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\markboth{\qquad Essentials of the method of maximal singularities \hfill}
{\hfill Aleksandr V. Pukhlikov \qquad}
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\begin{document}
\title{Essentials of the method\\ of maximal singularities}
\author{Aleksandr V. Pukhlikov}
\maketitle
\section{Historical introduction}
\subsection*{Noether's theorem} We start with a brief description of the
principal events in the history of our subject. Around 1870, M.~Noether
\cite{N} discovered that the group of birational automorphisms of the
projective plane $\Bir\PP^2$, which is also known as the {\em Cremona
group} $\Cr\PP^2$, is generated by its subgroup $\Aut\PP^2=\Aut\C^3/\C^*$
together with any {\em standard quadratic Cremona transformation} $\tau$,
that is, the map
\[
\tau\colon(x_0 : x_1 : x_2) \mapsto (x_1x_2 : x_0x_2 : x_0x_1)
\]
in any system of homogeneous coordinates.
Noether's argument is as follows: take any Cremona transformation
\[
\chi\colon\PP^2\broken\PP^2.
\]
Then either $\chi$ is a projective isomorphism, or the proper inverse
image of the linear system\index{mobile linear system} of lines of $\PP^2$
is a linear system $|\chi|$ of curves of degree $n=n(\chi)\ge2$ with
assigned base points $a_1,\dots,a_N$ (possibly including infinitely near
points).\index{infinitely near maximal singularity} Let
$\nu_1,\dots,\nu_N$ be their multiplicities with respect to the linear
system $|\chi|$, and assume that $\nu_1\ge\nu_2\ge\dots\ge\nu_N$. Then,
because two lines intersect in one point, the {\em free intersection} of
two curves of $|\chi|$ (that is, their intersection outside the base
locus) equals 1. Hence
\[
n^2=\sum^N_{i=1}\nu^2_i+1.
\]
Moreover, the curves in $|\chi|$ are rational and nonsingular outside the
base locus, and so, computing their geometric genus in terms of their
arithmetic genus, we get
\[
(n-1)(n-2)=
\sum^N_{i=1}\nu_i(\nu_i-1).
\]
It is easy to deduce from these two equalities that $N\ge3$ and that the
three maximal multiplicities\index{maximal!multiplicity}\index{Noether's
inequality} satisfy {\em Noether's inequality}
\[
\nu_1+\nu_2+\nu_3 > n.
\]
If $a_1,a_2,a_3$ are actual points of the plane $\PP^2$ (that is, not
infinitely near points), we can then consider the composite
\[
\chi\circ\tau\colon\PP^2\broken\PP^2,
\]
where $\tau$ is the standard quadratic transformation constructed from
these three points $(a_1,a_2,a_3)$.
Let us prove that $n(\chi\circ\tau)2n$,
\item {\em or} there is a curve $C\subset V$ such that $\mult_C|\chi|>n$.
\end{enumerate}
We admit that Fano never asserted that these two cases are the {\em
only\/} possibilities; thus, for example, he mentions the following
additional case: a base point $x\in V$ and a base line $L\subset E$, where
$E\subset\wave V$ is the exceptional divisor of the blowup of $x$,
satisfying the inequality
\[
\mult_x|\chi|+\mult_L|\wave\chi|>3n.
\]
But the general level of understanding and the technical weakness of his
time prevented him from giving a rigorous and complete description of what
happens when $n(\chi)\ge2$.
Fano then asserts that neither of the above cases can happen. Since
practically all of his arguments are absolutely invalid, it is really
amazing that this conclusion is true. (Still more amazingly, this was
invariably the case with Fano: wrong arguments often led him to true and
deep conclusions.) For instance, to exclude\index{excluding} the
possibility of a curve $C$ with $\mult_C|\chi|>n$, he attempts to argue on
the arithmetic genus of a general surface in $|\chi|$, apparently hoping
to reproduce Noether's arguments in terms of the genus of a curve in
$|\chi|$. However, Iskovskikh and Manin \cite{IM} found out that in fact
these arguments do not lead to any conclusion.
Having convinced himself that the case $n(\chi)\ge2$ is impossible, Fano
formulated one of his most impressive claims: any birational
transformation between two nonsingular quartics in $\PP^4$ is a projective
isomorphism. In particular, the group of birational automorphisms $\Bir
V=\Aut V$ is finite (trivial for sufficiently general $V$); therefore, $V$
is nonrational.
Fano did a lot of work in threefold birational geo\-metry along these lines
\cite{F3}. He gave a description (however incomplete and unsubstantiated
it may be) of birational transformations of cubic threefolds, complete
intersections $V_{2\cdot 3}$ in $\PP^5$, and many other varieties. Many of
his results have not been completed to this day. However, because of the
very style of Fano's work, his numerous mistakes and, generally speaking,
incompatibility of his geometry with the universally adopted standards of
mathematical arguments, his ideas and computations were abandoned for
about twenty years.
\subsection*{The work of Manin and Iskovskikh} In the late 1960s,
Yu.I.~Manin and V.A.~Iskovskikh in Moscow started their pioneering program
in threefold birational geo\-metry, after a series of papers on two
dimensional birational geometry. As a result, in 1970, they developed a
method strong enough to prove Fano's claim on the quartic threefold
\cite{IM}; we refer to this as the {\em method of maximal
singularities}.\index{maximal!singularity} Using this method, Iskovskikh
\cite{I} subsequently proved several more of Fano's claims and corrected
some of his mistakes. In the 1970s and 1980s, Iskovskikh and several of his
students -- A.A.~Zagorskii, V.G.~Sarkisov
\cite{S1,S2}, S.L.~Tregub \cite{T1,T2}, S.I.~Khashin \cite{Kh} and the
author \cite{P1}--\cite{P5} -- worked in this field, trying to describe
birational maps of certain classes of algebraic varieties. Their work was
often successful, but unfortunately not always: the method of maximal
singularities\index{maximal!singularity} was extended to a number of
classes of varieties, including some of arbitrary dimension and some
possibly singular, and including a large class of conic bundles. The
well-known Sarkisov program\index{Sarkisov!program} \cite{R1,C1} also has
its origins in the framework of this field. At the same time, the method
really only works for varieties of very small degree. We must admit that
at present we have no good method of studying the birational geometry of
higher dimensional Fano varieties and Fano fibrations in
general.\index{Fano!3-fold}\index{Fano!variety}\index{Fano!fibration}
Nevertheless, the results obtained using the method of maximal
singularities\index{maximal!singularity} cannot be proved at present in
any other way. (See \cite{K1}, \cite{K2} for an alternative approach in
the spirit of characteristic $p$ tricks.)
\subsection*{Acknowledgements} This paper is an extract from lectures
given by the author during his stay at the University of Warwick in
September--December 1995. Since \cite{IP} has been published, it does not
make sense to reproduce here all the details of the excluding/untwisting
procedures.\index{excluding}\index{untwisting} At the same time, \cite{IP}
was actually written in 1988. The real meaning of
the\index{test!class!method} ``test class'' construction has been
clarified in more recent work, and some new methods of excluding maximal
singularities\index{maximal!singularity} have appeared [P5--P7]. The aim
of the present paper is to give an easy introduction to the method of
maximal singularities.\index{maximal!singularity} We restrict ourselves to
explaining only the crucial points. The principal and most difficult part
of the method -- that is, excluding\index{infinitely near maximal singularity} infinitely near maximal
singularities\index{maximal!singularity} -- is presented here in a new
form, which is simple and easy; this version of the method has never been
published before.
I would like to express my gratitude to Professor M.~Reid who invited me
to the University of Warwick and arranged my lecture course on birational
geometry. I am thankful to all the staff of the Warwick Mathematics
Institute for hospitality.
I would like to thank Professor V.A.~Iskovskikh, who introduced me to the
problem of the quintic four-fold in 1982 and thus determined the direction
of my work in algebraic geometry. I am grateful to Professor Yu.I.~Manin
for constant and valuable support.
The author was financially supported by International Science Foundation,
grant M90000, by ISF and Government of Russia, grant M90300, and by Russian
Fundamental Research Fund, grant 93--011--1539.
\section{Maximal singularities of birational maps}
\subsection*{The first step}
\index{factorial@$\Q$-factorial} Fix a projective $\Q$-factorial variety
$V$ with at worst terminal singularities\index{terminal!singularities}
over the field $\C$ of complex numbers, and let $Y$ be a divisor on $V$
moving\index{mobile linear system} in a linear system $|Y|$ (for example,
an ample divisor). We assume that we are given a birational map
\[
\chi\colon V\broken W.
\]
Take the proper inverse image $|\chi|\subset|D|$ on $V$ of the linear
system $|Y|$. We write $\Bs|\chi|$ for its base subscheme. This linear
system $|\chi|$ and its base subscheme $\Bs |\chi|$ are our main objects
of study.
\begin{Exs} We list some of the main classes of varieties that have been
studied more or less successfully using the method of maximal
singularities\index{maximal!singularity} over the last 25 years:
\begin{enumerate}
\item smooth quartics $V_4\subset\PP^4$;
\item complete intersections $V_{2\cdot 3}\subset\PP^5$;
\item singular quartics $x\in V_4\subset\PP^4$;
\item smooth hypersurfaces $V_M\subset\PP^M$;
\item double projective spaces $\si\colon V\to\PP^n$ branched over a
smooth hyper\-surface $Z_{2n}\subset\PP^n$.
\end{enumerate}
\end{Exs}
\subsection*{Test pairs}\index{test!pair|(}
\begin{Defn}\label{Defn2.2} Let $W$ be a projective variety such that
$\dim W=\dim V$, $\codim\Sing W\ge2$, and $Y$ a divisor on $W$ that moves
in a\index{mobile linear system} linear system $|Y|$. We say that $(W,Y)$
is a {\em test pair} if the following conditions hold:
\begin{enumerate}
\renewcommand{\labelenumi}{(\alph{enumi})}
\item $|Y|$ is free from fixed components.
\item Termination of adjunction: there exists $\al\in\R_+$ such that
\[
|MY+NK_W| = \emptyset
\]
for all $M,N\in\Z_+$ with $N/M>\al$.
\end{enumerate}
The minimal $\al\in\R_+$ satisfying condition (b) is the {\em
index}\index{index} (or {\em threshold})\index{quasieffective!threshold}
of the pair $(W,Y)$. We denote it by $\al(W,Y)$.
\end{Defn}
\begin{Exs} We now give the main examples of test pairs, explaining
briefly the applications for which we need them.
\begin{enumerate}
\item $\PP^n$ with $Y=\text{hyperplane}$: used to decide whether $V$ is
rational.\index{rationality problem} This pair has index $1/(n+1)$.
\item A Fano fibration\index{Fano!fibration} $\fie\colon W\to S$ with
$Y=\fie\1$(very ample divisor on $S$): used to decide whether there are
structures of Fano\index{strict Mori fibre space}
fibrations\index{Fano!fibration} on $V$; for instance, take a conic
bundle\index{conic bundle} or Del Pezzo fibration.\index{del
Pezzo!fibration} The index here is obviously zero.
\item A Fano variety\index{Fano!variety} $V$ with $Y=-MK_V$: used to
describe the group $\Bir V$ and to give the birational classification
within the same family of Fano varieties.
\end{enumerate}
\end{Exs}
\subsection*{The language of discrete valuations}\index{valuation|(} We
recall briefly the definitions and facts we need about discrete
valuations; for more details see \cite{P5,P6}. For $X$ a quasiprojective
variety, we denote by $\cN(X)$ the set of {\em geometric} discrete
valuations
\[
\nu\colon \C(X)^*\to\Z,
\]
having a centre on $X$. If $X$ is complete, then $\cN(X)$ includes all the
geometric discrete valuations. The centre of\index{centre of a valuation} a
discrete valuation $\nu\in\cN(X)$ is denoted by $Z(X,\nu)$.
\begin{Exs}
\begin{enumerate}
\renewcommand{\labelenumi}{(\arabic{enumi})}
\item Let $D\subset X$ be a prime divisor, $D\not\subset\Sing X$. Then
$D$ determines a discrete valuation
\[
\nu_D=\ord_D.
\]
\item Let $B\subset X$ be an irreducible subvariety, $B\not\subset\Sing
X$. Then $B$ determines a discrete valuation:
\[
\nu_B(f)= \mult_B(f)_0- \mult_B(f)_{\infty}\,.
\]
Note that if $\si_B\colon X(B)\to X$ is the blowup of $B$ and
$E(B)=\si\1_B(B)$ its exceptional divisor, then
\[
\nu_B=\nu_{E(B)}
\]
under the natural identification of $\C(X)$ and $\C(X(B))$.
\end{enumerate}
\end{Exs}
\begin{Defn} Let $\nu\in\cN(X)$ be a discrete valuation. A {\em
realization} of\index{realisation of a valuation} $\nu$ is a triple $(\wave
X,\fie,H)$, where $\fie\colon \wave X\to X$ is a birational morphism and
$H\not\subset\Sing\wave X$ a prime divisor such that $\nu=\nu_H$.
\end{Defn}
\subsection*{Multiplicities} Let $\cJ\subset\Oh_X$ be a sheaf of ideals
and $\nu\in\cN(X)$.
\begin{Defn} The multiplicity of $\cJ$ at a discrete valuation $\nu$
is given by
\[
\nu(\cJ)=\mult_H\fie^*\cJ,
\]
where $(\wave X,\fie,H)$ is a realization of $\nu$.
\end{Defn}
Let $|\la|\subset|D|$ be a linear system\index{mobile linear system} of
Cartier divisors and $\cL_{|\la|}\subset\Oh_X(D)$ the subsheaf generated
by the global sections in $|\la|$. Set
\[
\cJ_{|\la|}=\cL_{|\la|}\tensor\Oh_X(-D)\subset\Oh_X.
\]
Obviously, $\cJ_{|\la|}$ is the ideal sheaf of the base subscheme
$\Bs|\la|$.
\begin{Defn} The multiplicity of $|\la|$ at a discrete valuation $\nu$
equals
\[
\nu(|\la|)=\nu(\cJ_{|\la|}).
\]
\end{Defn}
Now let $X$ be ($\Q$-)Gorenstein, and $\pi\colon X_1\to X$ a resolution.
Then
\[
K_{X_1}=\pi^*K_X+\sum_i d_iE_i
\]
where the $E_i\subset X_1$ are exceptional prime divisors. Consider a
realization $(\wave X,\fie,H)$ of $\nu\in\cN(X_1)=\cN(X)$. Then we get an
inclusion
\[
\fie^*\om_{X_1}\into\om_{\wave X},
\]
and the ideal sheaf
\[
K(X_1,\fie)= \fie^*\om_{X_1}\tensor\om_{\wave X}\1 \into\Oh_{\wave X}
\]
on $\wave X$.
\begin{Defn}\index{canonical!multiplicity (discrepancy)}
\index{discrepancy} The {\em canonical multiplicity (discrepancy)} of
$\nu$ is equal to $d_i$ if $\nu=\nu_{E_i}$, and is equal to
\[
K(X,\nu)=\mult_HK(X_1,\fie)+\sum_id_i\nu(E_i)
\]
otherwise.
\end{Defn}
\begin{Ex} Let $B\subset X$, $B\not\subset\Sing X$ be an irreducible
subvariety of codimension $\ge2$. Then
\begin{gather*}
\nu_B(\cJ)=\mult_B\cJ, \quad
\nu_B(|\la|)=\mult_B|\la|, \\
\text{and}\quad K(X,\nu_B)=\codim B-1.
\end{gather*}
\end{Ex}
\subsection*{Maximal singularities} We now return to our variety $V$ and
birational map $\chi\colon V\broken W$. Denote by $n(\chi)$ the index
(threshold)\index{quasieffective!threshold} of the pair $(V,D)$ (see
Definition~\ref{Defn2.2}).
\begin{Defn} A discrete valuation $\nu\in\cN(V)$ is said to be a {\em
maximal singularity}\index{maximal!singularity} of $\chi$ if the following
inequality holds:
\[
\nu(|\chi|)\,>\,n(\chi)K(V,\nu).
\]
\end{Defn}
\begin{Theorem}\label{1} Either $\al(V,D)\le\al(W,Y)$, or $\chi$ has a
maximal singularity.\index{maximal!singularity}
\end{Theorem}
\begin{pf} See \cite{P5,P6}; this is actually so easy that it can be left
as an exercise for the reader. The idea of the proof can be found in any
paper concerned with these problems (for instance, \cite{IM,I,P1,IP}).
However, please bear in mind that the proof should not depend upon
resolution\index{resolution!of singularities} of
singularities.\index{valuation|)} \end{pf}
\begin{Ex} Let $V$ be smooth with $\Pic\iso\Z K_V$, and assume that the
anticanonical\index{anticanonical!system} system $|\hbox{$-$}K_V|$ is free.
Then
\[
|\chi|\subset|\hbox{$-$}n(\chi)K_V|,
\]
and for a birational automorphism $\chi\in\Bir V$ either $n(\chi)=1$, or
$\chi$ has a maximal singularity.\index{maximal!singularity}
\end{Ex}
\subsection*{Maximal cycles} Suppose that $V$ is nonsingular.
\begin{Defn}\label{Defn:maxc} An irreducible subvariety $B\not\subset\Sing
V$ of codimension $\ge2$ is said to be a {\em maximal
cycle}\index{maximal!cycle} if $\nu_B$ is a maximal
singularity.\index{maximal!singularity} Explicitly:
\[
\mult_B|\chi|\,>\,n(\chi)(\codim B-1).
\]
A maximal singularity\index{infinitely near maximal singularity}
$\nu\in\cN(V)$ is said to be {\em infinitely near} if it is not a maximal
cycle. For singular $V$, these definitions should be modified slightly by
adding some valuations\index{valuation} sitting at the singularities (see
\cite{P3,P6}).
\end{Defn}
\section{The untwisting scheme}
\index{untwisting}
Assume that $\al(v,D)>\al(W,Y)$. Then $\chi$ has a maximal singularity.
The untwisting scheme is a strategy aiming to simplify $\chi$ according to
its maximal singularities.
\begin{Defn}[The basic conjecture]\label{Defn:Bconj} We say that $V$ {\em
satisfies the basic conjecture} if for any $\chi\colon V\broken W$ for
which the assumptions of Theorem~\ref{1} hold, we can replace ``maximal
singularity'' by\index{maximal!cycle} ``maximal cycle'': in other words,
$\chi$ has a maximal cycle whenever
\[
\al(v,D)\,>\,\al(W,Y).
\]
\end{Defn}
\begin{Rmk}\label{rem3.2} The point of Definition~\ref{Defn:maxc} is to
distinguish the maximal cycles,\index{maximal!cycle} which are ``shallow''
maximal singularities occurring at ground level on $V$, from the
``deeper'' infinitely near ones, which take several blowups to dig out.
The point of the basic conjecture is that it many cases, it divides our
study into treating the maximal cycles (which we exclude\index{excluding}
or untwist as\index{untwisting} appropriate), and the infinitely near
maximal singularities,\index{infinitely near maximal singularity} which
we exclude in good cases by the uniform method of
\S\S\ref{sec6}--\ref{sec7}.
\end{Rmk}
\subsection*{Excluding maximal cycles}\index{excluding} Assume that $V$
satisfies the basic conjecture. The first thing we must do is describe all
the subvarieties $B\subset V$ that can occur\index{maximal!cycle} as
maximal cycles, in other words, all $B$ such that
\[
|D-\nu B|
\]
is free from fixed components for some $D\in\Pic V$ and some
\[
\nu>(\codim B-1)\al(V,D).
\]
\subsection*{Untwisting maps}
\index{untwisting} The second step of the scheme is to construct a
birational automorphism $\tau_B\in\Bir V$ for each $B$ singled out at the
previous step; $B$ should be a maximal cycle\index{maximal!cycle} for
$\tau_B$.
If $B$ is a maximal cycle for $\chi\colon V\broken W$, take the composite
\[
\chi\circ\tau_B\colon V\broken W.
\]
Then we must be able to prove that
\[
n(\chi\circ\tau_B)2n$, a contradiction.
\QED \end{pf}
\begin{Prop} For any curve $C\subset V$,
\[
\mult_C|\chi|\le n.
\]
\end{Prop}
Our theorem is obviously an immediate consequence of this fact.
\begin{pfof}{the Proposition} We write $\Cbar=\pi(C)$ and consider the
following three cases:
\begin{enumerate}
\renewcommand{\labelenumi}{(\arabic{enumi})}
\item Easy case: $\pi\colon C\to\Cbar$ is a double cover and
$\Cbar\not\subset W$.
\item Moderately easy case: $\Cbar\subset W$.
\item Nontrivial case: $\pi\colon C\to\Cbar$ is birational and
$\Cbar\not\subset W$.
\end{enumerate}
\paragraph{Case (1)} Take a generic line $\Lbar$ intersecting $\Cbar$, so
that $L=\pi\1(\Lbar)$ is a smooth curve. The restricted linear series
\[
|\chi|\rest{L}
\]
has degree $2n$, but has as base points at least the two points
$C\cap\pi\1(\Lbar)$ with multiplicity $\mult_C|\chi|$.
\paragraph{Case (2)} Take a generic point $x\in\PP^m$ and consider the
cone $Z(x)$ over $\Cbar$ with vertex $x$. Then $Z(x)\cap
W=\Cbar\cup\Rbar(x)$, where the residual curve $\Rbar(x)$ intersects
$\Cbar$ at $\deg\Rbar(x)$ distinct points (see \cite{P5}). Let $R(x)$ be
the curve $\pi\1(\Rbar(x))$; then $\pi\colon R(x)\to\Rbar(x)$ is an
isomorphism, and
\[
|\chi|\rest{R(x)}
\]
is a linear series of degree $n\times\deg\Rbar(x)$ which has
$\deg\Rbar(x)$ base points of multi\-plicity $\mult_C|\chi|$.
\paragraph{Case (3)} Again, take a generic point $x\in\PP^m$ and consider
the cone $Z(x)$ over $\Cbar$ with the vertex $x$. Let
\[
\fie\colon X\to\PP^m
\]
be the blowup at $x$ with exceptional divisor $E$, so that the
projection\index{projection}
\[
\pi\colon X\to\PP^{m-1}=E
\]
is a regular map, making $X$ into a $\PP^1$-bundle over $E$. Let
\[
\al\colon Q\to\Cbar
\]
be the desingularization of $\Cbar$, and
\[
\Sbar=Q\times_{\pi(\Cbar)}X,
\]
which is a $\PP^1$-bundle over $Q$. Obviously, $\Num\Sbar=A^1(S)=\Z
f\oplus\Z e$, where $f$ is the class of a fiber and $e$ the class of the
exceptional section coming from the vertex of the cone. Obviously,
$f^2=0$, $f\cdot e=1$, and $e^2=-d$, where $d=\deg\Cbar=\deg\pi(\Cbar)$.
Let $h$ be the class of a hyperplane section; then $h=e+df$, so that
$h^2=d$.
Denote by $\wave C$ the inverse image of $\Cbar$ on $\Sbar$. Obviously,
the class $\wave c$ of $\wave C$ equals $h$. For generic $x$, the set
$\pi\1(Z(x))\cap\Bs|\chi|$ contains at most two curves: $C$ itself and
possibly the other component of $\pi\1(\Cbar)$; moreover, the inverse
image $\Wbar$ of $W$ on $\Sbar$ is a nonsingular curve.
Now let us take the surface $S=\Sbar\times_{Z(x)}V$, the double cover of
$\Sbar$ with the smooth branch divisor $\Wbar$. Denote the image of $C$ on
$S$ by $C$ again, and the other component of $\pi\1(\Cbar)$ on $S$ by
$C^*$. The inverse image of the\index{mobile linear system} linear system
$|\chi|$ on $S$ has at most two fixed components $C$, $C^*$ of
multiplicities $\nu,\nu^*$ respectively. Therefore the system $|nh-\nu
c-\nu^* c^*|$ on $S$ is free from fixed components, and we get the
following inequalities:
\[
\bigl(nh-\nu c-\nu^*c^*\bigr)\cdot c\ge0 \quad\text{and}\quad
\bigl(nh-\nu c-\nu^*c^*\bigr)\cdot c^*\ge0.
\]
It is easy to compute the multiplication table for the classes $h,c$ and
$c^*$ on $S$. The only intersection number we need is
\[
(c\cdot c^*)_S=\frac12(\wave c\cdot\wbar)_{\Sbar}=md,
\]
the others being obvious. Now we get the following system of linear
inequalities:
\[
(n-\nu^*)+(m-1)(\nu-\nu^*)\ge0 \quad\text{and}\quad
(n-\nu)+(m-1)(\nu^*-\nu)\ge0.
\]
If, for instance, $\nu\ge\nu^*$, then $\nu\le n$ by the second inequality.
By symmetry, we are done. \QED \end{pfof}
\subsection*{What do we know about maximal cycles?}
\index{maximal!cycle}
We list some of the known facts. First, maximal cycles do not exist in
the following 3 cases:
\begin{enumerate}
\item for a smooth hypersurface of degree $m$ in $\PP^m$ with $m\ge4$
\cite{P5};
\item for a smooth double space $V_2\to\PP^m\supset W_{2m}$ with $m\ge3$:
see \cite{I} for $m=3$, \cite{P2} for $m\ge4$, and see also \cite{IP} (and
also for a slightly singular $V_2$, \cite{P6});
\item for a smooth double quadric $V_4\to Q_2\subset\PP^{m+1}$, branched
over the intersection $Q_2\cap W_{2m-2}$ with $m\ge4$: see \cite{P2}, and
also \cite{IP}.
\end{enumerate}
Next, in many cases, there are strong restrictions on the type of maximal
singularities that can occur, for example:
\begin{enumerate}
\setcounter{enumi}{3}
\item For a singular quartic $V_4\subset\PP^4$ having a unique ordinary
double singular point $x$, there can be only 25 maximal cycles: the point
$x$ itself, and the 24 lines on $V$ passing through $x$ (see \cite{P3}).
Moreover, a maximal cycle\index{maximal!cycle} of a map $V\broken W$
is always unique.
\item For a double quadric $\pi\colon V_4\to Q_2\subset\PP^4$, branched
over $Q_2\cap W_4$, there can be at most one maximal cycle -- namely, a
line $L\subset V$, $L\cdot K_V=-1$, with $\pi(L)\not\subset W_4$ (see
\cite{I,IP}).
\item For a complete intersection $V=V_{2\cdot 3}=Q_2\cap
Q_3\subset\PP^5$, a maximal cycle $B$ is a curve: either a line $L$, or a
smooth conic $Y$ spanning a plane $\Pi(Y)$ contained in the quadric $Q_2$.
Moreover, a map $V\broken W$ can have at most two maximal curves, and if
there are exactly two, they are lines $L_1$ and $L_2$ spanning a plane
$\Pi(L_1\cup L_2)$ contained in $Q_2$ (see \cite{I,IP}).
\end{enumerate}
\section{Untwisting maximal cycles}
\index{maximal!cycle}
We give what is probably the simplest example of an
untwisting:\index{untwisting} the untwisting procedure of \cite{P3} for
the maximal singular point $x\in V_4\subset\PP^4$ on a quartic $V$ with an
ordinary double point.
\subsection*{Constructing the untwisting}
\index{untwisting}
Let $\pi\colon V\setminus\{x\}\to\PP^3$ be the
projection\index{projection} from $x$, so that $\deg\pi=2$. Then the
untwisting map $\tau\colon V\broken V$ interchanges the points in the
fibers of $\pi$.
Let $\si\colon V_0\to V$ be the blowup of $x$,
$E=\si\1(x)\iso\PP^1\times\PP^1$ its exceptional divisor, and $L_i$ for
$i=1,\dots,24$ the proper inverse images of the lines on $V$ passing
through $x$.
\begin{Lemma}\label{lem5.1} $\tau$ extends to a biregular automorphism of
\[
V_0\setminus \bigcup_{1\le i\le 24}L_i,
\]
so that it has a well-defined action $\tau^*$ on $\Pic V_0=\Z h\oplus\Z
e$. This action is given by the following relations:
\[
\tau^* h=3h-4e \quad\text{and}\quad
\tau^* e=2h-3e.
\]
\end{Lemma}
\begin{pf} $\pi$ extends to a morphism $V_0\to\PP^3$ of degree 2. The
covering involution\index{involution} $\tau$ is well defined away from the
one dimensional fibers, which are exactly the 24 lines $L_i$. Thus $\tau$
is an automorphism of the complement of a set of codimension 2 of $V_0$,
so that it has a well-defined action $\tau^*$ on $\Pic V_0$.
Obviously, for any plane $\Pi\subset\PP^3$ its proper inverse image
$\pi\1(\Pi)$ represents an invariant class, so that
\[
\tau^*(h-e)=h-e.
\]
Furthermore, $\pi(E)$ is a quadric in $\PP^3$ and $\pi(H)$ a quartic in
$\PP^3$, where $H\subset V$ is a hyperplane section disjoint from $E$. Thus
\[
e+\tau^* e=2(h-e)\quad\text{and}\quad
h+\tau^* h=4(h-e).
\QED
\]
\end{pf}
\subsection*{Untwisting} Let $\chi\colon V\broken W$ be our birational
map. We define the number $\nu_x(\chi)\in\Z_+$ as follows: the class of
the proper inverse image of the\index{mobile linear system} linear system
$|\chi|$ on $V_0$ is
\[
n(\chi)h-\nu_x(\chi)e.
\]
The condition that the singular point $x$ is a maximal
cycle\index{maximal!cycle} for $|\chi|$ means that
\[
\nu_x(\chi)>n(\chi).
\]
Now consider the composite $\chi\circ\tau\colon V\broken W$.
\begin{Lemma} \begin{enumerate}
\renewcommand{\labelenumi}{(\roman{enumi})}
\item $n(\chi\circ\tau)=3n(\chi)-2\nu_x(\chi)$.
\item $\nu_x(\chi\circ\tau)=4n(\chi)-3\nu_x(\chi)$.
\end{enumerate}
\end{Lemma}
\begin{pf}
Since $\tau$ is an automorphism in codimension 1, we can write
\[
n(\chi\circ\tau)h-\nu_x(\chi\circ\tau)e=
\tau^*\Bigl(n(\chi)h-\nu_x(\chi)e\Bigr).
\]
Applying the formulas obtained in the preceding Lemma~\ref{lem5.1} gives
the result.
\QED \end{pf}
Now if $x$ is a maximal point\index{maximal!point} for $\chi$, then
$\nu_x(\chi)>n(\chi)$, so that
\[
n(\chi\circ\tau)j$,}
\]
and $\fie_{i,i}=\id_{X_i}$. Note that $\fie_{i,j}(B_i)=B_j$ for $i\ge j$.
We denote the proper inverse image on $X_i$ of a subvariety or cycle
$(\mathrel{\dots})$ by a superscript $i$: $(\mathrel{\dots})^i$.
\begin{Prop} This sequence is finite: in other words, for some $k\in\Z_+$,
the triple $(X_k,\fie_{k,0},E_k)$ is a realization of $\nu$, that is,
$\nu=\nu_{E_k}$ (compare Definition~2.5.
\end{Prop}
\begin{pf} This is easy. See \cite{P6}, or prove it for yourself. \end{pf}
\begin{Defn} The sequence $\{\fie_{i,i-1}\}_{i=1,\dots,k}$ is called
the {\em resolution} of the discrete valuation\index{resolution!of a
valuation} $\nu$ (with respect to the model $X$).
\end{Defn}
\subsection*{The graph structure} \index{resolution!graph}
\begin{Defn}\label{Defn6.4} For $\mu,\nu\in \cN(X)$, we write
\[
\nu \,\geX \,\mu
\]
if $\nu$ is infinitely near to $\mu$, that is, if $\mu=\nu_{E_l}$
for some $l\le k$, and
\[
(X_l,\fie_{l,0},E_l)
\]
is a realization of $\mu$; in other words, the resolution of $\mu$ occurs
as an initial segment of the resolution of $\nu$.
We introduce an oriented graph\index{resolution!graph} structure on
$\cN(X)$, drawing an arrow
\[
\nu\stackrel{X}{\longrightarrow}\mu,
\]
if $\nu\geX\mu$ and $B_{k-1}\subset E^{k-1}_l$.
Denote by $P(\nu,\mu)$ the set of all paths from $\nu$ to $\mu$ in
$\cN(X)$, which is nonempty if and only if
$\nu\geX\mu$. Set
\[
p(\nu,\mu)=|P(\nu,\mu)| \quad\text{if $\nu\ne\mu$},
\]
and $p(\nu,\nu)=1$. We define $\cN(X,\nu)$ to be the subgraph of $\cN(X)$
with the set of vertices $\leX\nu$.
\end{Defn}
\subsection*{Intersections, degrees and multiplicities} Let $B\subset X$
with $B\not\subset\Sing X$ be an irreducible subvariety of codimension
$\ge2$; as usual, let $\si_B\colon X(B)\to X$ be its blowup, and
$E(B)=\si\1_B(B)$ the exceptional divisor. Let
\[
Z=\sum m_iZ_i, \quad\text{with}\quad Z_i\subset E(B)
\]
be a $k$-cycle for some $k\ge\dim B$. We define the {\em degree} of $Z$ as
\[
\deg Z=\sum_im_i\deg\left(Z_i\cap\si\1_B(b)\right),
\]
where $b\in B$ is a generic point, and the degree on the right-hand side
is the ordinary degree in the projective space $\si\1_B(b)\iso\PP^{\codim
B-1}$. Note that $\deg Z_i=0$ if and only if $\si_B(Z_i)$ is a proper
closed subset of $B$.
Our computations in what follows are based on the following statement.
\begin{Lemma} Let $D$ and $Q$ be distinct prime Weil divisors on $X$, and
write $D^B$ and $Q^B$ for their proper inverse images on $X(B)$. We write
$\bullet$ for the codimension\/ $2$ cycle of the scheme theoretic
intersection.
\begin{enumerate}
\renewcommand{\labelenumi}{(\roman{enumi})}
\item Assume that $\codim B\ge3$. Then
\[
D^B\bullet Q^B=(D\bullet Q)^B+Z,
\]
where $\Supp Z\subset E(B)$, and
\[
\mult_B(D\bullet Q)=(\mult_BD)(\mult_BQ)+\deg Z.
\]
\item Assume that $\codim B=2$. Then
\[
D^B\bullet Q^B=Z+Z_1,
\]
where $\Supp Z\subset E(B)$, $\Supp\si_B(Z_1)$ does not contain $B$, and
\[
D\bullet Q=\Bigl[ (\mult_BD)(\mult_BQ)+\deg Z \Bigr] B+ (\si_B)_*Z_1.
\]
\end{enumerate}
\end{Lemma}
\begin{pf} Let $b\in B$ be a generic point, $S\ni b$ a germ of a
nonsingular surface in general position with $B$, $S^B$ its proper inverse
image on $X(B)$. We get an elementary two dimensional problem: to compute
the intersection number of two different irreducible curves at a smooth
point on a surface in terms of its blowup. This is easy. \QED
\end{pf}
\subsection*{Multiplicities in terms of the resolution} We divide the
resolution $\fie_{i,i-1}\colon X_i\to X_{i-1}$ into the {\em lower part}
with $i=1,\dots,l\le k$, for which $\codim B_{i-1}\ge3$, and the {\em upper
part}, $i=l+1,\dots,k$, for which $\codim B_{i-1}=2$. It may occur that
$l=k$ and the upper part is empty.
Let $|\la|$ be a\index{mobile linear system} linear system on $X$ with no
fixed components, $|\la|^j$ its proper inverse image on $X_j$. Set
\[
\nu_j=\mult_{B_{j-1}}|\la|^{j-1}.
\]
Obviously,
\[
\nu_{E_j}(|\la|)=\sum^j_{i=1}p(\nu_{E_j},\nu_{E_i})\nu_i
\]
and
\[
K(X,\nu_{E_j})=\sum^j_{i=1}p(\nu_{E_j},\nu_{E_i})(\codim B_{i-1}-1).
\]
For simplicity of notation we write $i\to j$ instead of
\[
\nu_{E_i}\stackrel{X}\longrightarrow\nu_{E_j}
\]
in the graph\index{resolution!graph} of Definition~\ref{Defn6.4}.
Now everything is ready for the main step of the theory.
\section{Infinitely near maximal singularities. II. \\ The main
computation}\label{sec7}\index{maximal!singularity} \index{infinitely near
maximal singularity}
We now prove the crucial inequalities which enable us to
exclude\index{excluding} infinitely near maximal singularities in cases of
low degree.
\subsection*{Counting multiplicities} Let $D_1,D_2\in|\la|$ be generic
divisors. We define a sequence of codimension 2 cycles on the blowups
$X_i$, setting
\[
\renewcommand{\arraystretch}{1.3}
\begin{array}{rcl}
D_1\bullet D_2&=&Z_0,\\
D^1_1\bullet D^2_2&=&Z^1_0+Z_1,\\
&\vdots\\
D^i_1\bullet D^i_2&=&(D^{i-1}_1\bullet D^{i-1}_2)^i+Z_i,\\
&\vdots
\end{array}
\]
where $Z_i\subset E_i$. Thus for any $i\le l$ we get
\[
D^i_1\bullet D^i_2=Z^i_0+Z^i_1+\dots+Z^i_{i-1}+Z_i.
\]
For any $j$ with $i0$, then $i\to j$.
\end{Lemma}
\begin{pf} If $m_{i,j}>0$, then some component of $Z^{j-1}_i$ contains
$B_{j-1}$. But $Z^{j-1}_i\subset E^{j-1}_i$. \QED \end{pf}
\subsection*{Degree and multiplicity} Set $d_i=\deg Z_i$.
\begin{Lemma}
For any $i\ge1$ and $j\le l$ we have
\[
m_{i,j}\le d_i.
\]
\end{Lemma}
\begin{pf} Each $B_a$ is nonsingular at its generic point. But since
$\fie_{a,b}\colon B_a\to B_b$ is surjective, we can count multiplicities
at generic points. Now the multiplicities are nonincreasing with respect
to blowup of a nonsingular subvariety, so we are reduced to the obvious
case of a hypersurface in a projective space. \QED
\end{pf}
\subsection*{The computation} We get the following system of equalities:
\begin{equation}
\left.
\renewcommand{\arraystretch}{1.4}
\begin{array}{rcl}
\nu^2_1+d_1&=&m_{0,1},\\
\nu^2_2+d_2&=&m_{0,2}+m_{1,2},\\
&\vdots\\
\nu^2_i+d_i&=&m_{0,i}+\dots+m_{i-1,i},\\
&\vdots\\
\nu^2_l+d_l&=&m_{0,l}+\dots+m_{l-1,l}.
\end{array}
\right\}
\tag{$*$}
\end{equation}
Now
\[
d_l\,\ge\,\sum^k_{i=l+1}\nu^2_i\deg(\fie_{i-1,l})_*B_{i-1}
\,\ge\,\sum^k_{i=l+1}\nu^2_i.
\]
\begin{Defn} A function $a\colon \{1,\dots,l\}\to\R_+$ is {\em compatible}
with the graph\index{resolution!graph} structure if
\[
a(i)\,\ge\,\sum_{j\to i}a(j)
\]
for any $i=1,\dots,l$.
\end{Defn}
\begin{Exs} $a(i)=p(l,i)$, $a(i)=p(K,i)$.
\end{Exs}
\begin{Theorem} Let $a(\cdot)$ be any compatible function. Then
\[
\sum^l_{i=1}a(i)m_{0,i}\,\ge\,
\sum^l_{i=1}a(i)\nu^2_i+a(l)\sum^k_{i=l+1}\nu^2_i.
\]
\end{Theorem}
\begin{pf} Multiply the $i$th equality in $(*)$ by $a(i)$ and add them all
together: on the right-hand side, for any $i\ge1$, we get the expression
\[
\sum_{j\ge i+1}a(j)m_{i,j}\,=
\sum_{\begin{smallmatrix} j\ge i+1\\ m_{i,j}\ne 0 \end{smallmatrix}}
a(j)m_{i,j}\,\le\, d_i\sum_{j\to i}a(j)\,\le\, a(i)d_i.
\]
On the left-hand side, for any $i\ge1$, we get
\[
a(i)d_i.
\]
So we can throw away all the $m_{i,*}$ from the right-hand side for
$i\ge1$, and all the $d_i$ from the left-hand side for $i\ge1$, replacing
$=$ by $\le$.
\QED \end{pf}
\begin{Cor}
Set $m=m_{0,1}=\mult_{B_0}(D_1\bullet D_2), D_i\in |\chi|$. Then
\[
m\sum^l_{i=1}a(i)\,\ge\, \sum^l_{i=1}a(i)\nu^2_i+a(l)\sum^k_{i=l+1}\nu^2_i.
\]
\end{Cor}
\subsection*{Applications}
\begin{Cor}\label{cor7.7}
Set $r_i=p(K,i)$. Then
\[
m\sum^l_{i=1}r_i \,\ge\,\sum^k_{i=1}r_i\nu^2_i.
\]
\end{Cor}
\begin{pf} For $i\ge l+1$ obviously $r_i\le r_l$. \QED \end{pf}
\begin{Cor}[Iskovskikh and Manin \cite{IM}] Suppose that $\dim V=3$, and
let $\nu\in\cN(V)$ be a maximal singularity\index{maximal!singularity}
such that $Z(V,\nu)=x$ is a smooth point, $m=\mult_xC$, where the curve
$C=(D_1\bullet D_2)$ is the intersection of two generic divisors in
$|\chi|$, $n=n(\chi)$ and assume that $|\hbox{$-$}K_V|$ is free. Then
\[
m\left(\sum^l_{i=1}r_i\right)
\left(\sum^k_{i=1}r_i\right)\,>\,
n^2\left(2\sum^l_{i=1}r_i+\sum^k_{i=l+1}r_i\right)^2.
\]
In particular, $m>4n^2$.
\end{Cor}
\begin{pf} This follows immediately from the fact that $\nu$ is a maximal
singularity and from the preceding Corollary~\ref{cor7.7}. We prove the
final statement. Denoting
\[
\sum^l_{i=1}r_i,\quad \sum^k_{i=l+1}r_i
\]
by $\Si_0,\Si_1$, respectively, we get
\[
4\Si_0(\Si_0+\Si_1)\le (\Si_1+2\Si_0)^2,
\]
which is exactly what we want.
\end{pf}
\begin{Cor}[Iskovskikh and Manin \cite{IM}] The basic conjecture
(see Definition~\ref{Defn:Bconj}) holds for a smooth quartic
$V\subset\PP^4$.
\end{Cor}
\begin{pf} Obviously, $m\le 4n^2$. This contradicts the previous
corollary. \end{pf}
Since it is easy to show that $|\chi|$ has no maximal
cycles\index{maximal!cycle} on $V_4$ (\cite{IM} or \cite{P5}), we
get:
\begin{Cor}[Iskovskikh and Manin \cite{IM}] A smooth quartic threefold
$V\subset\PP^4$ is a birationally superrigid\index{birationally superrigid}
variety.
\end{Cor}
\section{Sarkisov's theorem on conic bundles} \index{conic bundle}
We give an extremely short version of the proof of Sarkisov's
theorem\index{Sarkisov!theorem on conic bundles}
\cite{S1,S2}. The idea of the proof is essentially the same as in these
well-known papers of Sarkisov. At the same time, our general viewpoint of
working in codimension 1 makes the arguments brief and very clear.
\subsection*{Statement of the theorem} Let $S$ be a smooth projective
variety of dimension $\dim S\ge2$, and let $\cE$ be a locally free sheaf of
rank 3 over $S$. Let
\[
X\subset\PP(\cE)\stackrel{\pi}{\to} S
\]
be a standard conic bundle,\index{standard conic bundle}\index{conic
bundle} that is, a smooth hypersurface with
\[
\Pic X=\Z K_X\oplus\pi^*\Pic S.
\]
Denote by $C\subset S$ the discriminant divisor. Recall that $C$ has at
most normal crossings, the fiber over any point outside $C$ is a smooth
conic, the fiber over the generic point of any component of $C$ is a pair
of distinct lines, and the inverse image of any component of $C$ on $X$ is
irreducible.
Let $\tau\colon V\to F$ be another conic bundle of the same dimension (not
necessarily smooth).
\begin{Theorem} If\/ $|4K_S+C|\ne\emptyset$, then any birational map
\[
\chi\colon X\broken V
\]
takes fibers into fibers, that is, there exists a map $\chibar\colon
S\broken F$ such that
\[
\tau\circ\chi=\chibar\circ\pi.
\]
\end{Theorem}
\subsection*{Start of the proof} We write
\[
\cF=\{C_u\bigm|u\in U\}
\]
for the proper inverse image of the family of conics $\tau\1(q)$ for $q\in
F$, and by
\[
\cFbar=\{\Cbar_u=\pi(C_u)\bigm|u\in U\}
\]
its image on the base $S$. When we perform a birational operation with
respect to these families, we sometimes replace the parametrizing set $U$
by some dense open subset; for brevity, we omit mention of this change,
and just bear in mind that we use the same symbol $U$, meaning it to be as
small as necessary.
Let $\si\colon S^*\to S$ be a birational morphism such that:
\begin{enumerate}
\renewcommand{\labelenumi}{(\arabic{enumi})}
\item $S^*$ is projective and nonsingular in codimension 1;
\item the proper inverse image
\[
\cF^*=\{L_u\bigm|u\in U\}
\]
of the family $\cFbar$ on $S^*$ is free in the following sense: for any
cycle $Z\subset S^*$ of codimension $\ge2$ a general curve $L_u$ does not
meet $Z$.
\end{enumerate}
The existence of such a morphism $\si$ can be proved by quite elementary
methods, without using Hironaka's results (see \cite{P6}). Set
\[
\PP^*=\PP(\si^*\cE) \quad\text{and}\quad
X^*=X\times_S S^*\subset\PP^*.
\]
Then $X^*$ is a singular conic bundle\index{conic bundle} over $S^*$. For
ease of notation, the natural morphisms of $X^*$ to $S^*$ and $X$ will be
denoted by $\pi,\si$ respectively, and the map $\chi\circ\si$ just by
$\chi$.
\begin{Prop} There exist: a closed subset $Y\subset S^*$ of codimension
$\ge2$, a nonsingular conic bundle\index{conic bundle}
\[
\pi\colon W\to S^*\setminus Y
\]
with nonsingular discriminant divisor
\[
C^*\subset S^*\setminus Y
\]
and
\[
\Pic W\iso\Z K_W\oplus\pi^*\Pic S^*,
\]
and a fiberwise map
\[
\la\colon X^*\broken W,
\]
such that $\pi\circ\la=\pi$. Moreover,
\[
|4K_{S^*}+C^*|\ne\emptyset.
\]
\end{Prop}
\begin{pf} We obtain $W$ by fiberwise restructuring of $X^*$ over the
prime divisors $T\subset S^*$ such that $\codim\si(T)\ge2$. If
$t\in\C(S^*)$ is a local equation of $T$ on $S^*$, then at the generic
point of $T$ the variety $X^*$ is given by one of the two following types
of equations:
\begin{align*}
\text{Case 1:}& \quad x^2+t^kay^2+t^lbz^2, \quad k\le l\\
\text{Case 2:}& \quad x^2+y^2+t^kaz^2,
\end{align*}
where $(x:y:z)$ are homogeneous coordinates on $\PP^2$, and $a,b$ are
regular and nonvanishing at a generic point of $T$. In Case~1 for $k\ge2$,
the variety $X^*$ has a whole divisor of singular points, that is,
$\pi\1(T)$. Blow it up $[k/2]$ times. Now in either case, the singularity
of our variety over $T$ is of type either $A_n$ or $D_n$. Blowing up the
singularities, {\em covering}\/ $T$, and contracting afterwards
$-1$-components in fibers, we get the proposition. The last statement is
easily obtained by computing the discrepancy\index{discrepancy} of $\nu_T$
on $S$. \QED \end{pf}
Denote $\chi\circ\la\1$ again by $\chi\colon W\broken V$.
Let $Z\subset W\times V$ be the (closed) graph of $\chi$, and let $\fie$
and $\psi$ be the projections\index{projection} (birational morphisms)
onto $W$ and $V$ respectively. Obviously, $Z$ is projective over $W$.
\begin{Prop}\label{Prop8.2}
For any closed set\/ $Y^*\supset Y$ of codimension $\ge2$ there exists an
open set $U\subset F$ such that
\[
\psi\1\tau\1(U) \,\subset\, \fie\1\pi\1(S^*\setminus Y^*)
\]
and $\psi\1\tau\1(U)$ is projective over $\tau\1(U)\subset V$.
\end{Prop}
\begin{pf} This follows immediately from the fact that the family of
curves $\cF^*$ is free on $S^*$. \QED \end{pf}
\subsection*{The test surface construction} Now let $|H^*|$ be any linear
system\index{mobile linear system} which is the inverse image of a very
ample linear system on $F$, and $|\chi|$ its proper inverse image on $W$.
Write
\[
|\chi|\subset
|\hbox{$-$}\mu K_W+\pi^*A|
\]
for some $\mu\in\Z_+$ and $A\in\Pic S^*$. If $\mu=0$, we get the statement
of the Theorem. So we assume that $\mu\ge1$. Let us show that this is
impossible.
In the notation of the preceding Proposition~\ref{Prop8.2}, set
$Q=\psi\1\tau\1(U)$. Obviously, we may assume that
\[
\psi\colon Q\to\tau\1(U)\subset V
\]
is an isomorphism. For a generic conic $R_u$ with $u\in U$, we have
\[
H^*\cdot R_u=0, \quad\text{and}\quad K_V\cdot R_u=-2.
\]
So the same is true on $Q$. Hence for some prime divisors
$T_1,\dots,T_m\subset Q$, we get
\[
\left(-\mu\fie^* K_W+\fie^*\pi^*A-\sum^m_{i=1}a_iT_i\right)
\cdot\psi\1(R_u)=0
\]
and
\[
\left(\fie^* K_W+\sum^m_{i=1} d_iT_i\right)\cdot\psi\1(R_u)=-2.
\]
Making the set $U$ smaller if necessary, we may assume that
\[
T_i\cdot\psi\1(R_u)\ge1 \quad\text{for all $i$.}
\]
Thus the cycles
\[
\pi\circ\fie(T_i)
\]
have codimension 1 in $S^*$ and the $T_i$ can be realized by the
successions of blowups
\[
\renewcommand{\arraystretch}{1.4}
\begin{matrix}
\fie^{(i)}_{j,j-1}: & X^{(i)}_j & \longrightarrow & X^{(i)}_{j-1} \\
& \bigcup & & \bigcup \\
& E^{(i)}_j & \longrightarrow & B^{(i)}_{j-1},
\end{matrix}
\]
where $B^{(i)}_0=\fie(T_i)$, $B^{(i)}_{j+1}$ covers $B^{(i)}_j$,
$E^{(i)}_{K(i)}=T_i$. Since $|\chi|$ has no fixed components,
$\deg(B^{(i)}_{j+1}\to B^{(i)}_j)=1$ and the corresponding graph of
discrete valuations\index{valuation}\index{resolution!graph} is a chain.
Taking the union of these blowups (that is, throwing away some more cycles
of codimension 2 from $S^*$), we get on $Q$ that
\[
|\wave\chi|\subset
\Bigl|
-\mu\fie^* K_W+\fie^*\pi^* A-
\sum_{i,j}\nu_{i,j}E^{(i)}_j
\Bigr|,
\]
whereas the canonical divisor on $Q$ equals
\[
\fie^*K_W+\sum_{i,j}E^{(i)}_j.
\]
Consequently, in view of $\mu\ge1$, the divisor
\[
\fie^*\pi^*A-\sum_{i,j}(\nu_{i,j}-\mu)E^{(i)}_j
\]
intersects $\psi\1(R_u)$ negatively. Of course, we may assume that
\[
\nu_{i,K(i)} \,\ge\, \mu+1 \quad\text{for all $i=1,\dots,m$.}
\]
Now consider the surface $\La_u=\pi\1(\pi\circ\fie(\psi\1(R_u))$ (the {\em
test surface}, see \cite{P5,P6}) and its proper inverse image $\La^*_u$
on $Q$. These surfaces are projective and, since $\cF^*$ is free, we get
\[
D^2\cdot\La^*\ge0,
\]
where $D$ is the class of $\psi\1(|H^*|)$. On the other hand, setting
$L=\psi\1(R_u)$, and $\Lbar=\pi(L)$, we can write $D^2\cdot\La^*$ as
\[
4\mu A\cdot\Lbar-\mu^2(4K_{S^*}+C^*)\cdot\Lbar-
\sum_{i,j}\nu^2_{i,j}E^{(i)}_j\cdot L
\]
(since for a generic $u\in U$ the curve $\psi\1(R_u)$ intersects all the
$T_i$ transversally). At the same time, according to the above remark,
\[
A\cdot\Lbar\,<\,\sum_{i,j}(\nu_{i,j}-\mu)E^{(i)}_j\cdot L,
\]
so that
\begin{align*}
4\mu A\cdot\Lbar & \,<\,
\sum_{i,j}4\mu(\nu_{i,j}-\mu)E^{(i)}_j\cdot L \\
& \,\le\,\sum_{i,j}\nu^2_{i,j}E^{(i)}_j\cdot L.
\end{align*}
Since the intersection
\[
(4K_{S^*}+C^*)\cdot\Lbar
\]
is obviously nonnegative, we get a contradiction:
\[
D^2\cdot\La^*<0. \QED
\]
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\end{thebibliography}
\bigskip
\noindent
Alexandr V. Pukhlikov,\\
Number Theory Section, Steklov Mathematics Institute,\\
Gubkina, 8,\\
117966 Moscow, Russia\\
e-mail: dost@dost.mccme.rssi.ru and pukh@mi.ras.ru
\end{document}