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% Alessio Corti
% Singularities of linear systems and 3-fold birational geometry
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\markboth{\qquad Singularities of linear systems and 3-fold birational
geometry \hfill}{\hfill Alessio Corti \qquad}
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systems and 3-fold \\ birational geometry}
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\begin{document}
\title{Singularities of linear systems and \\ 3-fold birational geometry}
\author{Alessio Corti}
\maketitle
\tableofcontents
\section{Introduction}
In this paper I explain some new techniques for studying singularities of
linear systems, with applications to birational maps between 3-fold Mori
fibre spaces,\index{Mori!fibre space} and especially the property of {\em
birational rigidity}, see Definition~\ref{dfn_rig} below. These techniques
are closely related and all have something to do with {\em Shokurov's
connectedness principle}\index{Shokurov connectedness} (see
Section~\ref{sec_sco}). Though they do not as yet form a coherent
``method'', they are intended to replace the combinatorial study of the
resolution graph,\index{resolution!graph} started by Iskovskikh and Manin
\cite{IM}.
My goal is to provide concise but complete proofs of the known criteria
for birational rigidity\index{birational rigidity|(} of 3-fold Mori fibre
spaces. I do not give the results in their most general or sharpest form,
leaving considerable scope for improvement in various places. For
instance, I do not discuss fibre spaces of cubic surfaces, the most
intricate parts of \cite{CPR}, or {\em relations} between generators of
$\Bir X$. My desire is rather to introduce the main ideas in the simplest
context in which they appear. I do, however, prove in Section~\ref{sec_ci}
the rigidity of a general smooth 3-fold complete intersection of type
$2,3$. I include this argument here because the result is important and
the method of proof is an excellent illustration of the power of the new
methods. A proof in the spirit of \cite{IM} is outlined in Iskovskikh and
Pukhlikov \cite{IP}.
I have tried to state many open questions, problems and conjectures, with
the aim of understanding how and why rigidity fails. On the other hand, I
have made a conscious decision to say nothing about dimension $\ge4$, even
though there have been some interesting recent developments (work of
Pukhlikov and Cheltsov).
The subject is classical. It was revived by the work of Iskovskikh and
Manin on the nonsingular quartic \hbox{3-fold} in the early 1970s, and it
is practised today by many followers of the Moscow school of birational
geometry. Mori theory\index{Mori!theory} came roughly a decade later,
initially a close but seemingly unrelated development, but now
increasingly relevant both as a foundational basis for classification and
a source of new techniques:
\paragraph{(1)} Mori fibre spaces\index{Mori!fibre space} are one possible
outcome of the minimal model program and are hence basic objects of the
classification of 3-folds. They provide a significant extension of the
classical world: for instance, there are 16 deformation families of smooth
and primitive Fano 3-folds, while there are several hundred known
deformation families of Fano \hbox{3-folds}\index{Fano!3-fold} with
terminal singularities\index{terminal!singularities} and Picard number 1
(the total number of families is known to be bounded).
\paragraph{(2)} Sarkisov realised\index{Sarkisov!program} that the method
going back to Noether, Castel\-nuovo and Fano of\index{untwisting} {\em
untwisting} or {\em factoring} birational maps as a composite of {\em
links},\index{link} which is the basis of all the results of the Moscow
school, can be extended to the category of Mori fibre spaces as an
abstract framework, the {\em Sarkisov program}.\index{Sarkisov!program}
\paragraph{(3)} The Sarkisov program has now been worked out in detail
\cite{CPR} for a significant class of singular Fano 3-folds, the 95
weighted\index{famous@``famous 95''} hypersurfaces
$X_d\subset\PP(1,a_1,\dots, a_4)$\index{weighted!hypersurfaces and c.i.}
with $-K_X=\Oh_X(1)$.
\paragraph{(4)} The techniques introduced in this paper to quantify and
study singularities of linear systems are based on the\index{log!category}
log category and Shokurov connected\-ness,\index{Shokurov connectedness}
important technical tools of Mori theory\index{Mori!theory} with
applications in other areas of classification theory. These techniques are
more powerful than the method of Iskovskikh and Manin, and easier to use.
Pukhlikov's article \cite{P4} contains an excellent and accessible
exposition of the method of Iskovskikh and Manin, which works well in many
cases, and is in some sense more elementary (for example, it does not
depend on Kodaira vanishing).
\paragraph{(5)} The links\index{link} of the Sarkisov program are
themselves made up of the divisorial contractions\index{divisorial
contraction} and flips\index{flip} associated to the extremal
rays\index{extremal!ray} of Mori theory.\index{Mori!theory} This point of
view has suggested some new ways of describing the links themselves, based
on methods and techniques of graded rings. This description is one of the
crucial innovations of \cite{CPR}. On the other hand, the classification
of \hbox{3-fold} divisorial contractions\index{divisorial contraction}
(Problem~\ref{prb_div}) is a beautiful open problem in Mori
theory\index{Mori!theory} whose solution would greatly enhance the
applicability of the Sarkisov\index{Sarkisov!program} program. The couple
of known cases of this problem are the basis of \cite{CPR} and of the
modern treatment of the rigidity of a quartic \hbox{3-fold} with a node,
see Theorem~\ref{thm_sq}.
In the remainder of the introduction, I give the most important
definitions, state the main results, and indicate briefly the contents of
the various chapters.
The Mori category\index{Mori!category} is the category of projective
varieties with $\Q$-factorial\index{factorial@$\Q$-factorial} terminal
singularities.\index{terminal!singularities} We always work in the Mori
category. Recall
\begin{dfn} A {\em Mori fibre space}\index{Mori!fibre space} is an
extremal contraction\index{extremal!contraction} $f\colon X\to S$ of fibre
type. In other words
$f_*\Oh_X=\Oh_S$ and:
\begin{enumerate}
\item $-K_X$ is relatively ample for $f$,
\item $\rk N^1 X=\rk N^1 S+1$,
\item $\dim S<\dim X$.
\end{enumerate}
\end{dfn}
If $\dim X=3$ there are 3 main cases
\begin{displaymath}
\begin{cases}
\dim S=0:& \text{$X$ is a Fano 3-fold,} \\
\dim S=1:& \text{$X$ is a Del Pezzo fibration,} \\
\dim S=2:& \text{$X$ is a conic bundle.}
\end{cases}
\end{displaymath}\index{del Pezzo!fibration}\index{conic bundle}
Note that this terminology is only used classically for nonsingular
varieties.
\begin{dfn}
\begin{enumerate}
\item The {\em Sarkisov category}\index{Sarkisov!category} is the category
whose objects are Mori fibre spaces and morphisms are birational maps
(regardless of the fibre structure).
\item Let $X\to S$ and $X'\to S'$ be Mori fibre spaces.\index{Mori!fibre
space} A morphism in the Sarkisov category, that is, a birational map
$f\colon X\broken X'$, is {\em square\/}\index{square birational map} if it
fits into a commutative square
\[
\mbox{\definemorphism{birto} \dashed \tip \notip
\diagram
X \rbirto^f \dto & X' \dto \\
S \rbirto^g & S' \enddiagram}
\]
where $g$ is a birational map (which thus identifies the function field
$L$ of $S$ with that of $S'$) and if, in addition, the induced birational
map of generic fibres $f_L\colon X_L\broken X'_L$ is biregular. In this
case, we say that $X\to S$ and $X'\to S'$ are {\em square
birational}.\index{square birational map}
\item A {\em Sarkisov isomorphism}\index{Sarkisov!isomorphism} is a
birational map
$f\colon X\broken X'$ which is biregular and square.
\end{enumerate}
\end{dfn}
\begin{dfn} \label{dfn_rig}
A Mori fibre space\index{Mori!fibre space} $X\to S$ is {\em birationally
rigid}\index{birational rigidity} if, given any birational map $\fie\colon
X\broken X'$ to another Mori fibre space $X'\to S'$, there exists a
birational selfmap $\al\colon X\broken X$ such that the composite
$\fie\circ\al\colon X\broken X'$ is square.\index{square birational map}
In other words, $X$ birational to $X'$ implies $X$ square birational to
$X'$. I say that $X\to S$ is {\em birationally rigid over\/
$S$}\index{birational rigidity!over a base $S$|(} if a map $\al$ as in the
definition can be taken to be defined over $S$.
\end{dfn}
Note that the definition does not say that every birational map $X\broken
X'$ is square.\index{square birational map} The main point is, of course,
that rigid implies nonrational\index{rationality problem} in a strong
sense. I believe the notion is reasonably well behaved in families (see
Chapter~\ref{cha_fa} for more remarks and examples):
\begin{con} \label{con_mod}
Birational rigidity is open in moduli. In other words, given any scheme
$T$, and a flat family of Mori fibre spaces\index{Mori!fibre space}
parametrised by $T$
\[
\mbox{\diagram
\sX\drto \rrto & & \sS \dlto \\
&T& \enddiagram}
\]
the set of all $t\in T$ such that the corresponding fibre $\sX_t\to \sS_t$
is birationally rigid is open in $T$ (possibly empty).
\end{con}
This paper contains proofs of the following results.
\begin{thm}[Sarkisov \cite{Sa1}, \cite{Sa2}] \label{thm_cb} Let $X\to S$
be a \hbox{$3$-fold} conic bundle.\index{conic bundle} Assume that the
morphism $X \to S$ is extremal;\index{extremal!contraction} that is, it is
a Mori fibre space,\index{Mori!fibre space} and
\begin{enumerate}
\item $X$ is smooth (this implies that $S$ is also smooth), and
\item the divisor $4K_S+\De$ is effective, where $\De\subset S$
denotes the discriminant of the conic bundle.\index{conic bundle}
Then $X$ is birationally rigid\index{birational rigidity!over a base $S$}
over $S$.
\end{enumerate}
\end{thm}
In Chapter~4, besides giving several proofs of this result, I explain why
the requirement that $X$ be smooth is not really a restriction, and also
indicate how one might hope to weaken the second assumption.
\begin{thm}[Pukhlikov \cite{P5}] \label{thm_dp}
Let $X\to S$ be a \hbox{$3$-fold} Del Pezzo fibration\index{del
Pezzo!fibration} of degree $d\le2$. Assume that the morphism $X \to S$ is
extremal; that is, it is a Mori fibre space,\index{Mori!fibre space} and
\begin{enumerate}
\item $X$ is smooth, and
\item the $1$-cycle $K_X^2\in N_1X$ does\index{K2@$K^2$ condition} not lie
in the interior $\Intr\NEbar X$ of the Mori cone.\index{Mori!cone}
Then $X$ is birationally rigid\index{birational rigidity!over a base $S$|)}
over $S$.
\end{enumerate}
\end{thm}
It is certainly too restrictive in this case to insist that $X$ be smooth,
and I think it is fair to say that the second assumption, although in some
sense analogous to the corresponding requirement for\index{conic bundle}
conic bundles, and easily verified in many natural examples, is poorly
understood. $K^2$ seems\index{K2@$K^2$ condition} to also control the
rigidity of Fano 3-folds,\index{Fano!3-fold}\index{birational rigidity} see
\cite{CPR}, Theorem~7.7.1 and Remark~7.7.2, especially (v).
Chapter~\ref{cha_dp} gives a proof of this result based on the same idea
as the original argument of Pukhlikov \cite{P5}, but technically simpler.
A similar result for Del Pezzo fibrations\index{del Pezzo!fibration} of
degree 3 \cite{P6} still seems rather hard (but should follow from the
methods of this paper).
Anthony Iano-Fletcher \cite{IF} wrote a list of 95
families\index{famous@``famous 95''} of Fano
\hbox{3-fold}\index{Fano!3-fold!hypersurface} hyper\-surfaces $X=X_d\subset
\PP^4_w$ in weighed projective $\PP^4_w=\PP(1,a_1,a_2,a_3,a_4)$, with
$-K_X=\Oh_X(1)$. Chapter~6 opens with an introduction to the following
theorem:
\begin{thm}[Corti, Pukhlikov, Reid \cite{CPR}] \label{thm_cpr}
Assume that $X$ is a general member of any of the $95$ families. Then
\begin{enumerate}
\item $X$ is birationally rigid,
\item $\Bir X/\Aut X$ is generated by a finite number of explicit
rational involutions.\index{involution}
\end{enumerate}
\end{thm}
Chapter~6 treats the special case $X=X_5\subset \PP(1,1,1,1,2)$ of the
theorem. We conjecture that the same conclusions hold if $X$ is
quasismooth.\index{quasismooth} The rest of Chapter~6 is devoted to some
extensions, generalisations, and related results.
Iano-Fletcher also constructed 86 families of Fano \hbox{3-fold}
codimension\index{Fano!3-fold} 2 weighted complete
intersections,\index{weighted!hypersurfaces and c.i.} the first example of
which is $X=X_{2,3}\subset \PP^5$, the complete intersection of a quadric
and a cubic in $\PP^5$. We have not as yet made any systematic study of
complete intersections, but I prove the following result:
\begin{thm}[Iskovskikh, Pukhlikov \cite{I1}, \cite{IP}] \label{thm_2,3}
A general $X=X_{2,3}\subset \PP^5$ is birationally rigid. Moreover, $\Bir
X/\Aut X$ is generated by rational involutions\index{involution} centred
on the lines and conics of $X$.
\end{thm}
The proof in \cite{IP} is long and complicated; that given here, while not
entirely painless, is quite a bit simpler, and is a nice application of
Shokurov connectedness.\index{Shokurov connectedness}
Selma Alt{\i}nok \cite{A} compiled a list of 70 families of codimension 3
Pfaffian weighted K3 surfaces, and 138 in codimension 4 (of which 115 are
firmly established, and most of the remaining 23 are conjectured to exist),
which are anticanonical sections of Fano 3-folds. She also produced 3
candidates for Fano \hbox{3-folds} of codimension $\ge4$ with
$|{-}K|=\emptyset$ that do not correspond to any K3 surfaces. It should be
possible, in principle, to study all these varieties by our methods.
Finally, I consider some cases where $X$ is not
quasismooth,\index{quasismooth} but acquires the simplest kind of
singularity.
\begin{thm}[Pukhlikov \cite{P3}] \label{thm_sq} Let $X=X_4\subset \PP^4$
be a quartic \hbox{$3$-fold} with a single ordinary double point $P\in X$.
Then $X$ is birationally rigid. Moreover, $\Bir X / \Aut X$ is generated
by the rational involution\index{involution} centred at the singular
point, and those centred at the lines of $X$ passing through the singular
point.
\end{thm}
I prove this result as a consequence of a classification of divisorial
contractions\index{divisorial contraction} $f\colon E\subset Y\to P\in X$
for $P\in X$ an ordinary \hbox{3-fold} double point $(xy+zt=0)\subset\C^4$
(more precisely, the germ of an ordinary double point); in fact, in
Theorem~\ref{thm_dc2}, I prove that $f$ must be the blowup of the maximal
ideal of $P$. The proof of Theorem~\ref{thm_dc2} is quite short, and the
resulting proof of Theorem~\ref{thm_sq} is much simpler and more local in
nature than the original \cite{P3}.
\subsection*{Overview of the methods} Before closing the Introduction, I
spend a few words describing the methods in very general terms. The
definitions and properties of the Sarkisov program\index{Sarkisov!program}
are recalled in Chapter~\ref{cha_sp}. Suppose that we wish to prove that a
given Mori fibre space\index{Mori!fibre space} $X\to S$ is rigid, or
otherwise restrict the possible models of $X$ as a Mori fibre space.
Let $X'\to S'$ be another Mori fibre space, $X\broken X'$ a birational
map. Choose a very ample complete linear system $\sH'$ on $X'$, and
denote its\index{mobile linear system} transform on $X$ by $\sH$. Roughly
speaking, the method is in 3 stages:
\scpa{Step 1: Set up} Necessarily $\sH\subset|{-}\mu K_X+A|$, where
$\mu>0$ is a rational number and $A$ a divisor pulled back from $X$.
Suitable biregular assumptions on $X$, such as the $K^2$
condition\index{K2@$K^2$ condition} of Theorem~\ref{thm_dp} or the
assumptions on the structure of the\index{anticanonical!ring}
anticanonical ring of Theorem~\ref{thm_cpr}, together with the yoga of
the\index{NFI@{Noether--Fano--Iskovskikh inequalities}}
Noether--Fano--Iskovskikh inequalities (Theorem~\ref{thm_nfi}), imply that
``$\sH$ has a base point with big multiplicity''. At first sight it may
not be obvious how to make this into a useful quantitative statement, but
our experience of working in Mori theory\index{Mori!theory} suggests the
following:
\[
K_X+\frac{1}{\mu} \sH \quad \text{is not canonical.}
\]
This means that there exists a\index{valuation}\index{centre of a
valuation} valuation $E$ with centre $C_E(X)$ on $X$ along which $\sH$
has multiplicity $m_E$ that is big compared to the degree $\mu$ and the
discrepancy;\index{discrepancy} or more precisely,
\[
m_E(\sH)>\mu a_E (K_X),
\]
where $a_E(K_X)$ is the discrepancy of $E$ with respect to the canonical
class of $X$. Alternatively, we can use a condition that $K_X -\sum\la_i
F_i+\frac{1}{\mu}\sH$ is not canonical; see for example the proof of the
rigidity theorem for Del Pezzo fibrations,\index{del Pezzo!fibration}
Theorems~\ref{thm_dp} and~\ref{thm_rdp}.
\scpa{Step 2: Exclusion} In the next stage, we try to use the information
that $K+\frac{1}{\mu}\sH$ is not canonical\index{canonical} to deduce
restrictions on the possible centres $\Ga=C_E(X)$ of the\index{valuation}
valuation $E$, in some cases even ruling out the possibility that any such
centres exist. This is clear in principle, but in practice various
approaches come to mind, and I don't know how to decide {\em a priori} which
works best, other than by experience with many calculations on a large
number of examples. Chapter~\ref{cha_sls} contains several variations,
refinements, small improvements and proofs of the following key result (see
Theorem~\ref{thm_2ds}, Corollary~\ref{cor_sco}, Corollary~\ref{cor_lcl},
Theorem~\ref{thm_3ds}).
\begin{lem}\label{lem_4n2} Let $X$ be a \hbox{$3$-fold}, $\sH$ a mobile
linear system\index{mobile linear system} on $X$, and suppose that the
closed point $X\ni P=C_E(X)$ is the centre of a valuation
$E$\index{valuation}\index{centre of a valuation} with
\[
m_E(\sH)>\mu a_E (K_X).
\]
Write $Z=H_1\cdot H_2$ for the cycle theoretic intersection of two general
members of $\sH$. Then
\[
\mult_P Z>4\mu^2.
\]
\end{lem}
For instance, if $X$ is a smooth quartic \hbox{3-fold} then
\[
4\mu^2=\deg Z\ge\mult_P Z>4\mu^2
\]
gives a contradiction; this is the hard case in the proof of \cite{IM}.
Traditionally, Lemma~\ref{lem_4n2} is proved via the combinatorial
analysis of the resolution graph\index{resolution!graph} of $E$ (the
Iskovskikh--Manin graph, see \cite{P4}). Although there are now 3 or 4
different ways of proving Lemma~\ref{lem_4n2}, it seems to me that we
still don't know a truly compelling reason why it holds.
\scpa{Step 3: Untwisting}\index{untwisting} Even when $X$ is rigid,
birational maps $X\broken X$ often exist. In this case Step~2 only gives
restrictions on what can happen, typically saying that no curve of large
enough degree and no nonsingular point can be a maximal centre, and
perhaps also excluding\index{excluding} some of the singular points.
Step~3 then consists of classifying all possible links\index{link}
$X\broken X'$, where $X'\to S'$ is ({\em a priori}) any\index{Mori!fibre space}
Mori fibre space. In fairly simple cases a little ingenuity is sufficient
to guess the correct answer but in more complicated situations, for
instance in \cite{CPR}, the simple-minded approach becomes computationally
intractable. In practice, it is often useful to calculate the
anticanonical ring\index{anticanonical!ring} $R(Y, -K_Y)$ for a large
number of extremal\index{extremal!contraction} divisorial
contractions\index{divisorial contraction} $E\subset Y\to P\in X$. I give
a very few simple examples in this paper and refer the interested reader to
\cite{CPR} for a fuller treatment.
Chapter~2 is an exposition of the Sarkisov program,\index{Sarkisov!program}
mostly without proofs. Chapter~3 is a study of singularities of linear
systems, based on Shokurov connectedness\index{Shokurov connectedness} and
inversion of adjunction, and represents the technical core of the whole
paper. Chapters~4, 5 and 6 give applications to conic bundles,\index{conic
bundle} Del Pezzo fibrations\index{del Pezzo!fibration} and Fano
3-folds.\index{Fano!3-fold} I have tried to cover all the 3-dimensional
results of \cite{IP}, including some of the more recent material.
These notes grew out of a lecture I gave at the Warwick Algebraic
Geo\-metry special year in November 1995. I then taught the material as
``Five lectures on \hbox{3-fold} birational geometry'' at RIMS in June
1997. I thank both institutions for their warm hospitality.
It is a great pleasure to acknowledge the influence that Miles Reid has had
in shaping my view of the world. For the dozens of hours of mathematical
conversations, ideas and criticism, this paper is as much his work as it is
mine. Almost all of the sharper results were originally proved by Alexandr
Pukhlikov, who is also coauthor of \cite{CPR}, and everything here grew out
of an attempt to understand his work. Finally, I would like to thank
Massimiliano Mella for checking many of the calculations in Chapters~3
and~6, and pointing out various mistakes in several of the earlier
versions, and Ivan Cheltsov for detecting an imbecility in one of the later
ones.
\section{The Sarkisov program} \label{cha_sp}\index{Sarkisov!program|(}
In this chapter, I explain the general structure of the Sarkisov program,
referring to Corti \cite{C2} and Bruno and Matsuki \cite{BM} for detailed
proofs. Apart from the\index{NFI@{Noether--Fano--Iskovskikh inequalities}}
Noether--Fano--Iskovskikh inequalities, the material here is not used
until Chapter~6.
Let $X\to S$, $X'\to S'$ be Mori fibre spaces\index{Mori!fibre space} and
\begin{displaymath}
\fie\colon X\broken X'
\end{displaymath}
a birational map. The Sarkisov program factors $\fie$ as a {\em chain of
links}: a link is an elementary birational map of one of four types
discussed and defined below. In favourable cases they can be classified
and the method proves that $X$ is birationally rigid, and gives explicit
generators of $\Bir X/\Aut X$.\index{birational rigidity|)}
The factorisation program begins by assigning $\fie$ a {\em Sarkisov
degree}\index{Sarkisov!degree} $\deg\fie$ (morally a discrete invariant, if
not in actual fact) and then {\em untwisting}\index{untwisting} $\fie$ by
a link\index{link} $\psi_1$ so that $\deg\fie \psi_1^{-1}<\deg\fie$. In
other words,
\begin{displaymath}
\fie=\fie_1 \psi_1,
\end{displaymath}
where $\fie_1=\fie \psi_1^{-1}$ has degree smaller than $\fie$; we then
continue inductively with $\fie_1$ in place of $\fie$.
In practice it is only possible to carry out the Sarkisov program
explicitly, in the original form discussed here, under suitable strong
restrictive assumptions on $X$---for example if $X$ is one of the 95 Fano
weighted hypersurfaces\index{weighted!hypersurfaces and c.i.}\index{famous@``famous 95''} (cf.\ \cite{CPR} and Chapter~6) or one of a
handful of the\index{weighted!hypersurfaces and c.i.} Fano weighted
complete intersections. The difficulty with {\em strict Mori fibrations},
that\index{strict Mori fibre space} is, conic bundles\index{conic bundle}
and Del Pezzo fibrations,\index{del Pezzo!fibration} is that they always
involve infinitely many links, which are very difficult to control.
Applications to these varieties therefore involve circumventing this
difficulty.
\subsection{The Sarkisov degree}\index{Sarkisov!degree|(}
Let $X$ be a variety and $\sH$ a mobile linear system (that is, without
fixed part or free in codimension 1).\index{mobile linear system}
A discrete\index{valuation} valuation $v$ of $k(X)$ is {\em geometric} if
the residue field $A_v/m_v$ has transcendence degree $\dim X-1$ or,
equivalently, if $X$ has a normal birational model $Z\to X$ containing a
prime divisor $E$ for which $v=v_E$ is the valuation along $E$. In the
language of Zariski, the model $E\subset Z\to X$ just discussed is a {\em
uniformisation} of $v$.\index{uniformisation of a valuation} We only
consider geometric discrete valuations. I often abuse notation and use $E$
for both the divisor itself and the valuation $v_E$. Write $a_E=a_E(K_X)$
for the discrepancy\index{discrepancy} and $m_E=m_E(\sH)$ for the
multiplicity of $\sH$ along $E$; in other words, if $E$ appears as a prime
divisor in $Z$, then $a_E$ and $m_E$ are defined by
\[
K_Z=K_X+a_E E\quad\text{and}\quad\sH_Z=\sH-m_E E,
\]
locally near the generic point of $E$. (We simplify the notation
throughout by suppressing the pullback $f^*$ of \hbox{$\Q$-divisors}.)
\begin{dfn}
Let $t\in\Q_+$ be a positive rational number. The pair $X,t\sH$ has {\em
canonical singularities} (respectively {\em terminal singularities})\index{canonical!singularities}---equivalently, I say that the
\hbox{$\Q$-divisor} $K_X+t\sH$ is {\em canonical} (or {\em terminal})---if
\[
m_E(\sH)\le \frac{1}{t} a_E(K_X) \quad\text{for all $E$,}
\]
(respectively, $<$ for all $E$ exceptional over $X$).
\end{dfn}
\begin{exe} \label{exe_cth}
\begin{enumerate}
\item For $X$ a surface, $X,t\sH$ is terminal if and only if $X$ is
nonsingular and
\[
\mult_P\sH<\frac{1}{t} \quad\text{for all points $P\in X$.}
\]
\item For $X$ a \hbox{3-fold}, $K+2\sH$ is terminal if and only if $X$ is
terminal and $\sH$ is free from base points. $K+\sH$ is terminal if and
only if $X$ is terminal and the scheme theoretic base locus of $\sH$ is a
finite set of nonsingular points.
\end{enumerate}
\end{exe}
Now fix Mori fibre spaces $X\to S$, $X'\to S'$, and a birational map
\[
\fie\colon X\broken X'.
\]
Our aim in this section is to define the Sarkisov degree $\deg\fie$ of
$\fie$. For this, choose a very ample complete linear system $\sH'=|{-}\mu'
K'+A'|$ on $X'$, where $A'$ is the pullback of a divisor ample on the base
$S'$. This choice is made once and for all at the beginning of the
factorisation process and $\sH'$ remains unchanged throughout the chain.
This is of course possible because the target Mori fibre space $X'\to S'$
remains unchanged throughout the chain. The definition of degree depends
upon this choice.
\begin{dfn} The {\em Sarkisov degree} $\deg\fie$ is the triple $(\mu,c,e)$
defined as follows:
\begin{enumerate}
\item If $\sH$ is the birational transform of $\sH'$ on $X$, then because
$X\to S$ is a Mori fibre space,\index{Mori!fibre space} we can write
\[
\sH\subset|{-}\mu K+A|,
\]
where $\mu$ is a positive rational number with bounded denominator and $A$
is the pullback of a divisor (not necessarily nef, or effective) on the
base $S$.
\item $c\in \Q_+$ is the canonical threshold\index{canonical!threshold}
of the pair $X,\sH$, that is,
\[
c=\max\{ t\mid K+t\sH \; \text{is canonical} \}.
\]\index{canonical}
\item $e\in\N$ is the (finite) number of crepant valuations
of\index{valuation} $K+c\sH$, that is,
\[
e=\#\Bigl\{E \Bigm| m_E (\sH)=\frac{1}{c} a_E (K_X) \Bigr\}.
\]
\end{enumerate}
We {\em order} such triples as follows: $(\mu_1,c_1,e_1)>(\mu_2,c_2,e_2)$
if either
\begin{enumerate}
\renewcommand{\labelenumi}{(\alph{enumi})}
\item $\mu_1>\mu_2$, or
\item $\mu_1=\mu_2$ and $c_1e_2$.
\end{enumerate}
\end{dfn}\index{Sarkisov!degree|)}
\subsection{Links and the 2-ray game}\index{link}\index{link|(}\index{link!of Type~I--IV|(}\index{2-ray game|(}
The references for this section are Koll\'ar \cite{Ko2} and \cite{C2}.
The 2-ray game is an inductive sequence of forced moves and
configurations, starting from a given configuration; it can be played in
various categories---terminal (Mori),\index{Mori!category} log
terminal,\index{log!category} etc. We only play it in the Mori category. I
want to emphasise that while the moves are uniquely determined by the
initial configuration, there is no guarantee at any point that the next
move exists.
Each configuration is a commutative diagram
\[
\mbox{\diagram
&Z\dlto_{\fie_P}\drto^{\fie_Q}& \\
X\drto& &\ Y\,, \dlto \\
&S & \enddiagram}
\]
consisting of 4 varieties and projective morphisms, in which the variety
on top swaps 2 extremal contractions\index{extremal!contraction} $\fie_P$
and $\fie_Q$ along (pseudo-) extremal rays\index{extremal!ray} labelled
$P$ and $Q$. The bottom variety $S$ is fixed throughout the game.
To ensure that the game is uniquely determined by the initial
configuration we always make the following assumptions:
\begin{enumerate}
\item $Z$ is projective over $S$ and $\rk N^1(Z/S)=2$.
This assumption implies that the Mori cone\index{Mori!cone} $\NEbar(Z/S)$
is a 2-dimensional closed cone in $N^1(Z/S)\iso\R^2$, so that there can be
at most 2 projective morphisms $Z\to\,?\to S$.
Because we insist that the game always remain within\index{Mori!category}
the Mori category we also assume that
\item $Z$ has $\Q$-factorial\index{factorial@$\Q$-factorial} terminal
singularities,\index{terminal!singularities}
\item for each ray $R\subset \NEbar(Z/S)$, either $K_Z\cdot R<0$, or
else the contraction $\fie_R$ of $R$ is (projective and)
small.\index{small!contraction}
Finally, since we are ultimately interested in links\index{link} between
Mori fibre spaces,\index{Mori!fibre space} we assume that
\item $\dim S<\dim Z$, and the Kodaira dimension $\kappa (Z/S)$ of $Z$
over $S$ is $-\infty$.
\end{enumerate}
Configurations can be classified according to the type of the contractions
of $P$, $Q$ into 9 types $df,dd,ds,ss,\dots$, where we write
$f=\text{fibre type}$, $d=\text{divisorial}$, $s=\text{small}$.
For\index{small!contraction} instance, a\index{divisorial contraction}
configuration of type $ds$ is one where $\fie_P$ is an extremal divisorial
contraction\index{extremal!contraction} in the Mori category,
and\index{Mori!category} $\fie_Q$ is a projective small contraction.
A move in the 2-ray game consists of creating a configuration $s*$
starting from a given configuration $* s$. Given
\[
\mbox{\diagram
&Z\dlto_{\fie_P}\drto^{\fie_Q}& \\
X\drto& &Y \dlto \\
&S & \enddiagram}
\]
with $\fie_Q$ small,\index{small!contraction} the new configuration
\[
\mbox{\diagram
&Z'\dlto_{\fie_{P'}}\drto^{\fie_{Q'}}& \\
X'\drto& &Y' \dlto \\
&S & \enddiagram}
\]
is uniquely determined by the requirements
\begin{enumerate}
\renewcommand{\labelenumi}{(\alph{enumi})}
\item $X'=Y$ and $\fie_{P'}\colon Z'\to X'$ is a small contraction that
is\index{small!contraction} the flip\index{flip} of $Q$. (Or the
``opposite'' of $Q$; here I use ``flip'' in the generalised sense of
\cite{Ko2}, Chapter~6. It can be a flip, or equally well, a flop or
inverse flip\index{inverse flip} of\index{Mori!category} the Mori
category.)
\item If $P'\subset\NEbar(Z'/S)$ is the flipped ray (whose contraction is
$\fie_{P'}$), we call the other ray $Q'\subset\NEbar(Z'/S)$; this exists
simply because $\NEbar(Z'/S)\subset\R^2$ is a two dimensional closed cone.
Then $Z'\to Y'$ is the contraction of $Q'$.
\end{enumerate}
When $K_Z\cdot Q>0$, there are several ways in which the game may end in
failure.\index{bad link} First, there is no guarantee that the flip $Z'\to
X'$ of $Q$ exists. Next, referring back to the above conditions (1--4),
even if $Z'$ exists, there is no guarantee that it has terminal
singularities\index{terminal!singularities} (2), that its other ray $Q'$
can be contracted or, if $Q'$ can be contracted, that it satisfies
condition (3). The game only continues if the new configuration exists and
satisfies conditions (2), (3); (1) and (4) are automatic.
A game with more than one move starts with a configuration $ds$ or $fs$,
and a winning game terminates with a configuration $sd$ or $sf$. There are
thus four types of winning games, modelled on the four possible one-move
games $dd$, $df$, $fd$, $ff$. We call these Type~I, II, III and~IV. A
winning game is also a winning game if we play all the moves in reverse.
In this sense a game of Type~III is the inverse of a game of Type~II.
Under the above assumptions consider for instance a winning game of Type~I.
By definition, the game begins with a configuration
\[
\mbox{\diagram
&Z\dlto_{d}\drto^{s}& \\
X\drto& &Y \dlto \\
&S & \enddiagram}
\quad
\raise-1.35cm\hbox{and ends with}
\quad
\mbox{\diagram
&Z'\dlto_{s}\drto^{d}& \\
X'\drto& &\ Y'\,. \dlto \\
&S & \enddiagram}
\]
Since $Z\to X$ is a divisorial contraction\index{divisorial contraction}
in the Mori category,\index{Mori!category} it follows that $X$ has
$\Q$-factorial\index{factorial@$\Q$-factorial} terminal
singularities\index{terminal!singularities} and, by assumption (4), $X\to
S$ is a Mori fibre space.\index{Mori!fibre space} Similarly, $Y'$ has
$\Q$-factorial\index{factorial@$\Q$-factorial} terminal singularities and
$Y'\to S$ is also a Mori fibre space. We call the resulting birational map
\[
\fie\colon X\broken Y'
\]
a {\em link of Type} I.
Links of Type~II, III and IV are defined in a similar way, modelled on
winning 2-ray games of Type~II, III, IV. We can visualise the four types
of links as in Figure~\ref{fig_Sark}:
\begin{figure}[ht]
\[
\begin{array}{lclc}
\raise-1cm\hbox{Type I:} &
\mbox{\definemorphism{birto}\dashed \tip \notip
\diagram
&Z\dlto\rbirto^{\text{flips}}&Z' \drto& \\
X\drto& & &X'\dlto\\
&S\rdouble &S' & \enddiagram}
& \quad
\raise-1cm\hbox{Type II:} &
\mbox{\definemorphism{birto}\dashed \tip \notip
\diagram
&Z\dlto\rbirto^{\text{flips}}&X' \dto \\
X\dto & &S' \dlto \\
S\rdouble &T & \enddiagram}
\\
\\[.5cm]
\raise-1cm\hbox{Type III:} &
\mbox{\definemorphism{birto}\dashed \tip \notip
\diagram
X \dto \rbirto^{\text{flips}}& Z \drto & \\
S \drto & &X'\dto \\
&T\rdouble&S' \enddiagram}
& \quad
\raise-1cm\hbox{Type IV:} &
\mbox{\definemorphism{birto}\dashed \tip \notip
\diagram
X\rrbirto^{\text{flips}}\dto& &X' \dto \\
S\drto & &S' \dlto\\
&T& \enddiagram}
\end{array}
\]
\caption{The links of the Sarkisov program} \label{fig_Sark}
\end{figure}\index{link}\index{link|)}\index{link!of Type~I--IV|)}\index{2-ray game|)}
\subsection{Construction of the untwisting link}
Fix Mori fibre spaces\index{Mori!fibre space} $X\to S$, $X'\to S'$ and a
birational map $\fie\colon X\broken X'$. If $\fie$ is\index{square
birational map}square biregular there is nothing to do. In this section,
assuming that $\fie$ is not square biregular, we show how to
untwist\index{untwisting} $\fie$ by making a Mori fibre space $X_1\to S_1$
and a link $\psi_1\colon X\broken X_1$ so that
$\deg\fie\psi_1^{-1}<\deg\fie$.
We use the following terminology. Let $U\to V$ be a projective morphism.
The Mori cone\index{Mori!cone} $\NEbar(U/V)=\NEbar_1(U/V)\subset N_1(U/V)$
is by definition the real closure of the cone of effective curves $C$
contained in fibres of $U\to V$. The dual cone $\NEbar{}^1(U/V)\subset N^1
(U/V)$ is the cone of\index{nef} {\em nef} classes. The real closure
$\NMbar{}^1(U/V)\subset N^1(U/V)$ of the cone $\NM^1(U/V)$ of effective
divisors is called the\index{quasieffective} quasieffective cone.\index{NFI@{Noether--Fano--Iskovskikh inequalities}}
\begin{thm}[Noether--Fano--Iskovskikh inequalities]
\label{thm_nfi} \
\begin{enumerate}
\item $\mu\ge \mu'$ and $\mu=\mu'$ if and only if $\fie$ is
square.
\item If $K+\frac{1}{\mu}\sH$ is canonical and quasieffective then
$\mu=\mu'$ and, in particular, $\fie$ is square.
\item If $K+\frac{1}{\mu}\sH$ is canonical and nef then $X\broken X'$ is
a Sarkisov isomorphism.\index{Sarkisov!isomorphism} \qed
\end{enumerate}
\end{thm}
A proof can be found in \cite{C2}, 4.2; the statement there that
corresponds to Theorem~\ref{thm_nfi}, (1) is slightly weaker, but the
claim here is easy enough to prove.
\begin{cor} \label{cor_nfi} If $\fie$ is not a Sarkisov isomorphism then
either:
\begin{enumerate}
\item $K+\frac{1}{\mu}\sH$ is not canonical, or
\item $K+\frac{1}{\mu}\sH$ is canonical but not nef.\index{nef} \qed
\end{enumerate}
\end{cor}
The construction of the untwisting\index{untwisting} link proceeds in 2
different ways, depending on whether we are in Case~(1) or~(2) of
Corollary~\ref{cor_nfi}. It is traditional and convenient to honour the
first case with the following definitions.
\begin{dfn} Let $X\to S$ be a Mori fibre space\index{Mori!fibre space}
and $\sH$ a mobile linear system\index{mobile linear system} on $X$.
Necessarily
\[
\sH \equiv -\mu K_X+A,
\]
where $A$ is a pullback from $S$. Let $c=c(X,\sH)$ be the canonical
threshold\index{canonical!threshold} of the pair $X,\sH$ and assume that
\[
c<\frac{1}{\mu}\,.
\]
\begin{enumerate}
\item A {\em maximal extraction} is an extremal
divisorial\index{divisorial contraction}
contraction\index{extremal!extraction}
\[
f\colon E\subset Y\to\Ga\subset X
\]
extracting a valuation\index{valuation} $E$ for which
$m_E(\sH)=\frac{a_E}{c}>\mu a_E(K_X)$, or equi\-valently
$K_Y+c\sH_Y=f^*(K_X+c\sH)$. It is a theorem that a maximal extraction
always exists; see \cite{C2}, Theorem~2.10, where this is called an {\em
extremal blowup} of the pair $X,\sH$.
\item An algebraic valuation $E\subset Z\to X$ of $X$ is a {\em maximal
singularity} of $\sH$ (or of $X$ itself) if $E$ is the exceptional divisor
of a maximal extraction. A {\em maximal centre} is the centre $C_E (X)$ on
$X$ of a maximal singularity. In \cite{CPR}, this is called a {\em strong}
maximal singularity.
\end{enumerate}
\end{dfn}
\begin{thm}
Assume that $\fie\colon X\broken X'$ is not a Sarkisov
isomorphism;\index{Sarkisov!isomorphism} then there exist a Mori fibre
space\index{Mori!fibre space} $X_1\to S_1$ and a link\index{link}
\[
\psi_1\colon X\broken X_1
\]
such that
\[
\deg\fie \psi_1^{-1}< \deg\fie.
\]
\end{thm}
\begin{proof} I briefly explain how to construct the link\index{link}
$\psi_1$, omitting the rather delicate verification that the Sarkisov
degree\index{Sarkisov!degree!decreases} decreases. The construction uses
the only known general method to guarantee from the start that a 2-ray
game\index{2-ray game} can be played to the end, namely when the 2-ray
game is a minimal model program for a (log) canonical divisor $K+B$.
According to Corollary~\ref{cor_nfi}, there are 2 cases:
\scpa{Case 1: $K+\frac{1}{\mu}{\sH}$ is not canonical} There is then a
canonical threshold\index{canonical!threshold} $c<\frac{1}{\mu}$, a
maximal centre $\Ga$ and a maximal extraction $f\colon E\subset
Z\to\Ga\subset X$ with
\[
K_Z+c\sH_Z=f^*(K_X+c\sH_X).
\]
The $K_Z+c\sH_Z$ minimal model program over $S$ is a winning 2-ray
game,\index{2-ray game} leading to a link\index{link} of Type~I
or~II.\index{link}\index{link!of Type~I--IV} It is shown in \cite{C2},
Theorem~5.4 that untwisting\index{untwisting} (strictly) decreases the
Sarkisov degree.
\scpa{Case 2: $K+\frac{1}{\mu}\sH$ is canonical but not nef} The link is
manufactured by first choosing a suitable contraction $S\to T$ as follows.
Let $P\subset\NEbar X$ be the extremal\index{extremal!ray} ray
corresponding to the Mori fibre space\index{Mori!fibre space} structure
$X\to S$. By the cone theorem for $K+\frac{1}{\mu}\sH$, for instance,
there exists an extremal ray $Q\subset \NEbar X$ with
$(K+\frac{1}{\mu}\sH)\cdot Q<0$. The contraction $X\to T$ of the
\hbox{2-dimensional} face $F=P+Q$ of $\NEbar X$ exists, for example by the
contraction theorem for $K+\bigl(\frac{1}{\mu}\pm\ep\bigr)\sH$ and
variations on it, and both the Mori fibration $X\to S$ and the contraction
$X\to Y$ of $Q$ factorise the morphism $X\to T$, giving a configuration
\[
\mbox{\diagram
&X\dlto \drto & \\
S\drto& &Y \dlto\\
&T& \enddiagram}
\]
of the 2-ray game.\index{2-ray game} Running a minimal model program
against the divisor $K+\frac{1}{\mu}\sH$ over $T$ wins the 2-ray game, and
we get a Mori fibration $X_1\to S_1$ and a link\index{link} $\psi_1\colon
X\broken X_1$ that is of Type~III if $S_1=T$ or of Type~IV if $S_1\to T$
is an extremal contraction.\index{extremal!contraction} When I wrote
\cite{C2} I did not realise that this kind of untwisting\index{untwisting}
decreases the Sarkisov degree,\index{Sarkisov!degree!decreases} but in
fact it does. \qed
\end{proof}
It is not known if the Sarkisov degree is a discrete invariant in general.
Nevertheless, it can be shown by a somewhat indirect and nonconstructive
argument that, in a chain factoring a given birational map, the degree
cannot decrease infinitely many times (\cite{C2}, pp.~246--248); thus the
Sarkisov program\index{Sarkisov!program} always terminates. In all
applications of the theory known to me termination is obvious, but it
seems that we still lack a compelling reason why it works in complete
generality.
\begin{prb} Find an effective proof of termination of the Sarkisov
program. In other words, find an {\em a priori} bound on the number of
links\index{link} needed to factor $\fie$, ideally depending only on the
Sarkisov degree.
\end{prb}
\begin{prb}
Extend the general framework of the Sarkisov program to factor the {\em
relations} in the Sarkisov category\index{Sarkisov!category} in terms of
``elementary relations'' or ``2-links'' (as yet undefined). There are a
handful of ad hoc cases in the literature (\cite{I1}, \cite{P3},
\cite{IP}), but a satisfactory general treatment is still lacking.
\end{prb}\index{Sarkisov!program|)}
\section{Singularities of linear systems} \label{cha_sls}
This chapter is the technical core of the whole paper. We consider a
surface or \hbox{3-fold} germ $P\in X$ together with a mobile linear
system\index{mobile linear system} $\sH$ on $X$. Assuming a condition
typically something like $K_X+\frac{1}{\mu}\sH$ not canonical, we derive
estimates on the singularity of the cycle $Z=H_1\cdot H_2$ (where
$H_i\in\sH$ are two general members) at the point $P$---typically
something like $\mult_P Z>4\mu^2$.
\subsection{Linear systems on surfaces}
In this section I prove the following strange-looking and deceptively
simple result which is used in combination with Shokurov
connectedness\index{Shokurov connectedness} in the proofs of many criteria
for birational rigidity.\index{birational rigidity}
Suppose that $P\in\De_1+\De_2\subset S$ is the analytic germ of a normal
crossing curve on a smooth surface, that is, isomorphic to
$0\in(xy=0)\subset\aff^2$. Let $\sL$ be a mobile linear
system\index{mobile linear system} on $S$ and denote by $\sL^2$ the local
intersection multiplicity $(L_1\cdot L_2)_P$ at $P$ of two general members
$L_1,L_2\in\sL$.
\begin{thm} \label{thm_2ds}
Fix rational numbers $a_1,a_2\ge0$ and suppose that
\[
K_S+(1-a_1)\De_1+(1-a_2)\De_2+\frac{1}{\mu} \sL
\]
is not log canonical for some $\mu\in\Q$, $\mu>0$.\index{log!canonical}
\begin{enumerate}
\item If either $a_1\le1$ or $a_2\le1$ then
\[
\sL^2>4a_1a_2\mu^2.
\]
\item If both $a_i>1$ then
\[
\sL^2>4(a_1+a_2-1)\mu^2.
\]
\end{enumerate}
\end{thm}
\begin{proof} For ease of notation, write
\[
D=(1-a_1)\De_1+(1-a_2)\De_2+\frac{1}{\mu} \sL.
\]
(Thus $D$ is a {\em subboundary} in the sense of Shokurov.) By assumption,
there is a geometric valuation\index{valuation} $E$ of $S$ with
discrepancy\index{discrepancy}
\[
a_E(K_S+D)<-1.
\]
\scpa{Step 1} I first show that (1) implies (2). Let $m_i=m_E (\De_i)$ be
the multiplicity of $\De_i$ along $E$ and assume, say, that $m_1\ge m_2$.
Then
\begin{align*}
-1>a_E(K_S+D)&=a_E (K_S) -m_E (D) \\
&=a_E (K_S)+m_1(a_1-1)+m_2(a_2-1)-\frac{m_E (\sL)}{\mu} \\
&\ge a_E (K_S)+m_2(a_1+a_2-2)-\frac{m_E (\sL)}{\mu} \\
&=a_E\Bigl(K_S+(2-a_1-a_2)\De_2+\frac{1}{\mu}\sL\Bigr).
\end{align*}
This means that the divisor
\[
K_S+D_2=K_S+(2-a_1-a_2)\De_2+\sL
\]
is not log canonical,\index{log!canonical} and (2) for $K_S+D$ is (1) for
$K_S+D_2$.
\scpa{Step 2} We assume that $a_1\le1$ and prove that (1) holds by
descending induction on the discrepancy\index{discrepancy} $a_E (K_S)$.
Let $F\subset T\to P\in S$ be the blowup of the maximal ideal of $P$. We
can write
\begin{align*}
K_S+D &=K_T+(1-a_1)\De_1'+(1-a_2)\De_2'+
\Bigl(1-a_1-a_2+\frac{m}{\mu}\Bigr)F+\frac{1}{\mu} \sL'\\
&=K_T+D_T,
\end{align*}
where $\De_i'$, $\sL'$ denote the proper transforms, $m=m_F (\sL)$ and
$D_T$ is defined by the formula. We have $a_E(K_T+D_T)<-1$ and we discuss
four cases, depending on the position of the centre $C_E T$ of $E$ on $T$:
\begin{enumerate}
\item[(a)] $C_E T\in F \cap\De_1'$,
\item[(b)] $C_E T\in F \cap\De_2'$,
\item[(c)] $C_E T\in F \setminus \{\De_1'+\De_2'\}$,
\item[(d)] $C_E T=F$, that is, $E=F$.
\end{enumerate}
By Step~1 and the inductive assumption, in the first three cases we may
assume that the result holds for $K_T+D_T$, since $a_E (K_T)4a_1\Bigl(a_1+a_2-\frac{m}{\mu}\Bigr)\mu^2+m^2 \\
&=4a_1^2\mu^2-4a_1m\mu+m^2+4a_1a_2\mu^2 \\
&=(2a_1\mu-m)^2+4a_1a_2\mu^2\ge4a_1a_2\mu^2.
\end{align*}
\scpa{Case~\rm (b)} If either $a_2\le1$ or $a_1+a_2-\frac{m}{\mu}\le1$,
same as (a), otherwise we assume that $a_2>1$ and
$a_1+a_2-\frac{m}{\mu}>1$.
\begin{align*}
\sL^2&>4\Bigl(a_2+a_1+a_2-\frac{m}{\mu}-1\Bigr)\mu^2+m^2 \\
&>4a_2\mu^2\ge4a_1a_2\mu^2.
\end{align*}
\scpa{Case~\rm (c)} By induction
\begin{align*}
\sL^2&>4\Bigl(a_1+a_2-\frac{m}{\mu}\Bigr)\mu^2+m^2 \\
&\ge4a_1\Bigl(a_1+a_2-\frac{b}{\mu}\Bigr)\mu^2+b^2 \\
&\ge4a_1a_2\mu^2,
\end{align*}
as in Case~(a).
\scpa{Case~\rm (d)} By assumption $m>(a_1+a_2)\mu$, so that
\[
\sL^2\ge m^2>(a_1+a_2)^2\mu^2\ge4a_1a_2\mu^2. \qed
\]
\end{proof}
\subsection{Shokurov connectedness and its implications} \label{sec_sco}\index{Shokurov connectedness|(}
I open this section by recalling the statement of Shokurov's connectedness
theorem, taken from \cite{Ko2}, 17.4 (see \cite{CPR}, proof of
Theorem~5.3.2 for an informal discussion) and then state some easy
consequences. In closing, I tie in with the previous section and prove
that a smooth point on a quartic \hbox{3-fold} cannot be a maximal centre.
Together with some easy arguments from \cite{IM}, \S5, which I do not
reproduce here, this implies that a smooth quartic \hbox{3-fold} is
birationally rigid.\index{birational rigidity}
\begin{thm}[Shokurov, Koll\'ar \cite{Ko2}, 17.4] \label{thm_sco}
Let $X$ and $Z$ be normal varieties and $h\colon X\to Z$ a proper
morphism such that $h_*\Oh_X=\Oh_Z$. Assume that a $\Q$-divisor $D=\sum
d_iD_i$ on $X$ satisfies
\begin{enumerate}
\item if $d_i<0$ then $\codim_{h(D_i)} Z\ge2$;
\item $-(K_X+D)$ is $\Q$-Cartier, $h$-nef\index{nef} and $h$-big.
\end{enumerate}
Let $g\colon Y\to X$ be a resolution of singularities such that the support
of $g^{-1} D\subset Y$ is a divisor with global normal crossings. We can
write
\[
K_Y=g^* (K_X+D)+\sum a_E(K_X+D) E.
\]
Let $f=h\circ g$:
\[
\mbox{\diagram
Y\drto_f \rrto^g& &X\dlto^h\\
&Z& \enddiagram}
\]
Then
\[
A\ =\! \sum_{a_E(K_X+D)\le-1}\! E
\]
is connected in a neighbourhood of any fibre of $f$.
\end{thm}
\begin{rem} \label{rem_sco}
The image $g(A)$ of $A$ in $X$ is sometimes called the {\em locus of log
canonical singularities}\index{log!canonical}\index{locus of log canonical
singularities} of $K+D$, and denoted by
\[
g(A)=\LC(X, K_X+D).
\]
The theorem implies that this set is connected in a neighbourhood of
any fibre of $h$.
\end{rem}
\begin{cor} \label{cor_sco}\index{log!surface method}
Let $P\in S_1+S_2\subset X\iso 0\in (xy=0)\subset \aff^3$ be a smooth
\hbox{$3$-fold} germ, $\sH$ a mobile linear system\index{mobile linear
system} on $X$, and $0\le a_1$, $a_2\le1$ rational numbers. Assume that
\[
K_X+(1-a_1)S_1+(1-a_2)S_2+\frac{1}{\mu}\sH
\]
is not canonical at $P$ for some $\mu\in\Q$, $\mu>0$.
\begin{enumerate}
\item If $P\in S$ is a hyperplane section through $P$,
$\De_1+\De_2=(S_1+S_2)\rest{S}$ and $\sL=\sH\rest{S}$ then
\[
K_S+(1-a_1)\De_1+(1-a_2)\De_2+\frac{1}{\mu}\sL
\]
is not log canonical.\index{log!canonical}
\item If $Z=H_1\cap H_2$ is the intersection of two general members of
$\sH$ then
\[
\mult_P Z>4 a_1 a_2 \mu^2.
\]
\end{enumerate}
\end{cor}
\begin{proof} By assumption there is a
valuation\index{valuation}\index{centre of a valuation} $E$ with centre
$C_E X=P$ with
\[
a_E\Bigl(K_X+(1-a_1)S_1+(1-a_2)S_2+\frac{1}{\mu}\sH\Bigr)<0.
\]
Clearly $m_E(S)$ is a strictly positive integer, so
\begin{multline*}
a_E\Bigl(K_X+S+(1-a_1)S_1+(1-a_2)S_2+\frac{1}{\mu}\sH\Bigr) \\
=a_E\Bigl(K_X+(1-a_1)S_1+(1-a_2)S_2+\frac{1}{\mu}\sH\Bigr)
-m_E (S)<-1
\end{multline*}
and $K_X+S+(1-a_1)S_1+(1-a_2)S_2+\frac{1}{\mu}\sH$ is not log
canonical.\index{log!canonical} Using \ref{thm_sco} as in \cite{Ko2},
17.7, we conclude that $K_S+(1-a_1)\De_1+(1-a_2)\De_2+\frac{1}{\mu}\sL$ is
not log canonical (this statement is usually called ``inversion of
adjunction'').
To prove (2), let $P\in S\subset X$ be a general hyperplane section, so
that $\sL=\sH\rest{S}$ is free from base curves and $\mult_P Z=\sL^2_S$,
and apply \ref{thm_2ds}. \qed \end{proof}
I stress that it is crucial in Corollary~\ref{cor_sco} to assume that
$a_1,a_2$ are {\em both} $\le1$; indeed the assumption $d_i\ge0$ is
absolutely crucial to the validity of \ref{thm_sco}.\index{Shokurov
connectedness|)}
For applications to rigidity criteria for Fano complete intersections I
need the following strange-looking corollary:
\begin{cor} \label{cor_lcl}
Let $P\in X$ be a smooth \hbox{$3$-fold} germ, and $\sH\subset X$ a
mobile linear system\index{mobile linear system} on $X$. Assume that
\[
K_X+\frac{1}{\mu}\sH
\]
is not canonical at $P$ and let $h\colon F\subset Y\to P\in X$ be the
blowup of the maximal ideal $m_P$ of $P$. Then either
\begin{enumerate}
\item $m=m_F(\sH)>2\mu$, or
\item there is a line $\Ga\subset F\iso \PP^2$ such that
\[
K_Y+\Bigl(\frac{m}{\mu}-1\Bigr)F+\frac{1}{\mu}\sH
\]
is not log canonical\index{log!canonical} at the generic point of\/ $\Ga$.
\end{enumerate}
\end{cor}
\begin{proof} If (1) fails, consider a general hyperplane section $P\in
S\subset X$ and let $T\subset Y$ be its proper transform. We may write
\[
h^* \Bigl(K_X+S+\frac{1}{\mu}\sH\Bigr)=K_Y+T+
\Bigl(\frac{m}{\mu}-1\Bigr)F+\frac{1}{\mu}\sH_Y,
\]
with
\[
\frac{m}{\mu}-1\le1.
\]
I assume, for simplicity, that $<$ holds (otherwise work
with $(1-\ep)\sH$ in place of $\sH$). By \ref{cor_sco}, we know
that
\[
K_T+\Bigl(\frac{m}{\mu}-1\Bigr)F\rest{T}+\frac{1}{\mu}{\sH_Y}\rest{T}
\]
is not log canonical.\index{log!canonical} Also, applying the
connectedness theorem to the morphism
\[
h\colon T\to S,
\]
we conclude that the log canonical\index{log!canonical} locus
\[
\LC\Bigl(T,K_T+\Bigl(\frac{m}{\mu}-1\Bigr)E\rest{T}
+\frac{1}{\mu}{\sH_Y}\rest{T}\Bigr)
\]
is a single isolated point, from which (2) follows. \qed \end{proof}
\begin{thm}[Iskovskikh, Manin \cite{IM}] \label{thm_im}
Let $X=X_4\subset \PP^4$ be a quartic \hbox{$3$-fold} and $P\in X$ a
nonsingular point. Then $P$ is not a maximal centre.
\end{thm}
\begin{proof} If the conclusion failed, we would have a mobile linear
system $\sH\subset |\Oh_X(\mu)|$\index{mobile linear system} on $X$ with
\[
K_X+\frac{1}{\mu}\sH\quad\hbox{not canonical at $P$.}
\]
If $P\in S\in |\Oh_X(1)|$ is a hyperplane section through $P$ and
$\sL=\sH\rest{S}$, we know from \ref{cor_sco} that
\[
K_S+\frac{1}{\mu}\sL
\]
is not log canonical\index{log!canonical} at $P$. Now, if $S$ is chosen
suitably, $\sL$ is free from base curves and, applying
Theorem~\ref{thm_2ds}, we obtain
\[
4\mu^2=L_1\cdot L_2\ge (L_1\cdot L_2)_P>4\mu^2
\quad\text{for general $L_1,L_2\in \sL$,}
\]
a contradiction. \qed \end{proof}
\subsection{Divisorial contractions}
\begin{dfn}
A \hbox{3-fold}\index{divisorial contraction} {\em divisorial contraction}
is a contraction
\[
f\colon E\subset Z\to\Ga\subset X,
\]
where
\begin{enumerate}
\item $E=f^{-1}\Ga$, $f\colon Z \setminus E\to X\setminus\Ga$ is an
isomorphism;
\item $Z$ has terminal singularities\index{terminal!singularities} and
$P\in\Ga\subset X$ is the germ of a \hbox{3-fold} terminal singularity
(it {\em may be} that $P=\Ga$);
\item $E$ is an irreducible divisor and $-K_Z$ is $f$-ample.
\end{enumerate}
\end{dfn}
\begin{prb} \label{prb_div}
Classify \hbox{3-fold} divisorial\index{divisorial contraction}
contractions up to local analytic iso\-morphism (over the base).
\end{prb}
This is a problem of \hbox{3-fold} biregular classification, analogous to
the classification of terminal singularities.\index{terminal!singularities} It seems clear that it is a fundamental problem, having
many potential applications to birational geometry. I now present the
known special cases of terminal\index{quotient singularity} quotient
singularities and ordinary nodes. Both results are extremely useful in the
applications to the 95 Fano\index{famous@``famous 95''} hypersurfaces and
to singular quartics.
The following result is due to Kawamata \cite{Ka}.
\begin{thm}[Kawamata \cite{Ka}] \label{thm_dc1}
Let $P\in X\iso\frac{1}{r}(1,a,-a)$ be a \hbox{$3$-fold} terminal
quotient singularity, with $r\ge2$, and
\[
f\colon E\subset Z\to\Ga\subset X
\]
an extremal\index{divisorial contraction}\index{Kawamata blowup}
divisorial\index{extremal!contraction}
contraction.\index{terminal!quotient sing.\ $\frac{1}{r}(1,a,r-a)$} Then
$f(E)=P$ and $f$ is the\index{weighted!blowup} weighted blowup with weights
$(1,a,r-a)$.
\end{thm}
\begin{proof} I only explain the idea, which is quite simple. There is a
unique valuation\index{valuation} $F$ with centre on $X$ such that
$a_F=\frac{1}{r}$. Now suppose that
\[
E\subset Z\to P\in X
\]
is an extraction with $E\ne F$; then $a_E\ge \frac{2}{r}$. The crucial
point here is that $F$ is still there, and has a centre somewhere on $Z$.
An easy calculation yields
\[
a(F,K_Z)\le0.
\]
In other words, $Z$ cannot have\index{terminal!singularities} terminal
singularities. \qed \end{proof}
\begin{thm} \label{thm_dc2}
Let $P\in X\iso (xy+zt=0)\subset\aff^4$, and
\[
f\colon E\subset Z\to P\in X
\]
a divisorial\index{divisorial contraction} contraction. Assuming that
$f(E)=P$, $f$ is the blowup of the maximal ideal $m_P\subset\Oh_X$.
\end{thm}
\begin{proof} Let $\sH_Z\subset |{-}\mu K_Z|$ be a finite dimensional
very ample linear system and $\sH_X=f_*(\sH_Z)$. Then
\[
K_Z+\frac{1}{\mu}\sH_Z=f^* \Bigl(K_X+\frac{1}{\mu}\sH_X \Bigr),
\]
so that
\[
m_E(\sH_X)=\mu a_E(K_X),
\]
while
\[
m_F(\sH_X)<\mu a_F(K_X) \quad\text{for all valuations $F\ne E$.}\index{valuation}
\]
Letting $\ep\colon F\subset Y\to P\in X$ be
the blowup of the maximal ideal $m_P$, we argue that $m_F(\sH_X)\ge \mu$,
thereby proving the result. We know that
\[
a_E\Bigl(K_Y+\Bigl(\frac{m_F}{\mu}-1\Bigr)F+\frac{1}{\mu}\sH_Y\Bigr)=0.
\]
Choose a reducible hyperplane section $P\in S_1+S_2\subset X$ isomorphic
to $0\in(t=0)\subset(xy+zt=0)$, and satisfying the following conditions:
\begin{enumerate}
\item the centre $C_E Y$ of $E$ on $Y$ does not lie on the proper
transform $S_1'+S_2'$ of $S_1+S_2$ on $Y$;
\item the curve $\Ga=S_1' \cap S_2'$ is disjoint from the general
member of $\sH_Y$.
\end{enumerate}
We have
\[
\ep^*\Bigl(K_X+S_1+S_2+\frac{1}{\mu}\sH_X\Bigr)
=K_Y+S_1'+S_2'+\frac{m_F}{\mu}F+\frac{1}{\mu}\sH_Y
\]
and
\[
\LC=\LC\Bigl(Y, K_Y+S_1'+S_2'+\frac{m_F}{\mu}F+\frac{1}{\mu}\sH_Y
\Bigr)\supset S_1'+S_2'+C_E Y.
\]
Now $F\subset Y$ can be contracted along the 2 rulings of $F\iso
\PP^1 \times \PP^1$
\[
\mbox{\diagram
&Y\dlto_{h_1}\drto^{h_2}& \\
Y_1\drto& &Y_2\dlto \\
&X & \enddiagram}
\]
so that passing from $Y_1$ to $Y_2$ is the familiar flop. By
Remark~\ref{rem_sco}, $\LC$ is connected in a neighbourhood of every fibre
of $h_1$ and every fibre of $h_2$. The only possibility is that $\LC=F$,
or, in other words, that $C_EY=F$ and $m_F\ge\mu$. \qed
\end{proof}
For smooth points all we have is the following
\begin{con}
Let $f\colon E\subset Z\to 0\in \aff^3$ be a\index{divisorial
contraction} divisorial contraction. Then $f$ is a\index{weighted!blowup}
weighted blowup with weights $1,m_1,m_2$ for some $(m_1,m_2)=1$.
\end{con}
I am convinced that the method of proof of Theorem~\ref{thm_dc2} can
eventually be used to tackle this conjecture.
\subsection{Linear systems on 3-folds}
Let $x\in X$ be a \hbox{3-fold} germ, $\sH$ a mobile linear system on $X$
and\index{mobile linear system} $Z=H_1\cdot H_2$ the cycle theoretic
intersection of 2 general members $H_1,H_2\in\sH$. A special case of
Corollary~\ref{cor_sco} states that
\[
K+\frac{1}{\mu}\sH \quad \text{not canonical} \quad
\Longrightarrow \quad \mult_x Z>4\mu^2.
\]
In Section~\ref{sec_sco} I reduce this to the surface result
Theorem~\ref{thm_2ds} by means of Shokurov connectedness\index{Shokurov
connectedness} (Theorem~\ref{thm_sco}). Recall that the surface case
\ref{thm_2ds} has a very elementary proof by induction on the
discrepancy\index{discrepancy} (with respect to the canonical class) of a
valuation\index{valuation} $E$ with $a_E(K_S+D)<-1$ (in the notation
introduced at the beginning of the proof of \ref{thm_2ds}). The purpose of
this section is to give a proof of the \hbox{3-fold} case in the same
spirit as in the surface case, that is, an elementary proof by induction on
the discrepancy of a valuation\index{valuation} $E$ with
$a_E(K_X+\frac{1}{\mu}\sH)<0$. There are several good reasons for wanting
to do this:
\begin{enumerate}
\item The proof by reduction to surfaces uses Shokurov's connectedness
principle \ref{thm_sco} which in turn depends on the Kodaira vanishing
theorem. The method of Iskovskikh and Manin is more ``elementary'' and
works in any characteristic.
\item In \ref{cor_sco}, (2), I insisted that $a_1$, $a_2$ are both $\le1$.
This assumption is indeed necessary to apply the connectedness theorem.
However, the surface statement Theorem~\ref{thm_2ds} does not need this.
The discussion here suggests that the following might be true. Let $x\in
S\subset X$ be a surface through $x$ (smooth, say). Then for $\la\ge0$,
if $K_X-\la S+\frac{1}{\mu}\sH$ is not canonical then $\mult_x
Z>4(1+\la)\mu^2$. This kind of statement may look rather artificial but
it is in fact quite natural and it would be very useful in the study of
rigidity of Del Pezzo fibrations.\index{del Pezzo!fibration}
Unfortunately I don't know if it is true; I prove a technical statement
which is similar and sufficient to establish the application to the
rigidity theorem for Del Pezzo fibrations in Section~\ref{sec_dp1}.
\end{enumerate}
\begin{thm} \label{thm_3ds} Let $x\in\sum S_i\subset X$ (for $i\ge1$) be
the germ of a smooth normal crossing surface $\sum S_i$ in a \hbox{$3$-fold}
$X$.
Let $\sH$ be a mobile linear system\index{mobile linear system} on $X$
and $Z=H_1\cdot H_2$ the cycle theoretic intersection of two general
members $H_1,H_2\in\sH$. Write
\[
Z=Z\pr+\sum Z_{S_i},
\]
where the support of $Z_{S_i}$ is contained in $S_i$ and $Z\pr$
intersects $\sum S_i$ properly. Note that $Z$ may have components contained
in several of the $S_i$, and as a consequence there may be a choice in the
decomposition $Z=Z\pr+\sum Z_{S_i}$. Let $\la_i\ge0$ be rational numbers
and assume that
\[
K_X-\sum\la_i S_i+\frac{1}{\mu}\sH \quad \text{is not canonical.}
\]
Then there are positive rational numbers $0\le t_i\le1$ (with $0\
4\bigl(1+\sum \la_i t_i\bigr) \mu^2.
\]
\end{thm}
\begin{rem} \label{rem_3ds} In fact the argument proves that, more
generally, for any number of surfaces $\sum S_i\subset X$, possibly
singular, $K_X -\sum \la_i S_i+\frac{1}{\mu}\sH$ not canonical implies
that there are rational numbers $0\le t_i\le1$ (with $0
4\Bigl(1+\sum \la_i t_i\nu_i\Bigr) \mu^2,
\]
where $\nu_i=\mult_x S_i$ is the multiplicity of $S_i$ at $x$.
\end{rem}
\begin{proof} The proof is very similar to the proof of
Theorem~\ref{thm_2ds} and I sketch it briefly, leaving some of the
details and calculations to the reader (the main difficulty here is to
make the statement, not the proof).
Let
\[
\ep\colon S_0'\subset X'\to x\in X
\]
be the blowup of the maximal ideal of $x\in X$. In other words, I depart
from usual practice, denoting by $S_0'$ the exceptional divisor of the
blowup. Write $S_i'=\ep^{-1}_* S_i$ (for $i\ge1$) and $\sH'=\ep^{-1}_*\sH$.
If $m=m_{S_0'}\sH$, we have that
\[
\ep^* \Bigl(K_X-\sum_{i\ge1} \la_i
S_i+\frac{1}{\mu}\sH\Bigr)=K_{X'}-\Bigl(-\frac{m}{\mu}+2+\sum_{i\ge1}
\la_i\Bigr)S_0' -\sum_{i\ge1}\la_iS_i'+\frac{1}{\mu}\sH'
\]
is not canonical. By induction, we may assume that the statement holds on
$X'$. The plan is to massage the ensuing inequality until we
get the result for $X$.
We consider 3 cases, depending on the nature of the centre $C_E X'\subset
X'$ of a valuation\index{valuation} $E$ with negative\index{discrepancy}
discrepancy $a_E(K_X-\sum \la_iS_i+\frac{1}{\mu}\sH)<0$:
\scpa{Case 1: $E=S_0'$} This is obvious.
\scpa{Case 2: $C_EX'\subset X'$ is a curve} In this case, the statement on
$X$ follows easily by applying the surface statement \ref{thm_2ds} to
$C_EX'\subset X'$. Details are left to the reader.
\scpa{Case 3: $C_EX'=x'\in X'$ is a closed point} I treat this case in some
detail. Let $Z'=H_1'\cdot H_2'$ be the intersection of 2 general members
of $\sH'$; note that this is neither $\ep^{-1}_*Z$ nor $\ep^* Z$. As
before for $Z$, write
\[
Z'=Z'\pr+\sum_{i\ge1} Z'_{S'_i}.
\]
I need the following, which are easy to see:
\begin{enumerate}
\item $\mult_x Z=\deg Z'_{S_0'}+m^2\ge \mult_{x'} Z'_{S_0'}+m^2$,
\item $\mult_x Z_{S_i}\ge \mult_{x'} Z'_{S_i'}$ if $i\ge1$,
\item $\mult_x Z\pr\ge \mult_{x'} Z'\pr$.
\end{enumerate}
There are rational numbers $0\le t_i'\le1$ such that
\[
\mult_{x'} Z'\pr+\sum_{i\ge0} t_i' \mult_{x'}
Z'_{S_i'}\ \ge\ 4\bigl(1+\sum_{i\ge0} \la'_i t_i'\bigr)\mu^2,
\]
where, of course, I have set $\la_i'=\la_i$ if $i\ge1$ and
\[
\la_0'=-\frac{m}{\mu}+2+\sum_{i\ge1} \la_i\ge0
\]
(if it were $<0$, we would be in Case~1). The statement follows by setting
\[
t_i=\frac{t_i'+t_0'}{1+t_0'} \quad\text{for $i>0$.}
\]
Indeed, first of all, by (1--3) above and the inductive hypothesis,
\begin{align*}
&\mult_x Z\pr\hbox{} +\sum_{i\ge1} t_i \mult_x Z_{S_i} \\
&\qquad =\frac{t_0'}{1+t_0'}
\Bigl(\mult_x Z\pr+\sum_{i\ge1}\mult_x Z_{S_i}\Bigr) \\
& \qquad\qquad
+ \frac{1}{1+t_0'}
\Bigl(\mult_x Z\pr+\sum_{i\ge1} t_i' \mult_x Z_{S_i}\Bigr) \\
&\qquad \ge\frac{t_0'}{1+t_0'}
(\mult_{x'} Z'_{S_0'}+m^2)+\frac{1}{1+t_0'}
\Bigl(\mult_{x'} Z'\pr+\sum_{i\ge1} t_i'
\mult_{x'} Z'_{S_i'}\Bigr) \\
&\qquad =\frac{1}{1+t_0'}\Bigl(\mult_{x'} Z'\pr+\sum_{i\ge0} t_i'
\mult_{x'} Z'_{S_i'}\Bigr)+\frac{t_0'}{1+t_0'}m^2 \\
&\qquad \ge \frac{4}{1+t_0'}
\Bigl(1+\sum_{i\ge0} \la'_i t'_i\Bigr)\mu^2+\frac{t_0'}{1+t_0'}m^2.
\end{align*}
An easy estimate allows us to complete the proof (recall the definition
of $\la_i'$):
\begin{align*}
&\frac{4}{1+t_0'}\Bigl(1+\sum_{i\ge0} \la'_i t'_i\Bigr)\mu^2+
\frac{t_0'}{1+t_0'}m^2 \\
&\qquad=\frac{4}{1+t_0'}\Bigl(1+\sum_{i\ge1}(t_0'+t_i')\la_i
\Bigr)\mu^2+\frac{t_0'}{1+t_0'}\Bigl(4\bigl(2-\frac{m}{\mu}\bigr)\mu^2+m^2
\Bigr) \\
&\qquad\ge4\mu^2\Bigl(1+\sum_{i\ge1}t_i\la_i\Bigr),
\end{align*}
where I have used $4\bigl(2-\frac{m}{\mu}\bigr)\mu^2+m^2\ge4\mu^2$.
\qed \end{proof}
\section{Conic bundles} \label{cha_cb}
This chapter is devoted to the study of conic bundles.\index{conic bundle}
In the first section I give several proofs and generalisations of the
known rigidity\index{birational rigidity} theorem, first shown by Sarkisov
\cite{Sa1}, \cite{Sa2}. Then I state some natural conjectures and discuss
possible approaches to them.
\subsection{Rigid Conic bundles}
\begin{dfn}
A {\em conic bundle}\index{conic bundle} is a \hbox{3-fold} Mori fibre
space\index{Mori!fibre space} $X\to S$ with (generic) fibre of dimension
1. In particular, it is always automatically assumed that $\rk\NS X-\rk\NS
S=1$. The generic fibre $X_\eta$ is a conic over the rational function
field $K(S)$ of $S$.
A conic bundle\index{conic bundle} $X\to S$\index{standard conic
bundle|(} is {\em standard} if the total space $X$ is smooth. It is easy
to see that then $S$ must itself be smooth and, given any $P\in S$, there
is an analytic neighbourhood $P\in U$ and analytic coordinates $s_1, s_2$
on $U$ such that $X\rest{U}\subset\PP^2 \times U$ is isomorphic to one of
the following models:
\begin{enumerate}
\renewcommand{\labelenumi}{(\alph{enumi})}
\item $x_0^2+x_1^2+x_2^2=0$, or
\item $x_0^2+x_1^2+s_2x_2^2=0$, or
\item $x_0^2+s_1x_1^2+s_2x_2^2=0$.
\end{enumerate}
It is easy to see that any conic bundle\index{conic bundle} can be put in
standard form; there is therefore no real loss of generality in
restricting attention to standard conic bundles.
\end{dfn}
\begin{thm}[Sarkisov \cite{Sa1}] \label{thm_rcb}
Let $X\to S$ be a standard conic bundle.\index{conic bundle} Denoting by
$\De\subset S$ the discriminant of the conic bundle, we assume that
$4K_S+\De$ is quasieffective.\index{quasieffective} If $X'\to S'$ is
another Mori fibre space,\index{Mori!fibre space} every birational map
$\fie\colon X\broken X'$ is square.\index{square birational map}
\end{thm}
To understand the result, it is useful to consider the following invariant.
\begin{dfn}
Let $X\to S$ be a conic bundle.\index{conic bundle} The {\em effective
threshold}\index{effective!threshold} $\tau=\tau (S,\De)$ is the
rational number
\[
\tau=\sup \bigl\{s \bigm| sK_S+\De\ge0 \bigr\}.
\]
\end{dfn}
\begin{pro}
Let $X\to S$ and $X'\to S'$ be conic bundles.\index{conic bundle} Assume
that $X\to S$ is square birational\index{square birational map} to $X'\to
S'$. Assume that
\begin{enumerate}
\item $X\to S$ is standard, and
\item $\tau\ge1$.
\end{enumerate}
Then $\tau'\ge\tau$. In particular, if in addition $X'\to S'$ is also
standard then $\tau=\tau'$.
\end{pro}
\begin{proof} The proof is a formal consequence of the following two
observations:
\begin{enumerate}
\renewcommand{\labelenumi}{(\alph{enumi})}
\item Given a standard conic bundle\index{conic bundle} $X\to S$ and the
blowup $C\subset S'\to P\in S$ of the maximal ideal of a point $P\in S$,
there is a (nonunique) standard conic bundle $X'\to S'$, square
birational\index{square birational map} to $X\to S$, with
$X'\rest{S'\setminus C}=X\rest{S\setminus P}$.
\item For a conic bundle $X\to S$, the discriminant only depends on the
generic fibre $X_\eta$. \qed
\end{enumerate}
\end{proof}
\begin{exa}
It is important to understand that $\tau$ is not a square birational
invariant\index{square birational map} for general (possibly nonstandard)
conic bundles\index{conic bundle}. Here is an example. Let $Z\subset
\PP^6$ be the cone over the Veronese surface and let $p\colon Z\broken
\PP^2$ be the projection\index{projection} from a general 3-plane
$H\subset\PP^6$. The blowup $X'\to Z$ of the 4 points $H\cap X$ resolves
the singularities of $p$, giving rise to a conic bundle $X'\to \PP^2$
\[
\mbox{\definemorphism{birto}\dashed \tip \notip
\diagram
Z\drbirto_p & X' \dto \lto \\
& \PP^2 \enddiagram}
\]
Note that $X'$ has an index 2 quotient\index{quotient singularity}
singular point $Q\in X'$ and the discriminant $\De'\subset \PP^2$ consists
of 3 lines meeting in a common point $P\in \PP^2$; hence $\tau'=1$. The
fibre over $P$ is the sum $\Ga_1+\Ga_2+\Ga_3+\Ga_4$ of $4$ curves
$\Ga_i\iso \PP^1$ meeting at $Q$ and $-K_{X'}\cdot\Ga_i=\frac{1}{2}$. If
$C\subset\FF_1\to P \in\PP^2$ is the blowup of $P$, $W\to X'$ the blowup
of $Q$, and $W\broken X$ the flop of the proper transforms of the curves
$\Ga_i$, we have a link of Type~II\index{link}\index{link!of Type~I--IV}
\[
\mbox{\definemorphism{birto}\dashed \tip \notip
\diagram
&W\dlto \rbirto^\flop& X\dto \\
X'\dto& & \FF_1\dllto\\
\PP^2& & \enddiagram}
\]
to a standard conic bundle\index{conic bundle} $X\to \FF_1$ whose
discriminant $\De\subset \FF_1$ consists of 3 disjoint fibres; hence
$\tau=0$.\index{standard conic bundle|)}
It is easy to imagine a link\index{link} $X\broken X'$ which looks like
that just constructed in a neighbourhood of the contraction $C\subset S\to
P\in S'$ of a \hbox{$-1$-curve}. For such a link we would have
$3K_S+\De=0$. In other words,
\[
\al K_S+\De=\al K_{S'}+\De'+(\al-3)C \quad\text{for all $\al$,}
\]
so that $\tau\ge3$ implies $\tau'=\tau$ but, if $\tau<3$ (or $\tau'<3$),
it might happen that $\tau<\tau'<3$. The reader is invited to construct
arbitrarily complicated examples using this idea.
\end{exa}
I leave the following well known lemma as an exercise.
\begin{lem} \label{lem_rcb} Let $\pi\colon X\to S$ be a conic
bundle.\index{conic bundle} Then $-\pi_*(K_X^2)=4K_S+\De$.\index{K2@$K^2$
condition} \qed
\end{lem}
\begin{pfof}{Theorem~\ref{thm_rcb}} I first prove the result in the case
of surfaces. In other words, I assume that $X$ is a smooth surface,
defined over an algebraically nonclosed field $k$, and $\pi\colon X\to S$
a conic bundle\index{conic bundle} structure. I think that this is the
natural context for the theorem: the proof generalises easily to
\hbox{3-folds} (and in fact also to higher dimensions, see below). In
closing, I give a second proof for the surface case which, while
technically more sophisticated than the first, has the advantage that it
generalises well to the case of Del Pezzo fibrations\index{del
Pezzo!fibration} (Chapter~\ref{cha_dp}).
I argue by contradiction in each case, assuming a Mori fibre space $X'\to
S'$\index{Mori!fibre space} and a\index{square birational map} nonsquare
map
\[
\mbox{\definemorphism{birto}\dashed \tip \notip
\diagram
X\dto_\pi \rbirto^\fie & X'\dto \\
S & \,S'. \enddiagram}
\]
Choose a very ample complete linear system
\[
\sH'=|{-}\mu' K'+A'|
\]
on $X'$, where $A'$ is the pullback of an ample divisor on the base $S'$.
The proper transform $\sH$ on $X$ is a mobile linear system\index{mobile
linear system} and, because $X\to S$ is a Mori fibre
space,\index{Mori!fibre space} there is a rational number $\mu$ (with
$2\mu\in\Z$) for which
\[
K+\frac{1}{\mu}\sH=A
\]
is a pullback from the base $S$. The assumption that $\fie$ is not
square,\index{square birational map} together with the
Noether--Fano--Iskovskikh\index{NFI@{Noether--Fano--Iskovskikh
inequalities}} inequalities (Theorem~\ref{thm_nfi}, (1)), implies that
$\mu>\mu'$. Basically, all the proofs rest on the following idea. Assume
that $K+\frac{1}{\mu}\sH$ is canonical.\index{canonical} Then, by
Theorem~\ref{thm_nfi}, (2), $A$ is not\index{quasieffective}
quasieffective. On the other hand a small calculation using
Lemma~\ref{lem_rcb} gives that
\[
A=\pi_* \frac{1}{\mu^2} Z+(4K_S+\De) \quad\text{is quasieffective,}
\]
where $Z=H_1\cdot H_2$ is the intersection of 2 general members of $\sH$,
a contradiction. Now $K+\frac{1}{\mu}\sH$ need not be canonical in general;
there are two ways around this. One way (the first proof) is to run the
Sarkisov program until\index{Sarkisov!program} it becomes canonical; the
other (the second proof) is to study $A$ more closely and show that it can
only be large at the expense of $\sH$ becoming too singular.
\scpa{(1) Surfaces (first proof)} By the Noether--Fano--Iskovskikh
inequalities\index{NFI@{Noether--Fano--Iskovskikh inequalities}}
(Theorem~\ref{thm_nfi}, (3)), running the Sarkisov program gives a chain of
square links
\[
X/S\broken X_1/S\broken\cdots\broken X_n/S
\]
until
\[
K_n+\frac{1}{\mu}\sH_n=A_n \quad\text{is not nef;}\index{nef}
\]
this amounts to saying that
\[
\deg A_n<0.
\]
We now choose general members $H_{1,n},H_{2,n}\in\sH$ and calculate the
{\em effective cycle} $Z_n=H_{1,n}\cdot H_{2,n}$ by
\[
\frac{1}{\mu^2}Z_n=(-K_n+A_n)^2=K_n^2-2K_n\cdot A_n.
\]
Taking the direct image under $\pi_n\colon X_n\to S$ gives
\[
A_n=\pi_* \frac{1}{\mu^2}Z_n+(4K_S+\De_n).
\]
The assumption and the birational invariance of $\tau$ imply that
$4K_{S_n}+\De_n$ is quasieffective, and hence also $A_n$,\index{quasieffective} a contradiction.
\scpa{(2) 3-folds} The proof is essentially the same.\index{NFI@{Noether--Fano--Iskovskikh inequalities}} By the
Noether--Fano--Iskovskikh inequalities (Theorem~\ref{thm_nfi}, (2)),
running the Sarkisov program\index{Sarkisov!program} gives a chain of
square links
\[
X/S\broken X_1/S_1\broken\cdots\broken X_n/S_n
\]
until $K_n+\frac{1}{\mu}\sH_n=A_n$ is {\em not quasieffective}. As
before, writing
\[
A_n=\pi_* \frac{1}{\mu^2}Z_n+(4K_S+\De_n)
\]
shows that $A_n$ is quasieffective,\index{quasieffective} a contradiction.
\scpa{(3) Surfaces (second proof)} In this proof I obtain a contradiction
from the cycle $Z=H_1\cdot H_2$ by arguing directly on $X$, without first
running the Sarkisov program.\index{Sarkisov!program} In doing so, I use
the assumption in the form $K^2\le0$. Here is the crucial point.
I claim that there are maximal singularities $E_i$ with centres $x_i\in
F_i\subset X$ lying in fibres $F_i$ over distinct points $s_i\in S$, and
positive rational numbers $\la_i>0$, such that
\begin{enumerate}
\item $a_{E_i}\Bigl(K-\la_iF_i+\frac{1}{\mu}\sH\Bigr)=a_{E_i}(K)+\la_i
m_{E_i} (F_i) -\frac{1}{\mu} m_{E_i}(\sH)=0$ and
\item $\sum \la_i>\deg A$.
\end{enumerate}
This is a small variation on the\index{NFI@{Noether--Fano--Iskovskikh
inequalities}} Noether--Fano--Iskovskikh inequalities. Indeed, choose a
common resolution
\[
\mbox{\diagram
& U \dlto_p \drto^q & \\
X & & \,X'. \enddiagram}
\]
Let $F_i\subset X$ be all the fibres containing the centre of some maximal
singularity of $\sH$ and $E_{ij}\subset U$ the $p$-exceptional divisors
with centre $C_{E_{ij}} X\in F_i$. Above a small neighbourhood of
$F_i\subset X$ we may write
\begin{align*}
K_U &=K+\sum a_{ij}E_{ij}, \\
\sH_U &=\sH-\sum m_{ij}E_{ij}, \\
F_i' &=F_i-\sum c_{ij} E_{ij},
\end{align*}
where $\sH_U$ and $F_i'$ are the proper transforms. By choice, for each
$i$ there is some $j=j(i)$ for which
\[
m_{ij(i)}> \mu a_{ij(i)}.
\]
Finally, suppose that
\[
\la_i=\max_j \Big\{ \frac{m_{ij}-\mu a_{ij}}{\mu c_{ij}}\Bigr\}>0.
\]
Then
\[
K_U-\sum \la_i F_i'+\frac{1}{\mu}\sH_U=
K -\sum \la_i F_i+\frac{1}{\mu}\sH+\sum \al_{ij}E_{ij},
\]
with
\[
\al_{ij}=a_{ij}+\la_ic_{ij}-\frac{1}{\mu}m_{ij}\ge0,
\]
and for each $i$ there is at least one index $j(i)$ for which
$\al_{ij(i)}=0$. To show that $\sum \la_i>\deg A$, I prove the equivalent
statement:
\[
D=K -\sum \la_i F_i+\frac{1}{\mu}\sH \quad\text{is not quasieffective.}
\]
{From} the last 4 displayed equations, if $D$ were quasieffective\index{quasieffective} then
\[
K_U+\frac{1}{\mu}\sH_U
\]
would also be (recall that $K_X+\frac{1}{\mu}\sH$ is canonical away from
$\bigcup F_i$), and then also
\[
q_* \Bigl(K_U+\frac{1}{\mu}\sH_U\Bigr)=
K'+\frac{1}{\mu}\sH'=K'+\frac{1}{\mu'}\sH' -\Bigl(\frac{1}{\mu'}
-\frac{1}{\mu}\Bigr)\sH'
\]
would be quasieffective; this is a contradiction because by
Theorem~\ref{thm_nfi}, (1),
\[
\frac{1}{\mu'}-\frac{1}{\mu}>0.
\]
This proves the claim.
I now prove that the claim implies the result. Indeed,
\[
K+\sum (1-\la_i)F_i+\frac{1}{\mu}\sH
\]
is not log terminal\index{log!category} at $x_i=C_{E_i}X$ and, by
Theorem~\ref{thm_2ds},
\[
K^2+4\deg A=(-K+A)^2=\frac{1}{\mu^2}H_1\cdot H_2=\frac{1}{\mu^2} \deg
Z\ge4\sum \la_i>4\deg A,
\]
a contradiction, given that $K^2\le0$ by assumption. \qed \end{pfof}
It seems to me that the rationally connected\index{rationally connected}
fibrations of Koll\'ar, Miyaoka and Mori \cite{KMM} provide a reasonable
framework to study birational rigidity\index{birational rigidity|(} of
higher dimensional varieties.
\begin{dfn}
\begin{enumerate}
\item A variety $V$ is {\em rationally connected}\index{rationally
connected} ({\em RC} for short) if any 2 general points on $V$ can be
joined by a rational curve.
\item A {\em rationally connected fibration} (or {\em RC fibration}),
written $X \supset U \overset{\pi}{\to} Z$, is a variety $X$, together
with a smooth open subset $X \supset U$ and a proper morphism $\pi\colon
U\to Z$, such that all fibres of $\pi$ are RC varieties.
\item A RC fibration is {\em extremal} (or an {\em ERC fibration})\index{extremal!RC fibration} if $\rk N^1(U/Z)=1$.
\end{enumerate}
\end{dfn}
\begin{exe} Use the above ideas to prove the following statement. Let $X
\supset U\overset{\pi}{\to} Z$ be an ERC fibration (with everything
defined over an arbitrary base field). Assume that the following 2
conditions hold:
\begin{enumerate}
\item $\dim X -\dim Z=1$;
\item if $\De\subset Z$ is the discriminant, there is a compactification
$(\Zbar,\Debar) \supset (Z,\De)$ such that $\codim_{\Zbar \setminus Z}
\Zbar\ge2$ and
\[
K_{\Zbar}+\frac{1}{4}\Debar\ge0
\]
is canonical and effective.
\end{enumerate}
Let $X' \supset U' \overset{\pi'}{\to} Z'$ be another ERC fibration and
$\fie\colon X\broken X'$ a birational map. Then $\fie$ is
square,\index{square birational map} that is, it is an isomorphism on
generic fibres.
\end{exe}
\begin{rem} If $W$ is a variety, let $\MV^1 W\subset N^1 W$ be the real
cone generated by the classes of mobile linear systems.\index{mobile
linear system} If $X\to S$ is a Mori fibre space,\index{Mori!fibre space}
it seems natural to consider the cone
\[
\KE^1 S=\{ A\in N^1 S \mid -K_X+A\in \MV^1 X\} \subset N^1S.
\]
It seems reasonable to expect that $\KE^1 S\subset\NE^1 S$ implies that
$X\to S$ is birationally rigid.\index{birational rigidity} It is of course
not clear what the assumption means, nor that it is invariant under square
birational\index{square birational map} operations. The argument just
given for conic bundles\index{conic bundle} says that
$4K_S+\De\ge0$ implies $\KE^1 S\subset\NE^1 S$, while\index{K2@$K^2$
condition} Chapter~\ref{cha_dp} says that $K_X^2\not\in\NE^2$ implies
$\KE^1 S\subset\NE^1 S$.
Does the cone $\KE^1 S$, or some other cone related to it, satisfy a
general structure theorem?
\end{rem}
\subsection{Open questions}
The following conjectures are due to Iskovskikh (\cite{I2}, \cite{I3}).
\begin{con}[Iskovskikh \cite{I2}] \label{con_cb1}
Let $X\to S$ be a standard conic bundle.\index{standard conic
bundle}\index{conic bundle} If $X$ is rational,
$\tau\le2$.\index{rationality problem}\index{rational!variety}
\end{con}
The near-converse is easy: if $\tau\le2$, $X$ is either rational or
birational to a cubic \hbox{3-fold} (and there is an excellent biregular
criterion to say which is which).
\begin{con}[Iskovskikh \cite{I2}] \label{con_cb2}
If $X$ is rational, there is a birational map $X\broken\PP^3$
transforming fibres into conics.
\end{con}
It is easy to see that the 2 conjectures are equivalent. This is quite
remarkable. Conjecture \ref{con_cb1} is a statement about conic bundles:\index{conic bundle} if the discriminant is large, the conic bundle is not
rational. Conjecture~\ref{con_cb2} on the other hand is a statement on the
size of the Cremona group of $\PP^3$: any net of rational curves in
$\PP^3$ is birational to a net of conics. It is natural to ask whether the
conjectures have a counterpart for Del Pezzo fibrations.\index{del
Pezzo!fibration} If so, this casts a new light on the meaning of rigidity:\index{birational rigidity} $\PP^3$ almost looks like a rigid variety
because it has a {\em large} birational group.
In the remainder of this section, I outline a possible approach to
Conjecture \ref{con_cb1}. As an intermediate step, I would like to pose
Problem~\ref{prb_chal} below as a challenge. First, note the following
result:
\begin{thm}
Let $X\to S$ be a surface conic bundle\index{conic bundle} defined over a
field $k$. If $3K_S+\De\ge0$ then $X\to S$ is birationally rigid.\index{birational rigidity|)}
\end{thm}
\begin{proof} The idea is as follows. Assume that there is a
link\index{link} $X/S\broken X' /S'$ of Type~III or~IV. This means that
$X$ is a Del Pezzo surface of degree
\[
K^2=8-\deg\De\le2
\]
having 2 extremal rays\index{extremal!ray} $R_1$ and $R_2$; suppose that
$R_1$ corresponds to the Mori fibre structure $X/S$. Let $\tau\colon X\to
X$ be the\index{involution!Geiser and Bertini} Geiser or Bertini
involution of $X$. I claim that $\tau R_1=R_2$. This proves the statement
since it follows that $\tau$ is a square birational\index{square
birational map} map from $X/S$ to $X'/S'$. If $d=2$ let $\pi\colon
X\to\PP^2$ be the 2-to-1 cover given by the complete linear system
$|{-}K_X|$, and if $d=1$ let $\pi\colon X\to Q$ be the 2-to-1 cover of the
singular quadric cone $Q\subset \PP^3$ given by the complete linear system
$|{-}2K_X|$. In either case $\tau$ is the involution exchanging the sheets
of $\pi$, and in either case $\NEbar{}^\tau(X)=\NEbar(X^\tau)=\Z$, which
implies $\tau R_1=R_2$.
\qed \end{proof}
\begin{prb}\label{prb_chal} Do something similar for 3-folds. I am almost
tempted to conjecture that if $X\to S$ is a standard (3-fold) conic bundle
and if\index{standard conic bundle}\index{conic bundle} $\tau\ge3$ then
$X\to S$ is birationally rigid.
\end{prb}
In closing, I wish to outline an approach to improving the notion of
standard models of conic bundles.\index{conic bundle} As we know, any
conic bundle $X\to S$ in the Mori category\index{Mori!category} is
birational to a standard conic bundle,\index{standard conic bundle} but
this involves blowing up the surface $S$. It would be desirable, even at
the expense of introducing some singularities, to construct distinguished
biregular models of conic bundles over $\PP^2$ or minimally ruled
surfaces. In addition to being an interesting question in its own right,
this might also be relevant for Conjectures~\ref{con_cb1}
and~\ref{con_cb2}.
Let $X\to S$ be a standard conic bundle\index{conic bundle} with
$1\le\tau<4$ (if $\tau<1$ it is easy to show that $X$ is
rational).\index{rational!variety} Now $K_S+\De$ is
quasieffective\index{quasieffective} and we assume, as we may (see for
example \cite{I3}, Lemma~4), that $K_S+\De$ is nef.\index{nef}
We first run an (ordinary) minimal model program for the base surface $S$
as follows. The nef threshold\index{nef!threshold} is
\[
t=t(S,\De)=\max \{s \mid \text{$sK+\De$ is nef} \}.
\]
Clearly $t\le \tau$. We define the chain
\[
(S, t)=(S_0, t_0)\to\cdots (S_i, t_i)\to (S_{i+1}, t_{i+1})\cdots\to (S_n, t_n)=(S', \tau)
\]
inductively by setting $t_i=t(S_i,\De_i)$, letting $\si_i\colon S_i\to
S_{i+1}$ be the contraction of an extremal\index{extremal!contraction}
rational curve $C_i$ with $(t_iK_i+\De_i)\cdot C_i=0$ (clearly $K_i\cdot
C_i<0$ and $C_i$ is a $-1$-curve) and setting $\De_{i+1}=\si_i(\De_i)$.
The program ends when the nef threshold finally catches up with the
effective threshold\index{effective!threshold} $t_n=\tau$, and we
meet a contraction $S_n=S'\to T$ of fibre type. Since $S'\to T$ is
$\PP^2$ or a minimally ruled surface $\FF_m$, we get the following result.
\begin{cor}
If $\tau<{4}$, then:
\[
\tau\in\left\{\frac{1}{2}\,,\frac{2}{2}\,,\dots,\frac{7}{2}\,,
\frac{1}{3}\,, \frac{2}{3}\,,\dots, \frac{11}{3} \right\}.
\]
\end{cor}
The typical cases are conic bundles\index{conic bundle} over $\PP^2$ with
discriminant of degree $<12$, or over a ruled surface of degree $<8$ over
the base.
As a second step, I propose the following analog of the results of
\cite{C1} and \cite{Ko3}.
\begin{con}
\begin{enumerate}
\item If $\si_i\colon S_i\to S_{i+1}$ is a contraction with $t_i\le2$,
there is a conic bundle $X_{i+1}\to S_{i+1}$ square
birational\index{square birational map} to $X_i\to S_i$. Moreover,
$X_{i+1}$ has terminal singularities\index{terminal!singularities} of
index $1$.
\item If $\tau\le3$, the above procedure eventually makes a
\hbox{$3$-fold} $X''$, with terminal singularities of index $1$, and a
conic bundle $X''\to S''$ over a smooth surface $S''$, where
$3K_{S''}+\De''$ is nef.\index{nef} I hope that one can do this with $S''$
a Del Pezzo surface or a conic bundle\index{conic bundle} $S''\to T''$.
\item If $3<\tau<4$, something nice can be done, e.g., a distinguished
model with terminal singularities\index{terminal!singularities} of index
one of\/ $\{1,2,3,4,6\}$.
\end{enumerate}
\end{con}
I hope that the singularities on the models (2) can be classified and will
turn out to be mild enough so that one can attach an intermediate Jacobian
and use it to show that $X$ is not rational,\index{nonrational variety}
as in the classical cases Beauville (\cite{Be1}, \cite{Be2}) and Shokurov
\cite{Sh}.
\section{Del Pezzo fibrations} \label{cha_dp}\index{del Pezzo!fibration|(}
In this chapter, devoted to Del Pezzo fibrations, I prove Pukhlikov's
rigidity criterion for fibrations of degree 1 and 2, explain Koll\'ar's
method of constructing semistable models\index{semistable!del Pezzo
fibration} of fibrations of cubic surfaces---this is a kind of analog of
the standard conic bundles\index{standard conic bundle} of
Chapter~\ref{cha_cb}---and, in the final section, raise some open
questions.
\subsection{Rigid Del Pezzo fibrations} \label{sec_dp1}
\begin{thm}[Pukhlikov \cite{P5}] \label{thm_rdp}
Let $X\to S$ be a \hbox{$3$-fold} Del Pezzo fibration of degree $d\le2$
and assume that
\begin{enumerate}
\item $X$ is smooth, and
\item the $1$-cycle $K_X^2\in N_1 X$ does not\index{K2@$K^2$ condition}
lie in the interior $\Intr\NEbar X$ of the Mori cone.\index{Mori!cone}
\end{enumerate}
Then $X$ is birationally rigid over $S$.\index{birational rigidity|(}\index{birational rigidity!over a base $S$}
\end{thm}
\begin{proof} The proof is essentially the same as the second proof of
the rigidity criterion for conic bundle\index{conic bundle} surfaces,
given in Chapter~\ref{cha_cb}, with Theorem~\ref{thm_3ds} replacing
Theorem~\ref{thm_2ds} in the endgame. {From} now on until the final
calculation in the last Step~4, I only assume $d\le3$.
The proof is by contradiction, assuming a Mori fibre space\index{Mori!fibre space} $X'\to S'$ and a map $\fie\colon X\broken X'$
\[
\mbox{\definemorphism{birto}\dashed \tip \notip
\diagram
X\dto_\pi \rbirto & X'\dto \\
S & S' \enddiagram}
\]
which is not birational on generic fibres. Choose a very ample complete
linear system
\[
\sH'=|{-}\mu' K'+A'|
\]
on $X'$, where $A'$ is the pullback of a divisor ample on the base $S'$.
The proper transform $\sH$ on $X$ is a mobile linear system\index{mobile
linear system} and, because $X\to S$ is a Mori fibre
space,\index{Mori!fibre space} there is a rational number $\mu$ such that
\[
K+\frac{1}{\mu}\sH=A
\]
is a pullback from the base $S$. The assumption $\fie$ not square, together
with\index{square birational map} the Noether--Fano--Iskovskikh
inequalities\index{NFI@{Noether--Fano--Iskovskikh inequalities}}
(Theorem~\ref{thm_nfi}, (1)), implies $\mu>\mu'$.
\scpa{Step 1} Using well known properties of the generic fibre (see for
instance \cite{C2}, Appendix), perhaps after composing $\fie$ with a
birational selfmap
\[
\mbox{\definemorphism{birto}\dashed \tip \notip
\diagram
X \drto \rrbirto^\al& &\,X,\dlto \\
&S& \enddiagram}
\]
I may assume that no maximal centre is a curve dominating $S$.
I now claim that no (vertical) curve in $X$ can be a maximal centre.
Indeed, if for example $E$ is a maximal singularity having centre $C_E
X=\Ga\subset F_t$, a line contained in the fibre $F_t$ over
$t\in S$, let $C\subset F_t$ be a general conic; then
\[
2\mu=\sH\cdot C\ge2\mult_C\sH>2\mu,
\]
which is a contradiction. The case $\deg\Ga\ge2$ is easier and left to the
reader.
\scpa{Step 2} I claim that there are maximal singularities $E_i$, having
centres $x_i\in F_i\subset X$ lying in fibres $F_i$ over distinct points
$s_i\in S$, and positive rational numbers $\la_i>0$ such that
\begin{enumerate}
\item $a_{E_i}\Bigl(K-\la_iF_i+\frac{1}{\mu}\sH\Bigr)=a_{E_i}(K)+\la_i m_{E_i} (F_i) -\frac{1}{\mu}
m_{E_i}(\sH)=0$, and
\item $\sum \la_i>\deg A$.
\end{enumerate}
The proof is identical to that of the corresponding claim in the proof of
Theorem~\ref{thm_rcb} (surfaces, second proof) and is therefore omitted.
\scpa{Step 3} Let $H_1$, $H_2$ be general members of $\sH$ and $Z=H_1\cdot
H_2$. By Step~1, Theorem~\ref{thm_3ds} and Remark~\ref{rem_3ds}, there are
numbers $04\mu^2\deg A. \qed
\]
\end{proof}
\begin{rem} Pukhlikov \cite{P6} proves that the same statement also holds
for fibrations of cubic surfaces---with minor restrictions on the nature
of the singular fibres.
\end{rem}
\subsection{Semistable models}\index{semistable!del Pezzo fibration}
In this section, I copy from Koll\'ar \cite{Ko3} the theory of semistable
models for hypersurfaces, which improves upon \cite{C1}. Let $\Oh$ be a
DVR with parameter $p\in\Oh$ and field of fractions $K$.
\begin{dfn}
A {\em weight system} $(\bx,\bw)$ on $\Oh [y_0,\dots,y_n]$
is a choice of coordinates
\[
(x_0,\dots,x_n)^t=M(y_0,\dots,y_n)^t,\quad \text{where $M\in\SL(n+1,\Oh)$}
\]
and weights $x_i\mapsto w_i=w(x_i)\in \R$.
\end{dfn}
Let $f_K\in K[y_0,\dots,y_n]$ be a polynomial. One can always find
an integer $s$ such that
$f=p^{-s}f_K\in\Oh [y_0,\dots,y_n]$ ($f$ is called an $\Oh$-model of $f_K$).
The largest such $s$ is the {\em multiplicity} of $f_K$ at $p$; it is
denoted $\mult_p f_K$.
\begin{dfn}
Let $f\in\Oh [y_0,\dots,y_n]$ be a homogeneous polynomial and
$X\subset \PP^n_{\Oh}$
the hypersurface defined by the equation $f=0$.
\begin{enumerate}
\item A weight system $(\bx,\bw)$ over $\Oh$ is called
\[
\renewcommand{\arraystretch}{1.5}
\begin{cases}
\text{\em properly stable}
& \displaystyle\text{if $\mult_p f (p^{\bw}\bx) < \frac{\deg f}{n+1}\sum_i
w_i$,}
\\
\text{\em semistable}
& \displaystyle\text{if $\mult_p f (p^{\bw}\bx)\le \frac{\deg
f}{n+1}\sum_i w_i$,} \\
\text{\em unstable}
& \displaystyle\text{if $\mult_p f (p^{\bw}\bx)> \frac{\deg f}{n+1}\sum_i
w_i$.}
\end{cases}
\]\index{semistable!del Pezzo fibration}
\item $f$ (or $X$) is called {\em properly stable} (respectively {\em
semistable}) if every weight system is properly stable (or semistable).
\item $f$ (or $X$) is called {\em unstable} if there is an unstable
weight system.
\end{enumerate}
\end{dfn}
\paragraph{A procedure to find semistable models.} We start with a
homogeneous polynomial $f_K\in K [y_0,\dots,y_n]$.
\begin{enumerate}
\item Find any $\Oh$-model $f_1$ of $f_K$.
\item Assume that we already have $f_j$. If $f_j$ is semistable we are done.
\item Otherwise there is a weight system
$(\bx,\bw)$ which is unstable on $f_j$. Set
\[
f_{j+1}=p^{-s} f_j(p^{w_0}x_0,\dots,p^{w_n}x_n),
\quad\text{where}\; s=\mult_p f_j(p^{\bw}\bx),
\]
and go back to (2).
\end{enumerate}
The next statement is a special case of the main result of
\cite{Ko3}.\index{semistable!del Pezzo fibration}
\begin{thm}[Koll\'ar \cite{Ko3}]
Let $f_K\in K [x_0,\dots,x_n]$ be a homogeneous polynomial. If the
hypersurface $X_K=(f_K=0)\subset \PP^n_K$ is nonsingular, $f_K$ has a
semistable model over $\Oh$ if and only $f_K$ is semistable over $\Kbar$.
Moreover, $f_K$ has only finitely many semistable models over $\Oh$ up to
the action of $\SL(n+1,\Oh)$.
\end{thm}
\begin{proof} The idea is that the procedure stops, because some
invariant goes down every time the procedure is applied. \qed
\end{proof}
\begin{thm}[Koll\'ar \cite{Ko3}]
In order to check semistability of a family of cubic surfaces it is
sufficient to use weight systems with the following 5 weight sequences
\[
(0,0,0,1),\quad (0,0,1,1),\quad (0,1,1,1),\quad (0,1,2,2),\quad
(0,2,2,3). \qed
\]
\end{thm}
This implies that for cubic forms in 4 variables over $\Oh$ the above
procedure becomes an effective algorithm.
\subsection{Open questions}
\begin{exa} Let $\Oh=\C[t]$ and consider the cubic form
\[
f_1=x_0^3+x_1^3+x_2^2x_3+t^6x_3^3.
\]
It is easy to see that it is semistable with respect to every weight
system where all the weights are 0 or 1. $f_1$ is unstable with weights
$(0,0,1,-2)$ and we obtain the properly stable cubic form
$f_2=x_0^3+x_1^3+x_2^2x_3+x_3^3$, which is the unique semistable
model.\index{semistable!del Pezzo fibration}
\end{exa}
The above is also an example of a square\index{square birational map} link
of Del Pezzo fibrations of degree 3 that is centred at an Eckardt point of
a fibre.
\begin{rem} I have not as yet been able to carry the necessary
calculations to conclusion, but I have the feeling that square links
between Del Pezzo fibrations are relatively common.
For example, let $X\to\De$ be a smooth Del Pezzo fibration of degree 2
over a small disc $\De$ with central fibre $X_0$. The\index{anticanonical!map} anticanonical linear system defines a finite
2-to-1 morphism $\pi\colon X_0\to \PP^2$ ramified along a smooth quartic
curve $B\subset\PP^2$. Let $Q\in B$ be a point lying on 1 of the 28
bitangents of $B$ and $P\in X$ the inverse image. In suitable coordinates
around $P\in X$ the fibration is defined by the function
\[
f=x+y^2+z^3.
\]
Let $Y\to X$ be the\index{weighted!blowup} weighted blowup with weights
$3,2,1$. It seems quite possible to me that the 2-ray game\index{2-ray
game} starting with $Y\to X\to\De$ can be played to the very end, thus
making a link.
One should also be able to do this for Del Pezzo fibrations of degree 3
and points $P\in X_0$ where the curve $T_P X_0 \cap X_0$ is a cuspidal
plane cubic.
\end{rem}
These examples reflect the indirect nature of the proof of
Theorem~\ref{thm_rdp}. I would like to pose the following problems.
\begin{prb} \begin{enumerate}
\item Set up a notion of semistable models for Del Pezzo fibrations of
degree 2 and 1, and prove that they exist. The results of \cite{C1} are
encouraging.\index{semistable!del Pezzo fibration}
\item Prove \ref{thm_dp}---including the $d=3$ case---under the more
general assumption that $X\to \PP^1$ is semistable at every point $t\in
\PP^1$.
\item Does there exist an analog for Del Pezzo fibrations of the
invariant $\tau$ of conic\index{conic bundle} bundles? that is, a
rational number (perhaps obtained as a threshold) that is minimal on a
semistable model, detects rigidity, and perhaps explains the role
of\index{K2@$K^2$ condition} the $K^2$ assumption in
Theorem~\ref{thm_rdp}? For instance, assume that the $K^2$ condition holds
for a semistable $X/S$; does it then hold for every $X'/S'$
square\index{square birational map} to $X/S$?
\end{enumerate}
\end{prb}\index{del Pezzo!fibration|)}
\section{Fano 3-folds} \label{cha_fa}\index{Fano!3-fold}
I review the main known rigidity theorems for Fano 3-folds: hypersurfaces
and complete intersections. In the last section, I indicate possible
future directions. For more information and further discussion, see
\cite{CPR}.
\subsection{Rigid Fano 3-fold hypersurfaces}\index{birational rigidity}
Anthony Iano-Fletcher wrote a list of 95 families\index{famous@``famous
95''} of \hbox{3-fold} Fano weighted hypersurfaces $X=X_d\subset \PP^4_w$
with $-K_X=\Oh(1)$.
\begin{thm}[Corti, Pukhlikov, Reid \cite{CPR}] \label{thm_cr}
Assume that $X$ is a general member of any of the $95$ families. Then
\begin{enumerate}
\item $X$ is birationally rigid,
\item $\Bir X/\Aut X$ is generated by a finite number of explicit
rational involutions.\index{involution}
\end{enumerate}
\end{thm}
Rather than give an overview of the whole proof, which is too long to
summarise here, I treat a special case in detail.
Let $X=X_5\subset \PP (1,1,1,1,2)$, $Q=(0,0,0,0,1)\in X$. We can write the
equation of $X$ as
\[
F(y, x_i)=y^2 x_0+yf_3(x_i)+f_5(x_i)=0.
\]
Consider the projection\index{projection}
\[
\PP(1,1,1,1,2)\broken \PP^3
\]
on the first 4 coordinates: this is well defined on the unique divisorial
contraction\index{divisorial contraction} $\pi=\pi_Q\colon E\subset Y\to
Q\in X$ at $Q$, $-K_Y=-K_X-\frac{1}{2} E$ is nef\index{nef} and the
anticanonical\index{anticanonical!model} model of $Y$ is the variety
\[
\Ybar=\Proj\bigoplus_{n\ge0} H^0(-nK_Y)=\Ybar_6\subset \PP (1,1,1,1,3).
\]
All of this can be seen explicitly by completing the square in the defining
equation of $X$
\[
x_0 F(y, x_i)=\bigl(x_0y+\frac{1}{2}f_3\bigr)^2+x_0f_5-\frac{1}{4}f_3^2.
\]
The equation of $\Ybar_6\subset \PP(1,1,1,1,3)$ is then
\[
G(z,x_i)=z^2+x_0f_5-\frac{1}{4}f_3^2=0,
\]
and the birational map $X\broken \Ybar$ is given by
\[
z=x_0y+\frac{1}{2}f_3.
\]
The exceptional set of the\index{anticanonical!map} anticanonical map
$\fie\colon Y\to \Ybar$ consists of the proper inverse images of the 15
lines $\Ga_i\subset X$ for
$i=1,2,\dots,15$, given by
\[
x_0=f_3(x_i)=f_5(x_i)=0.
\]
Each of these lines on $X$ has anticanonical degree
\[
-K_X\cdot\Ga=\frac{1}{2}.
\]
Being a double cover of $\PP^3$, $\Ybar$ has a rational biregular
involution\index{involution} $\si\colon \Ybar\to \overline {Y}$. The
corresponding birational involution $\tau$ of $X$ is a link;\index{link}
indeed it can be written as
\[
\mbox{\definemorphism{birto}\dashed \tip \notip
\diagram
&Y\dlto_{\pi_Q}\drto_\fie\rrbirto^{\fie^{-1} \si\fie}&
&Y \dlto^\fie\drto^{\pi_Q} & \\
X & &\Ybar& & X \enddiagram}
\]
where $\fie^{-1}\si\fie$ is the flop of the 15 proper transforms
$\Ga_i'\subset Y$.
\begin{thm} \label{thm_crs}
If $X=X_5\subset \PP (1,1,1,1,2)$ is quasismooth,\index{quasismooth}
then
\begin{enumerate}
\item $X$ is birationally rigid,
\item $\Bir X /\Aut X$ is generated by the rational\index{involution}
involution $\tau$ just described.
\end{enumerate}
\end{thm}
\begin{proof} Let $X'\to S'$ be a Mori fibre space,\index{Mori!fibre
space} and $\fie\colon X\broken X'$ a birational map. Choose a very ample
complete linear system
\[
\sH'=|{-}\mu' K'+A'|
\]
on $X'$, where $A'$ is the pullback of a divisor ample on the base $S'$,
and let $\sH\subset |{-}\mu K|$ be the proper transform on $X$. In the
first part of the proof (Steps~1 and~2 below) I classify the possible
maximal centres of $\sH$; in the second part (Step~3) I finish by invoking
the main statement of the Sarkisov program.\index{Sarkisov!program}
\scpa{Step 1} No curve on $X$ is a maximal centre of $\sH$. If a curve $C$
is a maximal centre, there is a maximal extraction $E\subset Y\to C\subset
X$. By Exercise~\ref{exe_cth}, (1), $c=\mult_C\sH>\mu$ and, by Kawamata's
Theorem~\ref{thm_dc1}, $C$ is contained in the smooth locus of $X$. In
particular then $\deg C$ is a positive integer. Let $Z$ be the intersection
$H_1\cdot H_2$ of two general members $H_1$, $H_2$ of $\sH$. Intersecting
with a general surface $S\in |{-}K|$ gives
\[
\frac{5}{2}\mu^2=Z\cdot S\ge c^2 \deg C>\mu^2 \deg C,
\]
so that $\deg C=1$ or $2$.
If $\deg C=1$, we can choose coordinates on $\PP$ so that $C$ is given by
$y=x_0=x_1=0$. The blowup $f\colon E\subset Y\to C\subset X$ of the
ideal sheaf of $C$ is the only divisorial\index{divisorial contraction}
contraction centred at $C$, and $C$ is the base locus of $|I_C(-2K)|$;
therefore the class
\[
M=f^{-1}_* |I_C(-2K)|=-2K-E
\]
is nef\index{nef} on $Y$. An easy calculation shows
\begin{align*}
M\cdot Z&=(-2K-E)(-\mu K-cE)^2 \\
&=2\mu^2(-K)^3-
(4\mu c+\mu^2)(-K)^2\cdot E+(2c^2+2\mu c)(-K)\cdot E^2 -c^2E^3 \\
&=5\mu^2 -2c^2-2\mu c-c^2<0,
\end{align*}
a contradiction. In the calculation I used that $(-K)^3=\frac{5}{2}$,
$(-K)^2\cdot E=0$ (the projection formula), $(-K)\cdot E^2=-\deg C=-1$
(also by the projection formula) and $E^3=-\deg N_C X=-\deg
C+2-2p_a(C)=1$.
If $\deg C=2$ then $C$ is given in suitable coordinates by
\[
y=x_3=x_0x_1+x_2^2=0 \quad\text{or}\quad
y^2+f_4(x_0,x_1)=x_2=x_3=0.
\]
In the first case, we work as before with the class $M=-2K-E$, in the
second with $M=-4K-E$. In either case it is easy to check that $M\cdot
Z<0$ (I leave the numerics as an exercise to the reader) and, as above,
$C$ cannot be a maximal centre.
\scpa{Step 2} No smooth point $P\in X$ is a maximal centre of $\sH$.
Assume first that $P$ does not lie on any of the 15 lines $\Ga_i\subset X$
described above. Then the linear system
\[
|I_P(-K)|
\]
isolates $P$. Intersecting with a general member $S\in |I_P(-K)|$,
using Corollary~\ref{cor_sco}, (2), we get
\[
\frac{5}{2} \mu^2=Z\cdot S\ge (Z\cdot S)_P>4\mu^2
\]
(where, as usual, $Z=H_1\cdot H_2$ is the intersection of 2 general
members of $\sH$), which is a contradiction. Assume now that $P\in\Ga$,
one of the 15 lines. Even though the base locus of $|I_P(-K)|$ is $\Ga$,
we still choose a general surface
\[
S\in |I_P(-K)|
\]
and use it as a ``test surface''.\index{test!surface} Restricting
$\sH$ to $S$ we obtain
\[
\sH\rest{S}=c\Ga+\sL,
\]
where $\sL$ is a mobile linear system\index{mobile linear system} on $S$
and $c=\mult_\Ga\sH$.
The argument now proceeds in 3 straightforward steps as usual: first, by
\ref{cor_sco}, (1) and \ref{thm_2ds}, two general members $L_1$, $L_2$
have a large intersection at $P$; second, we estimate $\sL^2=L_1\cdot
L_2$ from above; third, we derive a contradiction from the 2 previous
steps.
(a) We are assuming that $P$ is a maximal centre; therefore, by
\ref{cor_sco}, (1), $K_S+\frac{c}{\mu}\Ga+\frac{1}{\mu}\sL$ is not log
canonical\index{log!canonical} and by \ref{thm_2ds}
\[
(L_1\cdot L_2)_P> 4\Bigl(1-\frac{c}{\mu}\Bigr)\mu^2.
\]
(b) $S$ has an ordinary double point at $Q=(0,0,0,0,1)$ and
\[
(K_S\cdot\Ga)_S=(K+S\cdot\Ga)_X=0,
\]
so that
\[
K_S+\Ga\rest{\Ga}=K_\Ga+\Diff=K_\Ga+\frac{1}{2}Q
\]
and
\[
(\Ga\cdot\Ga)_S=-2+\frac{1}{2}=-\frac{3}{2}\,.
\]
{From} this we can calculate
\[
L_1\cdot L_2=(-\mu K\rest{S}-c\Ga)^2_S=\frac{5}{2}\mu^2-\mu
c-\frac{3}{2}c^2.
\]
(c) {From} the 2 previous steps, since $\sL$ is free from base curves, we
conclude that
\[
4\Bigl(1-\frac{c}{\mu}\Bigr)\mu^2<\frac{5}{2}\mu^2-\mu c-\frac{3}{2}c^2,
\]
a contradiction.
\scpa{Step 3: conclusion} According to the Sarkisov program there is a
Mori fibre space\index{Mori!fibre space} $X_1\to S_1$ and a link\index{link} $\psi_1\colon X\broken X_1$ such that $\deg\fie
\psi_1^{-1}<\deg\fie$. Now $X$ is a Fano 3-fold,\index{Fano!3-fold} so we
must be in Case~(1) of the\index{NFI@{Noether--Fano--Iskovskikh
inequalities}} Noether--Fano--Iskovskikh inequalities \ref{thm_nfi}. Then
the link starts with a maximal extraction and, by what I have proved and
Theorem~\ref{thm_dc1}, the extraction in question is the Kawamata blowup\index{Kawamata blowup} $E\subset Y\to Q\in X$. The link, if it exists, is
the unique answer to the 2-ray game\index{2-ray game} starting with $Y\to
X\to\pt$. But we know that the link\index{link} $\tau$ is an answer to
this 2-ray game.\index{2-ray game} Therefore the link in question is
$\tau$.\qed \end{proof}
\subsection{Rigid Fano 3-fold complete intersections} \label{sec_ci}
Let $X=X_{2,3}\subset \PP^5$ be a smooth complete intersection of
a quadric and a cubic in $\PP^5$. Every line $\Ga\subset X$
is the centre of a link\index{link}
\[
\tau_\Ga\colon X\broken X
\]
that can be understood as follows. Let $\pi_\Ga\colon Y\to X$ be the
blowup of $\Ga$. The proper inverse image of the linear system $\de(\Ga)$
of hyperplane sections through $\Ga$ is free on $Y$ and defines a
generically 2-to-1 map $\fie_{\de (\Ga)}\colon Y\to\PP^3$
\[
\mbox{\diagram
&Y\dlto_{\pi_\Ga}\drto^{\fie_{\de (\Ga)}}& \\
X & &\,\PP^3. \enddiagram}
\]
If $\si\colon Y\to Y$ is the biregular\index{involution} involution of
$Y$ exchanging the two sheets of $\fie_{\de (\Ga)}$ then we define
\[
\tau_\Ga=\pi_\Ga \si \pi_\Ga^{-1}.
\]
We leave it to the reader to construct a link\index{link} $\tau_\Ga$
starting with {\em any conic} $\Ga\subset X$.
In this section, I prove the following result.
\begin{thm}[Iskovskikh, Pukhlikov \cite{I1}, \cite{IP}] \label{thm_r23} If
$X=X_{2,3}\subset \PP^5$ is sufficiently general (the precise condition
is stated in the beginning of the proof), then
\begin{enumerate}
\item $X$ is birationally rigid, and\index{birational rigidity|)}
\item $\Bir X /\Aut X$ is generated by the rational\index{involution}
involutions $\tau_\Ga$ centred on lines and conics.
\end{enumerate}
\end{thm}
\begin{proof} The logic of the proof is identical to \ref{thm_crs}. The
same method shows that if a curve $C\subset X$ is a maximal centre then it
must be a line or conic. In \ref{thm_23} below, which is the hardest part
of the proof, I prove that a closed point $P\in X$ cannot be a maximal
centre. The statement then follows from the Sarkisov program,\index{Sarkisov!program} in the same way as the conclusion of the proof of
Theorem~\ref{thm_crs}. \qed \end{proof}
\begin{thm}[Pukhlikov \cite{IP}] \label{thm_23}
If $X=X_{2,3}\subset \PP^5$ is sufficiently general, then no (smooth)
point $P\in X$ can be a maximal centre.
\end{thm}
\begin{proof} The proof is very similar to the proof of
Theorem~\ref{thm_crs}, Step~2, and is conceptually an easy application of
the technique of Chapter~\ref{cha_sls}: reduction to surfaces via
inversion of adjunction, and a calculation using Theorem~\ref{thm_2ds}.
The new element is the use of\index{bad line} the ``bad line'' of
\ref{cor_lcl} to improve some inequalities (see Step~4 of the proof).
\scpa{Step 1. Main division into cases} In this proof I take $X$ general
to mean the following. Let $P\in X$ be a point, $T_P=T_P X$ the tangent
plane to $X$ at $P$, $C=T_P\cap X$ a curve of degree 6 in $T_P\iso
\PP^3$ with multiplicity $4$ at $P$. Then we have one of the following 4
cases:
\begin{enumerate}
\renewcommand{\labelenumi}{(\alph{enumi})}
\item $P\in C=\De\subset T_P\iso \PP^3$ is an irreducible (rational)
sextic with a 4-tuple point at $P$.
\item $P\in C=\De+\Ga\subset T_P\iso \PP^3$ is the sum of an irreducible
(rational) quintic with a triple point at $P$ and a line $P\in\Ga$.
\item $P\in C=\De+\Ga_1+\Ga_2\subset T_P\iso\PP^3$ is the sum of an
irreducible (rational) quartic with a double point at $P$ and two lines
through $P$.
\item $P\in C=\De+\Ga_1+\Ga_2+\Ga_3\subset T_P\iso \PP^3$ is the sum of a
twisted cubic and three lines through $P$.
\end{enumerate}
In each case we may also assume that the singularity $P\in C\subset T_P$
is analytically equivalent to a cone over 4 general points in $\PP^2$.
\scpa{Step 2. Basic set up} For the rest of the proof, we let
\[
F\subset Y\to P\in X
\]
be the blowup of the maximal ideal; choose a general hyperplane section
$S$ containing $C=T_P\cap X$,
\[
P\in C\subset S\in |\Oh_X(1)|,
\]
and let $S'\subset Y$ be the proper transform. Also denote by $\De'$,
$\Ga_i'\subset S'$ the proper transforms of $\De$, $\Ga_i$ and by
$\Phi=F\rest{S'}$ the restriction of $F$ to $S'$. Then $\De'$, $\Ga_i'$ and
$\Phi$ all have self-intersection $-2$ on $S'$ and, in addition, it is
easy to compile a multiplication table in each of the 4 cases:
\begin{enumerate}
\renewcommand{\labelenumi}{(\alph{enumi})}
\item $\De'\cdot\Phi=4$.
\item $\De'\cdot\Ga'=1$, $\Ga'\cdot\Phi=1$ and $\De'\cdot\Phi=3$.
\item $\Ga'_1\cdot\Ga'_2=0$, $\De'\cdot\Ga'_i=1$, $\Ga'_i\cdot\Phi=1$
and $\De'\cdot\Phi=2$.
\item $\Ga'_i\cdot\Ga'_j=0$ (for $i\ne j$), $\De'\cdot\Ga'_i=1$,
$\Ga'_i\cdot\Phi=1$ and $\De'\cdot\Phi=1$.
\end{enumerate}
Assuming that $P\in X$ is a maximal centre of a linear system $\sH\subset
|\Oh_X(\mu)|$ let $\sH'\subset Y$ be the birational transform and
\[
\sH'\rest{S'}=\sL'+d\De'+\sum c_i\Ga_i'
\]
the restriction of $\sH'$ to $S'$, where $\sL'$ is free from base curves.
We reach a contradiction in three steps as usual. First, we show that,
because $\sH$ has a high singularity at $P$, $P$ contributes a large
amount to the intersection $\sL'{}^2=L_1'\cdot L_2'$ of two general
members $L_1'$, $L_2'$ of $\sL'$, which must therefore be large. Second,
we use the multiplication table to estimate $\sL'{}^2$ from above.
Third, a contradiction is drawn from the previous steps by comparison.
In each case, write
\[
\frac{1}{\mu} \sL' \sim\fie\Phi+\de\De+\sum\ga_i\Ga_i,
\]
with $\fie=2-\frac{m_E(\sH)}{\mu}\le1$, $\de=1-\frac{d}{\mu}\le1$ and
$\ga_i=1-\frac{c_i}{\mu}\le1$. Note that necessarily $\de\ge0$ (as
observed above, $\De$ cannot be a maximal centre) and $\ga_i<0$ can only
happen if $\Ga_i$ is a maximal centre. After untwisting\index{untwisting}
by finitely many\index{involution} involutions $\tau_\Ga$ centred on lines
and conics I may and will assume that no curve on $X$ is a maximal centre;
hence $\ga_i\ge0$. (This manoeuvre is not logically strictly necessary but
it cuts down the complexity of the calculations by a factor of 2.)
{From} this and the multiplication table we can calculate
$\frac{1}{\mu^2}\sL'{}^2$, in the four main cases, as follows:
\begin{enumerate}
\renewcommand{\labelenumi}{(\alph{enumi})}
\item $\frac{1}{\mu^2}\sL'{}^2=-2\fie^2-2\de^2+8\fie\de$;
\item $\frac{1}{\mu^2}\sL'{}^2=
-2\fie^2-2\de^2-2\ga^2+6\fie\de+2\fie\ga+2 \de\ga$;
\item $\frac{1}{\mu^2}\sL'{}^2=
-2\fie^2-2\de^2-2\ga_1^2-2\ga_2^2+4\fie\de+
2\fie\ga_1+2\fie\ga_2+2\de\ga_1+2\de\ga_2$;
\item $\frac{1}{\mu^2}\sL'{}^2=
-2\fie^2-2\de^2-2\ga_1^2-2\ga_2^2-2\ga_3^2$ \newline
\hbox{\qquad\qquad} $+2\fie\de+2\fie\ga_1+2\fie\ga_2+
2\fie\ga_3+2\de\ga_1+2\de\ga_2+2\de\ga_3$.
\end{enumerate}
\scpa{Step 3. $F$ is a maximal singularity} I do this case first as a
warm-up. The assumption means that $\fie<0$. I draw a contradiction from
the assumption that $\frac{1}{\mu^2}\sL'{}^2\ge0$, using the expressions
derived in the previous step. I do this only in Case~(d) since the
arithmetic in the other cases is a specialisation of this case (and, in
any event, it is easier). We have
\begin{align*}
-\frac{1}{\mu^2}\sL'{}^2
&=2\fie^2+2\de^2+2\ga_1^2+2\ga_2^2+2\ga_3^2-\\[-4pt]
&\qquad-2\fie\de-2\fie\ga_1 -2\fie\ga_2
-2\fie\ga_3-2\de\ga_1-2\de\ga_2-2\de\ga_3 \\[4pt]
&=(\fie+\de-\ga_1-\ga_2)^2+(\fie+\de-\ga_3)^2
+(\ga_1-\ga_2)^2+\ga_3^2-6\fie\de>0,
\end{align*}
a contradiction.
\scpa{Step 4. $F$ is not a maximal singularity: $\sL'{}^2$ must be large}
In the 4 cases we get
\[
\renewcommand{\arraystretch}{1.2}
\begin{array}{ll}
\text{(a)}\qquad & \frac{1}{\mu^2}\sL'{}^2>8\fie \de. \\[10pt]
\text{(b)} & \frac{1}{\mu^2}\sL'{}^2>
\begin{cases}
\text{either} \; &8\fie \de \\
\text{or} \; &4\fie \de+4\fie\ga.
\end{cases} \\ [20pt]
\text{(c)} & \frac{1}{\mu^2}\sL'{}^2>
\begin{cases}
\text{either}\; & 8\fie \de \\
\text{or} \; & 4\fie \de+4\fie\ga_1 \\
\text{or} \; & 4\fie\ga_1+ 4\fie\ga_2.
\end{cases} \\[30pt]
\text{(d)} & \text{Same as (c).}
\end{array}
\]
I prove this assuming that we are in Case~(d) (the other cases are similar
and left to the reader). Consider the divisor
\begin{align*}
K_{S'}+D&=\Bigl(K_Y+S'+(1-\fie)F+\frac{1}{\mu}\sH'\Bigr)\rest{S'} \\
&=K_{S'}+(1-\fie)\Phi+(1-\de)\De+\sum (1-\ga_i)\Ga'_i+\frac{1}{\mu}\sL'.
\end{align*}
The crucial point here is that $P\in S$ is a double point; hence $S' \cap
F\subset F\iso \PP^2$ is a conic. Therefore, by Corollary~\ref{cor_lcl},
there are {\em two} points $Q_1, Q_2\in\Phi\subset S'$ where $K_{S'}+D$
is not log canonical.\index{log!canonical} The statement follows from
\ref{thm_2ds} and a straightforward division into cases. In more detail,
up to renumbering, one of the following 3 cases occurs:
\begin{description}
\item[(d.1)] Neither of $Q_1, Q_2$ lies in any of the $\Ga_i'$. By
\ref{thm_2ds}, the contribution $\sL'{}^2_{Q_i}$ is either $> 4\mu^2\fie$
or $> 4\mu^2\fie \de$, and hence $\ge4\mu^2\fie \de$ anyway.
\item[(d.2)] $Q_1\in\Ga_1'$, while $Q_2$ is not on any $\Ga_i'$. Then,
by \ref{thm_2ds}, $\sL'{}^2_{Q_1}>4\mu^2\fie\ga_1$ and
$\sL'{}^2_{Q_2}>4\mu^2\fie \de$.
\item[(d.3)] $Q_1\in\Ga_1'$,
$Q_2\in\Ga_2'$. Then, by \ref{thm_2ds},
$\sL'{}^2_{Q_1}> 4\mu^2\fie\ga_1$, and
$\sL'{}^2_{Q_2}> 4\mu^2\fie\ga_2$.
\end{description}
\scpa{(5) $F$ is not a maximal singularity: conclusion} I now conclude the
proof in Case~(d) only, since the arithmetic in the other cases is a
specialisation of this case (and, in any event, it is easier). Using the
calculation for $\frac{1}{\mu^2}\sL'{}^2$ in Step~2 above we get, in
cases (d.1), (d.2)
\begin{align*}
0&>-\frac{1}{\mu^2}\sL'{}^2+4\fie\de \\
&=2\fie^2+2\de^2+2\ga_1^2+2\ga_2^2+2\ga_3^2+\\
&\qquad+2\fie\de -2\fie\ga_1-2\fie\ga_2-2\fie\ga_3
-2\de\ga_1-2\de\ga_2-2\de\ga_3 \\
&=(\fie+\de-\ga_2-\ga_3)^2+(\fie-\ga_1)^2 +(\de-\ga_1)^2+(\ga_2-\ga_3)^2,
\end{align*}
a contradiction. In Case~(d.3) we get
\begin{align*}
0&>-\frac{1}{\mu^2}\sL'{}^2+4\fie\ga_1+4\fie\ga_2 \\
&=2\fie^2+2\de^2+2\ga_1^2+2\ga_2^2+2\ga_3^2+\\
&\qquad-2\fie\de+2\fie\ga_1+2\fie\ga_2-2\fie\ga_3
-2\de\ga_1-2\de\ga_2-2\de\ga_3 \\
&=(\fie+\ga_1+\ga_2-\de)^2+(\ga_1-\ga_2)^2+ (\fie-\ga_3)^2+(\de-\ga_3)^2,
\end{align*}
a contradiction. \qed \end{proof}
\subsection{Failure of rigidity}\index{rigid!boundary|(}
Let $X=X_4\subset \PP^4$ be a quartic \hbox{3-fold} with a single ordinary
double point $P\in X$. Projection\index{projection} from the node
generates a link\index{link} (which is a birational involution) $\tau$ of
$X$. It is easy to see that there are 24 lines $P\in\Ga_i\subset X$ each
of which generates a link (which also turns out to be a birational
involution) $\si_i\colon X\to X$ from the 2-ray game\index{2-ray game}
beginning with the extraction $E_i\subset Y\to\Ga_i\subset X$.
\begin{thm}[Pukhlikov \cite{P3}] Let $X=X_4\subset \PP^4$ be a quartic
\hbox{$3$-fold} with a single ordinary double point $P\in X$.
\begin{enumerate}
\item $X$ is birationally rigid;
\item $\tau$ and the $\si_i$ generate $\Bir X / \Aut X$.
\end{enumerate}
\end{thm}
\begin{proof} Along the same lines as \ref{thm_crs} and \ref{thm_r23}.
Using the same method as Theorem~\ref{thm_crs}, Step~1, it is easy to see
that, if a curve $C$ is a maximal centre, then it must be a line through
$P$. By \ref{thm_im} no smooth point $Q\in X$ can be a maximal centre. We
then conclude verbatim as in the proof of \ref{thm_crs} (conclusion)
bearing in mind that, by \ref{thm_dc2}, the only divisorial contraction
$E\subset Y\to P\in X$,\index{divisorial contraction} landing the
exceptional divisor into $P$, is the blowup of the maximal ideal. \qed
\end{proof}
This is encouraging, but there is a point where things go wrong.
\begin{exa} Let $X=X_4\subset \PP^4$ be a quartic \hbox{3-fold} with a
singular point $P\in X$ of the form
\[
x^2+y^2+z^4+t^4=0.
\]
$\la x+\mu y=0$ defines a pencil of rational surfaces on $X$. The 2-ray
game\index{2-ray game} beginning with the\index{weighted!blowup} weighted
blowup
\[
\ep\colon E\subset W\to P\in X
\]
with weights 2,2,1,1 links $X$ to a Del Pezzo fibration\index{del
Pezzo!fibration} of degree 2 with two unremovable index 2 points.
When the singularity degenerates to
\[
x^2+y^3+z^4+t^4=0,
\]
the 2 singular points coalesce into an exotic index 2 terminal singularity
and\index{terminal!singularities} the 2 fibres they live in to an exotic
fibre; cf.\ \cite{C1}.
\end{exa}
\begin{exa} Consider a quasismooth \hbox{3-fold} $X_7\subset
\PP(1,1,1,2,3)$ with coordinates $x_i,y,z$. In \cite{CPR} we construct a
link\index{link} $\tau\colon X\broken X$ centred at the point
$P_3=(0,0,0,1,0)$ and show that, as the coefficient of the monomial $y^2z$
degenerates to zero, the birational involution $\tau$ disappears (see
\cite{CPR}, 7.4.2 for a further discussion of this phenomenon).
\end{exa}
I find the above examples relatively encouraging, since they seem to
suggest that birational rigidity\index{birational rigidity} is a fairly
stable property. This has led me to Conjecture~\ref{con_mod}, that the
property is open in moduli.
I would like to ask the following questions, all of which are accessible
in principle using the methods of this paper, aiming to explore ways in
which rigidity can break down.
\begin{enumerate}
\item Determine which singular quartics (with
$\Q$-factorial\index{factorial@$\Q$-factorial} terminal singularities)
are\index{terminal!singularities} birationally rigid, and describe
alternative Mori fibre space\index{Mori!fibre space} models for the ones
which are not (joint work in progress with Massimiliano Mella). Extend
this to all 95 Fano\index{famous@``famous 95''}
hypersurfaces.\index{Fano!3-fold!hypersurface}
\item Extend \cite{CPR} to all Fano \hbox{3-folds} in codimension~2
complete intersections \cite{IF}, codimension~3 Pfaffians and
codimension~4 (Alt{\i}nok \cite{A}).
\end{enumerate}
Basically, I want to see what happens when the singularities become worse,
or the structure of the\index{anticanonical!ring} anticanonical ring more
complicated. The reader of \cite{CPR} may feel overwhelmed by the large
volume of calculations needed (at least at present) to pursue this type of
question and wonder what the point of it all is. I believe that it is
important to understand how birational rigidity fails: for example, when
can it happen that a Fano \hbox{3-fold}\index{Fano!3-fold} is birational
to a strict Mori fibration,\index{strict Mori fibre space} and what is the
meaning of the condition\index{K2@$K^2$ condition} on $K^2$ in
Theorems~\ref{thm_rcb} and~\ref{thm_rdp}? It is certainly possible that
this corner of nature is just chaotic. My hope is that once sufficiently
many examples are understood it will be possible to see some
structure.\index{rigid!boundary|)}
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\bigskip
\noindent
Alessio Corti\\
DPMMS, University of Cambridge,\\
Centre for Mathematical Sciences,\\
Wilberforce Road, Cambridge CB3 0WB, U.K.\\
e-mail: a.corti@dpmms.cam.ac.uk
\end{document}