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\addcontentsline{toc}{chapter}{J\'anos Koll\'ar: Nonrational covers of
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\markboth{\qquad Nonrational covers of $\C\PP^m \times \C\PP^n$
\hfill}{\hfill J\'anos Koll\'ar \qquad}
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\begin{document}
\title{Nonrational covers of $\protect\C\protect\PP^m\times
\protect\C\protect\PP^n$}
\author{J\'anos Koll\'ar}
\maketitle
\section{Introduction}\label{sec1}
This article aims to provide further examples of higher dimensional
\hbox{varieties} that are\index{rationally connected} rationally
connected, but not rational,\index{nonrational variety} and not even
ruled.\index{nonruled variety} The original methods of
\cite{Iskovskikh-Manin71, Clemens-Griffiths72} have been further developed
by many authors (for example, \cite{Beauville77, Iskovskikh80,
Bardelli84}), and give a fairly complete picture in dimension three. On
the other hand, in higher dimension, only special examples were known until
recently \cite{Artin-Mumford72, Sarkisov81, Sarkisov82, Pukhlikov87,
CTO89}. The rationality question\index{rationality problem} for
hyper\-surfaces in $\PP^n$ was considered in \cite{Ko95}. There I proved
the following:
\begin{thm}{\!\bf\cite{Ko95}}\label{thm1.1}\enspace Let\/ $X_d\subset
\C\PP^{n+1}$ be a very general\index{very general} hypersurface of
degree\/ $d$. Assume that
\begin{equation}
\frac{2}{3}n+3\le d\le n+1. \tag{1.1.1}
\label{eq1.1.1}
\end{equation}
Then\/ $X_d$ is not rational.\index{nonrational variety}\qed
\end{thm}
Here {\em very general} means that the result holds for hypersurfaces
corresponding to a point in the complement of countably many closed
subvarieties in the space of all hypersurfaces.
The method applies to hypersurfaces in $\PP^m\times \PP^{n+1}$. Let
$X_{c,d}$ be such a hypersurface of bidegree $(c,d)$. Via the
projection\index{projection} $X_{c,d}\to \PP^m$, it can be viewed as a
family of degree $d$ hypersurfaces in $\PP^{n+1}$ parametrized by $\PP^m$.
$X_{c,d}$ is rationally connected\index{rationally connected} if $d\le
n+1$, no matter what $c$ is (cf.\ \cite[IV.6.5]{Ko96}).
A straightforward application of the method of \cite{Ko95} provides an
analog of Theorem~\ref{thm1.1} for hypersurfaces when $c\ge m+3$ and $d,n$
satisfy the inequalities (\ref{eq1.1.1}). However, it is more interesting
to study some cases where the fibers of $X_{c,d}\to \PP^m$ are rational.
Here I propose to work out two cases: conic bundles\index{conic bundle}
and families of cubic surfaces.\index{del Pezzo!fibration}
The method of \cite{Ko95} works naturally for\index{cyclic cover} cyclic
covers, and it is easier to formulate the results that way.
Fix a prime $p$ and let $X_{ap,bp}\to \C\PP^m\times \C\PP^n$ be a degree
$p$ cyclic cover\index{cyclic cover} ramified along a very general
hypersurface of bidegree $(ap,bp)$. \cite{Ko95} shows that $X_{ap,bp}$ is
not rational\index{nonrational variety} and not even ruled if $ap>m+1$ and
$bp>n+1$. In this paper, I study the case when $bp=n+1$ and $n=1,2$. The
main results are the following:
\begin{thm}\label{thm1.2} Let\/ $X_{2a,2}\to \C\PP^m\times \C\PP^1$ be a
double cover ramified along a very general hypersurface of bidegree\/
$(2a,2)$, where\/ $m\ge2$. Then\/ $X_{2a,2}$ is not rational if\/ $2a>m+1$.
\index{nonrational variety}
More precisely, if\/ $Y$ is any variety of dimension\/ $m$ and\/
$\fie\colon Y\times\PP^1\broken X_{2a,2}$ a dominant map then
$2\divides\deg\fie$.
\end{thm}
\begin{rmk} \ \par
\begin{description}
\item[1.2.1.1] $X_{2a,2}\to \C\PP^n$ is a conic bundle.\index{conic
bundle} For conic bundles\index{Sarkisov!theorem on conic bundles} we have
the\index{rationality problem} very strong results of
\cite{Sarkisov81,Sarkisov82}, saying that a conic bundle is not rational
if the locus of singular fibers plus 4 times the canonical class of the
base is effective. In our case the locus of singular fibers is a divisor
in $\PP^m$ of degree $4a>2m+2$. This is lower than Sarkisov's bound
$4m+4$. Sarkisov's normal crossing assumptions are also not satisfied by
$X_{2a,2}\to \C\PP^n$. Thus some of our cases are not covered by
Sarkisov's results. This suggests the possibility that the bounds of
\cite{Sarkisov81,Sarkisov82} can be improved considerably. This is not
clear even for conic bundles\index{conic bundle} over
surfaces.\footnote{Compare Corti's article, Section~4.2.}
\item[1.2.1.2] It is natural to ask whether the
projection\index{projection} $X_{2a,2}\to\C\PP^m$ is the only conic bundle
structure of $X_{2a,2}$. The proof gives such examples in characteristic
2, but not over $\C$.
\end{description}
\end{rmk}
For families of cubic surfaces the result is weaker:
\begin{thm}\label{thm1.3} Let\/ $X_{3a,3}\to \C\PP^m\times \C\PP^2$ be a
cyclic triple cover\index{cyclic cover} ramified along a very general
hypersurface of bidegree\/ $(3a,3)$, where\/ $m\ge1$. Then\/ $X_{3a,3}$ is
not rational and\index{nonrational variety}\index{nonruled variety} not
even ruled if\/
$3a>m+1$.
\end{thm}
\begin{rmk} Similar results for $m=1$ were proved by
\cite{Bardelli84}.
\end{rmk}
For hypersurfaces in $\C\PP^m\times \C\PP^{n+1}$ these imply the
following:
\begin{thm}\label{thm1.4} \ \par
\begin{description}
\item[1.4.1] Let\/ $X_{c,2}\subset \C\PP^m\times \C\PP^2$ be a very
general hypersurface of bidegree\/ $(c,2)$, where\/ $m\ge2$. Then\/
$X_{c,2}$ is not rational if\/ $c\ge m+3$.
\item[1.4.2] Let\/ $X_{c,3}\subset \C\PP^m\times \C\PP^3$ be a very general
hypersurface of bidegree\/ $(c,3)$, where\/ $m\ge1$. Then\/ $X_{c,3}$ is
not rational if\/ $c\ge m+4$.
\end{description}
\end{thm}
\begin{pf} By \cite[V.5.12--13]{Ko96}, it is sufficient to find an
algebraically closed field $k$ and a single example of a nonruled
hypersurface $X_{c',2}\subset \PP^m\times \PP^2$, respectively
$X_{c',3}\subset\PP^m\times \PP^3$ over $k$ for some $c'\le c$. We proceed
to construct such examples in characteristic two for conic
bundles\index{conic bundle} and in characteristic three for families of
cubic surfaces.
Consider $\aff^m\times\aff^n$ with coordinates $(u_1,\dots,u_m;
v_1,\dots,v_n)$. Let $f(u,v)$ be a polynomial of bidegree $(c,n+1)$.
We have a hypersurface
\[
Z\deq (y^{n+1}-f(u,v)=0)\subset V\deq \aff^1\times \aff^m\times\aff^n,
\]
where $y$ is a coordinate on $\aff^1$. There are several natural ways to
associate a projective variety to $Z$:
\begin{description}
\item[1.4.3.1] We can view $V$ as a coordinate chart in $\PP^m\times
\PP^{n+1}$. The closure $\Zbar_1$ of $Z$ is a hypersurface of bidegree
$(c,n+1)$. For $n=1,2$ this gives our examples $X_{c,2}$ and $X_{c,3}$.
\item[1.4.3.2] We can compactify $\aff^m\times\aff^n$ to
$\PP^m\times\PP^n$ and view $f$ as a section of the line bundle
$\Oh_{\PP^m\times\PP^n}(c,n+1)$. Assume that $(n+1)\divides c$ and let
$L=\Oh(\frac{c}{n+1},1)$. The corresponding\index{cyclic cover} cyclic
cover $\Zbar_2=\PP^m\times\PP^n[\root{n+1}\of{f}]$ (see
Definition~\ref{dfn2.3}) gives another compactification of $Z$.
\end{description}
Theorems~\ref{thm1.2} and \ref{thm1.3} show, using the second
representation, that $\Zbar_2$ is not ruled if $f$ is very general and
$c>n+1$. This implies Theorem~\ref{thm1.4}. \qed\end{pf}
\subsection{Generalizations} It is clear from the proof that it applies to
many different cases. The main point is to have a family of conics or
cubic surfaces whose branch divisor is sufficiently large, but I found it
hard to write down a reasonably general statement.
\subsection{Outline of the proof of Theorems~\ref{thm1.2}--\ref{thm1.3}} By
\cite[V.5.12--13]{Ko96} it is sufficient to find an algebraically
closed field $k$ and a single example of\index{nonruled variety} a
nonruled\index{cyclic cover} cyclic cover $X_{2a',2}\to \PP^m\times \PP^1$
(respectively, $X_{3a',3}\to \PP^m\times\PP^2$) for some $a'\le a$. We
proceed to construct such examples in characteristic two for conic
bundles\index{conic bundle} and in characteristic three for families of
cubic surfaces.
Fix an algebraically closed field $k$ of characteristic $p$. Let
$\pi\colon X_{ap,bp}\to \PP^m\times \PP^n$ be a degree $p$\index{cyclic
cover} cyclic cover corresponding to a very general hypersurface of
bidegree $(ap,bp)$ (cf.\ Definition~\ref{dfn2.3}). Since $\pi$ is purely
inseparable, $X_{ap,bp}$ is (purely inseparably) unirational. We intend to
show that it is frequently not ruled.
Section~2 is a review of the machinery of inseparable\index{cyclic cover}
cyclic covers and their applications to\index{nonrational variety}
nonrationality problems. $X_{ap,bp}$ has isolated singularities by (2.1.4)
and (2.2.3). These can be resolved; if $\pi_Y\colon Y\to X_{ap,bp}\to
\PP^m\times \PP^n$ is a resolution, then by Corollary~\ref{cor2.5} there
is a nonzero map
\[
\pi_Y^*\Oh(ap-m-1,bp-n-1)\to
\bigwedge^{m+n-1}\Omega_Y^1\iso\Omega_Y^{m+n-1}.
\]
Thus \cite{Ko95} shows that $X_{ap,bp}$ is not rational\index{nonrational
variety}\index{nonruled variety} and not even ruled if $ap>m+1$ and
$bp>n+1$. In this paper I study the case when $ap>m+1$ and $bp=n+1$.
Section~3 contains a nonruledness criterion. Let $f\colon Y\to \PP^m$ be
the composite of $\pi_Y$ with the first projection.\index{projection}
Corollary~\ref{cor3.2} shows that if $ap>m+1$ and $bp=n+1$, then $Y$ is
ruled if and only if the generic fiber of $f$ is ruled over the function
field $k(\PP^m)$. This is the main technical departure from \cite{Ko95}.
There I used varieties in positive characteristic which were shown to be
not even separably uniruled. The varieties here are separably uniruled,
but we are able to get a description of all separable unirulings.
Rationality\index{rationality problem} or ruledness over nonclosed fields
is a very interesting question. Unfortunately I cannot say much, except
for $n=1,2$.
When $n=1$, the generic fiber of $f$ is a plane conic. Conics over
nonclosed fields are considered in Section~4; Theorem~\ref{thm4.1} gives a
complete description of their unirulings. Theorem~\ref{thm1.2} is implied
by Corollary~\ref{cor2.5}, Theorem~\ref{thm3.1} and
Proposition~\ref{prop4.2}.
If $n=2$, the generic fiber of $f$ is a cubic surface $S$, given by an
equation $u^3=f_3(x,y,z)$. Since $X_{ap,bp}$ has only isolated
singularities, $S$ is regular\index{regular versus smooth} over $k(\PP^m)$,
though we will see that it is not smooth (Remark~\ref{rmk5.2}). Thus we
are led in Section~5 to investigate regular Del Pezzo surfaces over
arbitrary fields.\index{del Pezzo!surface over a nonclosed field} We are
forced to study the situation over nonperfect fields; this introduces
several new features. The main result is Theorem~\ref{thm5.7} which
generalizes results of Segre and Manin to nonperfect fields. Together with
Corollary~\ref{cor2.5} and Theorem~\ref{thm3.1}, this result implies
Theorem~\ref{thm1.3}.
\subsection{Terminology} I follow the terminology of \cite{Ko96}.
If a scheme $X$ is defined over a field $F$, and $E$ is a field extension
of $F$, then $X_E$ denotes the scheme obtained by base extension.
For a field $F$, $\Fbar$ denotes an algebraic closure.
$X_F$ is called {\em rational}\index{rational!variety}\index{rational!over
$F$} if it is birational to $\PP^n_F$, and the birational map is defined
over $F$. Sometimes for emphasis I say that $X_F$ is rational over $F$.
The same convention applies to other notions, such as irreducible, etc.
We say that $X_F$ is ruled (respectively uniruled) if there is a scheme
$Y_F$ with $\dim Y_F=\dim X_F-1$ and a birational (respectively dominant)
map $Y_F\times\PP^1\broken X_F$. In the uniruled case, it may happen that
$Y_F$ is reducible over $\Fbar$.
If $X_{\Fbar}$ is rational\index{geometrically!rational} then we say that
$X_F$ is geometrically rational. Similarly for other notions (ruled,
uniruled, irreducible etc.).\index{geometrically!ruled, uniruled, etc.}
Following standard terminology, we say that a scheme is\index{regular
versus smooth} {\em regular} (or {\em nonsingular}) if all of its local
rings are regular. Over nonperfect fields there are schemes that are
nonsingular but not smooth.
\subsection{Acknowledgements} Partial financial support was provided by the
NSF under grant number DMS-9102866. I thank M.\ Grinenko and M.\ Reid for
several useful comments.
\section{Inseparable cyclic covers}\label{sec2}
\index{critical point!in positive characteristic}
\index{cyclic cover}
We first recall the definitions and basic properties\index{critical point}
of critical points of sections of line bundles in positive characteristic.
For proofs, see \cite{Ko95} or \cite[V.5]{Ko96}.
\begin{dfn} Let $X$ be a smooth variety over an algebraically closed field
$k$ and $f$ a function on $X$. Let $x\in X$ be a closed point and assume
that $f$ has a critical point\index{critical point} at $x$. Choose local
coordinates $x_1,\dots, x_n$ at $x$.
\begin{description}
\item[2.1.1] $f$ has a {\em nondegenerate critical point} at $x$ if
$\partial f/\partial x_1,\dots, \partial f/\partial x_n$ generate the
maximal ideal of the local ring $\Oh_{x,X}$. This notion is independent of
the local coordinates chosen.
If $\chr k\ne2$ or $\chr k=2$ and $n$ is even then $f$ has a nondegenerate
critical point at $x$ if and only if $f$ can be written in suitable local
coordinates as
\[
f=c+
\cases x_1x_2+x_3x_4+\dots+x_{n-1}x_n+f_3 &\text{if $n$ is even,}\\
x_1^2+x_2x_3+\dots+x_{n-1}x_n+f_3 &\text{if $n$ is odd,}\\
\endcases
\]
with $f_3\in m_x^3$.
\item[2.1.2] If $\chr k=2$ and $\dim X$ is odd, then every\index{critical
point!degenerate} critical point is degenerate.
\item[2.1.3] Assume that $\chr k=2$ and $\dim X$ is odd. A\index{critical
point} critical point of $f$ is called {\em almost nondegenerate} if $\len
\Oh_{x,X}/(\partial f/\partial x_1,\dots, \partial f/\partial x_n)=2$.
Equivalently, $f$ can be written in suitable local coordinates as
\[
f=c+ax_1^2+x_2x_3+\dots+x_{n-1}x_n+bx_1^3+f_3,\qtq{where $b\ne 0$.}
\]
\item[2.1.4] Assume that $\chr k\divides d$. Then the hypersurface
\[
Z=(y^d-f(x_1,\dots, x_n)=0)
\]
is singular at the point $(y,x)\in Z$ if and only if $x\in X$ is
a\index{critical point} critical point of $f$.
\item[2.1.5] Let $L$ be a line bundle on $X$ and $s\in H^0(X,L^d)$ a
section. Let $U\subset X$ be an affine open subset such that $L\rest{U}\iso
\Oh_U$. Choose such an isomorphism. Then $s\rest{U}$ can be viewed as
section of $\Oh_U^{\tensor d}\iso \Oh_U$. Thus it makes sense to talk
about its\index{critical point} critical points. If $\chr k\divides d$
then this is independent of the choice of $U$ and of the trivialization
$L\rest{U}\iso \Oh_U$. (This fails if the characteristic does not divide
$d$.)
\end{description}
\end{dfn}
The usual Morse lemma\index{Morse lemma in positive characteristic} can be
generalized to positive characteristic. We use it in a somewhat technical
form.
\begin{prop}\label{prop2.2} Let\/ $X$ be a smooth variety over a field of
characteristic\/ $p$, and\/ $L$ a line bundle on\/ $X$. Let\/ $d$ be an
integer divisible by\/ $p$ and\/ $W\subset H^0(X,L^d)$ a finite
dimensional vector subspace. Let\/ $m_x$ denote the ideal sheaf of\/ $x\in
X$. Assume that:
\begin{description}
\item[2.2.1] For every closed point\/ $x\in X$ the restriction map\/ $W\to
(\Oh_X/m_x^2)\tensor L^k$ is surjective,
\item[2.2.2] For every closed point\/ $x\in X$ there is an\/ $f_x\in W$
which has an (almost) nondegenerate\index{critical point} critical point
at\/ $x$.
\end{description}
Then a general section\/ $f\in W$ has only (almost) nondegenerate critical
points.\index{critical point}
\end{prop}
\begin{pf} This is a simple dimension count. Fix $x\in X$ and let
$W_x\subset W$ be the set of functions with a\index{critical point}
critical point at $x$. By (2.2.1), $W_x$ has codimension $n$. In $W_x$ the
functions with an (almost) nondegenerate\index{critical point} critical
point at $x$ form an open subset $W_x^0$ which is nonempty by (2.2.2).
Thus the set of functions with a degenerate critical point is $\bigcup_x
(W_x-W_x^0)$ and it has codimension at least one in $W$.
\qed
\end{pf}
\stepcounter{slem}
\stepcounter{slem}
\begin{slem}\label{slem2.2.3} Let\/ $X_1,X_2$ be smooth varieties over a
field of characteristic\/ $p$ and\/ $L_i$ very ample line bundles on\/
$X_i$. Let\/ $L=p_1^*L_1\tensor p_2^*L_2$ be the corresponding line bundle
on\/ $X=X_1\times X_2$. If\/ $p\divides d$ then a general section\/ $f\in
H^0(X,L)$ has only (almost) nondegenerate critical points.
\end{slem}
\begin{pf} Pick a point $x=(x_1,x_2)$. The condition (2.2.1) is clearly
satisfied. To check (2.2.2), choose global sections $u_i\in H^0(X_1,L_1)$
and $v_j\in H^0(X_2,L_2)$ that give local coordinates at $x_1$,
respectively $x_2$.
If $p\ne 2$ then $\sum u_i^2+\sum v_j^2$ gives a section of $L^d$ with a
nondegenerate\index{critical point} critical point at $x$.
If $p=2$ we need to consider a few cases. Set $n_i=\dim X_i$. We plan
to use the function
\[
g=\sum_{i=1}^{n_i/2}u_{2i-1}u_{2i} + \sum_{j=1}^{n_2/2} v_{2j-1}v_{2j}.
\]
If both of the $n_i$ are even, then we can take $f=g$. If both of
the $n_i$ are odd, then we can use $f=g+u_{n_1}v_{n_2}$. Otherwise
we may assume that
$n_1$ is odd and $n_2$ is even. Then we use
\[
f=g+u_{n_1}v_{n_2-1}+u_{n_1}^2v_{n_2}.
\]
Explicit computation shows that $f$ has an almost nondegenerate
critical point.\qed\end{pf}
\begin{dfn}[Cyclic covers]\label{dfn2.3}
Let $X$ be a scheme, $L$ a line bundle on $X$ and $s\in H^0(X, L^d)$ a
section. Assume for simplicity that its divisor of zeros $(s=0)$ is
reduced. The\index{cyclic cover} cyclic cover of $X$ obtained by taking a
$d$th root of $s$, denoted by $X[\root{d}\of{s}]$ is the scheme locally
constructed as follows:
Let $U\subset X$ be an open set such that $L\rest{U}\iso \Oh_U$. Then
$s\rest{U}$ can be identified with a function $f\in H^0(U,\Oh_U)$. Let
$V\subset\aff^1\times U$ be the closed subset defined by the equation
$y^d-f=0$ where $y$ is the coordinate on $\aff^1$. The resulting schemes
can be patched together in a natural way to get a scheme
$X[\root{d}\of{s}]$; cf.\ \cite[II.6.1]{Ko96}. We are only interested in
it up to birational equivalence, so the precise definitions are
unimportant.
\end{dfn}
The only result about\index{cyclic cover} cyclic covers we need is the
following special case of \cite[V.5.10]{Ko96}:
\begin{prop}\label{prop2.4} Let\/ $X$ be a smooth variety of dimension\/
$n$ over a field\/ $k$ of characteristic\/ $p$, $L$ a line bundle on\/ $X$
and\/ $d$ an integer divisible by\/ $p$. Suppose that\/ $s\in H^0(X, L^d)$
is a section with (almost) nondegenerate\index{critical point} critical
points, and let\/ $\pi\colon Y\to X$ be a smooth projective model of\/
$X[\root{d}\of{s}]$ ($Y$ always exists).
Then there is a nonzero map
\[
\pi^*(K_X\tensor L^d)\to
\bigwedge^{n-1}\Omega_Y^1\iso\Omega_Y^{n-1}.\qed
\]
\end{prop}
Applied to the\index{cyclic cover} cyclic covers
$\Zbar_2=\PP^m\times\PP^n[\root{p}\of{s}]$ from the introduction (see
1.4.3.2), we get the following:
\begin{cor}\label{cor2.5} Fix a prime\/ $p$ and let\/ $k$ be an
algebraically closed field of characteristic\/ $p$. Let\/ $s\in
H^0(\PP^m\times\PP^n,\Oh(a,b)^{\tensor p})$ be a general section and\/
$q\colon Y\to \PP^m\times\PP^n$ a smooth projective model of\/
$\PP^m\times\PP^n[\root{p}\of{s}]$.
Then there is a nonzero map
\[
q^*\Oh(ap-m-1,bp-n-1)\to
\bigwedge^{m+n-1}\Omega_Y^1\iso\Omega_Y^{m+n-1}.\qed
\]
\end{cor}
\section{A nonruledness criterion}\label{sec3}
In this section we prove the following generalization of
\cite[V.5.11]{Ko96}.
\begin{thm}\label{thm3.1} Let\/ $X,Y$ be smooth proper varieties and\/
$f\colon Y\to X$ a surjective morphism, where\/ $n=\dim Y$. Let\/ $M$ be a
big line bundle on\/ $X$ and assume that for some\/ $i>0$ there is a
nonzero map
\[
h\colon f^*M\to \bigwedge^i\Omega_Y^1.
\]
\begin{description}
\item[3.1.1] Let\/ $Z$ be an affine variety of dimension\/ $n-1$ and\/
$\fie\colon Z\times \PP^1\to Y$ a dominant and separable morphism. Then
there is a morphism\/ $\psi\colon Z\to X$ which fits in the commutative
diagram
\[
\CD Z\times \PP^1 @>\fie>> Y\\ @VVV @VVfV\\ Z @>\psi>> X.
\endCD
\]
\item[3.1.2] Let\/ $F=k(X)$ be the field of rational functions on\/ $X$
and\/ $Y_F$ the generic fiber of\/ $f$. There is a one-to-one correspondence
\[
\left\{
\begin{matrix}
\hbox{degree $d$ separable}\\
\hbox{unirulings of\/ $Y$}
\end{matrix}
\right\} \bij
\left\{
\begin{matrix}
\hbox{degree $d$ separable}\\
\hbox{unirulings of\/ $Y_F$}
\end{matrix}
\right\}.
\]
In particular, $Y$ is ruled if and only if\/ $Y_F$ is ruled over\/ $F$.
\item[3.1.3] Assume that for any two general points\/ $y_1,y_2\in
Y_{\Fbar}$ there is a morphism\/ $f=f(y_1,y_2)\colon \PP^1\to Y_{\Fbar}$
such that\/ $y_1,y_2\in \im f$, $Y_{\Fbar}$ is smooth along\/ $\im f$
and\/ $f^*T_{Y_{\Fbar}}$ is semipositive. Then any birational selfmap of\/
$Y$ preserves\/ $f$. Hence there is an exact sequence
\[
1\to \Bir(Y_F)\to \Bir(Y)\to \Bir (X).
\]
\end{description}
\end{thm}
\setcounter{slem}{3}
\begin{rmk} The assumption (3.1.3) is satisfied if $Y_{\Fbar}$ is
separably rationally connected\index{rationally connected} (cf.\
\cite[IV.3.2]{Ko96}). More generally, it also holds for the\index{cyclic
cover} cyclic covers of $\PP^n$ that we are considering
\cite[V.5.19]{Ko96}.
\end{rmk}
\begin{pf} $M$ is big, hence there is an open set $U\subset X$ such
that sections of $M^k$ separate points of $U$ for $k\gg 1$. In
particular, if $g\colon C\to X$ is a nonconstant morphism from a smooth
proper curve to $X$ whose image intersects $U$, then $\deg g^*M>0$.
Let $g\colon C\to Y$ be a morphism such that $g^*\Omega^1_Y$ is
seminegative. We have a map
\[
g^*h\colon g^*f^*M\to \bigwedge^i g^*\Omega^1_Y.
\]
Thus either $(f\circ g)(C)\subset X\setminus U$ or $(f\circ g)(C)$ is a
single point. This will allow us to identify the fibers of $f$.
To prove (3.1.1), pick a general point $z\in Z$ and let $\fie_z\colon
\PP^1\to Y$ be the restriction of $\fie$ to $\{z\}\times \PP^1$. Then
\[
\Omega_{Z\times \PP^1}^1\rest{\{z\}\times \PP^1} \iso
\Oh_{\PP^1}^{n-1}+\Oh_{\PP^1}(-2),
\]
and $\fie$ gives a map
\[
\Phi\colon (f\circ \fie_z)^*M \xrightarrow{\ \fie_z^*h\ }
\fie_z^*\bigwedge^i\Omega_Y^1 \xrightarrow{\ \bigwedge^id\fie\ }
\bigwedge^i\Bigl(\Oh_{\PP^1}^{n-1}+\Oh_{\PP^1}(-2)\Bigr),
\]
which is nonzero for general $z$ since $\fie$ is separable. Thus
$\deg(f\circ \fie_z)^*M\le 0$. By the above remarks, this implies that
$f\circ \fie_z$ is a constant morphism.
Pick a point $0\in \PP^1$ and define $\psi\colon Z\to X$ by $\psi(z)\deq
f\circ \fie(z,0)$. This proves (3.1.1).
Let $Z_F$ be the generic fiber of $\psi$. We obtain a dominant $F$-morphism
$\fie_F\colon Z_F\times \PP^1\to Y_F$ which is birational (respectively,
separable) if and only if $\fie$ is birational (respectively, separable).
Conversely, if $W_F$ is any variety and $W_F\to Y_F$ a morphism, then it
extends to a map $W\broken Y$ of the same degree. This proves (3.1.2).
Finally assume (3.1.3). Then there is an open set $Y^0\subset Y$ with the
property that if $y_1,y_2\in Y^0$ and $f(y_1)=f(y_2)$, then there is a
morphism $f=f_{(y_1,y_2)}\colon \PP^1\to Y$ such that $y_1,y_2\in \im f$,
$Y$ is smooth along $\im f$ and $f^*T_Y$ is semipositive.
Let $\fie\colon Y\broken Y$ be a birational selfmap of $Y$; $\fie$ is defined
outside a codimension 2 set $Z\subset Y$. By \cite[II.3.7]{Ko96}, the image
of the general $f_{(y_1,y_2)}$ is disjoint from $Z$. Thus we have an
injection
\[
f_{(y_1,y_2)}^*T_Y\hookrightarrow (\fie\circ f_{(y_1,y_2)})^*T_Y,
\]
which shows that the latter is also semipositive. Thus $\fie(y_1)$ and
$\fie(y_2)$ are in the same fiber of $f$. Therefore $\fie$ preserves $f$,
which gives the exact sequence
\[
1\to \Bir(Y_F)\to \Bir(Y)\to \Bir(X).\qed
\]
\end{pf}
Applying Theorem~\ref{thm3.1} to the projection\index{projection} $Y\to
\PP^m\times\PP^n\to \PP^m$ of Corollary~\ref{cor2.5} we obtain:
\begin{cor}\label{cor3.2} Fix a prime $p$ and let $k$ be an algebraically
closed field of characteristic $p$. Let $s\in
H^0(\PP^m\times\PP^n,\Oh(a,b)^{\tensor p})$ be a general section. Assume
that $ap>m+10$ and $bp=n+1$.
Then $Y'\deq \PP^m\times\PP^n[\root{p}\of{s}]$ is ruled if and only if the
generic fiber of\/ $Y'\to \PP^m$ is ruled over the field\/
$k(x_1,\dots,x_m)$.
Furthermore, $Y'$ has a degree $d$ separable uniruling if and only if the
generic fiber of\/ $Y'\to \PP^m$ has a degree $d$ separable uniruling over
the field\/ $k(x_1,\dots,x_m)$.
\qed
\end{cor}
Corollary~\ref{cor3.2} naturally leads to the following:
\begin{quest} Let $X_F$ be a variety over a field. When is
$X_F$ ruled over $F$?
\end{quest}
The problem is mainly interesting when $X_F$ is geometrically
ruled.\index{geometrically!ruled} I cannot say much in general, so I
consider only two simple examples: conics and Del Pezzo surfaces. One
should bear in mind that in our applications, $F$ is the function field of
a variety in positive characteristic, so is not perfect. Also, we need
these results in characteristic 2 for conics and in characteristic 3 for
cubic surfaces. These are the most unusual cases.
\begin{exa} If $X_F$ is an arbitrary variety having no $F$-points, then
$X_F$ is not rational, but it can happen that it is ruled. For instance,
if $Y_F$ has no $F$-points then $Y\times\PP^1$ is ruled and has no
$F$-points.
This can happen even for a quadric in $\PP^3$. For example, choose $a,b\in
F$ such that $C=(x_0^2+ax_1^2+bx_2^2=0)$ has no $F$-points (say $F=\R$ and
$a=b=1$). Then $C\times\PP^1$ is birational to the quadric
$Q=(y_0^2+ay_1^2+by_2^2+aby_3^2=0)$ via the map
\[
\fie\colon(x_0:x_1:x_2,s:t)\mapsto (sx_0+atx_1:sx_1-tx_0:sx_2:tx_2);
\]
but $Q$ clearly has no $F$-points.
\end{exa}
\section{Conics over nonclosed fields}\label{sec4}
The aim of this section is to study conics over arbitrary fields. We study
when they are ruled or uniruled. The main result is Theorem~\ref{thm4.1},
but for the applications we also need Proposition~\ref{prop4.2}.
\begin{thm}\label{thm4.1} Let\/ $F$ be a field and\/ $C_F\subset\PP^2_F$
an irreducible and reduced conic (which may be reducible or nonreduced
over $\Fbar$). The following are equivalent:
\begin{description}
\item[4.1.1] $C_F$ has a point in\/ $F$.
\item[4.1.2] $C_F$ is ruled.
\item[4.1.3] $C_F$ has an odd degree uniruling.
\end{description}
If\/ $C_F$ is smooth then these are also equivalent to
\begin{description}
\item[4.1.4] $C_F\iso\PP^1_F$.
\end{description}
\end{thm}
\begin{pf} Let $P$ be an $F$-point of $C$. If $C$ is geometrically
irreducible,\index{geometrically!irreducible} then projecting it from $P$
gives a birational map $C\to \PP^1_F$; hence $C$ is ruled. Otherwise,
projection exhibits $C$ as a cone over a length two subscheme of
$\PP^1_F$, thus again $C$ is ruled. This shows that (4.1.1) $\Rightarrow$
(4.1.2) while (4.1.2) $\Rightarrow$ (4.1.3) is clear.
The proof of (4.1.3) $\Rightarrow$ (4.1.1) is longer. Let $A$ be a
zero dimensional $F$-scheme and $\fie_F\colon A\times_F\PP^1\to C_F$ an odd
degree uniruling. Let $A_i\subset A$ be the irreducible components.
$\deg \fie_F=\sum_i\deg (\fie_F\rest{A_i\times\PP^1})$, thus one of the
$\deg (\fie_F\rest{A_i\times\PP^1})$ is odd. Thus we may assume that $A$
is irreducible. $\deg (\fie_F\rest{\red A\times\PP^1})$ divides $\deg
\fie_F$, hence we may also assume that $A=\spec_FF'$ where $F'\supset F$
is a field extension.
By base change to $F'$ we obtain
\[
\PP^1_{F'}\hookrightarrow \spec_{F'}( F'\tensor_FF')\times \PP^1_F\to
C_{F'}.
\]
Thus by the L\"uroth theorem, every irreducible component of $\red
C_{F'}$ is birational to $\PP^1_{F'}$. Passing to the algebraic
closure we obtain
\[
\fie_{\Fbar}\colon \spec_{\Fbar}(\Fbar\tensor_FF')\times \PP^1\to C_{\Fbar}.
\]
The left hand side may have several irreducible components, conjugate to
each other. Let $\fiebar\colon \PP^1\to \PP^1$ be the induced map between
any of the irreducible and reduced components. By counting degrees we
obtain the following:
\setcounter{slem}{4}
\begin{claim}\label{claim4.1.5} Notation as above. Then
\begin{description}
\item[4.1.5.1] $\deg \fie_F=\deg(F'/F)\deg (\fiebar)$ if\/ $C_{\Fbar}$ is
a smooth conic, and
\item[4.1.5.2] $2\deg \fie_F=\deg(F'/F)\deg (\fiebar)$ if\/ $C_{\Fbar}$
is a singular conic.
\end{description}
Thus\/ $\deg (F'/F)$ is odd in the first case, and not divisible by\/ $4$ in
the second.\qed
\end{claim}
Assume first that $C_{\Fbar}$ is a smooth conic. $A\times_F\spec F'$ has
a closed point, thus we get a morphism $\PP^1_{F'}\to C_{F'}$. Hence
$C_{F'}$ has a point in $F'$. This in turn gives an odd degree point on
$C_F$; let $L$ be the corresponding line bundle. The restriction of
$\Oh_{\PP^2}(1)$ to $C_F$ is a line bundle of degree 2. We conclude that
$C_F$ has a line bundle of degree 1. Any of its sections gives an
$F$-point on $C_F$.
If $C_{\Fbar}$ is a pair of intersecting lines, then the intersection
point is defined over $F$.
Finally consider the case when $C_{\Fbar}$ is a double line; this can
happen only in characteristic 2. The equation of $C_F$ is $\sum
b_ix_i^2=0$.
\begin{claim}\label{claim4.1.6} Let\/ $C_F=(\sum b_ix_i^2=0)$ be an
irreducible conic over a field of characteristic\/ $2$. Let\/ $E/F$ be a
separable extension. Then any\/ $E$-point of\/ $C_F$ is an\/ $F$-point.
\end{claim}
\begin{pf} We may assume that $E/F$ is Galois. Assume that $P$ is an
$E$-point which is not an $F$-point. Conjugates of $P$ over $F$ also give
$E$-points, thus we obtain that $\red C_{\Fbar}$ is defined over $E$.
$\red C_{\Fbar}$ is also defined over the purely inseparable extension
$F^i=F(\sqrt{b_0},\sqrt{b_1},\sqrt{b_2})$, hence also over the intersection
$F''\cap F^i=F$ (cf.\ \cite[I.3.5]{Ko96}). This is a contradiction.
\qed\end{pf}
As in the irreducible case, we know that $C_{F'}$ has an $F'$-point. Thus
by Claim~4.1.6, $F'/F$ is not separable. In view of (4.1.5.2) we conclude
that there is a subextension $F'\supset F''\supset F$ such that $F'/F$ has
odd degree (hence is separable) and $F'=F''(\sqrt{s})$ for some $s\in
F''$. By Claim~4.1.6 it is enough to show that $C$ has an $F''$-point. As
we mentioned earlier, $\red C_{F'}$ is birational to $\PP^1_{F'}$, thus
$\red C_{F'}$ is a line in $\PP^2$. Therefore $C_{F'}$ is a double line
with equation $(\sum a_ix_i)^2=0$ where $a_i\in F'$. $a_i^2\in F''$, thus
the equation of $C$ over $F''$ is $\sum a_i^2x_i^2=0$. Write $a_i=c_i+sd_i$
where $c_i,d_i\in F''$. The equation of $C$ is
\[
\sum a_i^2x_i^2=(\sum c_ix_i)^2+s^2(\sum d_ix_i)^2=0.
\]
The solution of $\sum c_ix_i=\sum d_ix_i=0$ gives an $F''$-point
on
$C$.
The equivalence with (4.1.4) was established in the course of the
proof. \qed\end{pf}
\begin{rmk} If $C_{\Fbar}$ is a smooth conic, then $C_F$ has a degree 2
separable uniruling. Take any general line in $\PP^2$. Its intersection
points with $C_F$ are in a separable extension $E\supset F$ of degree 2.
Thus $C_E\iso \PP^1_E$.
\end{rmk}
It remains to establish that the generic fibers appearing in
Theorem~\ref{thm3.1} are not ruled. There should be a general result about
conics over function fields, but I could not find a simple proof. For our
applications the following is sufficient:
\begin{prop}\label{prop4.2} Let\/ $k$ be a field of characteristic\/ $2$
and\/ $F=k(x_1,\dots,x_m)$ the field of rational functions in\/ $m\ge2$
variables. Fix an even integer\/ $d\ge2$ and let\/ $a,b,c\in
k[x_1,\dots,x_m]$ be general (inhomogeneous) polynomials of degree\/ $d$.
Then the conic
\[
C=(y_0^2=ay_1^2+by_1y_2+cy_2^2)\subset
\PP^2_F
\]
is not ruled (over\/ $F$). Moreover\/ $C$ does not have any odd degree
uniruling.
\end{prop}
\begin{coms} It is worth remarking that for Proposition~\ref{prop4.2} to
hold, it is essential that $k(x_1,\dots,x_m)$ is not perfect. A point on
$C$ is given by $P=(\sqrt{a},1,0)$. If $F$ is perfect of characteristic 2,
then $P$ is an $F$-point and $C$ is rational.
\end{coms}
\begin{pf} By Theorem~\ref{thm4.1}, it is sufficient to establish that
$C$ has no $F$-points. We can identify $ay_1^2+by_1y_2+cy_2^2$ with a
section $s$ of $\Oh_{\PP^{m}\times \PP^1}(d,2)$. Let $Y\deq \PP^{m}\times
\PP^1[\sqrt{s}]$ be the corresponding double cover. The generic fiber of
$\pi\colon Y\to \PP^{m}$ is $C$; thus it is sufficient to prove that $\pi$
has no sections. By Lemma~\ref{slem2.2.3}, $Y$ has only isolated
singularities. The fibers of $\pi$ over $b=0$ are double lines. This shows
that $\pi$ does not even have local sections at the generic point of
$b=0$.\qed\end{pf}
\section{Del Pezzo surfaces over a nonclosed field}\label{sec5}
\index{del Pezzo!surface over a nonclosed field|(}
\subsection{Del Pezzo surfaces}
To complete the proof of Theorem~\ref{thm1.3}, we need to study the Del
Pezzo\index{del Pezzo!surface over a nonclosed field} surfaces:
\begin{description}
\item[5.1.1] $S_F$ with equation $u^3=f_3(x,y,z)$ in characteristic 3.
\noindent Although we do not need it, it is very natural to study
also the surfaces:
\item[5.1.2] $T_F$ with equation $u^2=f_4(x,y,z)$ in characteristic 2.
\end{description}
To get an idea of the geometry of these surfaces, we study them first over
perfect fields.
\begin{srmk}[The case of perfect fields]\label{rmk5.2} Assume first that
our base field is algebraically closed. It is easy to see that we can
write
\[
f_3=l_1l_2l_3+l_4^3,\qtq{and} f_4=l_1l_2l_3l_4+q^2
\]
where the $l_i$ are linear and $q$ quadratic. Thus we can make
coordinate changes $u=u-l_4$ (respectively, $u=u-q$) and then a suitable
coordinate change among the $x,y,z$ to reduce the equations to the
form
\[
u^3=xyz,\qtq{and} u^2=xyz(x+y+z).
\]
{From} this we see that the cubic $S$ has three singular points of type
$A_2$. The degree two Del Pezzo $T$ has seven singular points of type
$A_1$ at the seven points of\/ $\F_2\PP^2$.
Consider next the case when the base field $F$ is perfect. Then these
singular points are defined over $F$. For the cubic $S_F$ resolve the
singular points, and contract the birational transforms of the 3
coordinate axes to get a Del Pezzo surface of degree~6. Over a perfect
field $F$, a Del Pezzo surface of degree~6 is rational\index{rationality
problem} if and only if it has an $F$-point \cite[IV.7.8]{Manin72}. Our
surface $S_F$ does have $F$-points over perfect fields of characteristic
3, for example $P=(1,0,0,\root{3}\of{f_3(1,0,0)})$.
For the surface $T_F$, resolve the singular points, and contract the
birational transforms of the seven lines in $\F_2\PP^2$ to get a
Brauer--Severi variety. It has a point in a degree 7 extension, hence it is
isomorphic to $\PP^2$. Thus our surfaces $S_F$ and $T_F$ are rational over
any perfect field.
\end{srmk}
In our case the field $F$ is not perfect, the surfaces $S_F$ and $T_F$ are
regular\index{regular versus smooth} over $F$ but they are not smooth. In
order to show that they are not ruled, we have to understand how the
presence of regular but nonsmooth points effects the birational geometry
of a surface.
The crucial result is the following:
\begin{thm}\label{thm5.3} Let\/ $F$ be a field and\/ $S,T$ regular and
proper surfaces over\/ $F$. Assume that\/ $T$ is smooth except possibly at
finitely many points. Let\/ $f\colon S\broken T$ be a birational map.
Then\/ $f$ is defined at all nonsmooth points of\/ $S$.
\end{thm}
\begin{pf} There is a sequence $p\colon S'\to S$ of blowups of closed
points such that $f\circ p\colon S'\to T$ is a morphism. Let $P\in S$ be a
closed nonsmooth point, and assume that $f$ is not defined at $P$. Then
$p^{-1}(P)$ is 1-dimensional and there is an irreducible component
$E\subset p^{-1}(P)$ such that $p\circ f$ is a local isomorphism at the
generic point of $E$. This implies that $S'$ is smooth at the generic
point of $E$. This contradicts the following lemma.\qed\end{pf}
\begin{slem}\label{slem5.3.1} Let\/ $F$ be a field and\/ $S,S'$ regular
surfaces\index{regular versus smooth} over\/ $F$. Suppose that\/ $p\colon
S'\to S$ is a proper and birational morphism and\/ $P\in S$ a closed
nonsmooth point.
Then\/ $S$ is not smooth at all points of\/ $p^{-1}(P)$.
\end{slem}
\begin{pf} By induction, it is sufficient to consider the case when $p$ is
the blowup of $P$. We may also assume that $S$ is an affine neighborhood
of $P$ such that the maximal ideal $m_P$ is generated by two global
sections $u,v\in m_P\subset \Oh_S$. The blowup can be described by two
affine charts, one of which is
\[
S'_1\deq (u-vs=0)\subset S\times \aff^1, \qtq{where $s$ is a global
coordinate on $\aff^1$.}
\]
By assumption $S$ is not smooth at $P$, so $S\times \aff^1$ is not smooth
along $P\times \aff^1$. $S'_1$ is a Cartier divisor on $S\times\aff^1$,
thus is it also not smooth along $P\times\aff^1$. This is what we had to
prove. \qed\end{pf}
In the course of the proof we used the following results about birational
transformations of regular surfaces. See, for example,
\cite{Zariski58} for a proof.
\begin{sprop}\label{sprop5.3.2} Let\/ $S,T$ be proper and regular
surfaces over a field\/ $F$ and\/ $\fie\colon S\broken T$ a birational map.
Then\/ $\fie$ is a composite of blowups and blowdowns (of closed points).
In particular, $h^i(S,\Oh_S)=h^i(T,\Oh_T)$. \qed
\end{sprop}
\begin{scor}\label{scor5.3.3} Let\/ $F$ be a field and\/ $S$ a
regular\index{regular versus smooth} and proper surface over\/ $F$. Assume
that\/ $S$ is smooth except possibly at finitely many points. Let\/
$f\colon S\broken S$ be a birational map.
Then\/ $f$ is a local isomorphism at all nonsmooth points of\/ $S$.
\end{scor}
\begin{pf} Factor $f$ as
\[
f\colon S\xleftarrow{\ p\ } S' \xrightarrow{\ p'\ } S
\]
where $p,p'$ are birational morphisms. By Theorem~\ref{thm5.3} we see
that $p$ and $p'$ are both local isomorphisms at the nonsmooth points of
$S$.\qed\end{pf}
This gives the following rationality criterion:\index{rationality problem}
\begin{cor}\label{cor5.4} Let\/ $F$ be a field and\/ $S$ a proper
regular\index{regular versus smooth} surface over\/ $F$. Assume that\/ $S$
is generically smooth. The following are equivalent:
\begin{description}
\item[5.4.1] $S$ is rational (over\/ $F$).
\item[5.4.2] There is a two dimensional linear system\/ $L=|C|$ on\/ $S$
with (infinitely near) base points\/ $P_i$ of multiplicity\/ $m_i$ such
that
\begin{description}
\item[5.4.2.1] a general\/ $C\in L$ is birational to\/ $\PP^1$;
\item[5.4.2.2] $S$ is smooth along a general\/ $C\in L$;
\item[5.4.2.3] $C\cdot K_S+\sum m_i=-3$ and\/ $C^2-\sum m_i^2=1$.
\end{description}
\end{description}
\end{cor}
\begin{pf} Assume that there is a birational map $f\colon S\broken
\PP^2_F$. Let $Z\subset S$ be the locus of nonsmooth points. By
Theorem~\ref{thm5.3}, $f$ is defined along $Z$ and $f(Z)$ is zero
dimensional. Set $L=f^{-1}_*|\Oh_{\PP^2}(1)|$. In (5.4.2.1--2) are clear,
and (5.4.2.3) is the usual equalities (5.4.3.1).
Conversely, assume (5.4.2). Resolve the base points of $L$ to obtain a
base point free linear system $L'$ on $S'$. {From} (5.4.2.3) we obtain that
\[
C'\cdot K_{S'}=-3\qtq{and}{C'}^2=1\qtq{ for $C'\in L'$.}
\]
Thus the linear system $L'$ maps $S'$ birationally to $\PP^2_F$.
\qed\end{pf}
What we really need is a ruledness criterion. If $S$ is ruled, it can be
birational to a surface which is the product of $\PP^1$ with a conic which
has no $F$-points. Thus the natural linear system obtained on $S$ is two
dimensional and its general member is geometrically reducible. I found it
clearer to concentrate instead on a single curve, which may not be defined
over $F$. We do not get an equivalence any longer, but for our applications
this does not matter.
\setcounter{slem}{2}
\begin{rmk} For linear systems we considered base points with
multiplicities. If we look at a general member, the corresponding notion
is a curve with assigned multiplicities. All infinitely near multiple
points are assigned with their multiplicity, but we may also have some
smooth points assigned with multiplicity one.
If $C\subset S$ is a curve on a smooth surface with assigned multiplicities
$m_i$ at the points $P_i$ and $\fie\colon S\to S'$ is a birational map,
there is a natural birational transform $C'\subset S'$ with assigned
multiplicities $m'_i$ at the points $P'_i$. To define this, we need only
the case when $\fie$ is the blowup of a point or its inverse, where the
definition is the obvious one. The values of the expressions
\begin{equation}
C\cdot K_S+\sum m_i\qtq{and}C^2-\sum m_i^2 \tag{5.4.3.1}
\end{equation}
are birational invariants, cf.\ \cite[p.~5]{Hudson27}.
\end{rmk}
\begin{cor}\label{cor5.5} Let\/ $F$ be a field and\/ $S$ a regular
and\index{regular versus smooth} proper surface over\/ $F$ such that\/ $S$
is generically smooth, geometrically
irreducible\index{geometrically!irreducible} and\/ $h^1(S,\Oh_S)=0$.
Assume that\/ $S$ is ruled (over\/ $F$).
Then there is a rational curve\/ $C\subset S_{\Fbar}$ with assigned
(infinitely near) multiple points\/ $P_i$ of multiplicity\/ $m_i$ such
that
\begin{description}
\item[5.5.1] $S_{\Fbar}$ is smooth along\/ $C$.
\item[5.5.2] $C\cdot K_S+\sum m_i=-2$ and\/ $C^2-\sum m_i^2=0$.
\item[5.5.3] $\Oh_{S_{\Fbar}}(2C)\in \Pic S$.
\end{description}
\end{cor}
\begin{pf} By assumption there is a regular, geometrically integral curve
$D$ and a birational map $f\colon S\broken D\times \PP^1$. By
Proposition~\ref{sprop5.3.2},
\[
0=h^1(S,\Oh_S)=h^1(D\times \PP^1,\Oh_{D\times \PP^1})=h^1(D,\Oh_D).
\]
Thus $D$ is isomorphic to a smooth conic.
Let $d\in D_{\Fbar}$ be a general point, $C'= d\times \PP^1$ and
$C=f^{-1}_*(C')\subset S_{\Fbar}$ the corresponding birational transform.
$S$ is smooth along $C$ by Theorem~\ref{thm5.3}, and (5.5.2) is the usual
equalities (5.4.3.1).
Finally, $D$ has a degree 2 point defined over $F$, thus the line bundle
$\Oh_{D\times \PP^1}(2C')$ is defined over $F$. Therefore
$\Oh_{S_{\Fbar}}(2C)$ is also defined over $F$. \qed\end{pf}
The main result of this section is Theorem~\ref{thm5.7}. Over a perfect
field it is a special case of more general results of Segre and Manin (see
\cite{Manin72}). The proofs in \cite{Manin72} use the structure of the
Picard group of smooth Del Pezzo surfaces to compute the action of certain
involutions.\index{involution} In our case the Picard groups are small by
Lemma~\ref{lem5.6}, and it is easier to use the geometric ideas of
\cite{Segre43,Segre51} to analyze the involutions. Theorem~\ref{thm5.3}
essentially says that all the relevant geometry takes place inside the
smooth locus, where the geometric description of the involutions works
well.
\begin{lem}\label{lem5.6} Let\/ $S_F$ denote an integral cubic
surface\/ $u^3=f_3(x,y,z)\subset \PP^3$ for\/ $\chr F=3$, or an integral double
plane with equation\/ $u^2=f_4(x,y,z)$ for\/ $\chr F=2$.
Then\/ $\Pic(S_F)=\Z[-K_S]$.
\end{lem}
\begin{pf} Let $\pi\colon S_F\to \PP^2$ be the
projection\index{projection} to the $(x,y,z)$-plane. $\pi$ is purely
inseparable, thus if $C\subset S_F$ is any divisor then
$\pi^*(\pi_*(C))=(\deg \pi)C$. Therefore $-K_S=\pi^*(\Oh_{\PP^2}(1))$
generates $\Pic(S_F)\tensor\Q$. Since $K_S^2\le 3$, we get that $K_S$ is
not divisible in $\Pic(S_F)$, hence $-K_S$ generates
$\Pic(S_F)/(\text{torsion})$. Thus we are left to prove that $\Pic S$ has
no torsion.
Let $[C]\in \Pic S$ be a numerically trivial Cartier divisor. By
Riemann-Roch,
\[
h^0(\Oh_S(C))+h^0(\Oh_S(K_S-C))\ge\chi(\Oh_S)=1.
\]
$-K_S$ is ample, so $h^0(\Oh_S(K_S-C))=0$. Thus $\Oh_S(C)$ has a section
and $\Oh_S(C)\iso \Oh_S$.\qed\end{pf}
\begin{thm}\label{thm5.7} Let\/ $F$ be a field and\/ $S$ a regular Del
Pezzo surface over\/ $F$. Assume that\/ $S_{\Fbar}$ is integral,\/
$\chi(\Oh_S)=1$, $\Pic S=\Z[-K_S]$ and\/ $K_S^2=1,2,3$.
Then\/ $S$ is not ruled (over\/ $F$).
\end{thm}
\begin{rmk} Let $S$ be a regular Del Pezzo surface over $F$ such that
$S_{\Fbar}$ is integral, $\chi(\Oh_S)=1$, and $K_S^2=1,2,3$. The structure
theory of Del Pezzo surfaces shows that $S$ is a cubic surface for
$K_S^2=3$, a double plane for $K_S^2=2$ and as expected for $K_S^2=1$
(cf.\ \cite[III.3]{Ko96}).
\cite{Reid94} contains examples of nonnormal Del Pezzo surfaces for which
$\chi(\Oh_S)<1$. Some of these may have regular\index{regular versus
smooth} models over nonperfect fields. I have not checked if this indeed
happens or whether anything can be said about their arithmetic properties.
\end{rmk}
\begin{pf} Assuming that $S$ is ruled, we derive a contradiction.
Corollary~\ref{cor5.5} guarantees the existence of a rational curve
$C\subset S_{\Fbar}$ satisfying the properties (5.5.1--3). By (5.5.3)
$\Oh(2C)$ is in $\Pic S$. Since $K_S^2\le 3$, $K_S$ is not divisible by 2
in
$\Pic(S_{\Fbar})$ and therefore $\Oh(C)$ is in $\Pic S$. This implies that
$C\in|{-}dK_{S_{\Fbar}}|$ for some $d$. We show that there cannot be such
a curve:
\end{pf}
\begin{prop}\label{prop5.8} Let\/ $S$ be an integral Del Pezzo surface
over an algebraically closed field such that\/ $\chi(\Oh_S)=1$, and\/
$K_S^2=1,2,3$. Then\/ $S$ does not contain any curve $C$ satisfying the
following properties:
\begin{description}
\item[5.8.1] $C$ is birational to $\PP^1$;
\item[5.8.2] $S$ is smooth along $C$;
\item[5.8.3] $C\in|{-}dK_S|$ for some $d$.
\item[5.8.4] $\sum_i m_i=dK_S^2-2$ and $\sum_i m_i^2=d^2K_S^2$, where
$P_i$ are the (assigned and infinitely near) multiple points of $C$ with
multiplicity $m_i$.
\end{description}
\end{prop}
\begin{pf} If $m_i\le d$ for every $i$ then
\[
\sum_i m_i^2\le d\sum_i m_i=d^2K_S^2-2dd$.
Let $P\in L\subset S$ be a line (that is, $-K_S\cdot L=1$). Then
$m\le (C\cdot L)=d$. Therefore $P$ cannot lie on any line.
If $K_S^2=1$ then any member of $|{-}K_S|$ is a line, hence there is a
line through any point. Hence we are done if $K_S^2=1$.
Next consider the case when $K_S^2=3$. That is, $S\subset \PP^3$ is
a cubic surface.
Let $D\subset S$ be the intersection of $S$ with the tangent plane at
$P$. Since there is no line through $P$, $D$ is an irreducible cubic
whose unique singular point is at $P$. In particular, $D$ is
contained in the smooth locus of $S$.
The point $P$ determines a birational selfmap $\tau$ of $S$ as
follows. Take any point $Q\in S$, connect $P,Q$ with a line and let
$\tau(Q)$ be the third intersection point of the line with $S$.
$\tau$ is an automorphism of $S-D$. Another way of describing $\tau$
is the following. Projecting $S$ from $P$ gives a diagram
\[
\CD B_PS@>q>> \PP^2\\ @VpVV @.\\ S @.
\endCD
\]
Let $E\subset B_PS$ be the exceptional curve; $C', D'\subset B_PS$ the
birational transforms of $C,D$. Then $q$ is a degree two morphism and
$\tau$ is the involution interchanging the two sheets. (Computing with the
local equation at $P$ shows that $q$ is separable in characteristic 2 if
$D$ is irreducible, thus $\tau$ always exists.) Furthermore, $\tau(E)=D'$
and $p^*\Oh_S(1)(-E)=q^*\Oh_{\PP^2}(1)$. Thus
\[
C'+(m-d)E\in |q^*\Oh_{\PP^2}(d)|,\qtq{hence}
\tau(C'+(m-d)E)\in |q^*\Oh_{\PP^2}(d)|.
\]
Pushing this down to $S$ we obtain that
\[
\tau(C)+(m-d)D\in |\Oh_S(d)|,\qtq{hence} \tau(C)\in |\Oh_S(d-(m-d))|.
\]
Thus $\tau(C)$ satisfies all the properties (5.8.1--4) and its
degree is lower than the degree of $C$. We obtain a contradiction by
induction on $d$.
A similar argument works if $K_S^2=2$, but the details are a little
more complicated. I just outline the arguments, leaving out some
simple details.
We already proved that $P$ is not on any line, and a similar argument
shows that $P$ cannot be a singular point of a member of $|{-}K_S|$.
Since $h^0(S,-2K_S)=7$, there is a curve $D\in |{-}2K_S|$ which has a
triple point at $P$. In fact, $D$ is unique, it is a rational curve
and $P$ is its only singular point. Thus $S$ is smooth along $D$. As
before we look at the blowup diagram
\[
\CD B_PS@>q>> Q\subset \PP^3\\ @VpVV @.\\ S @.
\endCD
\]
where $Q$ is a quadric cone, the image of $B_PS$ by the linear system
$|{-}2K_{B_PS}|$. Now $q$ is a degree two morphism and $\tau$ is the
involution interchanging the two sheets. (Again one can see that $q$ is
separable in characteristic 2 if $|{-}K_S|$ does not have a member which
is singular at $P$.) Let $E\subset B_PS$ be the exceptional curve; $C',
D'\subset B_PS$ the birational transforms of $C,D$. $\tau(E)=D'$ and
$p^*\Oh_S(2)(-2E)=q^*\Oh_{\PP^3}(1)$. As before we obtain that
\[
\tau(C)\in |\Oh_S(d-2(m-d))|.
\]
We obtain a contradiction by induction on $d$. \qed\end{pf}
\index{del Pezzo!surface over a nonclosed field|)}
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\bigskip
\noindent
J\'anos Koll\'ar,\\
Princeton University, Princeton, NJ 08544-1000\\
e-mail: kollar@math.princeton.edu
\end{document}