 \chapter{Regular surfaces: the minimum for the genera and rationality
conditions}\label{ch:6}
 \markboth{\qquad 6. \quad Regular surfaces \hfill}{}

In this chapter we want to consider for simplicity regular surfaces
with genus
 \[
 p_g=p_a=p,
 \]
although many results concerning these surfaces may be extended
without difficulties to irregular surfaces, as we will often point out.

 \section{The lower bound for the linear genus}\label{sec:6.1}
 \markright{\hfill The lower bound for the linear genus \quad 6.1\quad}
 Let us consider a surface $F$ of genus $p>0$, which we may assume to
have no exceptional curves (Chapter~IV, Section~\ref{sec:4.9}). In view
of this hypothesis, $F$ possesses some canonical curves, which, to
begin with, we will suppose to be of order greater than zero. We want
to prove that the linear genus of $F$ (which here is just the absolute
linear genus) is
 \[
 p^{(1)}>0.
 \]

We need to show that the hypothesis $p^{(1)}\le0$ leads to a
contradiction. Therefore let $p^{(1)}\le0$, hence let the degree of the
canonical system $|K|$ be
 \[
 p^{(2)}=p^{(1)}-1<0.
 \]

So we will have (at least) one canonical curve $K$ with negative
degree, which may consist of several, simple or multiple, parts:
 \[
 K=s_1K_1+s_2K_2+\cdots+s_rK_r,
 \]
where the $K_i$ cannot be curves with virtual genus $\rho_i=0$ and
degree $\nu_i=-1$, which would be exceptional curves of the first kind
on $F$. Let us intersect one of the components of $K$, for instance
$K_1$, with $K$. The number of intersections will be
 \[
 2\rho_1-2-\nu_1
 \]

This number cannot be negative. In fact, since $\rho_1\ge0$,
 \[
 2\rho_1-2-\nu_1<0,
 \]
yields
 \[
 \nu_1\ge-1
 \]
and thus
 \[
 \nu_1\ge0.
 \]
(since the values $\nu_1=-1$ and $\rho_1=0$ are excluded).

But this is a contradiction, because a component $K_1$ of degree
$\nu_1\ge0$ cannot have a negative number of intersections with
$K=s_1K_1+s_2K_2+\cdots$. Now, since we have proved that every component
$K_i$ of $K$ must have a nonnegative number of intersections with $K$,
one easily deduces that $K$ itself must have degree $p^{(2)}\ge0$.

Indeed, denoting by $n_{ij}$ the number of the intersections of two
components $K_i$ and $K_j$ of $K$, the degree of $K$ is given by
 \[
 p^{(2)}=\sum s_i^2\nu_i+2\sum s_is_jn_{ij}.
 \]

The expression of $p^{(2)}$ appears now as a quadratic form in the
$s_i$, and, taking into account Euler's theorem on homogeneous
functions, one sees that it has to be essentially nonnegative, since
its partial derivatives, which represent the double of the number of
intersections $(K_iK_j)$, are
 \[
 \frac{\partial p^{(2)}}{\partial s_i}=2(s_i\nu_i+\sum s_jn_{ij})\ge0.
 \]

The inequality $p^{(1)}>0$ still holds when the surface $F$ has a single
canonical curve of order zero ($p=1$). Consider in fact on the surface
$F$, which has no exceptional curves, an irreducible linear system
$|C|$ of genus $\pi$ and degree $n$, with no base points; we have in
this case
 \[
 n=2\pi-2,
 \]
since the number of intersections is $(CK)=2\pi-2-n=0$. And, denoting
by $|C'|$ the adjoint system of $|C|$, of genus $\pi'$ and degree $n'$,
we have
 \[
 |C'|=|C|.
 \]

Now, by computing the genus $\pi'=\pi$ of $|C'|$, we find
 \[
 \pi'=\pi=\pi+p^{(1)}+2\pi-2-n-1
 \]
 and so
 \[
 p^{(1)}=1.
 \]

The theorem that we have proved ($p^{(1)}\ge1$) holds also for all
surfaces of genus $p=0$ which have a plurigenus $P_i>0$. In fact one may
repeat the above argument, where the canonical curve is replaced by an
$i$th canonical curve $iK$; were $p^{(1)}\le0$, then this curve would
have a nonnegative number of intersections with each one of its
components (of genus $\rho$ and degree $\nu$). This number would be
given by
 \[
 i(2\rho-2-\nu).
 \]

Also in the case ($P_i=1$) in which the $i$th canonical curve is of
order zero, and thus a system $|C|$ coincides with its $i$th adjoint
system, one may conclude, as in the case $p=1$, that
 \[
 p^{(1)}=1.
 \]

Finally we notice that the proof of our theorem is completely
independent of the hypothesis that $F$ is regular. Therefore we can
state, without any restriction, that:
 \begin{proc}
 A surface with genus $p_g>0$, or even with $p_g=0$ but with a
plurigenus
$P_i>0$, has the absolute linear genus
 \[
 p^{(1)}\ge1.
 \]
 \end{proc}
We want to present another proof of the theorem we just stated; this
proof relies on a lemma which will have important applications later
on. Let $F$ be a surface, which we assume to have no exceptional curves,
with linear genus $p^{(1)}\le0$, and let us consider an irreducible,
regular linear system on it, of genus $\pi$ and degree $n\le2\pi-2$. The $C$ meet a virtual canonical
curve $K$ at
 \[
 2\pi-2-n\ge0
 \]
points. The curves $C'$ of the adjoint system $(|C'|=|C+K|)$ meet $K$
at
 \[
 2\pi-2-n+p^{(1)}-1<2\pi-2-n
 \]
points. So the number of intersections of the curves of the system
with the canonical curve decreases in passing from a linear system to
its adjoint. But this implies that either the adjunction process must
have an end, or that one eventually obtains linear systems of genus
$\pi$ and degree $N>2\pi-2$. Now again the existence of such a system
$|L|$ necessarily implies that the sequence of the adjoint systems
$|L'|$, $|L''|$, etc. must have an end, since the numbers of
intersections are $(L\cdot L)>(L'\cdot L)>(L''\cdot L)>\cdots$.

Therefore the assumption $p^{(1)}\le0$ yields that, starting from a
linear system $|C|$, one obtains only a finite sequence of successive
adjoint systems; thus a system $|C|$ cannot be contained in any one of
its adjoint systems $|C^{(i)}|$, namely all the plurigenera of the
surface are $P_i=0$.

Here two remarks are in order. The first one is that the assumption $F$
without exceptional curves is introduced only in order to find a
contradiction; if this assumption is removed then one is led at once to
the conclusion that $F$ has a system with $n>2\pi-2$, and that the
adjunction process terminates as soon as one obtains that system.

The second remark concerns the possibility that the adjoint system
$|C'|$ of the irreducible system $|C|$ from which we started is
reducible. If the surface $F$ is regular and $|C|$ has no fundamental
curves, then it is clear that $|C'|$ cannot be reducible, and so, in
particular, it cannot be reduced to the system $|C|$ plus a fixed
canonical curve. But, independently of the irreducibility of the
successive adjoint systems of $|C|$, it is sufficient to obtain a system
$|L|$, however reducible, of genus $\pi$ and degree $N>2\pi-2$; any
multiple of this again gives a linear system a fortiori satisfying the
same inequality. Therefore it is easy to construct an irreducible
system which again satisfies an analogous inequality; this can be done
just by adding $|sL|$ to a suitable irreducible linear system. We do
not believe it necessary to spend more time on this.

Meanwhile we may conclude by confirming the above stated result.

 \section{The bigenus and the plurigenera}\label{sec:6.2}
 \markright{\hfill The bigenus and the plurigenera \quad 6.2\quad}
 Let us consider a surface $F$ such that for any linear system of genus
$\pi$ the degree is $n\le2\pi-2$; as we saw (Chapter~IV,
Section~\ref{sec:4.9}), $F$ can be transformed in such a way that its
exceptional curves are eliminated. Starting from an irreducible linear
system $|C|$, which can be the hyperplane section system of a surface
$F$ without singularities, let us try to construct the bicanonical
system of $F$. For this, we need to subtract a curve $C$ from its
second adjoint system $|C''|$. Now if we assume that
 \[
 d=2\pi-2-n\ge0,
 \]
then the $C''$ cut out on $C$ a series of order $2\pi-2+d$ which is
certainly nonspecial; therefore one has to impose at most $\pi-1+d$
conditions on the curves of $|C''|$ in order that they contain $C$. Thus
by computing the dimension of $|C''|$ by the Riemann--Roch theorem, one
finds that the dimension of the bicanonical system $|C''-C|$ is given by
 \[
P-1\ge p_a+p^{(1)}-1;
 \]
thus we have
 \[
P\ge p_a+p^{(1)}.
 \]

This result does not depend on the effective existence of the
canonical curves (of order $d\ge0$), that is, it holds for $p_g\ge0$. In
particular, {\em for regular surfaces ($p=p_a=p_g$), without
exceptional curves, with virtual canonical curves of order $d$
($\ge0$), the bigenus is always
 \begin{equation}
 P\ge p+p^{(1)}.
 \label{eq:6.1}
 \end{equation}}

If one has irreducible canonical curves of genus $p^{(1)}>1$, on which
the bicanonical system cuts out the complete canonical series, then one
has exactly:
 \begin{equation}
P=p+p^{(1)}.
 \label{eq:6.2}
 \end{equation}

Instead of equality (\ref{eq:6.2}) one has in general the inequality
 \begin{equation}
P\ge p+1 \quad\hbox{for}\quad p^{(1)}=1,
 \label{eq:6.3}
 \end{equation}
since the {\em canonical system} of degree $p^{(1)}-1=0$ is then {\em
composed with the elliptic curves $k$ of a pencil} (which is linear,
since $F$ is regular): if a canonical curve is composed of $p-1$ curves
$k$ of such a pencil, then the bicanonical curves are formed by $p-2$
curves $k$ and therefore they form a linear system of dimension
 \[
 P-1=2p-2.
 \]
If a fixed curve has to be added to the $p-1$ curves $k$, then the
double of such a fixed curve may be a curve $k$ susceptible of varying
in the pencil; thus it may happen that
 \[
 P-1>2p-2.
 \]

The above results have to be modified if $p=1$, when there is an {\em
effective canonical curve of order $d=0$}: then, instead of the
inequality (\ref{eq:6.1}) one has:
 \[
 P=1
 \]
 and
 \[
 p^{(1)}=1 \quad (\hbox{that is, } P<p+p^{(1)}).
 \]

In fact, if $d=0$, then the system $|C|$ that we considered on the
surface $F$ has degree
 \[
 n=2\pi-2,
 \]
 and, since $p=1$, one has
 \[
 |C|=|C'|=|C''|.
 \]
Thus {\em the bicanonical system coincides with the system of order zero
which gives the canonical system and has dimension
 \[
 p-1=P-1=0;
 \]
 the genus of such a system $|K|$ is
 \[
 p^{(1)}=1,
 \]
 since, by adding a $K$ of order zero to a $C$ one has a reducible curve of
genus
 \[
 \pi=\pi+p^{(1)}+0-1.
 \] }

So the regular surfaces of genus $p=1$ with a canonical curve of order
zero are different from the surfaces with canonical curve of order
greater than zero in this respect: the former have
 \[
 P=1,
 \]
 whereas the latter have
 \[
 P\ge p+p^{(1)}\ge 2.
 \]

For a regular surface of genus $p=0$, without exceptional curves
($n\le2\pi-2$), with a virtual canonical curve of order $d=0$
($d=2\pi-2-n$), the inequality (\ref{eq:6.1}) still holds, as in the
case $d>0$, namely
 \[
 P\ge p^{(1)},
 \]
 since
 \[
 P=1, \quad p^{(1)}=1.
 \]

In fact, since $|C'|$ is different from $|C|$, $|C''|$ is distinct from
$|C'|$, too, and it cuts out on a $C$ a nonspecial series $g_{2\pi-2}$;
accordingly, subtracting $C$ from $|C''|$ imposes at most $\pi-1$
linear conditions: whence we deduce that it imposes exactly $\pi-1$
conditions, and we have
 \[
 |C''|=|C|,\quad P=1,\quad p^{(1)}=1.
 \]

 \begin{Remark} It is important to notice that the above results,
especially the fundamental inequality
 \begin{equation}
 P\ge p+p^{(1)},
 \label{eq:1}
 \end{equation}
{\em fail} absolutely for surfaces which contain linear systems
$|C|$ of genus $\pi$ and degree $n>2\pi-2$, that is, having virtual
canonical curves of negative order
 \[
 -d=n-(2\pi-2).
 \]

Indeed the system $|C''|$ will now cut out on a $C$ a series of order
$2\pi-2-d$ that may be special.
 \end{Remark}

As an example, look at the case of the plane, where we may consider
the complete linear system $|C_7|$ formed by the curves of the 7th
order. The adjoint curves are the quartics $C_4$, and the second adjoint
curves are the lines $C_1$, which cut out on a $C_7$ a series $g^7_2$,
which is special with index of speciality
 \[
 i=10.
 \]

Here we can compute the (linear) genus $p^{(1)}$ of the virtual
canonical system
 \[
 |K|=|C_1-C_4|.
 \]
 Since the intersection number of the $K$ with the $C_1$ is $-d=-3$ and
with the $C_4$ (of genus 3) is $-12$, we have
 \[
 0=p^{(1)}+3-12-1
 \]
whence we find again that the (relative) linear genus of the plane is:
 \[
 p^{(1)}=10.
 \]

Meanwhile we note that, since it is not possible to subtract $|C|$
from $|C_1|$, there is no bicanonical system and therefore
 \[
 P=0.
 \]

The formulas which express the bigenus in terms of the linear and
geometric genera can be extended to the plurigenera. If a surface $F$,
without exceptional curves, has virtual canonical curves of order $d>0$
($d=2\pi-2-n$), then one has
 \[
P_i\ge p_a+\frac{i(i-1)}{2}(p^{(1)}-1)+1
 \]
and for a regular surface ($p_a=p_g=p$) with linear genus $p^{(1)}>1$
and with irreducible $(i-1)$th canonical curves, one has
 \[
P_i\ge p_a+\frac{i(i-1)}{2}(p^{(1)}-1)+1.
 \]

The equality becomes an inequality in the case $p^{(1)}=1$, where the
pluricanonical curves of degree $i^2(p^{(1)}-1)=0$, are all composed
with the elliptic curves of one and the same pencil:
 \[
 P_i\ge p+1.
 \]

An exception to the fundamental inequality
 \[
 P_i\ge p_a+\frac{i(i-1)}{2}(p^{(1)}-1)+1.
 \]
is given by the surfaces with bigenus $P=P_2=1$ which possess a virtual
canonical curve of order zero and an effective bicanonical curve also
of order zero. In this case the series cut out by the third adjoint
system $|C'''|=|C'|$ on a $C$ is the canonical series (of dimension
$\pi-1$); then there is no curve of $|C'''|$ containing $C$, since $C$
imposes $\pi$ conditions on $|C'''|$, hence
 \[
 P_3=0.
 \]

Therefore the regular surfaces of genus zero and bigenus one with a
bicanonical curve of order zero differ from the surfaces having a
bicanonical curve of order greater than zero in this respect: that the
former have trigenus
 \[
 P_3=0,
 \]
 whereas the latter have
 \[
 P_3\ge1.
 \]

Furthermore the former have
 \[
 P_{2i}=1,\quad P_{2i+1}=0,
 \]
 whereas for the latter all plurigenera are essentially positive. And
moreover they are greater than one even in the case $p^{(1)}=1$. In
fact, by combining a bicanonical curve counted three times with a
tricanonical one counted twice, one obtains a pencil of sexticanonical
curves, so that
 \[
 P_6\ge 2.
 \]

 \section[Examples: double planes with linear genus
$p^{(1)}=1$]{Examples: double planes with linear\\ genus
$p^{(1)}=1$}\label{sec:6.3}
 \markright{\hfill Examples: double planes with linear genus
$p^{(1)}=1$ \quad 6.3\quad}
 An interesting illustration of the formulas of the above section is
given by the double planes with linear genus $p^{(1)}=1$\footnote{See
F.~Enriques, Sui piani doppi di genere lineare $p^{(1)=1}$, {\it
Rendic.\ Acc.\ Lincei}, April and May 1898.}. One proves here that, with
the exception of the double plane branched on a completely general
sextic
 \[
 z^2=D_6(x,y)
 \]
which has a canonical curve of order zero (genus and plurigenera
$p=P_i=1$), all the other double planes for which $p^{(1)}=1$ and some
plurigenus $P_i>0$, contain a pencil (rational or not) of elliptic curves $L$, which is transformed into itself by the involution $I$ whose pairs
correspond to the points of the plane. Therefore one is led to two
families of surfaces which are representable as a double plane of this
sort:

 \begin{enumerate}
 \renewcommand{\labelenumi}{(\Roman{enumi})}
 \item surfaces with a linear pencil of (irreducible) elliptic curves
$L$ fixed by the involution $I$ which induces a linear involution
$g_2^1$ on them; these surfaces are represented as double planes with
branch curve $D_{2n}$ of order $2n$ ($n>3$) having a point $O$ of
multiplicity $2n-4$ or $2n-3$, and possibly other singularities which
let the genus drop: quadruple points and $[3,3]$ points; the lines
through $O$ are the images of the curves $L$ on the double plane;

 \item surfaces with a pencil of irreducible elliptic curves $L$, which
may either be the curves of a rational pencil, fixed by the involution
$I$ which induces an elliptic involution $\ga_2^1$ on them, or the
curves of a pencil (rational or not), pairwise associated by the
involution $I$ which induces a rational involution in the pencil; these
surfaces are represented as double planes in such a way that the $L$
correspond to the elliptic curves of a pencil of Halphen\footnote{See,
for instance, Enriques--Conforto, {\it Le superficie razionali},
Zanichelli, Bologna.} (curves $C_{3s}$ with nine $s$-tuple points);
therefore (provided they have no exceptional curves) the branch curves
of these double planes are composed with the curves of the
above-mentioned pencil and possibly also with the cubic curve which
passes through its nine base points.
 \end{enumerate}

Among the double planes of class I, it is worth mentioning those
having a branch curve $D_{2n}$ with pairs of infinitely near triple
points ($[3,3]$ points) collinear with the $(2n-4)$-tuple point $O$.

For $n=4$ we have the double plane with a branch curve $D_8$ having a
4-tuple point $O$, and a pair of infinitely near triple points $A$ and
$A_1$, collinear with $O$; $D_8$ is composed of a $D_7$ with
$O^3A^2A_1^2$ and of the line $r=OA$. The genus of this double plane is
 \[
 p_g=p_a=p=1,
 \]
since there is one canonical curve represented by the line $r=OA$
(which is subject to the condition of passing simply through $O$ and
A). Less easy is the determination of the linear genus $p^{(1)}$, since
the double line $r=OA$ seems to correspond to a disconnected curve
formed by a rational curve counted twice. This difficulty can be
overcome is one understands that the surface $F$, represented as a
double plane by projection, possesses an impure canonical curve, formed
by two infinitely near exceptional curves whose points correspond to the
generic points of $r$, and by an infinitesimal curve represented by the
neighbourhood of the point $A_1$; the latter curve may be transformed
into a curve of order greater than zero and genus $p^{(1)}=1$, by means
of a transformation of $F$ into a surface without singularities.
Accordingly, $F$ has the bigenus
 \[
 P=p+p^{(1)}=2,
 \]
 and there is in fact a pencil of elliptic bicanonical curves
represented on the double plane by the lines through $O$: if we add the
exceptional curves, then we have, as images of the impure bicanonical
curves, the pairs of lines formed by $r$ and by a variable line through
$O$. What we have just said is made clear by considering the $\infty^3$
linear system of the conics through $AA_1$, which determines the
transformation of $F$ into a double quadric cone without exceptional
curves: the images of these double conics with 10 branch points are
hyperelliptic curves of genus 4, forming a linear system of degree 4
($<2\cdot4-2$); they are met at two (conjugate) points by the canonical
curve which corresponds to the neighbourhood of $A_1$ and at 4 points,
consisting of two pairs of points that are conjugate in the $g_2^1$, by
the bicanonical curves; therefore the latter are represented on the
cone by the plane sections of a pencil, and on the plane by the lines
through $O$. Let us notice that the quadric cone has a branch curve
$D_{10}$ of the 10th order; the $D_7$ may be regarded as a projection of
this curve from a triple point $P$ which has another infinitely near
triple point $P_1$ whose projection is $O$. The line $PP_1$ lies in the
tangent plane to the cone along the generator $k$ through $P$, and the
planes containing $k$ cut out on the cone the images of the bicanonical
curves.

Beside the above-mentioned double plane, with branch curve $D_8=rD_7$,
with $O^4A^3A_1^3$, one can also consider the double plane with branch
curve $D_8$ passing through $O^4$ and through two pairs of infinitely
near triple points $A^3A_1^3$ and $B^3B_1^3$ collinear with $O$. This
$D_8$ breaks up into the two lines $r=OA$ and $s=OB$ and into a
$D_6(O^2A^2A_1^2B^2B_1^2)$. The genus of such a double plane is
 \[
 p=0,
 \]
 and its bigenus
 \[
P=1,
 \]
 since there is an impure bicanonical curve, represented by the pair of
lines $r$ and $s$. This curve consists of a pair of exceptional curves
each counted twice. Computing the trigenus, one finds $P_3=0$. This also
indirectly stems from the following remarks.

If one transforms the double plane into a surface $F$ without
exceptional curves, then the bicanonical curve appears to be of order
zero: one may construct $F$ by means of the $\infty^4$ linear system of the
cubic curves through $OAA_1BB_1$, which are the images of curves of
genus 4 and degree 6; one is therefore led to a double cubic surface
(normal in $S_4$) with branch curve of the 10th order.

We will see later on that $F$ may also be transformed into the surface $F_6$ of the 6th order which passes doubly through the edges of a
tetrahedron.

Let us consider now the double planes (again in class I) with branch
curve $D_{10}$ of the 10th order and two pairs of double points $AA_1$
and $BB_1$, collinear with $O$: $D_{10}$ is composed of the lines
$r=OA$ and $s=OB$ and of a $D_8(O^4A^2A_1^2B^2B_1^2)$. We have here an
impure canonical curve, represented by the two exceptional lines $r$ and
$s$, and by a residual curve (which becomes of order $>0$ on a
transformed surface without singularities): this is composed of two
elliptic components corresponding to the neighbourhoods of the points
$A_1$ and $B_1$, and thus
 \[
 p_g=p_a=p=1,\quad p^{(1)}=1.
 \]

The bigenus $P$ has to be greater than one, and in fact there is a net
of bicanonical curves whose images on the double plane are the pairs of
lines through $O$; so we may conclude that
 \[
 P=3 \quad (>p+p^{(1)}).
 \]

Beside the aforesaid example, consider the double plane branched on a
curve $D_{10}$ with a 6-tuple point $O$ and 3 pairs of infinitely near
triple points $AA_1$, $BB_1$, $EE_1$, collinear with $O$. The surface
$F$ thus defined has no canonical curves, since there is no conic
passing doubly through $O$ and simply through $A,B,E$; therefore
 \[
 p_a=p_g=p=0.
 \]

We have instead elliptic bicanonical curves, represented by the lines
through $O$, plus the exceptional curves which are the images of the
branch lines
 \[
 r=OA, s=OB, t=OE;
 \]
 so
 \[
 p^{(1)}=1
 \]
 and
 \[
 P=2 \quad (>p^{(1)}).
 \]

Among these bicanonical curves there are three double curves (which
correspond to the three lines $r,s,t$, or rather to the neighbourhoods
of $A_1,B_1$ and $E_1$), which, taken together and with multiplicity
one, form a tricanonical curve of the surface $F$:
 \[
 P_3=1\quad (=p^{(1)}).
 \]

The double of this tricanonical curve belongs to the triple of the
bicanonical system; in this way one obtains the linear system of
sexticanonical curves of $F$, whose images are the triples of lines
through $O$. One has therefore:
 \[
 P_6=4.
 \]

We also find interesting examples for our theory in Class~II. Let us
consider a double plane with a branch curve $D_{6n}$ ($n>1$) formed by
an irreducible elliptic curve with 9 points of multiplicity $2n$,
belonging -- as it is well known -- to a pencil of Halphen\footnote{Any
(double) curve of the pencil of Halphen represents two distinct curves
$L$ which do not belong to the involution I, since this is the case for
$D_{6n}$; however this agrees with Comessatti's criterion, since the
$D_{6n}$ may be reduced by continuity to a double $\fie^3_n$. See for
instance Enriques--Chisini, Libro~V, Chapter~IV, \S38 (Vol.~III,
page~445).}. The surface thus defined possesses a canonical curve whose
image is a curve of order $3n-3$ subject to the condition of passing
$n-1$ times through the 9 base points of the pencil; so it turns out to
be the cubic curve $\fie_3$ through the 9 points, counted $n-1$ times,
or, as we shall say, the $\fie_3^{n-1}$. We have therefore
 \[
 p_a=p_g=p=1,\quad p^{(1)}=1.
 \]

Now the branch curve $D_{6n}$, together with the $\fie_3^{n-2}$, adjoint
to $\fie_3^{n-1}$, (to be counted twice) gives a bicanonical curve of
$F$ (see Chapter~V, Section~\ref{sec:5.7}), whereas another bicanonical
curve is given by the double $\fie_3^{2n-2}$. Whence we infer that the
curves of the pencil of Halphen defined by the $D_{6n}$, taken together
with the fixed $\fie_3^{n-2}$, represent a pencil of bicanonical
curves; so these are formed by irreducible elliptic curves
$L$\footnote{According with Comessatti's criterion, each curve $L$ of
this pencil is the image of two distinct elliptic curves, because the
group of apparent critical points cut out by $D_{6n}$ on $L$ is also
cut out by the $D_3^{2n}$, the double of the $D_{3n}$. See
Enriques--Chisini, Ibidem} (which do not belong to the involution $I$)
and by a fixed part, which is a $(n-2)$-tuple elliptic curve, whose
component belongs to the involution $I$ and is equivalent to an
$(n-1)$st submultiple of $L$. So we have
 \[
 P=2 (=p+p^{(1)}).
 \]

Finally we easily find that the tricanonical curves are given by the
pairs of curves $L$, so that
 \[
 P_3=3.
 \]

Together with the above example we also find in class II the double
plane with branch curve $D_{6n}=D_{6n-3}+D_3$, which is formed by an
irreducible Halphen curve $D_{6n-3}$, with 9 points of multiplicity
$2n-1$, and by the cubic curve $D_3$ through these 9 points.

The surface thus defined possesses one canonical curve represented by
the $D_3$ counted $n-1$ times, or, as we shall say, by the $D_3^{n-1}$,
so that
 \[
 p_a=p_g=p=1,\quad p^{(1)}=1.
 \]

The bicanonical curves which are fixed by the involution $I$ are: the
double $D_3^{n-1}$, counted twice, and the branch curve $D_{6n-3}+D_3$
together with the adjoint curve to the image of the canonical curve,
which is the double $D_3^{n-2}$. Therefore one has a pencil of elliptic
bicanonical curves, represented by the simple\footnote{See the previous
note.} curves of order $6n-3$ of the pencil of Halphen, taken together
with the multiple curve, represented by the double $D_3^{n-2}$:
 \[
 P=2.
 \]

We will not insist on the double planes of class II with branch curve
however composed with the curves of a pencil of Halphen: these lead in
general to irregular surfaces with
 \[
 p_a=1,p_g>1,\quad p^{(1)}=1.
 \]

Still we want to point out a double plane with $p^{(1)}=1$, $p=0$, $P=1$
and with a bicanonical curve of order greater than zero, possessing a
pencil of elliptic curves (not selfconjugate in $I$), represented by the
curves of a pencil of Halphen. Nevertheless this plane does not belong
to class II because there exists a line, the image of an exceptional
curve, which is a component of the branch curve. This is a double plane
which already occurred to Enriques, whose branch curve $D_{10}$ has
order 10, and it has been met by L.~Campedelli\footnote{L. Campedelli,
Sui piani doppi con curva di diramazione del 10$^o$ ordine and Sopra
alcuni piani doppi notevoli con curva di diramazione del 10$^o$ ordine,
{\it Rendic.\ Acc.\ Lincei}, 1932.} in the classification of these
double planes: the $D_{10}$ is composed of a $D_9+r$, where the $D_9$
is an (irreducible) Halphen elliptic curve with 9 triple points, six of
which
$AA_1$, $BB_1$, $EE_1$ are pairwise infinitely near, while the other
(distinct) three lie on the line $r$.

The surface represented on this double plane has no canonical curves,
since a conic passing through the three triple points of $D_9$ and
through the three points $A, B$, and $E$, would be reducible into the
exceptional curve corresponding to the line $r$\footnote{It is easy to
see that the curve corresponding to this branch line of genus 0 and
degree $-2$ has genus 0 and degree $-1$ on $F$.} and into a line through
$A$, $B$ and E, which, by contrast, are not collinear.

We thus have
 \[
 p=0.
 \]

On the other hand if a double line through $A$, $B$ and $E$ existed,
then it would have 4 branch points (three at the aforementioned three
points and one on $r$), thus the absolute linear genus of $F$ (the genus
of the canonical curves from which the exceptional curve has been
subtracted) would be
 \[
 p^{(1)}=1.
 \]

Now we have on $F$ one irreducible\footnote{The irreducibility of the
double cubic $D_3$ stems from Comessatti's criterion, taking into
account the construction of the pencil of Halphen to which the $D_9$
belongs.} bicanonical curve, represented on the double plane by the
cubic curve
$D_3$ passing through the 9 triple points of $D_9$ plus the exceptional
curve represented by the line $r$. Therefore the bigenus of $F$ is
 \[
 P=1.
 \]

According to our theorem one must have the trigenus $P_3\ge1$ and the
sextic genus $P_6\ge2$.

In order to verify this we observe that (since $D_3$ possesses an
adjoint curve of order $0$) a tricanonical curve of $F$ is given by the
branch curve $D_9$ plus the exceptional curve $r$. This tricanonical curve
is isolated, that is, it does not vary in a pencil. In fact, if this were
not the case, the bicanonical curve represented by $D_3$ would be
contained in it, since it has no common points with this tricanonical
curve, and we would have $p_g>0$. This means that the double elliptic
curves of the 9th order of the pencil of Halphen defined by the $D_9$
are the images of irreducible curves (which are selfconjugate in the
involution $I$ defined on $F$ by the double plane). This conclusion may
also be directly confirmed by the criterion of Comessatti. Finally we
have one tricanonical curve $D_3$ and a pencil of sexticanonical
irreducible curves represented by curves of the pencil of Halphen,
which contains the tricanonical curve counted twice and the bicanonical
curve counted three times. Then one has:
 \[
 P_3=1, P_6=2.
 \]

 \section{Rationality conditions for a surface}\label{sec:6.4}
 \markright{\hfill Rationality conditions for a surface \quad 6.4\quad}
 A rational surface, that is, a surface which can be put in one-to-one
correspondence with a plane, must have, like the plane, the genus
$p=p_a=p_g=0$, and all its plurigenera must vanish together with the
genus. The examples that we have considered (in particular the surface
of order 6 which passes through the edges of a tetrahedron with
multiplicity two) show that a regular surface of genus $p=0$ may have the
bigenus $P>0$, hence the vanishing of the genus does not give a
sufficient condition for the rationality of a surface. Now it is
extremely interesting the result that this condition can be expressed
by imposing that, beside the genus, also the bigenus vanishes, namely
by writing
 \[
 p_a=P=0
 \]
(whence also $p_g=0$ necessarily follows). This theorem is due to G.~Castelnuovo who wrote it in a memoir of 1896\footnote{G.~Castelnuovo,
Sulle superficie di genere zero. Soc.\ It.\ delle Scienze (detta dei
XL), see memorie scelte, page~307.}.

The proof of our theorem can be reduced to the rationality criteria
of Noether and Clebsch--Noether, concerning the surfaces with a linear
pencil of rational curves and the types of double planes which they
determine. With this in mind, one starts from an irreducible linear
system $|C|$ on the given surface and one successively constructs its
adjoint system $|C'|$, the second adjoint $|C''|$ and so on. Then one
realizes that, since one has $p_a=P=0$, the sequence of these successive
adjoint systems terminates, and the last adjoint system that is at
least $\infty^1$ is composed of either rational or elliptic curves;
therefore it leads to one of the cases of rational surfaces which are
contemplated in the above mentioned criteria of Noether and of
Noether-Clebsch.

The discussion can be carried out in a very simple way by
distinguishing between two assumptions:
 \begin{enumerate}
 \renewcommand{\labelenumi}{(\arabic{enumi})}

 \item the given surface $F$ contains an irreducible linear system $|C|$
of genus $\pi$ and degree $n>2\pi-2$, which may be assumed to be the
plane or hyperplane section system of a surface $F$ having no
singularities at all;

 \item for any given irreducible linear system on the surface $F$ the
inequality $n\le2\pi-2$ holds, where $n$ and $\pi$ denote the degree and
the genus of the system.
 \end{enumerate}

\paragraph{Hypothesis I.} It is clear that, starting from a given
system $|C|$ on $F$ and constructing the successive adjoint systems
$|C'|,|C''|$, etc. one obtains a finite number of systems; in other
words the adjunction process terminates. In fact the curves
$C',C'',\dots$ have decreasing order
 \[
 2\pi-2,\quad 2\pi-2-d,\quad 2\pi-2-2d,\dots
 \]
where
 \[
 d=n-(2\pi-2).
 \]

Therefore we want to consider the last adjoint system $|C^{(i)}|$ of
dimension $r_i\ge1$.

First we assume the adjoint systems $|C'|,|C''|,\dots,|C^{(i)}|$ of the
above sequence to be irreducible. The system $|C^{(i)}|$, the first
one of the sequence which has not an adjoint system of dimension
greater than zero, is formed by curves whose genus is either zero or
one. If the system $|C^{(i)}|$ is formed by curves of genus zero, then
the theorem of Noether\footnote{See for instance Enriques--Conforto,
Ibidem} about surfaces containing a linear pencil of rational curves
enables us to deduce immediately that $F$ is rational.

If by contrast the system $|C^{(i)}|$ is of genus 1, we need to
distinguish between the assumptions that its dimension is $r_i\ge2$, or
$r_i=1$. If $r_i\ge2$ then the complete system of elliptic curves
$|C^{(i)}|$, which, by the regularity of the surface, has a complete
characteristic series, has degree $n_i=r_i$, and therefore it contains a
net of degree 2; by means of this net the surface $F$ can be
represented on the double plane with a quartic branch curve: whence it
follows\footnote{Theorem of Clebsch, See Enriques--Conforto, Ibidem}
that our surface is rational.

On the other hand, if $r_i=1$, then the system $|C^{(i-1)}|$ which
appears before $|C^{(i)}|$ in the sequence of the adjoints has to be of
genus two, and may be supposed of dimension $r_{i-1}\ge2$, since
otherwise we may replace it with one of the preceding systems, again of
genus two. Now we can see that the system $|C^{(i-1)}|$ is not simple:
every $C^{(i-1)}$ passing through a point $P$ of the surface $F$ passes
as a consequence through the point $P'$ which is the conjugate of $P$
in the $g_2^1$ on the curve; in fact the conjugate of $P$ on a variable
$C^{(i-1)}$ through $P$ has to be fixed, otherwise it would describe a
curve $C^{(i)}$, or a component of $C^{(i)}$, of genus zero, contrary to
our assumptions.

Now, since $F$ is a regular surface, the adjoint system $C^{(i-1)}$ to
$C^{(i-2)}$ is a regular system, of dimension $p_{i-2}-1$, where
$p_{i-2}$ denotes the genus of $|C^{(i-2)}|$, and its characteristic
series is complete and nonspecial. Therefore this series, which, as we
have seen, is composed with the $g_2^1$, is precisely the double of
this $g_2^1$. Then the system $|C^{(i-1)}|$ is an $\infty^3$ system of
curves of genus 2, which belongs to an involution of pairs of points,
and our surface $F$ is represented by means of this system on a double
quadric $Q$, whose branch curve $D_6$ is of order six. Since the
generators of $Q$ cannot have an odd number of branch points, we deduce
that $Q$ is a cone, whose vertex is a branch point that has to be added
to $D_6$, which cuts out three points on the generators. The double
cone thus defined may be represented on Noether's double plane with a
sextic branch curve having two infinitely near triple points, whence it
follows that the double plane is rational\footnote{Theorem of Noether,
See Enriques--Conforto, Ibidem}.

We have assumed that the sequence of successive adjoint systems
$|C'|,|C''|$, \dots is formed by irreducible systems. If in this series
one finds a system $|C^{(h)}|$ having a fixed curve $\theta$ and an
irreducible variable part, which belongs to a system of dimension
$r_h\ge2$, then one may in general go on in the construction of our
sequence of adjoint systems replacing $|C^{(h)}|$ with the irreducible
system $|C^{(h)}-\theta|$, from which the fixed component $\theta$ has
been subtracted. And if this system has genus 0 or 1, therefore (by the
completeness of the characteristic series) has degree $n_h=r_h-1$ or
$n_h=r_h$, then one may conclude (like before) that the surface $F$ is
rational.

We still have to investigate carefully the case in which the system
$|C^{(h)}-\theta|$, without the fixed part $\theta$, is a pencil or is
composed with the curves $L$ of a pencil that, by the regularity of the
surface, has to be linear.

To start with, if the curves $L$ of the pencil are rational then the
aforementioned theorem of Noether enables us to say that the surface is
rational.

Let us assume by contrast that the curves $L$ have the genus $\rho>0$.
Then we notice that $\theta$, the fixed part of $|C^{(h)}|$, has to be a
fundamental curve for the irreducible system $|C^{(h-1)}|$, since $|C^{(h)}|$ cuts out on a $C^{(h-1)}$ the complete canonical series, which
has no base points.

Then the curves $L$ meet the $C^{(h-1)}$ at $\nu>1$ variable points
(since from the hypothesis $\nu=1$ we would deduce that the $L$ are
rational), and the complete canonical series cut out by
$|C^{(h)}-\theta|$ on a $C^{(h-1)}$ is composed with the groups of the
involution of order $\nu$ cut out by the pencil of the L; whence we
deduce that the $C^{(h-1)}$ are hyperelliptic curves and that $\nu=2$:
the pencil of the $L$, with which the curves of $|C^{(h)}-\theta|$ are
composed, is a linear pencil of bisecants.

Now we recall that the complete system $|C^{(h-1)}|$, adjoint to the
system $|C^{(h-2)}|$ (on the regular surface $F$ of genus $p=0$), is a
regular system (of dimension $\pi_{h-2}-1$) and therefore its
characteristic series is complete and nonspecial. This series is
composed with the $g_2^1$, since otherwise the conjugate of a point $P$
on a $C^{(h-1)}$ through $P$ would describe a rational curve, which
would be the curve $L$ through $P$ or one of its components.
Accordingly the system $|C^{(h-1)}|$ is not simple, but composed with
an involution of order 2 and the degree $n_{h-1}$ of this system of
hyperelliptic curves is equal to the double of the dimension minus one:
 \[
 n_{h-1}=2(r_{h-1}-1);
 \]
 whereas its genus $\pi_{h-1}$ is
 \[
 \pi_{h-1}=r_{h-1}-1.
 \]

We can achieve a more precise determination of these characters with
the following argument. If we subtract from the system $|C^{(h-1)}|$ a
curve $L$ (on which the $C^{(h-1)}$ cut out the pairs of a $g_2^1$) then we
obtain a linear system of dimension $r_{h-1}-2$, whose adjoint system
must be the adjoint system of $|C^{(h-1)}|$, minus the curve $L$, and
therefore the system formed by all the groups of $\pi_{h-1}-2$ curves
$L$; so the subtraction of $L$ from $|C^{(h-1)}|$ lets its genus drop by
one only. On the other hand the $C^{(h-1)}$ which are reducible into the
curve $L$ and a residual curve must be connected, thus $L$ has to meet the
residual curves at $i\ge0$ points; so, since the genus of these curves
is $\pi_{h-1}-1$, one has
 \[
 \pi_{h-1}=(\pi_{h-1}-1)+\rho +i-1
 \]
 whence (since $\rho>0$)
 \[
 \rho=1, i=1.
 \]

So the residual curves of a $L$ with respect to $|C^{(h-1)}|$ cut out on $L$ one point which is necessarily fixed and is also a common point of
the $L$ of our pencil; this yields that these curves are reducible in $L$ of the same pencil.

Now, using the system $|C^{(h-1)}|$, we transform the surface $F$ into a
double surface of order $m=r_{h-1}-1$ in a space of dimension
$m+1=r_{h-1}$. This surface, on which the curves $L$ are represented by
lines, has to be a cone with a certain vertex $O$ and must have a branch
curve $d$ of order
 \[
 2+2\pi_{h-1}=2m+2.
 \]

Since this curve has to intersect the generators of the surface in at
least 3 variable points (in order that the genus of the $L$ be
$\rho>0$), we must have $m=2$; in fact a hyperplane through $O$ cuts
out $m$ generators on our cone, and therefore one has
 \[
3m\le2m+2.
 \]

In this way we find the conclusion that the system $|C^{(h-1)}|$ leads
to the representation of our surface $F$ on a double quadric cone with a
sextic branch curve, which, as we have already recalled, is rational
(Noether).

\paragraph{Hypothesis II} We still have to examine the second
hypothesis: every irreducible linear system on the surface $F$ has the
genus $\pi$ and the degree $n$ which satisfy the inequality
 \[
 n\le2\pi-2.
 \]

Let us make this hypothesis, and we will derive from it a
contradiction. By the assumption, we may suppose that the surface $F$ is
transformed into a surface of a suitable space in such a way that it
contains no exceptional curves, and, if we like it, that it has no
singularities. Its hyperplane sections form a linear system $|C|$, whose
degree we denote by $n$ and whose genus by $\pi$; as we have said, one
has
 \[
 n\le2\pi-2.
 \]

Then the bigenus $P$ of $F$ may be computed by increasing by one the
dimension of the bicanonical system, which can be obtained by
subtracting a $C$ from the second adjoint system $|C''|$; as we have
seen, one finds that on a regular surface of geometric genus
$p=p_a=p_g=0$, and linear genus $p^{(1)}$, the bigenus is
 \[
 P\ge p^{(1)}.
 \]

Accordingly, if $P=0$, then one has
 \[
 p^{(1)}\le0.
 \]

Consider then the sequence of successive adjoint systems to $|C|$,
$|C'|,|C''|$, \dots; the number of intersections ($2\pi-2-n$) which $C$
has with the virtual canonical curves, increases by $p^{(1)}-1$ in
passing to the adjoint system, that is, it actually decreases by one at
least. Therefore at least one of the following facts must be true:
 \begin{enumerate}
 \renewcommand{\labelenumi}{(\arabic{enumi})}
 \item either the difference $n-(2\pi-2)$ which is obtained from the
characters of a system $|C|$ becomes positive for some of the adjoint
systems, and this directly contradicts our hypothesis;

 \item or the above sequence of adjoint systems is finite.
 \end{enumerate}

In this case, as it happened in Hypothesis~I, one either obtains a
final linear system, which is composed with the rational curves of a
linear pencil, or a pencil of elliptic curves which is the adjoint
system of the hyperelliptic curves of genus 2 of a linear $\infty^2$
system of degree at least 4. Whence, with the same argument as we have
developed in Hypothesis I, one deduces the rationality of $F$. But we do
not even need to repeat that argument; it is sufficient to point out
that, in contrast with our Hypothesis~II, we have achieved the
construction of a linear system of genus $\pi$ and degree $n>2\pi-2$ on
$F$, thus finding a contradiction. It is convenient to remark at this
point that, if one has on $F$ a system $|L|$ however reducible of genus
$\pi$ and degree $n>2\pi-2$, which does not contain any exceptional
component, then by adding a multiple of this system to a regular
system, one can always obtain an irreducible linear system whose
characters satisfy the same inequality. In conclusion we may state the
theorem: the necessary and sufficient conditions for a surface to be
rational are expressed by the vanishing of the numerical genus and of
the bigenus:
 \[
 p_a=P=0;
 \]
where the condition $P=0$ a priori yields also $p_g=0$. So in other
words one can say that the rational surfaces are characterized as
``regular surfaces with bigenus zero''.

This theorem is due to G.~Castelnuovo who proved it essentially by the
same method that we have used here: we have only introduced
simplifications of some details.
