fixed by an $\infty^q$ subgroup of the $\infty^p$ group of
$V_p$, also has irregularity $q$''.

Since a $V_{q-1}$ of $V_q$ is transformed by the corresponding Abelian
group $\Ga_q$ in the $V_{q-1}$ of an $\infty^q$ series, the irregularity
of $V_q$ is certainly $\ge q$, and it suffices to prove that it cannot
be $>q$.

Indeed if a $V_p$ contains such a $V_q$, then the group $\Ga_p$ of the
$\infty^p$ permutable transformations of $V_p$ contains an $\infty^q$
algebraic subgroup $\Ga_q$ fixing the varieties V of an $\infty^{p-q}$
series of imprimitivity; then $\Ga_p$ contains also an algebraic
$\infty^{p-q}$ subgroup $\Ga_{p-q}$, fixing the varieties $V_{p-q}$ of
an $\infty^q$ series\footnote{This theorem (already stated in Chapter~IX,
Section~\ref{sec:9.9}) follows, as we have said, from the geometric
interpretation, due to Castelnuovo, of a theorem by Picard--Poincar\'e
concerning the reducible Abelian integrals on a curve. See
Enriques--Chisini, {\it Lezioni} L.~VI, Chapter~III, \S38 (Vol.~IV,
pages~237--241).}.

Now, if one of the aforementioned $V_q$ in $V_p$ has an $\infty^r$
continuous system of inequivalent $V_{q-1}$, then these generate, by
means of the $\infty^{p-q}$ transformations of $\Ga_{p-q}$, a
continuous system of dimension $r+p$-$q$ of inequivalent $V_{p-1}$, thus
we have
 \[
r+p-q\le p,
 \quad
r\le q.
 \]

So: the $V_q$ contained in $V_p$ has the irregularity $q$ and not $>q$.

In particular for $q=2$, an Abelian $V_2$, that is, a hyperelliptic surface
$F$ imprimitively belonging to a Jacobi $V_p$ ($p\ge2$) and
transitively transformed into itself by $\infty^2$ transformations of
the corresponding $\Ga_p$, has the irregularity
 \[
 p_g-p_a=2
 \]
and therefore
 \[
 p_a=-1.
 \]

But (Chapter~X, Section~\ref{sec:10.9}), any hyperelliptic surface $F'$
can always be reduced to such a simple or multiple $F$, imprimitively
contained in
$V_p$, and if one has a multiple $F$ according to a certain number
$n>1$, then in any case this has no branch curve; therefore we can compute
the arithmetic genus of $F'$ using Severi's formula that relates the
arithmetic genera $p_a=-1$ and $p_a'$ of the two surfaces $F$ and $F'$
in a $(1,n)$ correspondence (Chapter~V, Section~\ref{sec:5.8}); since
there is no branch curve of the multiple $F$, this formula can be
written as
 \[
 24(p_a'+1)=24n(p_a+1),
 \]
 and therefore it gives
 \[
 p_a'=p_a=-1.
 \]

In conclusion a hyperelliptic surface has the characters
 \[
 p_g=1, \quad p^{(1)}=1, \quad p_a=-1.
 \]

Moreover, since it has only one canonical curve of order zero, all
its plurigenera are equal to one.

 \section{Characterization of elliptic and hyperelliptic surfaces by
means of the values of the genera}\label{sec:11.4}
 \markright{\hfill Characterization of elliptic and hyperelliptic
surfaces by genera \quad 11.4\quad}
 Conversely the values of the genera $(p_a,p_g,p^{(1)},P_4)$ that we
have found for the elliptic and hyperelliptic surfaces enable us to
characterize these surfaces (not birational to ruled surfaces) and
therefore the whole family of surfaces possessing a continuous group of
birational transformations into themselves.

Firstly we prove that a surface $F$ with the genera
 \[
 p_a=-1, \quad p_g>1, \quad p^{(1)}=1,
 \]
 is an elliptic surface.

In fact the $\infty^{p_g-1}$ canonical curves of $F$ are composed with
the elliptic curves $K$ of a pencil $(K)$, of a certain genus $\rho$;
and a priori the $\infty^q$ ($q=p_g-p_a$) inequivalent curves $L$ of
an $\infty^q$ complete system constructed on $F$ either cut out
$\infty^q$ equivalent groups on the $K$ or they cut out groups
belonging to a series formed by $\infty^1$ linear series: in the first
case the curves $L$ of $\{L\}$ are obtained from one another by sum and
subtraction of curves $K$ of $(K)$, whence we deduce
 \[
 \rho=q=p_g+1;
 \]
in the second case the system $\{L\}$ is formed by $\infty^1$ continuous
systems of dimension $q-1$, and each of these can be deduced from one of
the curves of $\{L\}$ by summing and subtracting curves $K$ of $(K)$, so
that
 \[
 \rho=p_g
 \]
 So-as we have done many times before-we may construct on $F$, besides
$(K)$, another irrational, elliptic, pencil of directrices of the $K$.
Accordingly the surface $F$ turns out to be elliptic (Chapter~X,
Section~\ref{sec:10.12}).

Now we shall see that in fact only the latter case occurs, and
not the former.

To see this, we compute the Zeuthen--Segre invariant of $F$ by using the
pencil of elliptic curves K: we have
 \[
 \De-4=12p_a-p^{(1)}+9=-4
 \]
 and therefore
 \[
 \De=0;
 \]
 whence it follows that the $K$ of $(K)$ never acquire double points and
thus they are all birationally equivalent to one another; in $(K)$ there
are a certain number of multiple elliptic curves with the same modulus
 \[
 s_1K_1, s_2K_2,\dots
 \]
 where
 \[
s_i>1.
 \]

In this setting, with the procedure that we have explained in
Chapter~X, Section~\ref{sec:10.11} (in the case of a pencil of genus
$\rho=1$) we may construct a base point free pencil (which is a priori
either rational or elliptic) of directrices $C$, of genus $\pi>\rho$.
The canonical curves of $F$ cut out the canonical series $g_{2\pi-2}$
(but not in general the complete series) on a $C$. Since the $K$ cut
out an involution of genus $\rho$ on
$C$, whose intersections with the $K_i$ are $s_i$-tuple points, the
canonical curves of $F$ are formed by groups of the
$g_{2\rho-2}^{\rho-1}$ in the pencil $(K)$ of genus $\rho$, plus the
curves $(s_i-1)K_i$. Whence we deduce that the genus of $F$ is
 \[
 p_g=\rho.
\qed
 \]

Also for $p_g=0$ and $p_a=-1$, $p^{(1)}=1$, as well as for $p_g>1$ and
$p^{(1)}=1$, the surface $F$ is elliptic (see Chapter~X,
Section~\ref{sec:10.12}), except that it can be an improper elliptic
surface, that is, an elliptic ruled surface ($P_{12}=0$)\footnote{See
Chapter~X, Section~\ref{sec:10.14}.}.

But the case $p_g=1$ gives rise to an exception: the surfaces whose
genera are
 \[
 p_a=-1, \quad p_g=1, \quad p^{(1)}=1,
 \]
 form two families:
 \begin{enumerate}
 \renewcommand{\labelenumi}{(\arabic{enumi})}
 \item surfaces with elliptic canonical curve of order greater than zero;

 \item surfaces with pure canonical curve of order zero.
 \end{enumerate}

The surfaces of the first family are elliptic, that is, they admit an $\infty^1$ elliptic group of transformations into themselves, whereas the
surfaces of the latter family are hyperelliptic, and in general not
elliptic: this means that they admit an $\infty^2$ Abelian group $\Ga$ of
transformations into themselves, but only for particular moduli there
is in $\Ga$ an $\infty^1$ (elliptic) algebraic subgroup.

In order to justify what we have said about Cases~1 and~2, it is
convenient to prove a

Lemma on the surfaces with irregularity $q=2$: ``a surface $F$ of
irregularity $q=p_g-p_a=2$, not containing a pencil of genus two of
curves, can be represented on a simple or multiple hyperelliptic
surface''.

In order to prove this lemma, let us consider on $F$ a (simple)
continuous system $\{C\}$, formed by $\infty^2$ inequivalent curves,
and let us consider the elements (curves) of $\{C\}$ as ``points'' of a
surface $\Phi$: this is a hyperelliptic surface (that is, the Picard
variety of $F$). There is now a correspondence between the surfaces $F$
and $\Phi$, in which the curves $L$ of an $\infty^2$ system $\{L\}$ on
$\Phi$ correspond to the points of $F$ (that is, the image of a point is
the $\infty^1$ system of curves $C$ passing through it). We claim that
the $\infty^2$ curves $L$ on $\Phi$ cannot be equivalent. Otherwise the
$\infty^1$ curves $C$ of $\{C\}$ passing through a point $O$ of $F$
would cut out equivalent groups on another $C=\Cbar$, and, by the third
equivalence criterion (Chapter~III, Section~\ref{sec:3.13}) they would
be equivalent.

In order to justify this assertion, it is convenient to consider the
series of groups of points $G$, on the surface $F$, cut out on a
particular $C=\Cbar$, not passing through $O$, by the $\infty^1$ curves
$C$ through $O$; if we denote the index of this series by $i$, then we
have $i$ groups of $G$ having a point $P$ of $\Cbar$ in common and $i$
groups $G$ having another point $P'$ of the same curve in common; and,
by virtue of a criterion by Severi--Castelnuovo\footnote{See
Enriques--Chisini, {\it Lezioni} Libro~V, Chapter IV, \S42 (Vol.~III,
page~483).}, it is necessary and sufficient to show that the two sets
of groups $G$ thus defined on $\Cbar$, P(G) and $P'(G)$, are
equivalent. To prove this, let us consider the curve $\Lbar$ on $\Phi$
corresponding to the point $O$ (or rather to the series C(O) of the
curves $C$ through $O$) and the point $A$ corresponding to the curve
$\Cbar$: if, as we assume, the $\infty^1$ curves $L$ through $A$ are
equivalent on $\Phi$, then they cut out equivalent groups of points on
$\Lbar$; so in particular the groups cut out on the aforementioned
curve by the two curves $L$ -- say $L_P$ and $L_{P'}$ -- corresponding
to the two points $P$ and $P'$ of $F$ are equivalent. But -- going back
to the surface $F$ -- this means that the two groups of curves $C$
through $O$ and $P$ and through $O$ and $P'$, that cut out the groups
of points P(G) and $P'(G)$ on $\Cbar$, are equivalent within the series
of the C through $O$ and so they belong to an $\infty^1$ rational
series of groups of curves $C$; if we recall that a rational series of
groups of points on a curve is always formed by equivalent
groups\footnote{See Enriques--Chisini, {\it Lezioni}, Libro~V,
Chapter~I, \S10, Nota (Vol.~III, page~78).}, then we deduce that $P(G)$
and $P'(G)$, cut out on $\Cbar$ by these groups of $C$, are equivalent
on the aforementioned curve $\Cbar$.

Now that we have proved that the $\infty^2$ curves $L$ of $\Phi$
(corresponding to the points of $F$) cannot be all equivalent, we still
have to examine the hypothesis that any $L$ of $\{L\}$ belongs to an
$\infty^1$ series of equivalent curves $L$; in this case we shall
see that the surface $F$ contains a pencil of genus 2. Indeed under
this assumption we find on $F$ a pencil of curves $K$ whose points
correspond to equivalent curves $L$, and we can see that the $C$ of a
series $C(O)$, passing through any given point $O$ on the surface cut
out equivalent groups on a $K$, and therefore the same happens for all
the curves $C$: to prove this, it suffices to repeat the previous
argument, showing that the groups of curves common to $C(O)C(P)$ and to
$C(O)C(P')$ are equivalent (inside the series $C(O)$), and therefore
that also the groups of points $P(G)$ and $P'(G)$ cut out on $K$ by the
$C$ of $C(O)$ are equivalent on the curve $K$.

In this way we verify that the $\infty^2$ inequivalent curves $C$ of
$F$ cut out equivalent groups on any K; hence the $\infty^2$ continuous
system $\{C\}$ can be constructed from a linear system $|C|$, or from a curve
$C$, by summing and subtracting curves $K$ of the pencil $(K)$: therefore
this turns out to be of genus ($\rho\ge2$, thus) $p_g-p_a=2$.

So the statement of the Lemma is justified.

Now we can add that: the surfaces of irregularity $q=2$, with $p_g>0$,
containing a pencil of genus two of curves (of genus $\pi>0$) have $p_a\ge0$.

In fact let $F$ be a surface with arithmetic genus $p_a=-1$ containing a
pencil of genus two of curves K; we prove that one has $p_g>1$.

Computing the Zeuthen--Segre invariant with the pencil $(K)$, we find
 \[
 \De-4+8(\pi-1)=12p_a+p^{(1)}+9=-3-p^{(1)}
 \]
 and so
 \[
 p^{(1)}=1, \quad \De=0, \quad \pi=1;
 \]
therefore the elliptic curves of the pencil $(K)$ are birationally
equivalent to one another. Then (as in Chapter~X,
Section~\ref{sec:10.11}) we can construct a base point free pencil of
directrices curves $C$ of the $K$ (which a priori could either be
rational or elliptic, but it is elliptic, as we shall see):
so\footnote{See the remark to Chapter~IX, Section~\ref{sec:9.8}.} the
canonical curves of $F$ are composed with canonical groups of the $K$
of $(K)$ and, possibly, also with fixed parts (more precisely by
$(s-1)C_s$ where $sC_s=C$ is a multiple $s$-tuple curve of the pencil
$(C)$); whence
 \[
 p_g\ge2.
\qed
 \]

By the lemma on the surfaces with irregularity $q=2$ and the remark
that the surfaces with $q=2$ and a pencil of genus two have $p_g\ge2$,
hence $p_a\ge0$, we can say that a surface $F$ with $p_a=-1$ and $p_g=1$
can be represented on a simple or multiple hyperelliptic surface $\Phi$,
having a certain branch curve $D$. The curve of $F$ corresponding to
$d$ is a canonical curve; thus, if the linear genus of $F$ is
$p^{(1)}=1$, one of the following two cases, corresponding to the above
Cases~1 and~2, must occur:

 \begin{enumerate}
 \renewcommand{\labelenumi}{(\Alph{enumi})}
 \item either $d$ is an elliptic curve of order greater than zero, or

 \item $d$ is a curve of order zero.
 \end{enumerate}

 (1) In the first case the $\infty^2$ transformations of the
hyperelliptic surface $\Phi$, belonging to the continuous group $\Ga$,
send the curve $d$ (of degree zero) to the curves of a pencil, and each
curve $d$ of this pencil (or each irreducible component of $D$) is
invariant for an $\infty^1$ elliptic subgroup of $\Ga$. Whence it follows
that the multiple surface $\Phi$ admits an $\infty^1$ elliptic group of
transformations, hence it is an elliptic surface. \qed

(2) If, by contrast, there is no branch curve on the multiple surface
$\Phi$ (without exceptional curves), then $F$ is an Abelian surface, as
well as $\Phi$, hence it is hyperelliptic (Chapter~IX,
Section~\ref{sec:9.9}); so we are in case (2).

We may distinguish between the two cases 1) and 2) by remarking that,
in case 1) (since $p_g=1$ and not $p_g>1$) the elliptic curves of the
pencil (D) of $\Phi$, or rather their irreducible components, correspond
to curves $K$ of an elliptic pencil of elliptic curves on $F$, in which
the number of curves having a double point is $\De=0$.

Therefore in this pencil $(K)$, formed by birationally equivalent
curves, there is at most a finite number of multiple curves $D_i$ each
to be counted $s_i$ ($>1$) times, and the particular curve $d$ giving
the branch curve of the multiple $\Phi$ is formed by a certain number
$r$ of these curves:
 \[
 D=\sum (s_i-1)D_i.
 \]

Whence it follows that the canonical curve, corresponding to $D$, in
the pencil $(K)$ on $F$, is
 \[
 \sum [(s_i-1)/s_i]K.
 \]

So, if $r>1$, we have on $F$ a bicanonical system of dimension $\ge
r-1$:
 \[
 |rK+\sum [(s_i-2)/s_i]K|,
 \]
having as fixed components the curves corresponding to the $D_i$ of
multiplicity $s_i>2$, each counted $s_i-2$ times; in any case, if $r=1$,
then there is a quadricanonical system, at least $\infty^1$, whose
variable components are pairs of curves $K$, forming a $g_2^1$ inside the
elliptic $(K)$.

In conclusion, in case 1), the elliptic surfaces $F$ ($p_a=-1$, $p_g=1$,
$p^{(1)}=1$) has the quadrigenus
 \[
P_4>1,
 \]
while the hyperelliptic surface (with the same $p_a=-1$, $p_g=1$ and
$p^{(1)}=1$, but with canonical curve of order zero) has all plurigenera
equal to one, and, in particular, the quadrigenus:
 \[
P_4=1.
 \]

Let us summarize the results that we have obtained:

The nonruled surfaces ($P_4+P_6>0$ or $P_{12}>0$), admitting a
continuous group of birational transformations into themselves, are
globally characterized by the values of the genera:
 \[
 p_a=-1, p^{(1)}=1;
 \]
 they are always either elliptic surfaces (with an $\infty^1$ group) and
$p_g>1$, or $p_g=1$ and $P_4>1$; or hyperelliptic surfaces (with an
$\infty^2$ group) if
 \[
 p_g=P_4=1.
 \]

In general the hyperelliptic surfaces are not also elliptic, that is,
their $\infty^2$ group does not contain in general any $\infty^1$ algebraic
(elliptic) subgroup. This can be seen by considering the Jacobi surface
corresponding to a curve of genus two (the surface $F$ that represents
the pairs of points of $C$), or similarly the Abelian involutions on $F$ or
the multiple $F'$ without branch curve.

The hyperelliptic $F$ constructed this way contains no elliptic curves,
unless $C$ either contains an elliptic involution $\ga_n^1$ of a certain
order $n>1$, or it degenerates into a pair of elliptic curves; in the
particular case in which $F$ contains an elliptic curve $C$, this is
carried by the transformations of the group $\Ga$ of $F$ into the curves of
an elliptic pencil $(C)$ and (according to the Picard--Poincar\'e
theorem on reducible integrals, geometrically interpreted by
Castelnuovo) there is another elliptic pencil $(K)$ of elliptic curves
on $F$: these can be constructed as directrices of the pencil $(C)$ in
the way we have indicated in Chapter~X, Section~\ref{sec:10.11}.
Therefore $F$ can be considered an elliptic surface in two different
ways, since it admits two $\infty^1$ elliptic groups of
transformations, subgroups of the
$\infty^2$ group $\Ga$, whose trajectories are the C and the $K$. Let
us finally add that such a twice elliptic $F$, of determinant $n>1$ (if
the C and the $K$ meet at $n$ points), can be represented on that
particular hyperelliptic surface (also twice elliptic of determinant
one) which is the Jacobi surface corresponding to the degenerate curve
of genus two formed by two elliptic curves; such a surface has to be
taken as multiple of order $n$.

\begin{Remark} The conditions that we have used to define the family of
elliptic and hyperelliptic surfaces $(p_a=-1,p^{(1)}=1)$ are
superfluous: {\em the condition $p^{(1)}=1$ is a consequence of
$p_a=-1$.} We shall prove this for the elliptic surfaces with any $p_g$
in Section~\ref{sec:11.8}; here we shall justify the assertion only for
the surfaces with $p_g=1$, that is, essentially for the hyperelliptic
case.

Let us start from the conclusion we have found above that: ``the
surfaces $F$ with the genera $p_a=-1$ and $p_g=1$ can be represented on
a, simple or multiple, hyperelliptic surface $\Phi$''. If we have a
multiple $\Phi$, then there is a branch curve $d$ on it, of genus
$p^{(1)}$ equal to the linear genus of $F$, and we must ascertain that
$p^{(1)}>1$ yields
 \[
 p_a\ge0.
 \]

Let us prove this in the simplest case, in which the branch curve $d$
of $\Phi$ is a simple curve, of genus $\rho=p^{(1)}$, having a certain
number $\de$ of nodes and a certain number $\tau$ of cusps, so that we
may apply Severi's formula\footnote{So, in order to complete the proof,
one has to extend this formula to the case in which the branch curve is
however composed of multiple parts.

 We know a priori that the claim holds thanks to a result (partly
relying on the theory of simple integrals of the first kind belonging
to a surface), which has been kindly communicated to us by
G.~Castelnuovo. It is the following: for the irregular surfaces which
are not birational to ruled surfaces, one has
 \[
 I+4=13+12p_a-p^{(1)}\ge0
 \]
where $I$ is the Zeuthen--Segre invariant. In fact for $p_a<0$ (that is,
$p_a=-1$), this yields $p^{(1)}=1$.} giving the relation between the
arithmetic genera $p_a'$ and $p_a$ of the two surfaces $\Phi$ and $F$ in
a $(1,n)$ correspondence (see Chapter~V, Section~\ref{sec:5.8}). This
formula gives us\footnote{The case in which the strict inequality holds
corresponds to the hypothesis in which there are on $d$ some
fundamental points for the $(1,n)$ correspondence.}
 \[
 24(p_a+1)\ge 24n(p_{a}'+1)+6(\rho -1)-2\tau,
 \]
where
 \[
 p_a'=-1,\quad \rho>1,
 \]
that is,
 \[
 24(p_a+1)\ge 6(\rho-1)-2\tau.
 \]

But the curve $d$ belongs to a system, at least $\infty^2$, on the
hyperelliptic surface $\Phi$, and it has a characteristic series (cut out
by the infinitely near curves) of degree
 \[
 2\de+3\tau\le2(\rho+\de+\tau-1).
 \]

Therefore one has
 \[
 \tau\le2(\rho-1)
 \]
 and
 \[
 24(p_a+1)>0,
 \]
 that is,
 \[
 p_a\ge0.
\qed
 \]
 \end{Remark}

Summarizing: The surfaces with genera
 \[
 p_a=-1 \quad\hbox{and}\quad p_g=1
 \]
have the linear genus
 \[
 p^{(1)}=1
 \]
and, if they have a canonical curve of order zero (as happens for
$P_4=1$), then they are hyperelliptic surfaces.

\begin{NOTE} The analysis of the nonruled surfaces admitting a
continuous group of transformations into themselves starts with the
study of hyperelliptic surfaces made by Picard from the transcendental
point of view\footnote{E. Picard, M\'emoire sur la th\'eorie des
fonctions alg\'ebriques de deux variables ind\'ependantes. J. de Math.,
1889. (Chapter~III) See the Vol.~III of the Trait\'e by Picard and
Simart, Paris 1906.}.
\end{NOTE}

A brilliant application of the theory of simple integrals of total
differentials introduced by him, leads the author to the
characterization of the surfaces of the aforementiond type, possessing
an $\infty^2$ group of permutable transformations, by means of:

 \begin{enumerate}
 \renewcommand{\labelenumi}{(\arabic{enumi})}
 \item the existence of two simple integrals of the first kind;

 \item the existence of a double integral of the first kind (that,
according to Clebsch and Noether, means $p_g=1$);

 \item the nonexistence of a (pure) canonical curve of order greater
than zero.
 \end{enumerate}

The condition 3) (pointed out by Picard, who, deceived by a mistake
of Noether, believed it to be superfluous) yields the inversion of
the two simple integrals and therefore the parametric representation of
the surface with (hyperelliptic) four times periodic functions of two
independent variables.

{From} this result Enriques\footnote{Sulle superficie algebriche che
ammettono un gruppo continuo di trasformazioni birazionali in se
stesse. {\it Rendic.\ Circ.\ Mat.\ di Palermo}, 1905.} drew the
characterization of the hyperelliptic surfaces by means of the values $p_a=-1$,
$p_g=P_4=1$ of the genera.

Later on Severi\footnote{Sulle superficie algebriche che ammettono un
gruppo continuo permutabile a due parametri di trasformazioni
birazionali, {\it Atti Ist.\ Veneto}, 1908.} translated the proof of
Picard's theorem in a geometric form, deducing the existence of the
$\infty^2$ group of transformations directly from the conditions
$p_a=-1$, $p_g=1$ and canonical curve of order zero, as we have also
done here.

The case of surfaces with an $\infty^1$ elliptic group, already met by
Picard, has been exhausted, from the transcendental point of view, by
Painlev\'e's analysis\footnote{P. Painlev\'e, Le\c{c}ons sur la th\'eorie
analytique des \'equations diff\'erentielles. Stockholm, 1895 (page~270).}.
He defines the elliptic surfaces by means of a parametric
representation with elliptic functions of one parameter and algebraic
functions of another, and he succeeds then in classifying all nonruled
surfaces possessing a continuous group of birational transformations
into themselves. The geometric translation of this result and the
classification of the whole family of surfaces by means of the values
of the genera was obtained by Enriques in the quoted Memoir of
1905\footnote{Sulle superficie algebriche che ammettono un gruppo
continuo ecc., and partially already in the previous Memoir, also
published in {\it Rendic.\ Circ.\ Mat.\ di Palermo}: Sulle superficie
di genere zero (1905).}.

 \section{Surfaces with genera $p_g=1$ and $p^{(1)}=1$}\label{sec:11.5}
 \markright{\hfill Surfaces with the genera $p_g=1$ and
$p^{(1)}=1$ \quad 11.5\quad}
 The results that we have established in the previous sections enable
us to classify now the surfaces of geometric genus $p_g=1$ and linear
genus $p^{(1)}=1$. These surfaces can be arranged in three families:

 \begin{enumerate}
 \renewcommand{\labelenumi}{(\arabic{enumi})}
 \item Regular surfaces of genus $p_a=p_g=1$ with canonical curve of
order zero (characterized by the value of the bigenus $P_2=1$), which
give rise to infinitely many classes of surfaces (with 19 moduli) with
canonical hyperplane sections of genus $\pi$ and order $n=2\pi-2$, in
the space $S_\pi$ ($\pi=2,3,\dots$) (See Chapter~VII,
Section~\ref{sec:8.2}).

 \item Hyperelliptic surfaces with genera $p_a=-1$ and $p_g=1$, also
possessing a canonical curve of order zero (characterized by the value
of the quadrigenus $P_4=1$), which give rise again to infinitely many
classes of normal surfaces with canonical hyperplane sections of genus $\pi$ and order
$n=2\pi-2$ in the space $S_{\pi-2}$ (as we shall see each of
these classes depends on three moduli).

 \item Surfaces possessing a (rational or elliptic) pencil of elliptic
curves: this property is shared by the surfaces of types (1) and (2)
only for particular values of their moduli.
 \end{enumerate}

In order to prove that ``the three families of surfaces (1), (2) and (3)
exhaust the surfaces with genera $p_g=1$ and $p^{(1)}=1$'' it is
convenient on the one hand to recall some remarks already made before
in this treatise and on the other hand to point out a few facts we have
not remarked yet. We mean the following propositions:

 \begin{enumerate}
 \renewcommand{\labelenumi}{(\alph{enumi})}
 \item the surfaces with geometric genus $p_g=1$ have the arithmetic genus
 \[
 p_a\ge-1,
 \]
 that is,
 \[
 \hbox{either $p_a=1$} \quad \hbox{or $p_a=0$} \quad \hbox{or $p_a=-1$;}
 \]

 \item the (regular) surfaces with $p_g=p_a=1$ and canonical curve of
order zero are defined by the value of the bigenus $P_2=1$ (Chapter~VI,
Section~\ref{sec:6.2}); by contrast, the surfaces of this family having
the linear genus $p^{(1)}=1$ and possessing a (pure) canonical curve of
order greater than zero have the bigenus $P_2>1$ and contain a (linear)
pencil of elliptic curves, that are either bicanonical curves or
components of bicanonical curves;

 \item the surfaces with $p_g=1$ and with a canonical curve of order
zero have the arithmetic genus either $p_a=1$ or $p_a=-1$ (Chapter~IX,
Section~\ref{sec:9.8}) and, if $p_a=-1$, then they have also the linear
genus $p^{(1)}=1$, and they are either hyperelliptic surfaces or
elliptic surfaces, with a pencil of elliptic curves, according to the
value of the quadrigenus $P_4=1$ or $P_4>1$ (Section~\ref{sec:11.4});

 \item the surfaces for which $p_g=1$, $p_a=0$ and $p^{(1)}=1$ (which
by c) must contain a canonical curve of order greater than zero)
possess an elliptic pencil of paracanonical curves, and therefore have
the bigenus
$P_2>1$.
 \end{enumerate}

The propositions b) and c) just recall facts we have already seen;
therefore it suffices to prove a) and d), stating something new.

Since the proposition a) has been already established for the
elliptic and hyperelliptic surfaces, it suffices to prove it under the
hypothesis the surface $F$ (of genus $p_g=1$) does not possess a
continuous group of transformations into itself (later on, in
Section~\ref{sec:11.7}, this proposition will appear as a particular
case of a more general theorem to the effect that the arithmetic genus
of nonruled surfaces always is
$p_a\ge-1$).

Let $F$ be a surface of genus $p_g=1$, which does not admit a continuous
group of transformations, and let $q=p_g-p_a$ be its irregularity: we
want to deduce a contradiction from the hypothesis
 \[
 q>2,
 \]
 that is,
 \[
 p_a<-1.
 \]

To do this, we assume $q>2$ and consider the Picard variety $V_q$
corresponding to an $\infty^q$ continuous system of inequivalent
curves $C$ on $F$. As we have seen (Chapter~IX,
Section~\ref{sec:9.11}), if $F$ does not contain an irrational pencil
of curves $K$, then an
$\infty^1$ series of curves $C$ of $\{C\}$ leads to the construction of
a, simple or multiple, surface identical with $F$ inside $V_q$. And
more than that, since we have in mind to prove that $p_g\ge q$, we may
as well assume that we have in $V_q$ a simple surface identical with
$F$: in fact in the case of a multiple surface we recall the remark of
Section~\ref{sec:11.4} and see that, if this surface admits a continuous
group of transformations into itself, whereas $F$ does not, then
$p_a\ge0$, and accordingly $p_g\ge q$ for $F$ itself.

Thus let us have in $V_q$ a surface identical with $F$ (denoted again by
$F$) which does not admit a continuous group of transformations into
itself; by means of the $\infty^q$ transformations of $V_q$ our $F$ is
transformed into an $\infty^q$ series, say $\{F\}$, of analogous surfaces.
For simplicity we assume first of all $q=3$. Since $V_q=V_3$ has the
geometric genus equal to one, the $\infty^2$ characteristic system of
$F$, cut out on a given $F$ by the infinitely near surfaces, is formed by
special curves (and in fact canonical curves, since they are residual
with respect to the intersection with the canonical variety of $V_3$);
thus one finds
 \[
 p_g\ge 3
 \]
for $F$.

The argument can be extended to the case $q>3$. In this case we may
take an $\infty^{q-3}$ series of surfaces inside the series $\{F\}$:
this forms a variety $V_{q-1}$ and we may assume in general that such a
variety is contained in a linear system $|V_{q-1}|$ of dimension $r$ (as
large as we want). Then we construct the $\infty^{r+q}$ continuous
system $\{V_{q-1}\}$ generated by $|V_{q-1}|$ via the $\infty^q$
transformations of $V_q$. This continuous system has an $\infty^{r+q-1}$
characteristic linear system of special or canonical varieties, cut out
on a $V_{q-1}$ by the infinitely near varieties; therefore these
$V_{q-1}$ have the genus
 \[
P\ge r+q.
 \]

Now we compute the parameters on which the $V_{q-1}$ of $\{V_{q-1}\}$
containing an $F$ depend: there are $\infty^q$ surfaces $F'$ and
$\infty^{r+q}$ $V_{q-1}$ each containing $\infty^{q-3}$ surfaces $F'$,
thus the dimension of the system $\{F\}$ is (in the worst case):
 \[
r+2q-3-x=q
 \]
where $x$ denotes the dimension of the system of the $V_{q-1}$
containing an $F$: $x=r+q-3$. Whence it follows that, in a given $V_q$,
the characteristic system of $\{V_{q-1}\}$ cuts out on a $F$ contained
in $V_q$ a (special or canonical) linear system whose dimension is at
least:
 \[
r+q-1-(x-1)=3.
 \]

And therefore we deduce that the genus of $F$ is
 \[
 p_g\ge 4.
 \]

In conclusion we have proved that a surface $F$ of genus $p_g<3$, which
does not contain an irrational pencil of curves, cannot have the
irregularity
 \[
q>2;
 \]
in particular an $F$ of genus $p_g=1$ must have the arithmetic genus
 \[
 p_a\ge-1.
 \]

The conclusion can be extended to the hypothesis that $F$ contains an
irrational pencil of curves (C), since the computation of the
Zeuthen--Segre invariant by means of this pencil gives
 \[
 \De-4\ge12p_a-p^{(1)}+9.
 \quad
12(p_a+1)-p^{(1)}\le0,
 \quad
p_a\ge-1
 \]
and for $p_a=-1$, $p^{(1)}=1$.

Theorem d) can be proved as follows.

Let us consider an (irreducible) linear system $|C|$, of a certain
genus $\pi$, on $F$.

The adjoint system $|C'|$ has the dimension
 \[
 p_a+\pi-1=\pi-1,
 \]
 and cuts out on a $C$ a series $g_{2\pi-2}^{\pi-2}$ of deficiency one,
contained in the canonical series. Moreover $|C'|$ is contained in an
$\infty^1$ series of paradjoint linear systems $|\Cbar'|$, which also cut
out a series $g_{2\pi-2}^{\pi-2}$ on $C$. Therefore there is an elliptic
paracanonical curve defined as the difference
 \[
 \Cbar'-C,
 \]
and the analogous paracanonical curves $K$ form an elliptic pencil on
$F$: let us add that any curve $2K$ belongs to a linear system, at least
$\infty^1$, and therefore $P_2>1$. \qed

 \begin{Remark} An example of surfaces with the genera $p_a=0$, $p_g=1$,
$p^{(1)}=1$, and possessing an elliptic pencil of elliptic curves, is
given by the simplest paraelliptic surfaces, that we have met in
Chapter~X, Section~\ref{sec:10.11}.
 \end{Remark}

It could be interesting to study and classify the surfaces with $p_a=0$
and $p_g=1$, having the linear genus $p^{(1)}=1$, or also
$p^{(1)}=2$,\dots

The results we have found can be summarized by saying: The nonruled
surfaces ($P_{12}>0$) possessing a canonical or pluricanonical curve of
order zero (and therefore base point free linear systems of genus $\pi$ and degree
$n=2\pi-2$) are defined by the condition
 \[
P_{12}=1,
 \]
 which yields
 \[
 p^{(1)}=1
 \]
and they give rise to the following families:

 \begin{enumerate}
 \renewcommand{\labelenumi}{(\arabic{enumi})}
 \item $p_g=0$ and $p_a=0$, regular surfaces with bicanonical curve of
order zero ($P_2=1$), represented by the surface of order 6 possessing
with multiplicity two through the edges of a tetrahedron (Chapter~VII,
Section~\ref{sec:7.1});

 \item $p_g=0$ and $p_a=-1$, elliptic surfaces of the types I$_a$ I$_b$
II$_a$ II$_b$ III$_a$ III$_b$ III$_c$ (Chapter~X, Sections~16--19);

 \item $p_g=1$ and $p_a=1$ ($P_2=1$), regular surfaces with all the
genera equal to one (Chapter~VII, Section~\ref{sec:7.2});

 \item $p_g=1$ and $p_a=-1$ ($P_4=1$) hyperelliptic surfaces
(Sections~4 and~6 of this chapter).
 \end{enumerate}

 \section{A note on the geometric theory of hyperelliptic
surfaces}\label{sec:11.6}
 \markright{\hfill A note on the geometric theory of hyperelliptic
surfaces \quad 11.6\quad}
 Among the families with $P_{12}=1$, listed with the numbers 1), 2), 3),
4) at the end of the previous section, one may distinguish: on the
one hand the surfaces of genus $p_g=1$, of the types 3) and 4) ($p_g=1$,
$p_a=1$ and $p_a=-1$), and on the other hand the surfaces of genus
$p_g=0$ (the regular surfaces with $P_{12}=1$, and the elliptic ones)
of the types 1) and 2). The latter types form a finite number of
families each having a certain number of moduli: and precisely 10
moduli those of type 1), 2 moduli or 1 modulus those of type 2). By
contrast the surfaces 3) and 4) give rise to infinitely many families
distinguished by the values of (at least) one arithmetic character,
each having a certain number of moduli.

What we just said has been proved for the surfaces with all the
genera equal to one (type 3)) in Chapter~VII, Section~\ref{sec:7.2};
but it is also true for the hyperelliptic surfaces of type 4), as we
shall explain now.

{From} the transcendental point of view, the classification of the
hyperelliptic surfaces stems from the parametric representation of
these surfaces by means of four times periodic functions of two
independent variables\footnote{See for instance Enriques--Chisini,
{\it Lezioni} Libro~VI, Chapter~III, \S39 (Vol.~IV, page~242).}. Their
primitive normal periods can be reduced to the form
 \[
 \begin{array}{cccc}
 1 & 0 & g & h\\
 0 & 1/\de & h & g',
 \end{array}
 \]
 where $\de$ is an integral character (the so called divisor of the
corresponding surfaces). Now for $\de=1$ one has the standard surface,
that is, the Jacobi surface $F_j$, representing the variety of the pairs of
points of a curve of genus two; by contrast, for $\de>1$ one has a
multiple $F_j$ of order $\de$ without branch curve, or also an
involution of order $\de$ on the surface $F_j$.

The classification of the hyperelliptic surfaces obtained in this way
has an algebro-geometric meaning, and therefore the problem arises of
justifying it from this point of view. We say briefly how this can be
done.

To start with we can determine the characters of the Jacobi surface
$F_j$:
 \[
 p_g=1, \quad p_a=-1
 \]
 and canonical curve of order zero (thus $P_4=1$). This can be obtained
by representing $F_j$ on the double Kummer surface of order 4 with 16
double points, as we have seen in Chapter~V,
Section~\ref{sec:5.8}\footnote{See also Enriques--Chisini, {\it
Lezioni}, L.~V, Chapter~III, \S40 (Vol.~IV, page~24).}. This
representation is based on the knowledge of the (involutory)
transformations of the 2nd kind of the variety of the pairs of points
of a curve of genus two; these are obtained by associating the
coresidual pairs with respect to a given $g_4^2$ of
$C$: such an involution is birationally equivalent to a Kummer surface
of order four, having sixteen double points corresponding to the
sixteen pairs of points of $C$ coming from the division by two of the
$g_4^2$.

Now the products of these involutions $I$ on the Jacobi surface $F_j$
give rise to a continuous Abelian group $\Ga$; and there are cyclic
transformations in $\Ga$ generating finite groups $G_\de$ contained in
it: such a $G_\de$ (let for instance $G_\de$ be cyclic) gives rise to an
involution $I_\de$ of order $\de$ on $F_j$, without double points,
corresponding in general to a hyperelliptic surface $F_\de$ distinct
from $F_j$. In this way one obtains the hyperelliptic surfaces of
divisor $\de>1$. And we can see that these can also be produced as
multiple $F_j$ of order $\de$, without branch curve; to do this we need
to see that it is possible to construct with the groups of an
involution $I_\de$ another involution of higher order, say $r\de$,
which is birationally identical with $F_j$.

In what we have said there is a sketch of a geometric theory of the
hyperelliptic surfaces, based on the study of cyclic transformations
belonging to the group $\Ga$ of the Jacobi surface $F_j$: such a study
appears as an extension of the study of the cyclic involutions on the
elliptic curves, developed in Enriques--Chisini, {\it
Lezioni}\footnote{Libro~V, Chapter~III, \S\S27--28 (Vol.~III,
pages~251--298).}.

But the consideration of the hyperelliptic surfaces from the geometric
point of view also confronts us with other interesting problems.

Indeed we have proved (Chapter~IX, Section~\ref{sec:9.9}) that an
Abelian variety $V_q$ can be regarded as identical with a simple or
multiple variety imprimitively contained in a Jacobi variety $V_p$
corresponding to a curve of genus $p\ge q$; but it is not true in
general that one can always take $p=q$, because the number of moduli on
which $V_q$ depends is greater than the number $3q-3$ of moduli of the
curves of genus
$q$\footnote{See Enriques--Chisini, {\it Lezioni}, Libro~V, Chapter~III,
\S33 (Vol.~III, page~355) and Libro~VI, Chapter~III, \S35 (Vol.~IV,
page~216).}. Now the transcendental theory shows that the hyperelliptic
surfaces (with a given divisor) depend on three moduli, and therefore
they can all be reduced to the type of a simple or multiple Jacobi
surface
$F_j$, or also to the (fixed point free) involutions on the $F_j$.

This can be seen from the geometric point of view, in a way that we
want to briefly sketch. On a hyperelliptic surface $F$ (for which
$p_a=-1$) a curve of genus $\pi$ belongs, in general, to a linear
system of dimension $\pi-2$, in which there is a finite number of curves
$C$ with $\pi-2$ double points, and therefore of genus two.

By means of the transformations of the second kind of $F$, belonging to
the group $\Ga$, we therefore obtain an $\infty^2$ continuous system
$\{C\}$ of curves of genus two on $F$, birationally equivalent to one
another, whose three moduli give the moduli of $F$. Now a $C$ of this
system is invariant for a single (involutory) transformation of the
second kind\footnote{If we refer to the parametric representation of
the group $G$, then a transformation of the second kind corresponds to the
transformation sending a $\pi$ of $G$ in the inverse $\pi^{-1}$. It is
easy to see that, even if the curve $C$ of genus two has singular
transformations into itself, there cannot be more than one
transformation of the second kind of $F$ leaving it fixed.} of $F$,
inducing the $g_2^1$ on $C$; therefore the group $\Ga$ is birationally
equivalent both to the surface $F$ (regarded as a set of points) and to
the system $\{C\}$ regarded as an $\infty^2$ variety of elements (curves $C$).
Whence it follows that $F$ is birationally equivalent to the $\infty^2$
series of groups of points cut out by the curves of $\{C\}$ on a particular $C$: if the groups of this series are not equivalent to one another, then
$F$ is identical with the Jacobi surface of $C$; otherwise, if for any
group of this series there is a certain number $\de$ of groups equivalent
to it in the series, then the hyperelliptic surface $F$ is birationally
identical with an involution $I_\de$ of order $\de$ on the Jacobi
surface.

The theorem we have established takes an interesting geometric
significance in relation with the various constructions of the
hyperelliptic surfaces that one can give. Indeed one can construct a
hyperelliptic surface starting with a Jacobi variety corresponding to a
curve $C_p$ of genus $p$. If on such a $C_p$ there are two Abelian
integrals which are reducible to four periods (according to
Castelnuovo's geometric interpretation of the Picard--Poincar\'e
theorem)\footnote{See Enriques--Chisini, {\it Lezioni}, Libro~VI,
Chapter~III, \S38 (Vol.~IV, page~234).} then the corresponding Jacobi
variety $V_p$ contains an $\infty^2$ algebraic subgroup of the group
$\Ga$ of the transformations of the first kind, and therefore one has
an Abelian variety (that is, a hyperelliptic surface) imprimitively
contained in $V_p$. The previous theorem tells us that the
hyperelliptic surfaces constructed in this way can also be found as
simple or multiple Jacobi surfaces, or also as involutions on the
Jacobi surfaces corresponding to a curve of genus $p=2$.

It is convenient to illustrate this property by referring to the
simplest case, in which we deal with the hyperelliptic surfaces
imprimitively contained in a Jacobi variety $V_3$, corresponding to a
curve $C_3$ of genus 3. Here the group $\Ga$ of the transformations of
the first kind of $V_3$ contains two complementary algebraic subgroups:
an
$\infty^2$ $\Ga_2$, giving rise to an imprimitivity series formed by
$\infty^1$ hyperelliptic surfaces $F_2$, and an $\infty^1$ $\Ga_1$
giving rise to $\infty^2$ elliptic curves $C_1$; the surfaces $F_2$ and
the curves $C_1$ meet one another at a certain number of points. And it
is easy to prove that each $C_1$ is a unirational transform of $C_3$,
that is, it corresponds to an elliptic involution $\ga_n^1$ (of a
certain order $n)$ belonging to $C_3$.

Therefore one can construct a hyperelliptic surface $F_2$,
imprimitively contained in a Jacobi variety $V_3$, corresponding to a
curve $C_3$ of genus 3 possessing an elliptic involution $\ga_n^1$; to
do this, it suffices to construct the curve $C_3$ corresponding to a
multiple cubic of multiplicity $n$, with an arbitrary choice of the four
branch points and of the corresponding substitutions on the
sheets\footnote{See the existence theorem for the multiple curves in
Enriques--Chisini, {\it Lezioni}, Libro~V, Chapter~IV, \S38 (Vol.~III,
page~427).}. We notice that the $C_3$ containing an elliptic $\ga_n^1$
depend on four moduli: the absolute invariant of the $K$ and the
distances of 3 branch points from the remaining point on the elliptic
curve $K$.

The absolute invariant of $K$ corresponds to the modulus of the $C_1$ on
the $V_3$, and the remaining 3 moduli belong to the complementary
surfaces $F_2$.

Let us add that the Jacobi's $V_3$ we have just constructed contains,
in general, an involution given by the groups of points cut out on the
$C_1$ by the $F_2$: the $\Vbar_3$ representing these groups, which is
a unirational transform of $V_3$, is the Jacobi variety of a degenerate
curve of genus 3 formed by an elliptic curve and a curve of genus two;
so then (according to the theorem we have just treated), the
hyperelliptic surfaces $F_2$, forming an imprimitivity system for the
group of the transformations of the first kind of the $V_3$, turn out to
be birationally equivalent to an involution on one of the analogous
$\Fbar_2$, which is the Jacobi surface corresponding to the
aforementioned curve of genus two.

We think that the short account we have just given suffice to indicate
the interest, the beauty and the richness of the properties of the
hyperelliptic surfaces.

We only add that the study of the irregular hyperelliptic involutions
on a hyperelliptic surface $F$ (and in particular on a Jacobi surface
$F_j$) can be naturally extended to the study of the other irregular
elliptic involutions or regular involutions with the genera equal to
one that may belong to $F$; these lead to surfaces still having a
parametric representation by hyperelliptic functions of {\em rank}
$r>1$\footnote{Where each point of the surface corresponds to $r$ points
in the prismatoid of the periods.}.

The irregular elliptic involutions (belonging to $F$ with special
moduli) give rise to elliptic surfaces with $p_a=-1$, $p_g=0$ and
$P_{12}=1$, possessing a virtual canonical curve of order zero, as
already mentioned in Chapter~X, Sections~16--19.

The regular involutions of order 2 belonging to a hyperelliptic surface
of divisor $\de\ge1$ lead in general to surfaces of rank $r=2$ (that
are hyperelliptic in an extended sense), and more precisely: for $\de=1$
to the Kummer surfaces of order four with 16 double points (Chapter~V,
Section~\ref{sec:5.8}); and for $\de>1$ to a series of analogous
surfaces of order $n=2\pi-2$ with canonical hyperplane sections of
genus $\pi$ in the space $S_\pi$ ($\pi=4,5,6,\dots$), with sixteen
double points. But for special moduli there can also be {\em singular
regular involutions} on an $F$, generated by a finite group of
birational transformations of $F$; and, looking at this way of
generating them (as proved by Enriques--Severi), they have been
completely classified in the memoirs by Enriques and
Severi\footnote{M\'emoire sur les surfaces hyperelliptiques, {\it Acta
Math.}, t.~32 and 33 (1908).} quoted above, and by Bagnera and De
Franchis\footnote{Le superficie algebriche le quali ammettono una
rappresentazione parametrica mediante funzioni iperellittiche di due
argomenti, {\it Mem.\ Soc.\ It.\ delle Scienze} (detta dei XL), 1908.},
which complement each other.

Finally we recall that this circle of ideas can be extended from the
hyperelliptic surfaces to the regular surfaces with all the genera
equal to one; on these it is also possible to find regular involutions
of the same genus $p_g=1$ (or with $p_g=0$ and bigenus $P_2=1$),
generated by a finite group of birational transformations. In relation
to this topic we want to mention the interesting papers of the Belgian
school, and particularly of Godeaux.

 \section{The general classification of algebraic
surfaces}\label{sec:11.7}
 \markright{\hfill The general classification of the algebraic
surfaces \quad 11.7\quad}
 We are now in a position to put together a synthesis\footnote{See
Enriques, Sulla classificazione delle superficie algebriche in
particolare di genere lineare $p^{(1)}=1$, {\it Rendic.\ Acc.\ Lincei},
Febbraio 1914.} of the main results of the theory of surfaces, giving a
general classification of the algebraic surfaces, according to the
value of the 12-genus $P_{12}$: in what follows, this classification
will be made more precise, as we shall see in Sections~8--9.

We distinguish between the algebraic surfaces according to the
value of the $P_{12}$ and -- in the second instance -- of the (absolute)
linear genus $p^{(1)}$.

We have {\em four large families}\index{classification of surfaces} of
surfaces:

 \begin{enumerate}
 \renewcommand{\labelenumi}{(\Alph{enumi})}
 \item for $P_{12}=0$, the surfaces birational to ruled surfaces
(rational or otherwise);

 \item for $P_{12}=1$, the surfaces with effective or virtual
canonical curve of order zero;

 \item for $P_{12}>1$ and $p^{(1)}=1$, the surfaces containing
infinitely many linear systems of canonical or pluricanonical curves
composed with the elliptic curves of a pencil;

 \item for $P_{12}>1$ and $p^{(1)}>1$, the surfaces admitting canonical
or pluricanonical surfaces as a projective model.
 \end{enumerate}

It is convenient to recall and explain the {\em fundamental and
characteristic} properties that distinguish these four families of
surfaces:

 \paragraph{(A) $P_{12}=0$} Surfaces birational to ruled surfaces
(rational or otherwise): these possess linear systems of curves of
genus $\pi$ and degree $n>2\pi-2$, an infinite number of exceptional
curves that cannot be eliminated, $\infty^1$ rational groups of
transformations and continuous series of birational transformations
which do not form a group. They can be divided into:

\paragraph{$(\mathrm A')$} rational (regular) surfaces, characterized by
the conditions
 \[
 p_a=P_2=0,
 \]
 and

\paragraph{$(\mathrm A'')$} irrational ruled surfaces, for which
 \[
 p_a<0;
 \]
the ruled surfaces of genus $p=-p_a>1$ are characterized (as we shall
see in Section~\ref{sec:11.8}) by the simple condition
 \[
 p_a<-1,
 \]
whereas the ruled surfaces of genus $p=1$ are distinguished from the
properly elliptic surfaces with $p_a=-1$ by the conditions
$P_4=P_6=P_{12}=0$).

 \paragraph{(B) $P_{12}=1$, $p^{(1)}=1$ ($p_a\ge-1$)} Surfaces having a
canonical or pluricanonical curve of order zero. For any base point
free linear system of genus $\pi$ and degree $n$ on the surface without
exceptional curves, one has
 \[
n=2\pi-2;
 \]
thus one in general obtains as projective images of the surfaces of this
type surfaces $F_n$ with plane or hyperplane sections of genus $\pi$ and
order $n=2\pi-2$, in a space $S_r$ with $r=\pi$, $\pi-1$ or $\pi-2$
dimensions (the three values of $r$ correspond to the three cases:
$p_a=p_g=1$, $p_a=p_g=0$, and $p_a=-1$).

The surfaces $F_n$ of the family (B) divide in two categories:

 \paragraph{$(\mathrm B')$} surfaces with (plane or hyperplane)
canonical sections, having geometric genus $p_g=1$ and arithmetic genus
$p_a=1$ or $p_a=-1$, that is, regular surfaces with all the genera equal
to one (Chapter~VII, Section~\ref{sec:7.2}) and hyperelliptic surfaces;

 \paragraph{$(\mathrm B'')$} surfaces whose sections are not canonical
curves, but curves representing linear series coming from the division
of the $s$-canonical series by $s$ (for $s=2,3,4,6$).

These are the regular surfaces of genus $p_g=p_a=0$ and $P_2=1$
($P_3=0$) which reduce to the sextic surfaces $F_6$ passing with
multiplicity two through the edges of a tetrahedron (Chapter~VII,
Section~\ref{sec:7.1}) and the elliptic surfaces with $p_a=-1$ and
$p_g=0$ containing a linear pencil of elliptic curves, which we have
divided in the seven types I$_a$, I$_b$, II$_a$, II$_b$, III$_a$,
I$_b$, III$_c$ in Chapter~X, Sections~16--19.

The difference between the surfaces (B$'$) and B$''$) is this: while the
(B$''$) give rise to a finite number of types, the (B$'$) give rise to
infinitely many families of surfaces, depending on (at least) one
integer character, which is susceptible of assuming infinitely many
values.

This has been proved for the regular surfaces with all the genera equal
to one in Chapter~VII, Section~\ref{sec:7.2} (where we also saw that
each family depends on 19 moduli), but it is also true for the
hyperelliptic surfaces; in that case, together with the Jacobi surface
$F_j$ corresponding to a curve of genus two, there are the infinitely
many other surfaces corresponding to involutions of order $\de$ on
$F_j$, or more precisely to multiple covers of the $F_j$ of order $\de$
without branch curve: each of these families depends on three moduli
(see Section~\ref{sec:11.6}).

By contrast the surfaces (B$'$) give rise, as we have said, to a finite
number, and precisely to 8 families, which are clearly characterized,
and having: the first family (regular surfaces) 10 moduli (Chapter~VII,
Section~\ref{sec:7.1}) and the others (elliptic surfaces) 2 moduli (the
I$_a$ and I$_b$) or 1 modulus (the remaining families, of the harmonic
or equianharmonic type).

Let us add that the surfaces (B$'$) always contain pencils of elliptic
curves, whereas the surfaces (B$''$) in general do not contain such
pencils; therefore they have a closer similarity with the surfaces of
type (C).

 \paragraph{(C) $P_{12}>1$ and $p^{(1)}=1$} Surfaces possessing a
(rational or irrational) pencil of elliptic curves $K$, components of
canonical or pluricanonical curves. Together with the (B$''$), these
surfaces form the vast family of {\em surfaces containing a pencil of
elliptic curves.}\index{surface!with pencil of elliptic curves}

These surfaces can be classified, as we have seen in the regular case
(Chapter~VII, Section~\ref{sec:7.4}), according to some integral
characters, that is:
 \begin{enumerate}
 \renewcommand{\labelenumi}{(\arabic{enumi})}
 \item the arithmetic genus $p_a$;

 \item the determinant $n$, that is, the order of the smallest group of
points that can be rationally determined on a $K$;

 \item the number of the curves of the pencil that reduce to a multiple
curve; and moreover, in the irregular case

 \item the genus $\rho$ of the pencil, which is
 \[
 \rho=p_g-p_a
 \quad
 (p_a\ge0)
 \]
in general (Remark, Chapter~IX, Section~\ref{sec:11.8}), and
 \[
 \rho=p_g-p_a-1
 \]
in the case that the surface contains, besides $(K)$, another elliptic
pencil of curves; this case yields $p_a=-1$ and leads to the elliptic
surfaces.
 \end{enumerate}

For the surfaces C) that are regular, we have seen that their families,
depending on characters mentioned above, can be also characterized by
the values of the plurigenera (Chapter~VII, Section~\ref{sec:7.4}).

It seems that these results also extend, to a great extent, to the {\em
irregular surfaces}:\index{irregular!surfaces} $p_a<p_g$. In any case
this is an {\em object of study}, which we recommend to the attention
of the young researchers.

\paragraph{(D) $P_{12}$, $p^{(1)}>1$} These surfaces have to be thought
of as forming the most general type; in contrast with the previous cases
(A), (B) and (C), these give rise, for given values of the characters
$p_a$, $p_g$, $p^{(1)}$, to a {\em finite number of families}, each
containing a continuous system of classes, depending on a {\em certain
number of moduli}.\index{number of moduli} In fact the surfaces (D)
correspond to a type of canonical or pluricanonical surfaces, of order
$p^{(1)}-1$ or a multiple of $p^{(1)}-1$, respectively, which give a
projective birational model of the surfaces themselves; the
classification of the surfaces of given order, with some other given
characters, can only lead to a finite number of arithmetically distinct
types.

We have already had the opportunity of studying the {\em general problem
of the classification of the surfaces with $p^{(1)}>1$ in the regular
case} (Chapter~VII). The considerations we have developed extend, to a
great extent, to the case in which one drops the hypothesis $p_a=p_g$;
in any event this is an {\em object of research} for those interested.
It is worth remarking that when one proceeds to the construction of
canonical or pluricanonical surfaces with given $p_g$ and $p^{(1)}$ one
usually meets regular surfaces, and the possibility of irregular
surfaces only appears as an exception; thus the canonical surfaces with
$p_a\ge4$, which are constructed in the first cases $p^{(1)}=5, 6,
7,\dots$ (Chapter~VIII), turn out to be regular, and also regular appear
to be the tricanonical surfaces for $p^{(1)}=2$ and the bicanonical
ones for $p^{(1)}=3$, which are constructed in correspondence with
$p_a>0$ following the procedure of Chapter~VIII,
Section~\ref{sec:8.14}--\ref{sec:8.20}\footnote{Some considerations
which can be completed with the theorems on the simple integrals
belonging to a surface lead us to state what follows: in general the
surfaces of irregularity $q\ge3$ and
$p_a\ge0$, whose canonical system is not composed with the curves of a
pencil, have the linear genus $p^{(1)}\ge6p_a+4$; if $q=2$ and
$p_a\ge0$, then $p^{(1)}\ge3p_a+4$.

Therefore we would be left with the problem of examining the surfaces
of irregularity $q=1$, with an elliptic pencil of curves (this was
kindly communicated to us by G. Castelnuovo, November 1942).}.

It is worth adding here that, in force of an inequality which we shall
talk about in the next section, all the surfaces (D) (with
$p^{(1)}>1$) have the arithmetic genus $p_a\ge0$.

 \section{Surfaces with negative arithmetic genus}\label{sec:11.8}
 \markright{\hfill Surfaces with negative arithmetic genus \quad
11.8\quad}
 The classification we have treated in the previous section can be
made more precise on the basis of an inequality between the
irregularity and the arithmetic genus, that so far has not been proved
in an algebro-geometric way, and which therefore we need to take from
the transcendental theory (see Note, page~\pageref{note:transc}). The
following theorem holds: For any surface of genus $p_g$ and irregularity
$q=p_g-p_a$ which does not contain an irrational pencil of curves, one
has
 \[
 p_g>2(q-2)
 \]
or also
 \[
 p_g<2p_a+4.
 \]
 So: if $p_g\ge2p_a+4$, then the surface contains an irrational pencil
of curves $(C)$.

The previous inequality is certainly satisfied if the arithmetic
genus is
 \[
 p_a<-1 \quad (p_g\ge0).
 \]

Therefore we can say that the surfaces $F$ of arithmetic genus $p_a<-1$
contain an irrational pencil (C): a pencil of genus $p>0$ formed by
curves $C$ of a certain genus $\pi$. But it is easy to see that $\pi=0$, and
therefore $F$ can be transformed into a ruled surface (of genus
$p=-p_a$).

To see this, we argue by contradiction: if $\pi>0$, then $F$ can be
reduced to a surface without exceptional curves, with absolute linear
genus $p^{(1)}\ge1$; thus -- by computing its Zeuthen--Segre invariant in
relation to (C) -- we find (as in Chapter~X, Section~\ref{sec:10.4})
 \[
 \De-4=12p_a-p^{(1)}+9
 \quad
\De\ge0
 \quad
p_a\ge-1!
 \]

Therefore we conclude that: the surfaces of arithmetic genus $p_a<-1$
are all birational to ruled surfaces (of genus $p=-p_a$): in the theorem
(Chapter~X, Section~\ref{sec:10.4}) which gives $p_a=-p$ and $p_g=0$ as
conditions for a surface to be ruled, the condition $p_g=0$ is
superfluous, since it follows from the other as a consequence.

Let $p_a=-1$. If the surface has geometric genus
 \[
 p_g>1,
 \]
 then the inequality
 \[
 p_g\ge 2p_a+4
 \]
is still satisfied and therefore $F$ possesses an irrational pencil of
curves $C$ (of genus $\pi>0$). But, in relation to this pencil one finds
the Zeuthen--Segre invariant
 \[
 \De-4=12p_a-p^{(1)}+9
 \quad
 p_a=-1, \De\ge0
 \]
 and therefore
 \[
 p^{(1)}=1.
 \]

So the condition $p^{(1)}=1$, which -- in addition to $p_a=-1$ and
$p_g>1$ -- characterizes the elliptic ruled surfaces (Chapter~XI,
Section~\ref{sec:11.4}) is a consequence of the other two conditions.
Since for $p_g=0$ and $p_a=-1$ one still has $p^{(1)}=1$ and $F$ turns
out to be a (proper or improper) elliptic surface, we may state that:
the surfaces with arithmetic genus $p_a=-1$ and geometric genus
$p_g\ne1$ are elliptic surfaces (either proper elliptic surfaces or
ruled surfaces of genus one).

Let us finally examine the case:
 \[
 p_a=-1,\quad p_g=1.
 \]

In this case we have seen (Chapter~X, Section~\ref{sec:10.4}) that
either $F$ is hyperelliptic (with $P_4=1$) or it can be reduced to a
multiple hyperelliptic surface $\Phi$, with an elliptic branch curves,
and then it is an elliptic surface (with $P_4>1$).

So we can complete the theorems established in Chapter~X,
Section~\ref{sec:10.4}, in the sense that we have already announced
there, by saying that:

The elliptic and hyperelliptic surfaces are globally characterized by
the condition
 \[
 p_a=-1
 \]
(together with the conditions excluding the improper elliptic surfaces,
birational to ruled surfaces: $P_4+P_6=0$ or $P_{12}=0$).

The hyperelliptic surfaces (which only for particular moduli are also
-- in a double way -- elliptic) can be distinguished from the elliptic
ones by the conditions
 \[
 p_g=P_4=1
 \]
 (instead of
 \[
 p_g=1, P_4>1).
 \]

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

The surfaces with negative arithmetic genus ($p_a<0$) are:
 \begin{enumerate}
 \renewcommand{\labelenumi}{(\arabic{enumi})}
 \item either birational to (irrational) ruled surfaces,

 \item or elliptic or hyperelliptic surfaces (in the latter case:
$p_g=1$).
 \end{enumerate}

Or also:

The negative arithmetic genus defines the whole family of irrational
surfaces admitting a continuous group of birational transformations.

 \begin{NOTE}\label{note:transc} The fundamental inequality
$p_g<2p_a+4$, for the surfaces which do not contain an irrational
pencil of curves, follows -- as we have said -- from considerations of
the transcendental theory. Let us firstly say that the genus $p_g$ of a
surface $F$ is the number of the linearly independent double integrals
of the first kind belonging to $F$, and, on the other hand, the
irregularity is the number of the simple integrals of the first kind of
$F$; moreover we recall that Noether pointed the way to ``construct a
double integral of the first kind starting with two simple integrals of
the first kind which are not one a function of the other''.

Now the fact that the surface $F$ possesses two simple integrals, $I_1$
and $I_2$, one a function of the other, means that $F$ contains an
irrational pencil of algebraic curves $C$, on which $I_1$ and $I_2$ are
constant.

This remark occurred for the first time to Castelnuovo, in relation to
the surfaces of genus $p_g=0$ (see Chapter~IX, Section~\ref{sec:9.6});
later (1905), for the surfaces of any genus $p_g$, to De
Franchis\footnote{Sulle superficie algebriche le quali contengono un
fascio irrazionale di curve, {\it Rendic.\ Circ.\ Mat.\ di Palermo}
(April 25th, 1905).}, Castelnuovo\footnote{Sulle superficie aventi il
genere aritmetico negativo. Ibidem, May 14th, 1905 ({\it Memorie
scelte}, XXVII, page~501).} and Enriques\footnote{Sulle superficie
algebriche che ammettono un gruppo continuo di trasformazioni ecc.,
(Ibidem, May 14th, 1905).}, who discovered in this way the appropriate
tool for the classification of the surfaces of genus $p_a<0$.

The relation between the genus $p_g$ and the irregularity $q$, or
between $p_g$ and $p_a$, has been clarified and proved in the simplest
way by Castelnuovo with the aforementioned inequality
 \[
 p_g<2p_a+4,
 \]
holding for the surfaces $F$ which do not contain an irrational pencil of
curves. Meanwhile, on the basis of this relation, De Franchis stated that
the surfaces $F$ of genus $p_a<-1$ possess an irrational pencil of
curves\footnote{Let us also add that De Franchis (clarifying a previous
result of 1904) proves that ``every irregular double plane possesses an
elliptic or hyperelliptic pencil of curves''.} while Castelnuovo and
Enriques -- making use of an argument already employed in the hypothesis
$p_g=0$ (Chapter~X, Section~\ref{sec:10.4}) -- saw that these surfaces
are birational to ruled surfaces of genus $p=-p_a$ ($>1$).

Finally Enriques made a deeper analysis of the surfaces for which
$p_a=-1$, succeeding in proving that they are either elliptic or
hyperelliptic surfaces.

The developments we have just mentioned have been followed by some
important papers by A. Rosenblatt\footnote{Sur les surfaces
irr\'eguli\`eres etc. {\it Rendic.\ Circ.\ Mat.\ di Palermo}, 1913.} and
A. Comessatti\footnote{Rendic.\ Acc.\ Lincei, 1919 and Intorno alle
superficie algebriche irregolari ecc., {\it Rendic.\ Circ.\ Mat.\ di
Palermo}, t.~XLVI, 1922.} that led to the classification of the
surfaces for which
 \[
 p_g\ge 2p_a+4.
 \]

Already Rosenblatt (1913) saw that there are only few types of these.
Comessatti (1919 and 1922) takes up again Castelnuovo's analysis,
dealing with the more general problem of ``determining which relations
there are between some functions of two variables when their mutual
Jacobians satisfy a certain number of linear homogeneous equations with
constant coefficients''.

Whence one, in particular, draws the classification of the algebraic
surfaces for which
 \[
 p_g\ge 2p_a+4,
 \]
proving that they belong to the following types:

 \begin{enumerate}
 \renewcommand{\labelenumi}{(\arabic{enumi})}
 \item ruled surfaces of genus $p>1$;
 \item elliptic surfaces with $p_g>1$ and $p_a=-1$;
 \item surfaces representing the pairs of points of two curves of genus
2 and $\pi$: $p_g=2\pi$ and $p_a=\pi-2$;
 \item particular surfaces with linear genus $p^{(1)}=1$ and $p_a\ge0$.
 \end{enumerate}

It would be desirable to give an algebro-geometric proof of these
results, as well as of Castelnuovo's inequality
 \end{NOTE}

 \section{Summary of the above classification}\label{sec:11.9}
 \markright{\hfill Summary of the above classification \quad 11.9\quad}
 The theorems concerning the surfaces with negative arithmetic genus
that we have just treated enable us to give a more precise general
classification of the surfaces; in particular they imply that for
$p^{(1)}>1$ one must have $p_a\ge0$.

This deduction has a meaning of general interest: for the surfaces of
type D) ($P_{12}>1$, $p^{(1)}>1$), since $p_a\ge0$, one obtains a
bicanonical system of dimension $P_2-1=p_a+p^{(1)}-1$, and therefore -- for
$p^{(1)}>3$ -- one has $P_2-1\ge3$. So, except for an objection about the
reducibility of the bicanonical curves, which has to be critically
examined\footnote{As we did for the regular surfaces in Chapter~IV,
SectionS~18--20.}, the surfaces for which $p^{(1)}>3$ have as their
projective model a bicanonical surface (that, in some particular cases,
can be multiple).

And it is therefore apparent the special interest of having a
complete classification of the surfaces with $p^{(1)}=2$ and $p^{(1)}=3$,
to which we refer in the study we have treated in Chapter~VIII,
Section~\ref{sec:8.14}--\ref{sec:8.20}; and even more if one happens to
find that these surfaces have (except possibly some cases) the property
of regularity, which in our analysis has been assumed to be true a
priori\footnote{See the footnote at the end of Section~\ref{sec:11.7}
of this Chapter.}.

Let us now summarize the more precise results we have obtained in the
general classification of the algebraic surfaces, using also some
formula whose meaning has been explained before, in
Section~\ref{sec:11.7}.

\pagebreak
 %\begin{table}[hbt]
 \parbox{5.0cm}{$P_{12}=0$: ruled surfaces $n>2\pi-2$, non-eliminable
exceptional curves, continuous series of transformations not forming a
group.}
 \quad
 $\left\{
 \begin{tabular}{l}
$p_a=0$, rational surfaces \\ \qquad ($p_a=P_2=0$)\\
$p_a=-1$, elliptic ruled surfaces \\ \qquad ($p_a=-1$,$P_4=P_6=0$)\\
$p_a<-1$, ruled surfaces of \\ \qquad genus $p=-p_a>1$.
 \end{tabular}
 \right.$

\vskip1cm

\parbox{2.6cm}{$P_{12}=1$: virtual canonical curve of order 0,
$n=2\pi-2$ ($p^{(1)}=1$)}
 \quad
 $\left\{
 \begin{array}{l}
 p_g=1 \\ [3cm]
 p_g=0 \\
 \end{array}
 \right.$
 $
 \begin{array}{l}
 \left\{ \begin{tabular}{l}
$p_a=1$, infinitely many families of\\
 \qquad surfaces ($p_a=P_2=1$)\\
$p_a=-1$, infinitely many families of \\
 \qquad hyperelliptic surfaces of \\
 \qquad divisor $\de=1,2,3,\dots$ \\
 \qquad ($p_a=-1,p_g=P_4=1$).
 \end{tabular} \right.\\
 \\
 \left\{ \begin{tabular}{l}
$p_a=0$, surfaces of order 6 passing doubly\\
 \qquad through edges of tetrahedron \\
 \qquad ($p_a=P_3=0$, $P_2=1$).\\
 $p_a=-1$, elliptic surfaces I$_a$, I$_b$, II$_a$, II$_b$, \\
 \qquad III$_a$, III$_b$, III$_c$ ($P_2=0,1$, \\
 \qquad $P_4=0,1$, $P_3=0,1$, $P_4=0,1$).
 \end{tabular} \right.\\
 \end{array}
 $

\vskip1cm

 $\begin{array}{c}
 P_{12}>1 \\
 (n<2\pi-2)
 \end{array}$
 $\left\{
 \begin{array}{l}
 p^{(1)}=1 \\ [3cm]
 p^{(1)}>1 \\
 \end{array}
 \right.$
 $\begin{array}{l}
 \left\{
 \begin{tabular}{l}
 $p_a=-1$, elliptic surfaces with a pencil \\
 \qquad of genus $p_g$ of elliptic curves \\
 \qquad ($p_a=-1, P_4\ne1$)\\
$p_a\ge0$, surfaces whose canonical and pluri-\\
 \qquad canonical curves are composed with \\
 \qquad elliptic curves of a pencil of genus $p_g-p_a$:\\
 \qquad depend on more arbitrary integral characters
 \end{tabular}
 \right. \\
 \\
 \left\{
 \begin{tabular}{l}
 $p_a\ge0$, surfaces which admit a projective \\
 \qquad canonical ($p_g>3$) or bicanonical ($p^{(1)}>3$) \\
 \qquad model, etc.: a finite number of types for a \\
 \qquad given value of $p^{(1)}$.
 \end{tabular}
 \right.
 \end{array}$

 %\end{table}
\pagebreak

At this point we would like to be allowed to stop for a while, as when
one climbs a mountain and likes to rest for a moment on the peak he as
reached, whence admiring the view of the Nature which is offered to the
sight.

Fifty years ago the study of these theories started in Italy, after they
had been sketched by the genius of a precursor (Max Noether); at that
time, joking on the difficulties and the exceptions that we met at every
step, we used to say that while algebraic curves (whose theory was
already harmoniously settled) had been created by God, the surfaces
were, by contrast, the works of the Devil.

It is now apparent that God liked to create for the surfaces a more
harmonious order, where a wonderful beauty shines; and He wanted in them
-- as we say with the Poet\footnote{Note of the translators: ``to
impress a stronger footprint of his creating spirit'', from ``Il 5
Maggio'', by Alessandro Manzoni} --

\[
 \begin{array}{c}
 \hbox{del creator suo spirito}\\
 \hbox{piu' vasta orma stampar.}
\end{array}
\]

The richness of properties and the beauty, which were hidden for such a
long time but are now apparent, should not be a reason of vain pride for
the Italian school of geometry or for the foreign mathematicians who
contributed to discover them; they rather have to rouse a feeling of
reverence for the wonderful order of the mathematical objects, that the
thought encounters and almost collects, as the living species, from
Mother Nature; so we have to nourish the faith of the young scholars
that, behind the difficulties, the exceptions, the apparent
incongruities, there is actually in this world of objects a divine
harmony, on which the joint efforts of the researchers will succeed to
shed a clearer and clearer light.
