 \chapter{Adjoint surfaces}\label{ch:3}
 \markboth{\qquad 3. \quad Adjoint surfaces \hfill}{}
 \index{adjoint surfaces}

We have seen that on a surface $F$ the curves $C'$ adjoint to an
irreducible linear system $|C|$ cut out canonical groups on the curves
of $|C|$. Now we wants to investigate whether, and to what extent, this
property characterizes the curves $C'$. This question is related to
another question of projective type: how the adjoint curves to the
system of plane sections $C$ are cut out on a surface $F$, taken in
ordinary space, by certain surfaces, that we will call adjoint to $F$,
defined by the condition that they pass with multiplicity $i-1$ through
every $i$-tuple curve of $F$ and behave in a suitable way at the
isolated multiple points of $F$.

 \section{Surfaces of order $n-3$ adjoint to a surface of order $n$
without proper multiple points}\label{sec:3.1}
 \markright{\hfill Surfaces adjoint to a surface of order $n$ \quad
3.1\quad}
 Let us consider a surface $F$ of order $n$ in ordinary space and let
us denote by $C$ the plane sections of $F$. $F$ can have singularities
of two kinds:

 \begin{enumerate}
 \renewcommand{\labelenumi}{(\arabic{enumi})}
 \item one or several multiple curves, for instance a double curve that
corresponds on a transform of $F$ to a locus of pairs of points that are
neutral for the transform of the linear system $|C|$, or an $i$-tuple curve
corresponding to a locus of neutral groups of $i$ points and so on;
 \item some particularly notable multiple points, both isolated multiple
points and points of the multiple curves whose singularity is not due
to the fact that the multiple curve passes through them.
 \end{enumerate}

As far as the multiple curve is concerned, we shall as a rule address,
for the sake of simplicity, to the case of ordinary multiple curves and
sometimes even to the case of a nodal curve; but it will be easy to
understand that what we shall say holds also for an arbitrary curve,
however singular, provided one takes into account the infinitely near
multiple curves defining it.

As to the case of the notable isolated singularities, which have to be
taken into account in the considerations of this chapter, we shall make
a precise distinction by resorting to the following criterion: a
multiple point $O$ of $F$ will be said to be {\em improper} if the genus
of a plane section through $O$ equals the genus of the sections with
planes not passing through $O$; on the other hand, a singular point of
$F$ that {\em lowers the genus of plane sections} will be said to be a
proper multiple point.

If we refer to a transform of $F$ on which the multiple curve corresponds
to a locus of points that are neutral for $|C|$, then an improper
multiple point $O$ of $F$ corresponds in general to a neutral group $G$ of
points (imposing only one condition to the curves $C$ that must contain
them); so, for instance, a triple point for the surface $F$ and for its
double curve corresponds in general to a neutral triple of points to a
point that is $(i+1)$-tuple for $F$ and for its $i$-tuple curve corresponds
to a neutral group of $i+1$ points, and so on.

Yet it may happen that, in a transformation of $F$, the point $O$ gives
rise to a fundamental curve $\theta$ for the system $|C|$, which in general
can be composed of as many disconnected exceptional parts as the points
of the group $G$ mentioned above; and it is worth remarking that this
decomposition of $\theta$ can be derived from the property, which is
characteristic, of being an improper fundamental curve, not lowering
the genus of the $C$. In fact this property yields that every connected
part of $\theta$ is of genus zero and has only one common point with a
curve residual to $C$ (or to one of its multiples), and so this part is
characterized as an exceptional curve.

Now let us consider a surface $F$ that is not ruled and such that its
plane sections $C$ are of order $n$ and of genus $\pi>1$ and a curve
$L$ on $F$ that cuts canonical groups on the curves $C$. In a generic
plane the group of points cut out by $L$ belongs to a curve
$\fie_{n-3}$ of order $n-3$, adjoint to the plane section $C$ of $F$,
consisting of all the intersection points of $C$ and the $\fie_{n-3}$
away from the multiple points of $C$ (where an $i$-tuple point of $C$
counts for $i(i-1)$ intersections); the curve $C$ cannot have any other
intersection with the $\fie_{n-3}$ outside $L$, unless $C$ is reducible.

Let us take a generic space line $a$ and, in every plane through $a$, the
curve $\fie_{n-3}$ adjoint to the plane section that is determined by
the group of points where $L$ meets the plane; as the plane through a
varies, $\fie_{n-3}$ describes a surface $\Phi$, passing $i-1$ times
through every $i$-tuple curve of F\footnote{The statement can be verified
analytically by considering the case of a surface described by
translating a plane curve with an $s$-tuple point ($s=i-1$) in a normal
direction, in which case it is clear that the $s$-tuple point describes
an $s$-tuple line.}, and having order $n-3+h$, where $h$ is the
multiplicity with which it contains the line $a$.

Now if $h>0$, then the total intersection of $F$ and $\Phi$ must contain
all the common points of $a$ and $F$, which are not on $L$ (nor on the
multiple curve). If $A$ is one of these points, there is at least one
plane, tangent to $\Phi$ at $A$, whose section with $F$ meets the
adjoint curve $\fie_{n-3}$ in $A$ away from the canonical group on $L$
and therefore turns out to be reducible. But then by a well known
theorem of Kronecker--Castelnuovo\footnote{See Enriques--Conforto,
Ibidem, Libro~II, Chapter~II, \S7, page~273.} the surface $F$, thus
having $\infty^2$ reducible plane sections, should be a Steiner surface
with plane sections of genus zero, which contradicts our assumptions.
One concludes that the curves $L$ that cut out canonical groups on the
plane sections $C$ are the intersections of $F$, away from the multiple
curve, with adjoint surfaces $\Phi_{n-3}$ of order $n-3$. Therefore the
curves $C'$ adjoint to $|C|$ must be contained (totally or partially)
in the system $|L|$. Now the $C'$ and the $L$ are curves of the same
order $2\pi-2$, namely they have the same intersection number with the
$C$, and therefore they can only differ by fundamental curves of $|C|$;
this means that the linear system $|C'|$ (if less ample than $|L|$) is
obtained by imposing to $|L|$ some base point occurring at some multiple
point $O$ of $F$; by contrast no base point of the $C'$ occurs at an
improper multiple point $O$, because the adjoints to the plane sections
$C$ through $O$, that are again of genus $\pi$, also meet these sections
at $2\pi-2$ points (at least virtually) away from $O$.

We can be make clearer what we have just said by considering a
transformed surface of $F$, on which the point $O$ corresponds to a
neutral group $G$ of simple points: here the system $|C_1'|$ adjoint to
$|C_1|$ (that is obtained from $|C|$ by imposing simple points) does not
differ from $|C'|$. Therefore we may state the theorem:

{\em On a surface $F$, of order $n$, of ordinary space, not ruled and
without proper multiple points, the adjoint curves of the plane
sections $C$ are characterized by the property of cutting out canonical
groups on the above-mentioned curves $C$; and the complete linear
system of these adjoint curves is cut out on $F$ by the surfaces
$\Phi_{n-3}$ of order $n-3$ (adjoint to it) passing $i-1$ times through
every $i$-tuple curve of $F$.}

 \section{Adjoint surfaces of arbitrary order}\label{sec:3.2}
 \markright{\hfill Adjoint surfaces of arbitrary order \quad 3.2\quad}
 This result can be extended to the curves $L$ that cut out on the plane
sections $C$ of $F$ groups belonging to the series which is the sum or the
difference of the canonical series with a multiple, by any integer $r$,
of the characteristic series: under the above assumptions (F not ruled
and without proper multiple points) the $L$ are cut out on $F$ (away
from the multiple curve) by surfaces $\Phi_{n-3+r}$ of order $n-3+r$,
adjoint to it. To begin with, let $r>0$. Then it suffices to repeat the
above argument, with only one warning: when $r\ge3$ there are
infinitely many curves $\fie_{n-3+r}$ passing through the group $\fie$
cut out by $L$ on a plane through $a$, they are adjoint to the section
$C$ with the plane and form a linear system of dimension
$(r-1)(r-2)/2$; and it is convenient to determine one of these by
assigning, for instance, a suitable number of its points on a and then
a certain number of tangents and osculating curves. If instead $r<0$,
then add to the curve $L$ the curve H cut out on $F$ by a generic
surface $\Psi$ of order $|r|$. Then the curve $L+H$ cuts out on the
plane sections of $F$ groups belonging to the canonical series and
therefore there exists a $\Phi_{n-3}$, adjoint to $F$, that goes
through it. But such a $\Phi_{n-3}$, having the curve $H$ of order
$|r|n>|r|(n-3)$ in common with $\Psi$, breaks up as the union of
$\Psi$ itself and of a $\Phi_{n-3+r}$ still adjoint to $F$ and passing
through $L$.

In particular from the previous considerations it follows that:

{\em On a surface $F$ of order $n$ of ordinary space, not ruled and not
having any proper multiple point, the surfaces $\Phi_{n-3}$ passing
$i-1$ times through every $i$-tuple curve of $F$ (namely adjoint to it)
cut out the complete linear system $|K|$ of (impure) canonical curves,
{\em as already announced in Chapter~II, Section~\ref{sec:2.4};} and in
general the $\Phi_{n-4+r}$ with $r>0$ cut out on $F$ the complete
linear system adjoint to the $r$-tuple system of plane sections, which
usually is only partially cut out by the system of all surfaces of
order $r$.

Moreover, this system $|(rC)'|=|C'+(r-1)C|=|K+rC|$ is characterized by
the property that its curves cut out on the plane sections $C$ of $F$
groups belonging to the series sum of the canonical series and of the
$(r-1)$-tuple of the characteristic series.}

 \section{Ruled surfaces}\label{sec:3.3}
 \markright{\hfill Ruled surfaces \quad 3.3\quad}
 The previous theorem has been established under two restrictive
assumptions:

 \begin{enumerate}
 \renewcommand{\labelenumi}{(\arabic{enumi})}
 \item that $F$ is neither ruled nor a Steiner surface, and that
 \item $F$ has no proper multiple points.
 \end{enumerate}

We see at once that the restriction on $F$ to be nonruled is really
essential. If $F$ is a ruled surface of order $n$ and genus $\pi>0$ (that,
for the sake of simplicity we will assume to have only a double curve)
there are neither curves $C'$ adjoint to the plane sections $C$ nor adjoint
surfaces $\Phi_{n-3}$, but there are groups of $2\pi-2$ generators cutting
out canonical groups on the $C$.

In fact, introducing the virtual canonical system $|K|$, we may write:
 \[
 |C'|=|C+K|
 \]
The $C$ meet the generators at only one point, the $K$ meet them at $-2$
points\footnote{Since the curves of the system $|2C+K|$ split in
$2\pi-2+n$ generators, then they cut the generators in zero points:
therefore the curves $K$ cut the generators in $-2$ points.}, therefore
the $C'$ should cut the generators at
 \[
 1-2=-1
 \]
points, so they should contain them, which is a contradiction.

On the other hand, one can also see that there cannot be surfaces $\Phi_{n-3}$ of order $n-3$ passing through the double curve of $F$, since
this curve meets the generators at $n-2$ points.

Let us add that, on the ruled surface $F$, there exist also groups of
$2\pi-2+(r-1)n$ generators cutting out on the plane sections $C$ groups
belonging to the series sum of the canonical series and of the
$r$-tuple of the characteristic series, but except for the case $r=2$,
these groups are not adjoint curves to $|rC|$ and are not intersections
of adjoint surfaces $\Phi_{n-4+r}$, since the curves of $|(rC)'|$ have
to meet the generators of $F$ at $r-2$ (and not at zero) points.

On the contrary, for $r\ge2$, the surfaces $\Phi_{n-4+r}$ passing through
the double curve cut out on the ruled surface curves that are adjoint
to the $r$-tuple system of plane sections. This can be proved by
remarking that those curves must cut out canonical groups on the curves
of $|rC|$, at least on those that are cut out on $F$ by general surfaces
$\Phi_r$ of order $r$. Indeed on a $\Phi_r$ the curves intersection of
the $\Phi_{n-4+r}=\Phi_{r+n-4}$ are adjoint to the intersections with
$F$.

We will prove that the system cut out on the ruled surface $F$ by the
$\Phi_{n-4+r}$ is always complete later, when we will learn how to
estimate its dimension.

 \section[Characteristic properties of canonical curves and of the
curves adjoint to a linear system]{Characteristic properties of
canonical\\ curves and of adjoint curves to a linear
system}\label{sec:3.4}
 \markright{\hfill Characteristic properties of adjoint and canonical
curves \quad 3.4\quad}
 In order to state the results obtained in an invariant form, we have
to consider a surface $F$ and an irreducible linear system $|C|$ on it,
and characterize, with respect to $C$, the canonical curves and the
curves adjoint to the $C$. But before stating the result, it is
convenient to get rid of a superfluous restriction. Actually we have
assumed the system $|C|$ to be not only irreducible, but also simple of
dimension $r\ge3$, so that $\infty^3$ $C$ may be regarded as the plane
sections of a transformed surface.

We can do without this restriction.

If $|C|$ is (or contains) a net of irreducible curves of genus $\pi$ and
order $n$, we may assume that this net is cut out on a surface $F$, of
order $m>n$, by the planes of a star with centre in an $s$-tuple point
$O$ ($m=n+s$).

Then, starting from a curve $L$ that cuts canonical groups on the $C$, and
with respect to a pencil of planes whose base line goes through $O$, one
constructs, as before, an adjoint surface $\Phi_{m-3}$ cutting out $L$ on
$F$, passing $i-1$ times through every $i$-tuple curve of $F$, and having
multiplicity $s-1$ at the $s$-tuple point $O$; then one has $L=C'$.

However, the feasibility of this construction is based on two
assumptions:
 \begin{enumerate}
 \renewcommand{\labelenumi}{(\arabic{enumi})}
 \item there do not exist $\infty^1$ reducible plane sections through
$O$;
 \item $F$ has neither proper multiple points (not even infinitely near
$O$), nor multiple lines through $O$, that might turn out to have
smaller multiplicity for the $\Phi_{m-3}$ so constructed.
 \end{enumerate}

Considering the matter from the point of view of the invariant theory,
the two restrictions can be combined in only one requiring that: the
net of the $C$ does not contain $\infty^1$ reducible curves. Actually the
existence of a fundamental curve for the $\infty^r$ system $|C|$, means
that it contains $\infty^{r-1}$ reducible curves.

Now we may state the following theorem:

If on a surface $F$ one has an irreducible linear system $|C|$, of
dimension $r>1$, not containing $\infty^{r-1}$ reducible curves, then the
curves $C'$ adjoint to it are characterized by the property of cutting
out canonical groups on the $C$. And the canonical curves of $F$ are
defined by the property of cutting out on the $C$ groups residual to the
canonical series.

Also the curves adjoint to a multiple of the system $|C|$ can be
characterized by the groups that they cut out on the $C$; the matter is
clear and there is no need to insist on this point.

 \section{Isolated multiple points}\label{sec:3.5}
 \markright{\hfill Isolated multiple points \quad 3.5\quad}
 Let us go back to the projective consideration of the plane sections of
a surface in the ordinary space.

We want to examine to what extent the results obtained can be extended
to the case when $F$ has multiple points that lower the genus of the
sections through them. For the sake of simplicity it suffices to
consider the curves $C'$ adjoint to the system of plane sections $C$. So
the question is to what extent these curves are characterized by the
simple property of cutting out canonical groups on the plane sections
$C$ of $F$. On the surface $F$ of order $n$ considered in projective
space, a curve $L$ cutting out canonical groups on the $C$ is again, as
we have seen before, intersection with a $\Phi_{n-3}$, of order $n-3$,
passing $i-1$ times through every $i$-tuple curve of $F$. But the
system $|L|$, although containing $|C'|$ partially, may not coincide
with it; in any event one has $|L|=|C'+\theta|$, where $\theta$ denotes
a curve, or a set of proper fundamental curves of $|C|$. And, from the
projective point of view, the surfaces $\Phi_{n-3}$ cutting out the
$C'$ are different from the $\Phi_{n-3}$ that cut out the $L$ (subject
only to the condition of passing $i-1$ times through every $i$-tuple
curve of $F$) by the further condition of passing also through some
multiple points of $F$.

To clarify the matter, let us assume that $F$ has an isolated $s$-tuple
point, namely away from the multiple curve. Let us consider the planes
of the space passing through a point $P$; the sections of these planes
form a net of $|C|$, and, from the projective point of view, the
Jacobian of the net is cut out by the polar surface of $P$, of order
$n-1$, which in general passes $i-1$ times through every $i$-tuple
curve of $F$, and similarly with multiplicity $s-1$ through the
$s$-tuple point $O$. However beware that the neighbourhood of the
$s$-tuple point $O$ corresponds (on a transformed surface) to a
fundamental curve $\theta$ of $|C|$, which is a component (usually
counted with multiplicity one) of the Jacobian $C_j$\footnote{By
assumption $q$ has no variable intersections with the $C$. Take a
generic point $A$ on $q$; the $C$ which have to pass through $A$ split
as the union of $q$ and of the residual curves $C_1$ of a pencil; among
these there is a $C_1$ passing through $A$, that together with $q$ is a
curve of the net having a double point at $A$.}. As a consequence, in
order to obtain a whole Jacobian curve, one has to add the
neighbourhood of the point $O$ to the above polar. So the system
$|C_j|$ is actually cut out by the surfaces of order $n-1$ passing
$i-1$ times through every $i$-tuple curve of the surface $F$ and with
multiplicity $s-2$ through every $s$-tuple point $O$. It follows that the
curves $C'$ adjoint to the plane sections are cut out by the surfaces
$\Phi_{n-3}$ subject to the condition of passing $i-1$ times through
every $i$-tuple curve of $F$, and of passing $s-2$ times through the
isolated $s$-tuple point.

The same argument can be repeated for a proper $s$-tuple point belonging
to the multiple curve of $F$; the $s$-tuple point is (in general and at
least) $(s-2)$-tuple for the surfaces that cut out on $F$ the adjoints $C'$
to the plane sections.

 \section{Adjoints and subadjoints}\label{sec:3.6}
 \markright{\hfill Adjoints and subadjoints \quad 3.6\quad}
 The previous analysis, concerning a surface $F$ with improper multiple
points, leads us to distinguish between:

 \begin{enumerate}
 \renewcommand{\labelenumi}{(\arabic{enumi})}
 \item the surfaces passing $i-1$ times through every $i$-tuple curve of $F$,
and
 \item the surfaces that, besides the condition of passing $i-1$ times
through every $i$-tuple curve of $F$, also enjoy the particular property
of passing through the proper multiple points of $F$ in a way that (for
arbitrarily large order) they give rise, by intersecting, to adjoint
curves of a multiple of the plane sections.
 \end{enumerate}

Keeping the name of adjoint surfaces for the latter, we shall call the
former subadjoint surfaces, and therefore we shall call subadjoint
curves of the plane sections $C$ the curves defined by the property of
cutting out canonical groups on the $C$. Now one has to investigate, for
every particular kind of proper multiple points, the extra conditions
that they impose to the subadjoint surfaces in order that they become
adjoint. In relation with this subject, it is convenient to remark
that, for the adjoints of a certain order, the conditions concerning a
given multiple point may turn out to depend partly on the other
conditions imposed by the multiple curves or by other multiple points;
but we may assume (as this will be made clearer later) that this is not
the case for surfaces with large enough order. However, by computing
the number of these conditions without taking into account the possible
relations of dependence that we have mentioned above, we shall have a
number that will always be the same\footnote{When one imposes to the
surfaces $f$ of large enough degree to have a certain singular point $O$,
one is led to impose to the $f$ a certain multiplicity at the point $O$
and at a certain number of points infinitely near to $O$ and belonging
to its higher order neighbourhoods. As a consequence the $f$ may acquire
the prescribed multiplicities along infinitesimal curves of these
neighbourhoods. So the number of conditions imposed on the $f$ will be
independent of the order of these.} for the adjoint surfaces of any
order. If this number really corresponds to the number of adjoints
$\Phi_{n-4}$ of order $n-4$, then it expresses the genus drop of $F$
due to that multiple point. However, by saying that a multiple point
lets the genus of the surface drop of a certain amount, we shall mean
the virtual drop of this genus, a concept which we shall explain later
on, and meaning also that the virtual expression of the genus yields
another invariant character of the surface (its numerical genus) that
may be smaller than the (geometric) genus introduced before.

Using the language that we have introduced we may state:

A proper $s$-tuple point of the surface $F$ (either belonging or not to the
multiple curve) is (in general and at least) an $(s-2)$-tuple point for
the adjoint surfaces.

In the simplest case, when $O$ is isolated with osculating cone of genus
$(s-1)(s-2)/2$, the multiplicity of $O$ for the adjoint surfaces is
precisely $s-2$, while there is no vanishing condition for the
subadjoints; so one can say that a priori $O$ imposes $s(s-1)(s-2)/6$
linear conditions on the adjoint surfaces of order $n-4$, and so it makes
the genus drop of $s(s-1)(s-2)/6$.

In particular a conical double point does not affect the genus of the
surface.

\begin{EXAMPLES} The nonsingular surface $F_4$ of order 4 has
(superficial) genus $p=1$, since it has one canonical curve of order
zero, section with the adjoint surface of order $n-4=0$. If $F$
acquires a triple point, then its genus drops by one and so it becomes
$p=0$; according to the fact that the $F_4$ with a triple point is
rational.

The $F_4$ with a conical double point is still of genus 1, and therefore
certainly irrational. On the other hand there are rational surfaces of
order 4 with (only) one double point\footnote{Surfaces determined by
M.~Noether. See Enriques--Conforto, Ibidem, Libro~I, Chapter~VIII.}: the
first type of such surfaces is the $F_4$ with a tacnode. {From} this
example it is already apparent that ``double points of particular type
(e.g., one whose osculating cone consists of a double plane) may cause
the genus of the surface to drop'': hence the need for a careful
analysis.

The nonsingular surface $F_5$ of order 5 ($n=5$) has genus $p=4$, since
it has $\infty^3$ canonical curves that are the sections of the planes
of the space (adjoint to $F_5$). If $F_5$ acquires a quadruple point
($s=4$) its genus drops by $s(s-1)(s-2)/6=4$, according to the fact that
it becomes rational.

The $F_5$ with a double line $a$ has genus $p=2$, since its canonical
curves are the $\infty^1$ cubic sections with the (adjoint) planes
passing through $a$. If it acquires a triple point outside $a$, then
its genus drops to 1. In this case there is one adjoint plane, namely
the plane $Oa$, that cuts on $F$ the impure canonical curve consisting
of three lines through $O$. These lines are exceptional for $F_5$; in
fact one sees that the quadrics through $O$ and a cut out on $F_5$ a
linear system $|C|$ of genus 5 and order 8, which determines a
transformation of $F_5$ into a $F_8$ of $S_5$ containing a plane cubic
$C_3$, the image of the point $O$, whereas the three lines of the
impure canonical curve are transformed into three collinear points of
$C_3$ that are simple for the surface $F_8$: and projecting $F_8$ from
the line containing these three points of $C_3$ one gets the $F_5$
back. Since the plane sections of $F_8$ are canonical curves of genus 5
in $S_4$, their system is selfadjoint, hence one verifies that the
genus of the surface is equal to 1.
 \end{EXAMPLES}

 \section{Multiple points of the first kind}\label{sec:3.7}
 \markright{\hfill Multiple points of the first kind \quad 3.7\quad}
 If one wants to better understand the conditions that a singular point
of $F$ imposes to the adjoint surfaces, it is convenient, to start with,
to distinguish the multiple points of the first kind from those of a
higher kind. A multiple point $O$ of $F$, that we assume to lower the
genus of the plane sections $C$, will be called a ``multiple point of the
first kind'' when near to it there is no other multiple point lowering
the genus of the $C$ through $O$. On a suitable transform of $F$ the point $O$ becomes a fundamental curve $\theta$ for the system $|C|$, and the property
that ``there is no part of it that is a fundamental curve of the
residual system $|C-\theta|$'' characterizes $\theta$ as a fundamental
curve of the first kind.

The conditions imposed on the adjoint surfaces $\Phi$ by a multiple
point $O$ of the first kind can be simply expressed by saying that: the
$\Phi$ must be cut by the planes through $O$ along curves that become
adjoint to the sections of $F$ if one adds to them a line through $O$.

The simplest way to prove this is to refer to the $\Phi_{n-4}$ of order
$n-4$ and to assume that the net of sections $C$ of $F$ with the planes
through $O$ does not contain $\infty^1$ reducible curves. Then it is
evident that the $\Phi_{n-4}$ mentioned above cut on the $C$ groups
residual to the characteristic series, and this property -- under our
assumptions -- suffices to define canonical curves. The argument can be
repeated in an analogous way for the $\Phi_{n-4+r}$ with $r>0$, even if
there are no $\Phi_{n-4}$.

 \begin{EXAMPLES} Let $F$ be a surface with a tacnode $O$, that is, a
point $O$ such that the plane sections through it have two infinitely
near double points at $O$\footnote{See Enriques--Chisini, Ibidem, Libro~IV}. Applying the previous criterion, one sees that the plane sections
of the $\Phi_{n-4}$, plus a line through $O$, must be adjoint curves of
the sections of $F$. This implies that the adjoints have to pass simply
through the tacnode $O$.

If instead one has a oscnode, that is, a point $O$ such that the plane
sections through it have three infinitely near double points at $O$, then
one sees in a similar way that the adjoints have to pass simply through
the oscnode and touch the tangent plane to $F$ there.
 \end{EXAMPLES}

 \begin{EXERCISE} Let $F_5$ be a surface of order 5 with a double line
$a$ and a tacnode $O$. The adjoints of order $n-4=1$ to this surface
are reduced to the plane $Oa$, hence one has $p=1$. The intersection of
$F_5$ with the plane $Oa$ (away from $a$) consists of two exceptional curves:
a conic and a line $b$, touching at $O$.

One can verify these assertions by transforming the $F_5$ into a
nonsingular $F_4$ of order 4, containing a line $a'$: this surface (on
which the system of plane sections is selfadjoint) has genus $p=1$ and
has a canonical curve of order zero.

The above-mentioned transformation can be performed with a special
quadratic transformation of the first kind of the space, defined by the
homaloidal net of the quadrics passing through two concurrent lines and
touching a given plane at a point of one of the lines: the lines in the
base locus are a and the line $b$ intersection of the tacnodal plane at $O$ with the plane $Oa$, and the tacnodal plane at $O$ is the fixed tangent
plane that we have mentioned above. With this transformation the
surface $F_5$ becomes a $F_4$ of order 4 and the quartic residual
intersection of the tacnodal plane of $F_5$, away from the line $b$, becomes a
line $a'$ of $F_4$. The exceptional conic of $F_5$ is carried to a point
$O'$ of $F_4$ and the exceptional line $b$ to a point $B$; the
transforms of the plane sections of $F_5$ are curves $C_7$ of order 7,
residual intersections of $F_4$ with the quadrics through $a'$, that
have a double base point at $O'$ and a simple base point at $B$.

Conversely, if one takes an arbitrary surface $F_4$ containing a line $a'$,
and a generic point $O'$ on $F_4$ (belonging to a cubic $K$ section with a
plane through $a$), one can transform $F_4$ in an $F_5$ with a double line
and a tacnode. In order to do this, one can use the homaloidal system
of the quadrics passing through $a'$ and through the line $b'$ tangent
to the cubic $K$ at $O'$, and touching $F_4$ at $O'$. One has to remark
here that by imposing a double point $O'$ to the curves $C_4$, partial
intersections of the quadrics through the line $a'$, gives as a
consequence also the existence of a simple base point at the tangential
point of $O'$ on $K$. Thus the base point free complete system of the
$C_4$, which is of genus 6, dimension 6 and order 10, gives rise to a
superabundant system with a double base point and a simple base point,
of genus 5, order 5 and dimension 3.
 \end{EXERCISE}

 \section{Double planes}\label{sec:3.8}
 \markright{\hfill Double planes \quad 3.8\quad}
 We make now a short digression in order to consider a remarkable family
of surfaces, which can be explained by using the previous
considerations. Let us consider the surface $F$ defined by the equation
$z^2=f(x,y)$, where $f$ is a plane curve of even order $2n$: by
projecting from the point at infinity $O$ on the $z$-axis,
$F$ is represented on the double plane with branch curve $f$. An $f$ of
even order arbitrarily taken, considered up to birational
transformations of the plane, defines a class of surfaces including
$F$; if $f$ is of odd order, one has to add to it the line at infinity
of the plane as a branch line\footnote{See Enriques--Chisini, {\it
Lezioni},  Libro~V, \S12, Vol.~III, page~90.}.

The surface $F$ is of order $2n$ and, if $f$ is nonsingular, it has only one
proper multiple point at the point $O$. The analysis of this singularity
can be reduced to the analysis of the singularity that the point at
infinity on the $z$-axis has for the hyperelliptic curve section of $F$ with a plane parallel to that axis, for instance the hyperelliptic
curve $z^2=f(x)$, with $f$ of order $2n$. It is well known\footnote{See
Enriques --chisini, Ibidem, Libro~V, \S12 note, Vol.~III, page~93.} that
such a curve has a $(2n-2)$-tuple point at $O$, and infinitely near to it
and facing it there are $n-1$ double points along a branch of order $n-1$
tangent to the line at infinity of the plane.

So the surface $F$ has a $(2n-2)$-tuple singular point $O$, defined by the
property that every plane section through it must have $n-1$ double
points near $O$.

Let us now consider the adjoint surfaces of $F$ of order $2n-4$. These
are cones (and, more properly, cylinders) of vertex $O$, since, when one
adds a plane to them, their sections are curves adjoint to the sections
of $F$; furthermore these cones are subject to the condition of
containing the double points of $F$ near $O$, and so they have as a
component the plane at infinity counted $n-1$ times. We conclude that
the surfaces of order $2n-4$ adjoint to $z^2=f(x,y)$ consist of the plane
at infinity counted $n-1$ times and of the cones of order $n-3$ with
vertex $O$. The plane at infinity intersects $F$ along $2n$ exceptional
lines. The canonical curves of the double plane that has a general
curve of order $2n$ as a branch curve are represented on the plane by the
curves $C_{n-3}$ of order $n-3$. The genus of the double plane is:
 \[
 p=\frac{n(n-3)}{2}+1=\frac{(n-1)(n-2)}{2}
 \]
Now remark that a quadruple point on the branch curve $f$ is a tacnode
for the surface $z^2=f(x,y)$, with the plane $z=0$ as the tacnodal
plane, whose lines through the origin have four intersections with the
surface. We deduce that: a quadruple point of the branch curve lowers
the genus of the double plane by one.

In a similar way a sextuple point of the branch curve gives rise to an
oscnode of $F$ and therefore it is double for the canonical $C_{n-3}$; in
general a 2$i$-tuple point of the branch curve is $(i-1)$-tuple for the
canonical $C_{n-3}$ and so it lowers the genus of the double plane by
$i(i-1)/2$.

A double point $A$ of $f$, of odd multiplicity $2i+1$, must be
considered, in general, as of multiplicity $2i$, since the birational
transformations of the plane that have $O$ as a fundamental point
transform the neighbourhood of this point into a line that is a
component of the transform of the branch curve. But, for the same
reason, a multiple point of $f$ of odd multiplicity $2s+1$ occurring
infinitely near $A$, must be regarded as a $(2s+2)$-tuple point; and so,
in particular, if one has two infinitely near multiple points $A$ and
$A_1$, with the same odd multiplicity $2i+1$, the canonical $C_{n-3}$
must pass $i$ times (instead of $i-1$) through $A$ and $i-1$ times
through $A_1$. One gets the first example for $i=1$: the $[3,3]$
singularity thus defined lowers the genus of the double plane by 1, as
it happens for quadruple point. In Section~\ref{sec:3.10} we shall find
again this result by examining the singular double point which
corresponds to it for the surface $F$.

Historical note.-The notion of double planes goes back to A.~Clebsch
Ueber den Zusammenhang einer Klasse von Fl\"achenabbildungen mit der
Zweitheilung der Abelschen Funktionen\footnote{{\it Math.\ Ann.},
Band~III, 1861. See also: Crelle's J., 63.}, who represented on the
simple plane the double planes with branch curve of order 2 and 4 (the
latter arising -- according to Geiser -- from a projection of the cubic
surface) and those with a sextic with a quadruple point as a branch
curve, arising by projection from a surface of the 4th order with a
double curve. After that (1878) M.~Noether\footnote{Ueber die
einzweideutigen Ebenentransformationen, {\it Sitzungsberichte der
Phys-med. Societ\"at zu Erlangen}, 10.} studied all the rational double
points in a systematic way, reducing them to three birationally
distinct types, that -- via a $(1,2)$-transformation of R.~De Paolis --
give rise to the types of involutions of E.~Bertini. The third type of
Noether is the double plane branched along a sextic with two infinitely
near double points.

Later on (1896) Enriques, starting the study of various classes of
irrational double planes\footnote{In particular in the memoire Sui piani
doppi di genere 1. {\it Mem.\ della Societ\`a It.\ delle Scienze (detta
dei XL)}, 1896.}, showed explicitly how to obtain the images of the
canonical and pluricanonical curves on the double plane (see
Chapter~III, Section~\ref{sec:3.11} and Chapter~V,
Section~\ref{sec:5.7}).

 \section{The genus of the fundamental curves and their influence on
the adjoints}\label{sec:3.9}
 \markright{\hfill The genus of the fundamental curves and their
influence \quad 3.9\quad}
 We resume the consideration of multiple points of the first kind
corresponding to irreducible fundamental curves of the system $|C|$ of
plane sections. Such a fundamental curve $\theta$ has (virtual) genus
$\rho$ and degree $-s$; as we shall see, $s$ coincides with the
multiplicity of the point $O$ for $F$ and $\rho$ is the genus of the
(simple or multiple) osculating cone at $O$ (where one takes into
account the branches of the double curve that go through $O$ and
regards the other possible double generators of the cone as virtually
nonexistent).

We know that, because of the existence of the multiple point $O$, the
system $|C'|$ adjoint to the system of plane sections $C$ is obtained
from the subadjoint system $|L|$ by imposing that its elements contain
the curve $\theta$ counted a certain number of times, and we intend to
determine the number $x$ such that $|L|=|C'+x\theta|$.

Expressing the condition for $\theta$ to be a fundamental curve, not
intersecting the curves of $|C|=|C_1+\theta|$, we have:
 \[
 (\theta C_1)+(\theta\theta)=0
 \]
and therefore
 \[
 (\theta C_1)=s.
 \]
The number $s$, essentially positive, is the number of intersections of
$\theta$ with the curves $C_1$ that are residual to it with respect to
$|C|$, that is, in general, the multiplicity of the point $O$ for $F$,
as we have said before. Now, in order to determine the number $x$
defined above we may start with $|C'|$ and try to successively extend
it by adding $\theta$, as many times as possible.

It is clear that these successive extensions will attain a limit, when
the number of intersections of $\theta$ with the curves that one adds to
it becomes zero or negative. So we shall have:
 \[
 (\theta C')+x (\theta \theta)\ge0
 \]
We write
 \[
 |C|=|C_1+\theta|,|C'|=|C_1+\theta '|
 (\theta C')=(\theta C_1)+(\theta\theta')=s+2\rho -2;
 \]
the previous inequality becomes
 \[
 2\rho -2-(x-1)s\ge0,
 \]
thus determining an upper bound for the value of $x$:
 \[
 x\le1+(2\rho -2)/s.
 \]
The first information that we obtain from the previous inequality is
that if $\rho=0$, then also $x=0$. So the irreducible fundamental curves
of the first kind with genus zero of a linear system $|C|$ do not
affect the definition of the adjoint curves. Or, in a projective form:
a multiple point of the first kind with (simple or multiple)
irreducible osculating cone of (virtual) genus zero does not affect the
genus of the surface.

This class of multiple points includes, as a first example, the conical
double point, which we mentioned before, and then a triple point that
is simple for the double curve of $F$, and so on. If in the previous
inequality we set $\rho=1$, then one has $x\le1$. Here one can always
take $x=1$. In fact let us consider a surface $F$ having a double curve
on which there is an $s$-tuple point through which there are
$(s-1)(s-2)/2-1$ branches of this curve. Now the osculating cone to $F$
at $O$ is a cone of order $s$ that has an adjoint cone of order $s-3$;
this means that the subadjoint surfaces (of very high order), which
have to go through the double curve, have multiplicity $s-3$ at $O$,
whereas the adjoints must have multiplicity $s-2$ at $O$.

The case of multiple points with $\rho=1$ includes also the tacnode,
where the irreducible curve $\theta$ reduces to the neighbourhood of $O$ on
the tacnodal plane counted twice. In this case in fact $\theta$ is
represented on the double pencil of tangents with four branch lines
that correspond to the cuspidal plane sections of the tacnode. Thus one
finds again the result proved in the previous section to the effect
that the tacnode lowers the genus of the surface by 1.

In a similar way one sees that the oscnode corresponds to a fundamental
curve $\theta$ of genus $\rho=2$, and here the value of the number $x$
is $x=2$, according to what we have already seen: the oscnode lowers the
genus of the surface by 3.

Now, referring to the case of an irreducible fundamental curve of the
first kind of arbitrary genus $\rho$, one may wonder whether, as we have
seen in the case of an isolated multiple point, the system $|C'|$ can
always be successively extended by adding $\theta$, as far as $x$ satisfies the previous inequality. The answer is negative: the upper
bound of $x$ is not always achieved. To see this it suffices to give an
example. Let us assume that $F$ has a double curve passing with six
branches through a sextuple point $O$, whose osculating cone at $O$ has
therefore genus $\rho=4$. In general the tangents to these branches do
not belong to a quadric cone and so a subadjoint surface of $F$ will
have multiplicity 3 at $O$, since it has there an osculating cone of the
3rd order adjoint to the osculating cone of $F$. The point $O$ will be
of multiplicity 4 for the adjoint surfaces, so that the system adjoint
to the plane sections is obtained from the larger subadjoint system by
imposing that it contains only one curve $\theta$ corresponding to the
neighbourhood of $O$, and the value 1 is strictly smaller than the upper
bound for $x$ indicated by the previous inequality:
 \[
 6-(x-1)6\ge0
 \]
which gives $x=2$ as the maximum for $x$.

 \section{Point singularities of a higher kind}\label{sec:3.10}
 \markright{\hfill Point singularities of a higher kind \quad 3.10\quad}
 The analysis of point singularities of a higher kind and of the effect
that they have on the genus of the surface can be reduced to the study
of the multiple points that may occur infinitely near to a proper
singular point and that are not defined by the fact that proper or
infinitesimal singular curves go through them.

If one imagines that the point singularity that a surface $F$ has at a
point $O$ is completely solved, then the system of plane sections of $F$,
or equivalently one of its multiples (that we may assume of arbitrarily
large dimension) gives us on the transform of $F$ a linear system $|C|$
having a fundamental curve corresponding to the neighbourhood of the
point $O$. If the singularity of $O$ is of the first kind, then this
fundamental curve is also of the first kind, hence subtracting it from
$|C|$ none of its components is again fundamental for the residual
system. By contrast we define fundamental curve of the second kind
(corresponding to a multiple point of the second kind) a fundamental
curve containing some parts that are fundamental curves of the first
kind for the residual system. And so, in general, one may define by
induction the fundamental curves of the $s$th kind, by reducing this
concept to that of curves of the $(s-1)$st kind.

{From} this definition it follows that the problem of determining the
effect of a multiple point of the $s$th kind on the genus of the surface $F$ can be reduced to the analogous problem for singularities of
the $(s-1)$st kind. Thus the determination of the conditions that a
fundamental curve of the system $|C|$ imposes on the the subadjoint
curves in order that they become adjoint to $|C|$ can be reduced to the
analogous problem with respect to the system that one obtains from
$|C|$ by subtracting that curve.

For the sake of simplicity we refer to the case of a fundamental curve
of the second kind of the system $|C|$.

On the surface $F$ whose plane sections are the $C$ we have a multiple
point $O$ and infinitely near to, in the first order neighbourhood, it
there are some (proper) multiple points $O_1,O_2,\dots$. On a
transformed surface, the point $O$ corresponds to a curve $\theta$ fundamental for $|C|$; $O_1$ to a fundamental
curve $\theta_1$, and so on. Now it is interesting to remark that the
curves $\theta_1,\theta_2,\dots$ must occur as parts of the fundamental
curve $\theta$, so that one has in general
 \[
 \theta=\theta_0+\theta_1+\theta_2+\cdots
 \]
In fact the transformation applied to the surface $F$ can be performed
step by step. First one considers a transforming system of surfaces
having $O$ as a simple base point, so that the neighbourhood of $O$ is
transformed into a (simple or multiple) curve $\theta_0$ containing the
singular points $O_1,O_2,\dots$. Then one transforms successively each
of the points $O_1,O_2,\dots$ into a curve that, in general, will be
connected with $\theta_0$. Because of this connectedness, when one
subtracts the curve $\theta_0$ from $|C|$, as a consequence one subtracts
also the curves $\theta_1,\theta_2,\dots$ which are also fundamental
curves (not intersecting the $C$). In other words the sum
 \[
 \theta=\theta_0+\theta_1+\cdots
 \]
is a univalent fundamental curve for $|C|$, that is, to subtract it which
from $|C|$ imposes only one condition. On the contrary the residual
system $(C-\theta)$ still has $\theta_1,\theta_2,\dots$ as fundamental
curves, so that we may say that the curves
 \[
 \theta_0+2\theta_1+\theta_2+\cdots
 \]
and
 \[
 \theta_0+\theta_1+2\theta_2+\cdots
 \]
and so on are bivalent fundamental curves of $|C|$, that is, to subtract it
from $|C|$ imposes two conditions.

For instance, let us consider the case of an (isolated) uniplanar
double point. This is a double point $O$ whose osculating cone is
reduced to a plane counted twice; the lines of this plane through $O$
have in general a contact of order 3 with the surface, except three
lines with a contact of order 4 that define three double points
$O_1,O_2,O_3$, infinitely near to $O$\footnote{See for instance
Enriques--Chisini, Ibidem, Libro~IV.}. For the sake of simplicity we
shall assume that the surface $F$ to which the uniplanar point $O$
belongs is a cubic surface $F_3$, and we shall elucidate what we have
said by referring to its plane representation.

To represent $F_3$ on a plane we may take the $\infty^3$ linear system
of the cubics $C_3$ passing through three collinear base points
$O_1,O_2,O_3$ and through three more points $O_1',O_2',O_3'$
infinitely near these, where the line containing $O_1,O_2,O_3$
corresponds to the uniplanar point $O$ and to the part $\theta_0$ of
his neighbourhood. Let us transform our system of $C_3$ via a Cremona
transformation of order 4, taking as transforming net a net of quartics
with a triple base point, three simple base points $O_1,O_2,O_3$ and
three more simple base points. The system $|C_3|$ will be transformed in
another system of curves $|C_9|$, with a sextuple base point $P$, three
collinear triple points $A_1,A_2,A_3$, three more double points
$B_1,B_2,B_3$, and three simple points $O_1,O_2,O_3$, that belong to
the lines $PB_1,PB_2,PB_3$ respectively. The line $A_1A_2A_3$ counted
twice represents the curve $\theta_0$, whereas the lines
$PB_1,PB_2,PB_3$ are the images of $\theta_1,\theta_2,\theta_3$. It is
immediately clear that the group of lines
$\theta_0+\theta_1+\theta_2+\theta_3$ is a univalent fundamental curve
for the system $|C|$. Indeed if one subtracts the line
$\theta_0$, then as a consequence the lines
$\theta_1,\theta_2,\theta_3$, each of which meets $\theta_0$ in a point
which is not a base point, must be a base curves of the residual
system, hence the above-mentioned line $A_1A_2A_3$ turns out to be
subtracted twice. Therefore the residual system of
$\theta=\theta_0+\theta_1+\theta_2+\theta_3$ consists of the quartics
$C_4$ passing three times through $P$, simply through $B_1,B_2,B_3$ and
simply also through $A_1,A_2,A_3$. $\theta_0$ is not a fundamental line
for this residual system any more; on the contrary, the lines
$PB_1,PB_2,PB_3$ are still fundamental.

The analysis of the singularity of the second kind that a surface $F$ has
at an ordinary uniplanar point $O$ makes it clear that such a point does
not affect neither the adjoint surfaces nor the genus.

In fact let us consider the system $|C|$ of plane sections of $F$ (which
one may replace with one of its multiples). The plane sections $C_1$
through $O$ form a linear system that has only fundamental curves of the
first kind and of genus zero $\theta_1,\theta_2,\theta_3$ corresponding
to the double points $O_1,O_2,O_3$, infinitely near $O$. So these
double points do not affect the canonical curves, that are
characterized by the property of cutting out on the $C_1$ residual
groups of the characteristic series, that is, groups belonging to a curve
of order $n-3$ adjoint to one of the $C_1$ formed by a $\fie_{n-4}$ and
a line through $O$. Then the curves adjoint to the $C_1$ are cut out by
the adjoint surfaces $\Phi_{n-3}$ that pass simply through $O$ and,
adding the curve $\theta$ that represents the neighbourhood of $O$, it
turns out that the curves $C'$ adjoint to the sections $C$ of $F$ are
cut out by $\Phi_{n-3}$ which do not have to pass through $O$ any more.

While the ordinary uniplanar double point does not affect the genus of
a surface $F$, there may be specializations of this singularity that do
affect the genus. We have already seen that this happens for the
tacnode, that is characterized by the property that the tangent lines
through $O$ have a quadruple contact with $F$ there and that therefore
makes the genus of the plane sections through it drop by two instead of
one. But even more surprising is the case of a uniplanar double point
that makes the genus of the plane sections through it drop by only one
and has an infinitely near tacnode $O_1$, so that the genus of the plane
sections through $OO_1$ drops by three instead of two. In this case the
system $|C_1|$ of plane sections through $O$ has a fundamental curve
$\theta$ of genus 1, whose image is the tacnode $O_1$, and this implies
that the surfaces $\Phi_{n-4}$ have to pass simply through $O_1$, that is,
as a consequence, through $O$. So one deduces that a uniplanar double
point, lowering the genus of the plane sections through it by only one,
with an infinitely near tacnode, lowers the genus of the surface by 1.

\begin{Remark} Noether pointed out two kinds of surfaces of order
four that turn out to be rational as a consequence of the possess of the
singularity described above\footnote{See Enriques--Conforto, {\it Le
superficie razionali}, Ibidem, pages~196 and~204.} and the rationality
of these surfaces is consistent with the fact that the above-mentioned
singular point lowers the genus of the surface of order four by one,
and so it reduces it to zero. The two types of surfaces of Noether give
rise to a plane representation respectively given by:

 \begin{enumerate}
 \renewcommand{\labelenumi}{(\arabic{enumi})}
 \item the $\infty^3$ system of curves $C_7$ of order seven having a triple
base point and nine double base points on a fundamental cubic; and
 \item the $\infty^3$ system of curves $C_9$ of order nine having eight
triple base points, a double point and a simple point, on a fundamental
cubic.
 \end{enumerate}

 \end{Remark}
In both cases the fundamental cubic of the representing system is a $\theta$ of the second kind that is fundamental also for the residual
system.

And it is evident from the above representation that this curve $\theta$ has genus 1 and degree
$-1$ on the surface $F_4$, or on a transform of $F$ on which the
singularity is resolved without introducing new exceptional curves: in
fact the plane cubic from which 10 base points have been removed is
precisely a curve of degree $-1$\footnote{At this point it is necessary
to warn the reader that second type of Noether's surfaces presents a
projective complication, i.e., the tacnodal plane of $O_1$ passes
through $O$, so that the plane sections through $OO_1$ contain a third
fixed double point collinear with $OO_1$ and become reducible in the
line $OO_1$ and a cubic of genus 1 for which the line is a flex
tangent. As a consequence the analysis of the singularity must be
carried out using a system of intersections with surfaces of higher
degree instead of the system of plane sections. See Enriques--Conforto,
Ibidem, pages~206--7.}.

Now the analysis of the singularity presented by Noether's surface $F_4$
can be extended to analogous singularities of an arbitrary surface: the
singularity of the second kind consisting of a double point with an
infinitely near tacnode corresponds in general to a fundamental curve
of genus 1 and of degree $-1$ for the linear system $|C|$ of the plane
sections of the surface\footnote{See D.~Pedoe, On the canonical system
of certain algebraic surfaces. Proc.\ Cam.\ Phil.\ Soc., Vol.~33
(1937).}.

We can give a direct proof of the statement (independent of the
possibility of approximating the given surface with an $F_4$ having the
same singularity). To start with one shows that a curve of genus 1 and
degree $-1$ fundamental for a linear system $|C|$ yields the above
mentioned singularity on a surface having $\infty^3$ $C$ as plane
sections; and the converse also holds, if one starts with an $F$ having
the stated singularity, resolves it and computes the characters of
the fundamental curve corresponding to it.

Let us assume that $\theta$ is a fundamental curve for a system $|C|$ of
genus $\pi$ and degree $n$, and that the curve $\theta$ has genus $\rho=1$ and
degree $\nu=-1$; it is clear that the number of intersections
 \[
 (C-\theta)\theta=C\theta-\theta\theta=-\nu=1,
 \]
namely the curves of the residual system $|C_1|=|C-\theta|$ have one
intersection, necessarily fixed, with $\theta$ (that is, $C_1\theta=1$). As
a consequence the genus $x$ of $C_1$ is given by the equation
 \[
 \pi=x+1+1-1=x+1,
 \]
 that is,
 \[
 x=\pi-1
 \]
and its (virtual) degree is computed by the formula
 \[
 n=CC=(C_1+\theta)(C_1+\theta)=C_1C_1+\theta\theta+2
 \]
that gives
 \[
 C_1C_1=n-1
 \]
(but the effective degree of $|C_1|$ is $n-2$, since there is a base
point on $\theta$).

Now the image of $\theta$ on the surface $F$ that has $\infty^3$ $C$ as
plane sections is a point $O$ that lowers the genus of the plane
sections through it by 1 and therefore it is double for $F$; and near
$O$ there is a singular point $O_1$ that lowers the genus of the (plane
or superficial) sections through it by two, and is therefore a tacnode.

Conversely let us start with a surface $F$ having a double point $O$
with an infinitely near tacnode $O_1$. The point $O$ corresponds on a
suitable transform of $F$ to a curve $\theta$ fundamental for the
system of plane sections $C$, which a priori consists of two parts
$\theta_0$ and $\theta_1$, the former of genus zero corresponding to
the first order neighbourhood of $O$ (not including the tacnode), the
latter of genus 1 corresponding to the neighbourhood of $O_1$: the curve
 \[
 \theta=\theta_0+\theta_1
 \]
is fundamental for $|C|$, and necessarily connected, and represents the
whole neighbourhood of the point $O$.

Now one can compute the genus $\rho$ and the degree $-\nu$ of the
curve. By expressing the fact that subtracting it from the system $|C|$
of genus $\pi$ lowers the genus of $C$ by 1, we have
 \[
 |C|=|C_1+\theta|
 \pi=\pi-1+\rho+x-1
 \]
where $x\ge1$ denotes the number of points of connection for the
reducible $C$, that are common to $\theta$ and $C_1$; whence one deduces
 \[
 x=1 \quad\hbox{and}\quad\rho=1.
 \]
So, computing the number of intersections $\theta C_1$ as a function of
the degree of $\theta$, one has
 \[
 C_1\theta=(C-\theta)\theta=-\theta \theta=-\nu=1
 \]
One computes next the degree $-\nu_1$ of $\theta_1$ by writing down that
imposing to the sections $C_1$ to pass through $O$ lets their genus
drop by 2: we have
 \[
 |C_1|=|C_2+\theta_1|
 \pi-1=\pi-3+1+y-1
 \]
where $y$ denotes the number of intersections
 \[
 C_2\theta_1=(C_1-\theta_1)\theta_1=-\theta_1\theta_1=-\nu_1,
 \]
whence one obtains
 \[
 \nu_1=y=2.
 \]
Now, denoting by $-\nu_0$ the degree of $\theta_0$ one has
 \[
 -\nu=-\nu_0-\nu_1+2=-1
 \]
thus
 \[
 \nu_0+\nu_1=3, \nu_0=1.
 \]
In conclusion the curve $\theta_0$ of genus zero and degree $-1$ turns
out to be an exceptional curve that can be transformed into a simple
point of the curve $\theta_1$ of genus 1, and therefore the latter
becomes a $\theta$ of degree
 \[
 \nu_0=-1
 \]
(and still has genus $\rho=1$).

The singularity consisting of a double point with infinitely near a
tacnode occurs in the study of double planes, when the branch curve has
two infinitely near triple points $O,O_1$, or, as somebody says, a
$[3,3]$ point. In this case consider for instance the system of curves
$C$ having a certain genus $\pi$ that is represented on the double
plane by the $\infty^5$ system of conics; the conics through $O$
represent curves $C_1$ of genus $\pi-1$, but those going through $O$ and $O_1$ represent curves of
genus $\pi-3$. So the point $O$ is a double point of the surface
$z^2=f(x,y)$ that makes the genus of the (plane or superficial) sections
drop by only 1, whereas the point $O_1$ infinitely near to it makes the
genus of these sections further drop by 2, hence it is a tacnode. So we
see again that: a $[3,3]$ point belonging to the branch curve of order
$2n$ of a double plane must be considered as a simple base point for
the curves of order $n-3$ that are the images of the canonical curves
(see Section~\ref{sec:3.7}) and so it lowers the genus of the double
plane by 1.

The two rational surfaces of order four of Noether's that we have
recalled above are projected onto a double plane from their singular
point. Their branch curve has order 6, and has a quadruple point, and a $[3,3]$ point for the two types respectively: so it is confirmed that the
singularity of those surfaces lowers by one the genus of the surfaces
of order 4.

 \begin{EXERCISE}
 \begin{enumerate}
 \item Show that the (isolated) singularity consisting of two infinitely
near triple points $OO_1$ imposes two simple base points to the adjoints
and therefore it has the same effect as two distinct triple points on
the genus of the surface.

Then compute the genus of a surface $F_5$ of order 5 with a double line
and two infinitely near triple points $OO_1$ away from the line: one finds that
 \[
 p=0
 \]
This result agrees with the fact that $F_5$ is rational, since it has a
pencil of rational quartic plane sections through $OO_1$.

It is possible to obtain the plane representation of the above
mentioned $F_5$ by means of a quadratic transformation of the second
kind, corresponding to the homaloidal system of the quadrics passing
through the double line of $F_5$ and through three more points of $F_5$,
two of which are the infinitely near triple points $O$ and $O_1$. In this
way $F_5$ will be transformed into a surface $F_4$ of order 4 with a double
line.

 \item Consider a nonsingular surface $F_4$ of order 4, containing a
line $a$: the sum of the linear system of plane sections and of the
double of the pencil of the cubic sections $K_3$ through a gives a
linear system of curves $C$ of genus $3+1+1+(3-1)+(3-1)=9$ and it is
easy to see that the degree of this system is 16 and the dimension is
equal to 9. Impose that the $C$ have a triple point at a point $P$ of
$F_4$ away from $a$, that is a flex for the $K_3$ passing through it.
Thus one obtains an $\infty^3$ linear system of curves of genus 6 and
degree 7, that has $K_3$ as a fundamental curve of genus 1 and degree
$-1$. Via this system $F_4$ can be transformed into a surface $F_7$ of
order 7 with a quadruple line, a triple line (that reduces to 6 the
genus of the plane sections) and a double point with an infinitely near
tacnode: so one finds for $F_7$ the same genus of $F_4$, that is,
$p=1$. There is an adjoint surface of order $7-4=3$, that is reduced to
the plane containing the quadruple and the triple line counted twice
and to the plane projecting the isolated double point from the
quadruple line, which cuts $F_7$ along an exceptional cubic.

Investigate the inverse transformation of $F_7$ into $F_4$.
 \end{enumerate}
 \end{EXERCISE}

 \begin{Remark} In this section we have made some remarks concerning
interesting examples, but we do not believe, in any way, that we have
exhausted the theory of fundamental curves of linear systems and its
relationships with the theory of singularities. Therefore it is
necessary to examine and to meditate on a larger class of examples and
to investigate the complications to which they give rise. In order to
have a deeper understanding of this type of questions, we advise the
scholar to consider some, even quite simple, singularities that may
belong to the branch curve of a double plane: infinitely near quadruple
points, series of four or six infinitely near triple points, series of
triple points infinitely near to quadruple points, and so on. The
question is to understand these cases by characterizing the fundamental
curve of the system of plane sections of the surface $F$, that
corresponds to the above multiple points, and the consequences of the
subtraction of it from the system\footnote{See: A.~Franchetta, Sui
punti doppi isolati delle superficie algebriche, {\it Rendic.\ Acc.\
Lincei}, 1945, Notes~I and~II.}.
 \end{Remark}

 \section[Bicanonical curves and biadjoint surfaces]{Bicanonical curves
and biadjoint\\ surfaces}\label{sec:3.11}
 \markright{\hfill Bicanonical curves and biadjoint surfaces \quad
3.11\quad}
 Given a surface $F$ of a certain order $n$ in ordinary space, we have
learned how to construct on it the canonical curves as intersections of
surfaces $\Phi_{n-4}$ of order $n-4$ adjoint to $F$, and more generally
the adjoint curves of the $r$-tuple system of plane sections $C$ of
$F$, via the adjoint surfaces $\Phi_{n-4+r}$, which always cut out on
$F$ a complete linear system. It is now easy to determine on $F$ the
bicanonical curves and also the biadjoint curves of the $r$-tuple
system of $|C|$, by making use for this of biadjoint surfaces of $F$,
$\psi_{2n-8}$ and $\psi_{2n-8+r}$ of order $2n-8$ and $2n-8+r$
respectively.

For the sake of simplicity, let us refer to the case of genus $p>0$, in
which the adjoints $\Phi_{n-4}$ and the canonical curves that are
sections with them actually exist. The behaviour of the biadjoint
surfaces $\psi_{2n-8}$ with respect to the singularities of $F$ is
defined as a rule by the behaviour of two $\Phi_{n-4}$ that, taken
together, form precisely a $\psi_{2n-8}$. Still it is necessary to
define this behaviour more precisely, as we shall do with the following
remarks.

If $F$ has a double curve and no proper multiple points, then the $\Phi_{n-4}$ pass through this curve simply and the $\psi_{2n-8}$ go
through it doubly: the linear system cut out by these $\psi_{2n-8}$ is
complete because it is deduced from the system cut out by the adjoints $\psi_{2n-8}$ (that we already know to be complete) by subtracting the
neighbourhood of the double curve.

But if $F$ has a curve of multiplicity $i>2$ then the biadjoint surfaces $\psi_{2n-8}$ (and similarly
the $\psi_{2n-8+r}$) should pass through this curve, not with
multiplicity $2i-2$, but with multiplicity $i$, and should have a
suitable contact with the branches of the surface along it; for
instance if $i=3$ then the $\psi$ should contain the triple curve of
$F$ as a triple curve and they have to touch simply the branches of the
surface $F$ along it. In fact a surface $\psi$ that passes four times
through a triple curve of $F$ may always be replaced by a linear
combination of $\psi$ and $F$ that passes three times through this
curve and touches the branches of $F$. So the $\psi$ are subject to
less restrictive conditions and one can say that they cut out on $F$ a
linear system obtained from the linear system cut out by the adjoints
of the same order, by subtracting a certain number of times the
neighbourhood of the multiple curve. By contrast the $\psi$ passing four
times through the triple curve of $F$ do not give in general a complete
system.

Similarly an isolated $s$-tuple point of the surface $F$ ($s>2$), which
is in general $(s-2)$-tuple for the adjoint surfaces $\Phi$, has to be a
multiple point for the biadjoint surfaces $\psi$, and its multiplicity
will not be $2s-4$ but $s$, and in addition the $\psi$ have to touch $F$
$s-4$ times at that point.

Thus if $O$ is an isolated triple point of $F$, then the $\psi$ have to
pass through it doubly. If instead $O$ is a quadruple point, then the $\psi$ must have only a quadruple point there: but if $O$ is a quintuple
point of $F$, then it has to be only quintuple for the $\psi$, that must
have the same osculating cone, though

Special attention must be paid to the behaviour of the biadjoint
surfaces at a tacnode of the surface $F$. At first sight one might think
that such a point, being simple for the adjoint surfaces $\Phi_{n-4}$,
should be double for the $\psi_{2n-8}$, as it is for the surfaces
consisting of two $\Phi_{n-4}$. But let us think about the way the
behaviour of the surfaces $\Phi_{n-4}$ at a tacnode $O$ is determined: a
plane section through $O$ has two infinitely near double points $O$ and $O_1$, and the section of a $\Phi_{n-4}$ is such that, together with a line
through $O$, it must form an adjoint $\fie_{n-3}$ of the section of $F$.
Therefore a priori $\Phi_{n-4}$ is forced to pass through the point $O_1$
infinitely near $O$, and only as a consequence it happens to pass through
the proper point $O$. Similarly, a plane through $O$ will be cut out by a
biadjoint surface $\psi_{2n-8}$ along a curve that a priori should pass
doubly through $O_1$, and only by an effect of transitivity will pass
simply through $O$ and $O_1$. So the conditions that a tacnode imposes on
the biadjoint surfaces consist only in passing through this point and
in touching the tacnodal plane there: hence the tacnode lowers the
bigenus by three. These conditions can be verified also by remarking
that they imply for the bicanonical curves sections of the $\psi_{2n-8}$
the characteristic property of cutting out on the generic curve of a
linear system $|C|$ (for instance of the net of plane sections through
$O$) groups belonging to the double of the residual series of the
characteristic series.

Let us now turn to the case of the Noetherian singularity consisting of
a double point $O$ with an infinitely near tacnode $O_1$. Also in this
case the vanishing of the $\Phi_{n-4}$ at $O$ is a consequence of the
principle of transitivity of singularities: a priori the $\Phi_{n-4}$
should pass through the infinitely near point $O_1$. Therefore the
$\psi_{2n-8}$ which have to pass through $O_1$ twice, are actually
determined by the condition of passing through the two infinitely near
points $O$ and $O_1$. Thus: {\em a double point with an infinitely near
tacnode imposes in general to the biadjoint surfaces to pass simply
through the two infinitely near points, $O$ and $O_1$, and therefore it
lowers the bigenus only by two.} It is easy to check these results by
referring to a double plane with branch curve $f(x,y)=0$ of a certain
order $2n$, namely to the surface $z^2=f(x,y)$. We have seen that a
quadruple point of $f=0$ is a tacnode for the surface $F$, and, in the
plane, the canonical curves of $F$ are represented by curves of order
$n-3$ passing simply through $O$. Now the curves of order $2n-6$ passing
doubly through $O$ represent bicanonical curves of $F$, although the
system of this curves is not complete, in general\footnote{A complete
linear system of curves $|C|$ on the double plane does not represent on
the surface $F$ a complete system, but a system of curves belonging to
the involution $I$ whose pairs correspond to the points of the double
plane; and in general this can be extended to a larger system whose
curves no longer belong to the involution $I$. This can be seen already
in the case of the double plane with branch curve $f_4=0$ of degree 4
obtained by projecting from the point at infinity $O$ of the $z$-axis
the surface $F_4$ of degree 4 of equation $z^2=f_4(x,y)$, that has a
tacnode at $O$ with the plane at infinity as the tacnodal plane. Here
the $\infty^5$ conics of the plane are the images of the curves cut out
on $F_4$ by the quadric cones through $O$, which are hyperelliptic
curves belonging to the involution $I$. But the $\infty^5$ system of
such curves is contained in the $\infty^6$ linear system cut out on
$F_4$ by the quadrics touching the tacnodal plane at $O$. Concerning the
possibility of extending the system of bicanonical curves whose images
on the double plane are curves of degree $2n-6$, the matter depends on
the behaviour of the biadjoint surfaces at the point at infinity of the
$z$-axis, where $F$ has a known singularity. It suffices to say that in
general the $y_{2n-8}$ will have multiplicity $2n-2$ there and a
suitable contact with the surface. Whence it is clear that, while the
canonical curves of a surface that can be represented on the double
plane always belong to the involution whose pairs correspond to the
points of that plane, in general the same does not happen for the
bicanonical curves. (See Chapter~V, Section~\ref{sec:5.7})}. So one can
check that a tacnode lowers the bigenus of $F$ by 3.

The next case is the case of the noetherian singularity that
corresponds to a $[3,3]$ point of the branch curve $f$. The $[3,3]$
point should be considered as a double point $O$, with an infinitely
near 4-tuple point $O_1$. The images of the bicanonical curves must
then pass doubly through $O_1$, namely -- by transitivity -- the images
of the bicanonical curves must pass simply through $O$ and $O_1$, and
so a $[3,3]$ point of the branch curve of the double plane lowers the
bigenus of the surface by 2.

 \section{Surfaces of genus $p=0$ and bigenus $P>0$}\label{sec:3.12}
 \markright{\hfill Surfaces of genus $p=0$ and bigenus $P>0$ \quad
3.12\quad}
 The previous results allow us to construct easily surfaces of genus
$p=0$ and bigenus $P>0$, which therefore are certainly irrational.

The first example was found by Enriques\index{Enriques surface} in
1896\footnote{Introduzione\dots, Ibidem}. It is the surface $F_6$ of order
6 that passes doubly through the edges of a tetrahedron. If we consider
a projective coordinate system in which the tetrahedron is fundamental, $F_6$ is represented by an equation of the form:
 \[
F_6=\fie(x_2x_3x_4,x_3x_4x_1,x_4x_1x_2,x_1x_2x_3)+
 x_1x_2x_3x_4f(x_1,x_2,x_3,x_4)=0
 \]
where $f$ and $\fie$ are quadratic forms in the variables
$x_1x_2x_3x_4$, and respectively in the products of triples of these
variables. In fact $F_6$ belongs to the double of the linear system
consisting of the cubic surfaces that pass simply through the edges of
the tetrahedron. $F_6$ has genus $p=0$, since there are no quadrics
adjoint to it, that is, passing through the edges of the tetrahedron. But
the bigenus of $F_6$ is $P=1$, because there exists a surface of order
$2\cdot 6-8=4$ biadjoint to it, that is, our tetrahedron. This surface
$\psi_4$ cuts out on $F_6$ a bicanonical curve of order 0. The linear
system $|C|$ of plane sections of $F_6$ is of genus $\pi=4$ and of
degree $n=2\pi-2=6$; its characteristic series is a complete and
nonspecial $g_6$, whose double is the double of the canonical series.
The system $|C'|$ adjoint to $|C|$ is cut out on $F_6$ by the cubic
surfaces
$\Phi_3$ passing simply through the edges of the tetrahedron; this is,
as well as $|C|$, a system of sextics of genus 4 and degree 6. It is
remarkable here that the double of the system $|C|$ and the double of
the system $|C'|$ coincide:
 \[
 |2C|=|2C'|
 \]
This equality just expresses the real meaning of the existence of the
bicanonical system $|2C-2C'|$ of order zero.

Furthermore, since
 \[
 |(2C)''|=|2C'|=|C+C''|
 \]
one deduces that
 \[
 |C''-C|=|2C'-2C|=|2C-2C|
 \]
and so
 \[
 |C''|=|C|;
 \]
in words: the system of plane sections of $F_6$ is in turn the adjoint of
its adjoint system, cut out by the surfaces $\Phi_3$ passing simply
through the edges of the tetrahedron.

Let us compute the linear genus $p^{(1)}$ of $F_6$ by calculating the
virtual genus of the curves of $|C'-C|$ according to Chapter I,
Section~\ref{sec:1.10}. Denoting by $\pi$ and $n$ the genus and the
degree of $|C|$, by $x$ and $y$ the genus and the degree of the virtual
curve $K=C'-C$ of order 0, that has $i=0$ intersections with the $C$,
we shall write the genus and the degree of $|C+K|=|C'|$:
 \[
 \pi'=\pi+x+0-1=\pi
 n'=n+y+2\cdot 0=n
 \]
whence
 \[
 x=1,\quad y=0:
 \]
the linear genus of $F_6$ is $p^{(1)}$=1.

Another example of surface of genus $p=0$ and of bigenus $P>0$ was
pointed out by Castelnuovo, again in 1896\footnote{G.~Castelnuovo,
Sulle superficie di genere zero. Mem.\ della Soc.\ It.\ delle Scienze
(detta dei XL), s.~III, t.~10, 1896.}. It is the surface $F_7$ of
order seven that passes three times through a line $r$, twice through a
conic $\Ga$ (not intersecting $r$) and moreover has three tacnodes whose
tacnodal planes contain $r$.

Such an (irreducible) surface actually exists and can be constructed as
follows: first of all take a cubic surface $F_3$ that passes simply
through the line $a$ and through the conic $\Ga$ and consider three planes
$a_1a_2a_3$ passing through the line $r$ and tangent to $F_3$ at the
points $A_1,A_2,A_3$ (away from $r$). It is easy to see that there
exists an irreducible surface $F_4$ of order 4 passing doubly through
the conic $\Ga$ and tangent to the planes $a_1a_2a_3$ at the points
$A_1,A_2,A_3$: this surface taken together with the three planes
$a_1a_2a_3$ is a particular reducible $F_7$ that passes twice through
the conic $\Ga$, three times through the line $r$, and has a tacnode at
the points $A_1,A_2,A_3$ with tacnodal planes $a_1a_2a_3$. But another
reducible surface satisfying the same conditions can be obtained by
adding to $F_3$ counted twice a plane through $r$; then, taking the
pencil generated by the two reducible $F_7$, one obtains an irreducible
$F_7$ with the singularities described above.

There are no surfaces $\Phi_3$ adjoint to $F_7$: in fact a $\Phi_3$
which should pass through the conic $\Ga$ and doubly through the line
$r$ that does not intersect $\Ga$, should be a priori reducible,
containing the plane of $\Ga$ as a component. But the residual quadric
should first of all pass doubly through $r$, thus it should split into
two planes through $r$, and besides it should pass through the tacnodes
$A_1,A_2,A_3$; this imposes three conditions impossible to be
satisfied. One concludes that the genus of $F_7$ is
 \[
 p=0.
 \]
Now we determine the surfaces $\psi_{2\cdot7-8}=\psi_6$ biadjoint to
$F_7$. Among these certainly there are the surfaces consisting of the
plane of $\Ga$ counted twice and of four planes through $r$, three of
which have to pass through the tacnodes $A_1,A_2,A_3$ and therefore
coincide with the planes $a_1a_2a_3$; thus one obtains $\infty^1$
biadjoint surfaces $\psi_6$ that contain a plane through $r$ as the
variable part and cut out $\infty^1$ bicanonical quartics of genus 1 on
$F_7$, forming a base point free pencil. It is clear that this
bicanonical system of order zero cannot be extended, and therefore the
condition that the biadjoint surfaces $\psi_6$ pass three times through
the line $r$ and touch the branches of $F_7$ through it, implies here, as
a consequence, that the line $r$ is quadruple for $\psi_6$.

Summing up we shall say that: the surface $F_7$ of Castelnuovo has genus $p=0$ and bigenus P=2, since it contains a pencil of plane bicanonical
quartics of genus 1 and degree 0.

It is easy to determine the linear genus of $F_7$, that is the (virtual)
genus $p^{(1)}$ of its virtual canonical curves $K$; in fact, since the
degree of $|K|$ is equal to $p^{(1)}-1$, the genus and the degree of
$|2K|$ are respectively given by
 \[
 3p^{(1)}-2 \quad\hbox{and}\quad 4p^{(1)}-4.
 \]
{From} either of of the relations $3p^{(1)}-2=1$ and $4p^{(1)}-4=0$ one
deduces that the linear genus is $p^{(1)}=1$.

No other example of surfaces of genus $p=0$ and of bigenus $P>0$ was
found for many years, and in particular, it remained an unanswered
question whether among such surfaces there are any for which the linear
genus is $p^{(1)}>1$, having an irreducible canonical system of
dimension $>1$. This question is now answered by the examples actually
constructed, simultaneously and independently of one another, by
L.~Godeaux\footnote{See L.~Godeaux, Sur une surface alg\'ebrique de
genre z\'ero et de bigenre deux, {\it Rendic.\ Acc.\ Lincei}, Vol.~XIV,
1931} and by L.~Campedelli\footnote{L.~Campedelli, Sui piani doppi con
curva di diramazione del decimo ordine, Ibidem, Vol.~XV, 1932.} in
1931--32.

We shall only describe the example constructed by Campedelli that
refers to a double plane with branch curve of order ten, having six
$[3,3]$ points (pairs of infinitely near triple points). It is possible
to actually construct such a curve $f_{10}$, at least as a curve formed
by three conics two by two bitangent and by a quartic $f_4$ that
touches them at their six contact points. Indeed one obtains three
mutually bitangent conics by taking for instance two concentric circles
(that are tangent at the cyclic points of the plane), and an ellipse
that touches the smaller circle at the ends of a diameter and the
larger one at the ends of an orthogonal diameter. The 6 contact points
of the conics thus obtained yield 12 linear conditions for a quartic.
Taking one of these quartics together with the 3 conics as the branch
curve of a double plane, we shall have that:

 \begin{enumerate}
 \renewcommand{\labelenumi}{(\arabic{enumi})}
 \item on the double plane there is no canonical curve, which should be
a conic passing through the six singular points of $f_{10}$, whereas one
sees that in the example we constructed those points do not belong to a
conic (circle); hence the genus of the double plane is
 \[
 p=0;
 \]

 \item there is a net of quartics that, like the quartic $f_4$ which is
a component of the branch curve $f_{10}$, pass through its twelve triple
points: they represent $\infty^2$ bicanonical curves of the surface,
belonging to the involution whose pairs of points correspond to points
of the plane, and one may see that they form the complete bicanonical
system, so that the bigenus of the surface is
 \[
 P=3.
 \]
 \end{enumerate}

 \begin{Remark} The proof that $P=3$, and not $P>3$ will follow later on
from the expression of $P$ via the linear genus, that here gives
$P=p+p^{(1)}=p^{(1)}=3$, where the value of $p^{(1)}$ stems directly by
remarking that the virtual double conic forming the canonical curve
would have 8 branch points: 6 at the triple points of $f_{10}$ and two
more outside. If one does not want to work with the virtual canonical
curve, then it also suffices to remark that the degree of the
bicanonical system is the double of the degree of the system of
quartics, so that one has
 \[
 4(p^{(1)}-1)=2\cdot 4=8,p^{(1)}=3.
 \]
If one does not want to use the relationship between the bigenus and
the linear genus, then one can nevertheless prove directly that, for
our surface, the net of double quartics that are the images of the
bicanonical curves is complete. In order to do this it suffices to
remark that the component $f_4$ of the branch curve is a part of one and
only one bicanonical curve, consisting of the curve itself counted
twice, and that the $\infty^2$ quartics through the triple points of
$f_{10}$ cut out on $f_4$ a complete and nonspecial $g_4$: in fact if
this series were the canonical series of $f_4$ then the 12 triple
points of $f_{10}$ should lie on a cubic, and it is easy to see that
this gives a contradiction.
 \end{Remark}

 \section{Equivalence criteria}\label{sec:3.13}
 \markright{\hfill Equivalence criteria \quad 3.13\quad}
 The construction of the subadjoint and adjoint surfaces of a nonruled
surface $F$ of ordinary space contains an equivalence criterion: two
curves $L$ and $M$ that cut out on the plane sections of $F$ groups
belonging to the sum (or difference) series of the canonical series of
$C$ and of a multiple of the characteristic series, are equivalent, up
to fundamental curves of the system $|C|$; they are certainly
equivalent if $|C|$ does not have any proper fundamental curve. And one
can remark that, when the system $|C|$, as we have said, has no proper
fundamental curve, the equivalence of the curve $L$ and $M$ can be
deduced just by the condition that they cut out on the $C$ equivalent
groups, even if the series containing them is not the sum of the
canonical series and of a multiple of the characteristic series.

In fact, taking an adjoint surface of $F$ of large enough order
containing $L$, one obtains a curve $D$ as the residual intersection.
Now $L+D$ and $M+D$ cut on a curve $C$ groups equivalent to the sum of
the canonical series and of a multiple of the characteristic series;
therefore they are equivalent, and so $L$ and $M$ also turn out to be
equivalent.

The equivalence criteria can be developed independently of the theory
of adjoint and subadjoint surfaces, as F.~Severi\footnote{See
F.~Severi, Il teorema di Abel sulle superficie algebriche, {\it Ann.\ di
Mat.}, (3), 12, 1905. Osservazioni varie di geometria sopra una
superficie algebrica, {\it Atti Ist. Veneto}, 65, parte II, 1906. Alcune
relazioni di equivalenza ecc., Ibidem, 70, 1911.} did, in an interesting
way. We shall briefly explain these developments. To start with we
consider on a surface $F$ a linear pencil of curves $C$ with at least
one simple base point $O$ and without reducible curves. Then if the two
curves $K_0$ and $K_\infty$ not passing through $O$ cut out equivalent
groups of points on the $C$, then they are equivalent.

In fact on any curve $C$ of the pencil we can assume that the groups cut
out by $K_0$ and $K_\infty$ are the groups of zeros and poles of a rational
function $\fie$ that is determined up to a nonzero constant, and we can
determine this constant too, by imposing that $\fie$ takes the value 1 at
the point $O$.

As $C$ varies in its pencil, $\fie$ gives us a rational function of the
points of the surface that has $K_0$ as the curve of zeros and
$K_\infty$ as the curve of poles, so that these curves turn out to be
equivalent, Q.E.D. But, in order that the argument be rigorous, one has
to prove that $K_0$ is the whole curve of zeros of our function and
$K_\infty$ is the whole curve of poles. To do this remark that in case
the opposite were true one should add to $K_0$ a curve that, having no
variable intersection with the irreducible $C$ of the pencil, must be
composed with one or more of the $C$; but then, on such a $C$, the
point $O$ would be an indeterminacy point of the function, where the
function would take the values zero and 1 at the same time, and this is
impossible, since $O$ belongs neither to the group of zeros nor to the
group of poles that determine the corresponding rational function on $C$.

The previous equivalence criterion can be generalized. Let $|C|$ be a
pencil of curves, rational or irrational, without reducible curves: two
curves $K_0$ and $K_\infty$ that cut out equivalent groups on the $C$
are equivalent up to curves of the pencil, that is, one has
 \[
 K_0+\sum C\equiv K_\infty+\sum\nolimits'C.
 \]
To prove this second equivalence criterion, we may proceed as before to
construct a rational function having $K_0$ and $K_\infty$ as the curves
of zeros and the curve of poles respectively. We know that $\fie$ is
defined up to a multiplicative constant on every $C$ of the pencil, and
(denoting by $t$ the parameter of the $C$ of this pencil) this way we
obtain a function of the points of the surface, $\la(t)\fie$, where
$\la$ has to be determined. In order to do this it is not necessary to
determine a point on every $C$, as we did in the proof of the first
criterion, but it suffices to determine a group of a certain number $m$
of points, consisting of the intersections that $C$ has with an
auxiliary curve $L$ lying on the surface; then one may prescribe the
sum of the values
 \[
 \la\fie_1+\la\fie_2+\cdots+\la\fie_m
 \]
that $\fie$ takes at the points of this group. Even more simply it
suffices to take instead of $\fie$ the function
 \[
 f=\fie/(\fie_1+\fie_2+\cdots+\fie_m)
 \]
that is a rational function of the points of the surface, independent
of $\la$.

It is obvious that $K_0$ is contained in the curve of zeros and
$K_\infty$ is contained in the curve of poles, but these are not the
entire curves of zeros and poles: one has to add to $K_0$ a group $\sum
C$ consisting of the $C$ that pass through the intersection points of
$L$ and $K_\infty$; in fact on these $C$ the denominator of $f$ becomes
$\infty$ and so $f$ is everywhere zero. Similarly, one has to add to
$K_\infty$ a group $\sum'C$ consisting of the $C$ that pass through the
intersections of $L$ and $K_0$, as one sees by interchanging the roles
of $K_0$ and $K_\infty$, that is, by replacing $f$ by $1/f$.

Then we shall have
 \[
K_0+\sum C\equiv K_\infty+\sum'C.
 \]
That $K_0+\sum C$ is actually the whole curve of zeros of $f$ can be
proved by remarking that this curve of zeros can contain, besides
$K_0$, only a certain number of curves having no variable intersections
with the $C$, and therefore a group of (irreducible) $C$; but $f$
cannot be identically zero on a $C$ unless the sum
$\fie_1+\fie_2+\cdots+\fie_m$ becomes infinite and so the same happens
for one of the $\fie_i$, which means that $C$ itself has to go through a
common point of $L$ and $K_\infty$.

Let us add two remarks. The first is that when the pencil of curves $C$
is rational, the $C$ are equivalent, and so there exists a difference
$\sum C-\sum'C$ consisting of a group of a certain number $s$ of
curves that has to be taken with a plus or minus sign, so that one has
 \[
 K_0\equiv K_\infty-sC \quad\hbox{or}\quad
 K_\infty \equiv K_0+sC.
 \]
And if in particular the pencil of the $C$ has base points, then one
certainly has the equivalence
 \[
 K_0\equiv K_\infty
 \]
when the equivalence of the two groups $(CK_0)$ and $(CK_\infty)$ holds,
also taking into account the points that occur at the base points.

In fact, if, for instance, one has
 \[
 K_0\equiv K_\infty-sC
 \]
then it follows that
 \[
 (CK_0) \equiv (CK_\infty)-s (CC);
 \]
but one has also
 \[
 (CK_0) \equiv (CK_\infty),
 \]
and so, if the group $(CC)$ is nonzero, one should have $s=0$.

Let us now turn to the third equivalence criterion: if on a surface $F$
it is given an $\infty^1$ system of index\footnote{This means that there
are $n$ curves of the system passing through a generic point of the
surface.} $\nu>1$ of curves all of which are irreducible, then two
curves $K_0$ and $K_\infty$ cutting out equivalent groups on the $C$
are equivalent.

In order to prove this, one constructs as before on every curve $C$ a
rational function $\fie$ having its zeros at the intersections with $K_0$
and its poles at the intersections with $K_\infty$, and this function is
determined up to a multiplicative constant $\la$. Then one take an
auxiliary curve $L$ on $F$ having a certain number $m$ of intersections with
the $C$, and -- for every $C$ -- one considers the sum of the values that
$\fie$ takes at these $m$ points:
 \[
 \fie_1+\fie_2+\cdots+\fie_m.
 \]
In this way, the rational function
 \[
 f=\frac{\fie}{\fie_1+\fie_2+\cdots+\fie_m}
 \]
is determined on every $C$. After this, we determine a rational
function $\psi$ of the points of the surface by assigning as its value
at a point $P$ the sum of the values $f_1,f_2,\dots f_\nu$ that $f$
takes on the $\nu$ curves $C$ passing through $P$:
 \[
 \psi=f_1+f_2+\cdots+f_\nu.
 \]
We may denote the above mentioned values of the $f_i$ by means of an
upper index; and then we shall write
 \[
 f_i=\frac{\fie^{(i)}}{\fie_1^{(i)}+\cdots+\fie_m^{(i)}}\,.
 \]
The rational function $\psi$ clearly vanishes on the curve $K_0$, where
all the $\fie^{(i)}$ are equal to zero, and its poles must necessarily
occur on $K_\infty$; but it is not evident a priori that all the points
of $K_\infty$ are poles of the rational function $\psi$, since a sum of
infinities is not always infinite. What we can say is that the curve
$\Kbar_0$ of zeros of $\psi$ is of the form $\Kbar_0=K_0+A$, whereas, if
$\Kbar_\infty$ is the curve of poles, then one has
$K_\infty=\Kbar_\infty+B$. Intersecting the curves $\Kbar_0$ and
$\Kbar_\infty$ with a generic $C$, one has:
 \[
 (K_0,C)+(A,C) \equiv (K_\infty,C)-(B,C)
 \]
But one has $(K_0,C)\equiv(K_\infty,C)$ by assumption, thus the curves
$A$ and $B$ would have no intersections with the generic curve $C$,
whereas our assumptions exclude the existence of such curves.
Concluding, one has
 \[
 K_0=K_0,\quad \Kbar_\infty=K_\infty
 \]
and so the equivalence of the curves $K_0$ and $K_\infty$ is proved.

 \begin{Remark} In the above discussion we have considered criteria that
enable us to say when two (isolated) curves on a surface $F$ are
equivalent. There exist other more significant criteria, concerning a
continuous (at least $\infty^1$) series of curves $|C|$ given on $F$.
The first criterion of this kind was established by Enriques in 1895. A
rational series of curves $|C|$ always consists of equivalent curves,
that is, it is contained in a linear system. The property can be shown
as for the rational series of groups of points on a curve\footnote{See
Enriques--Chisini, Ibidem, Libro~V, Chapter~I, note to \S10, Vol.~III,
page~78 (and page~485).}. But, if one does not want to reduce to that
case, it can also be proved using the criteria given before. Another
remarkable criterion was pointed out by Severi
(1905)\footnote{Osservazioni ecc., Ibidem.}. The result is that, in
general, an $\infty^1$ (at least) series of curves consists of
equivalent curves if the $C$ cut out equivalent groups on only one
generic curve of a continuous system $\{D\}$ of index $\nu>1$.

In fact the condition for an $\infty^1$ series of groups of points on a
curve to consist of equivalent groups (that is, to be contained in a
linear series) can be expressed, with Castelnuovo, via a numerical
criterion, in which appears the number of double points of the series
(that has to attain the maximum value)\footnote{See Enriques--Chisini,
Ibidem, Libro~V, Chapter~IV, 42 (Vol.~II, page~482).}. Now, if this
criterion is satisfied by the groups that are sections of one of the $D$
with the $C$, by continuity it will be satisfied also with respect to
the other $D$, and so one is reduced to the third criterion, given
before.
 \end{Remark}

We add that the equivalence criterion of the $\infty^1$ series of
groups of points on a curve, that, as we recalled above, was
established by Castelnuovo, can be extended also to surfaces, as
R.~Torelli showed. He writes an inequality that is satisfied by the
number $d$ of curves with a double point that belong to an $\infty^1$
series (where the Zeuthen--Segre invariant of the surface appears; see
Chapter~V); the maximum of this number $d$ corresponds to the case when
the series consists of equivalent curves\footnote{See R.!Torelli, Sui
sistemi algebrici di curve appartenenti ad una superficie algebrica.
{\it Atti Acc.\ Torino}, 1907.}.
