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\title{Twenty five years of 3-folds\\ -- an old person's view}
\author{Miles Reid}
\date{ d\`aq\`\i w\u an ch\'eng
\footnote{It takes a long time to make a big pot.}}
\makeindex
\begin{document}
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% \tableofcontents
\section{Introduction}
This paper is an expanded version of a historical and autobiographical
talk given at the closing conference of the Warwick 3-folds activity in
December 1995 (at the time still entitled ``Twenty years\dots''). It
discusses the development of 3-folds since the 1970s, particularly my work
on canonical singularities\index{canonical!singularities}
and\index{Mori!theory} Mori theory since around 1975.
Castelnuovo and Enriques's work establishing the foundational results on
algebraic surfaces took place over something like 12 years around 1900. The
classification of surfaces has undoubtedly progressed since then, and
remains a vital area of research, but the foundations of the theory were
laid during this relatively short period. The comparable foundational work
on 3-folds took place over the 20 years between 1970 and 1990, from the
first papers of Iitaka and Ueno on Kodaira dimension through to Mori's
solution of the\index{flip!theorem} flip problem and his Fields medal at
the Kyoto international congress. While future generations will
undoubtedly simplify and extend the theory of \hbox{3-folds}, and find new
applications, it seems to be a matter of historical record that the
primary issues were settled during the 1970s and 1980s.
A survey based on autobiography may be a relatively painless way to gain
insight into the subject, compared to the investment of effort involved in
working through a more serious account of\index{Mori!theory} Mori theory.
The narrative form allows me to stress that, while the subject is not
without its technical aspects, the key issues are mostly very simple once
grasped; like other revolutionary ideas, canonical singularities and Mori
theory\index{Mori!theory} build on classical foundations, and, with the
benefit of hindsight, stand out clearly as inevitable developments.
Looking through my old letters and papers for a retrospective sketch of my
early career in 3-folds has given me some new insights, a bit like
preparing the index to Volume~1: Foundations. Talk of ``inevitable lines
of thought'' prompts the hope that some bolshy youngsters will mount their
own revolution on my generation's orthodoxy (but I don't see them coming).
I don't really need to apologise for the self-indulgence of my
undertaking: a whole generation of algebraic geometers now work in the
classification of varieties, and it is hard to read a paper or hear a
lecture in the subject without recognising concepts and terminology that I
had a hand in shaping.
\paragraph{History} It is widely appreciated that mathematicians usually
treat history in a curiously dishonest way, rewriting the history of the
subject as it should have been discovered, aiming to exploit past
experience in building a modern worldview. The essential difficulty seems
to be that the story in strictly chronological order will not make sense
to anyone; the writer wants to give an explanation based on the logical
layout of the subject, whatever violence it does to historical truth.
Writing this paper has certainly confirmed this point for me: given the
choice between saying things chronologically or according to the divisions
of the subject matter, my sense of style always leads me to the latter.
Another inevitable slant of autobiographical writing is to exaggerate the
significance of the author's discoveries, and I also follow this tradition.
My first introduction to algebraic geometry took place in my first year as
a graduate student at Cambridge in 1969--70 under the benevolent and
liberal care of Peter Swinnerton-Dyer. I learned many things directly from
him: the factorisation of birational maps between del Pezzo surfaces,
Hodge theory and its application to Abelian varieties. A particular item
that stands out is the derivation of the Weierstrass model of an elliptic
curve from Riemann--Roch: the graded ring
$R(E,P)=k[x,y,z]/(z^2=y^3+ax^4y+bx^6)$ and its Proj, the weighted
hypersurface $E_6\subset\PP(1,2,3)$,\index{weighted!hypersurfaces and
c.i.} has served as the model for all my graded ring calculations.
On Swinnerton-Dyer's advice I studied Mumford's ``little red book'' in a
joint seminar with Jean-Louis Colliot-Th\'el\`ene and Barry Tennison. My
work on canonical 3-folds\index{canonical!3-fold} could reasonably be
traced back to Mumford's question (\cite{red}, Chapter Three, \S8; this
follows a treatment of normalisation as a method of resolving surface
singularities):
\begin{quote}
{\em The existence of such a simple way of ``making'' every variety
normal is one of the reasons why normal varieties are an important class.
Life would be much simpler if there were an analogous way of canonically
constructing a nonsingular variety birationally equivalent to any given
variety. }
\end{quote}
\section{Canonical models are inevitable}
The main discovery of my early career in 3-folds was the realisation that
singularities arise as an inevitable part of higher dimensional life. They
turn up in two different ways: by {\em equations} and by {\em cyclic
quotient constructions}.\index{quotient singularity} Already the simplest
cases are fun, and understanding them properly takes us a long way into the
substantial issues. The ordinary quadric cone arises both as the simplest
hypersurface singularity $X:(xy=z^2)\subset\C^3$ and as the simplest
quotient singularity, the quotient of $\C^2$ by $(u,v)\mapsto(-u,-v)$. Its
resolution $Y\to X$ is\index{resolution!of singularities} the ordinary
blowup, having exceptional locus $E\iso\PP^1$ with $E^2=-2$ (a
\hbox{$-2$-curve}), and it is well known that the canonical class of $Y$
is trivial. Because $K_Y=K_X=0$ this is a\index{crepant} crepant resolution
of a strictly canonical singularity.\index{canonical!singularity} The
3-fold ordinary double point $(x^2+y^2+z^2+t^2=0)$ or $(xy=zt)$ discussed
in Section~\ref{sec!A1} behaves in wholly different ways: it is not a
finite quotient singularity, it is terminal,\index{terminal!singularities}
its blowup does not have trivial canonical class, it has the
two\index{small!resolution} small resolutions discussed in
Section~\ref{sec!flop} that are related by the classic flop. It is
completely unrelated to the simplest 3-fold quotient\index{quotient
singularity}\index{terminal!quotient sing.\ $\frac{1}{r}(1,a,r-a)$}
singularity $\frac12(1,1,1)$, the quotient of $\C^3$ by
$(u,v,w)\mapsto(-u,-v,-w)$ or the cone over the Veronese surface discussed
in Section~\ref{sec!Ver}.
The next point concerns canonical rings.\index{canonical!ring} As I
explain in Section~\ref{sec!can}, my interpretation of Enriques' work on
surfaces of general type underlines the importance of working with
pluri\-canonical graded rings, rather than just the 1-canonical or the
individual $m$-canonical linear systems. Zariski and Mumford's proof of
the finite generation of canonical rings\index{canonical!ring} of surfaces
and Grothendieck's definition of Proj frees us from the obligation of
working only with varieties with a fixed embedding in projective space and
graded rings generated by elements of degree~1. The first canonical
\hbox{3-folds} generalise Enriques' models in a natural way. They make
clear, for example, that the birational invariant $K^3$ that controls the
asymptotic growth of the plurigenera is a rational number. Once this point
is recognised, there is really no turning back to the nonsingular category.
\subsection{Lefschetz theory and the ordinary double point}\label{sec!A1}
My first main point is the following:
\begin{quote}
{\em The ordinary double point has entirely different behaviour and
meaning in different dimensions.}
\end{quote}
The ordinary double point (ODP)
\[
X:\Bigl(\sum_{i=0}^{n+1}x_i^2=0\Bigr)\subset\C^{n+1}.
\]
is the simplest $n$-fold singularity. The function $\C^{n+1}\to\C$ given by
$f(\xu)=\sum x_i^2$ has a {\em nondegenerate critical point} at $0$:
critical\index{critical point} because all \hbox{$\partial f/\partial
x_i=0$}, and nondegenerate because the Hessian matrix of second derivatives
is a nondegenerate quadratic form. Over $\R$, these are just the
properties of a Morse function. Lefschetz theory is a complex analog of
Morse theory. However, in many contexts of classification, we may be more
interested in the singular $n$-fold $X$ than in the function $f(\xu)$.
The basic idea of Lefschetz theory is to argue on the neighbouring fibre
$X_t=f\1(t)$ for $t$ close to $0$. Let's work in a small neighbourhood
$\sum |x_i|^2\le\de$ of the singular point, and take $t=\ep e^{i\theta}$
where $0<\ep\ll\de\ll1$. Then from the $C^\infty$ point of view, $X_t$ and
$X_0$ practically coincide outside the $\de$-ball, and
$M=X_t\cap\hbox{$\de$-ball}$ (the {\em Milnor fibre} of the singularity)
is given by quadratic equations. Scaling and separating real and imaginary
parts puts the equations of $M$ in the form
\[
\sum u_i^2=1 \hbox{ and } \sum u_iv_j=0, \quad\hbox{where} \quad
x_i=(\hbox{modulus})\times e^{i\theta}\times (u_i+iv_i).
\]
First, the things that are common to all dimensions: the Milnor fibre is
the tangent bundle to $S^n$ (see Figure~\ref{fig!Lefsch}), the vanishing
cycle is $\de=S^n$, the monodromy as we follow the diffeomorphism of the
fibres around $e^{i\theta}$ is given by the Picard--Lefschetz formula
$x\mapsto x\pm(x\cdot\de)\de$.
\begin{figure}[ht]
\centerline{\epsfbox{Lefsch.ps}}
\caption{The ODP according to Lefschetz: the Milnor fibre is
diffeomorphic to the tangent bundle $T_{S^n}$ of the vanishing cycle
$\de=S^n$, and the monodromy action is given by the Picard--Lefschetz
formula $x\mapsto x\pm(x\cdot\de)\de$ }
\label{fig!Lefsch}
\end{figure}
For the purposes of birational geometry, we want to resolve the
singularity; then each dimension is quite different. In the curve case,
the ODP is the node, consisting of two branches meeting only at a point.
The resolution of singularities\index{resolution!of singularities} or
normalisation consists simply of separating the two branches. This leads
to the standard picture of genus change in a degeneration of Riemann
surfaces (see Figure~\ref{fig!curve}).
\begin{figure}[ht]
\centerline{\epsfbox{curve.ps}}
\caption{Picture of a curve losing 1 from its genus by acquiring an ODP,
then disconnecting}
\label{fig!curve}
\end{figure}
Something beautiful and completely unique happens in the surface case: the
Milnor fibre $M$ is diffeomorphic to the resolution $Y\to X$. The ODP is
the ordinary quadratic cone in $\C^3$. Its resolution is a neighbourhood
of a copy of $\PP^1_\C=S^2_\R$ with self-intersection $-2$. At the level
of topology, the diffeomorphism is clear from Figure~\ref{fig!Lefsch},
because the Milnor fibre is the tangent bundle to the vanishing cycle
$S^2_\R$, which also has self-intersection $-2$.
The 3-fold ODP demonstrates originality by having a nontrivial analytic
class group: up to the obvious change of coordinates, it is the cone over
the quadric surface $Q\iso\PP^1\times\PP^1: (xy=zt)\subset\PP^3$, and the
two linear systems of generators of $Q$ correspond to Weil divisors that
are not factorial; for example, the principal divisor $x=0$ breaks up as
the union of two planes $x=z=0$ and $x=t=0$, neither of which is
principal. The local class group can be viewed topologically as the second
cohomology of the link of the singularity. As I discuss in
Section~\ref{sec!min}, the two rulings of $Q$ give rise to the two small
resolutions\index{small!resolution} of the 3-fold ODP and the flop between
them.
It is also interesting to run through a parallel discussion of the $A_2$
singularity $\sum_{i=1}^{n+1}x_i^2+x_0^3=0$. In the curve case, this
singularity is a cusp, which is locally irreducible. The Milnor fibre has
two vanishing cycles, but in the 3-fold case, they have nondegenerate
intersection pairing, and the resulting singularity is factorial; in fact,
it is a homology manifold. It has no small modification. Its blowup is a
nonsingular 3-fold containing an exceptional surface $E$ isomorphic to the
quadric cone $Q\subset\PP^3$, and with $\Oh_E(-E)\iso\Oh_Q(1)$. See Mirel
Caib{\u a}r's thesis \cite{Cai} for a much more detailed discussion of the
relation between Milnor fibre, class group, and geometry of the resolution.
\paragraph{Problem on higher dimensional ODPs}
I propose as a challenge to try to understand the new characteristic
features of 4-fold and 5-fold ODPs. For example, the 5-fold ODP is the
cone over the Pl\"ucker quadric $Q_6=\Grass(2,4)\subset\PP^5$, and the
analog of the class group is the two tautological vector bundles of
$\Grass(2,4)$ or the spinor bundles of $Q_6$. My reason for saying this is
that a number of people have looked for simple-minded extensions to higher
dimensions of features such as\index{simultaneous resolution} simultaneous
resolution of Du Val surface and\index{small!resolution} small resolutions
of 3-fold cDV points, but I really don't know of any reasonable class of
higher dimensional examples where this is likely to give useful results.
For anyone wanting to study 4-fold or 5-fold analogs of the theory of
surfaces and 3-folds, my problem seems a more profitable direction.
\paragraph{History} The ODP $y_1y_2=y_3y_4$ and its two rational functions
$\frac{y_1}{y_3}=\frac{y_4}{y_2}$ and $\frac{y_1}{y_4}=\frac{y_3}{y_2}$
occur in Mumford \cite{red}, Chap.~I, \S4, Remark~IV after Proposition~1
in the discussion of rational functions. I was a graduate student at the
IHES between 1970 and 1972, and heard about the ideas of Picard--Lefschetz
from lectures by my great teacher Pierre Deligne, around the time he was
preparing Part~II of SGA7 for publication \cite{D}.
The first project that Deligne offered me, global Torelli and the
surjectivity of the period map for K3 surfaces of degree 2 or 4, turned
out to be beyond my technical capabilities, but I learned a vast amount in
my first year by studying it, among other things: classification of
surfaces, moduli spaces and G.I.T., Hodge theory including variation and
degeneration of Hodge structures, Satake compactifications. Deligne taught
me about Du Val singularities (including the name, in case you think I use
it out of patriotism to the UK or to Trinity College, Cambridge). An
important point in the Torelli project was the idea that K3s with only Du
Val singularities should be viewed as points ``at finite distance'' in the
moduli space, since they have monodromy of \hbox{finite} order, and their
resolutions are\index{resolution!of singularities} still K3s. Equally
well, you could start learning to live with the singular variety.
\subsection{The Veronese cone}\label{sec!Ver}
I change tack, to discuss the first type of canonical
singularity\index{canonical!singularities} arising as a\index{quotient
singularity|(} quotient. Many classical construction of surfaces, such as
those of Kummer surfaces, Enriques surfaces or Godeaux surfaces, involve a
group action having no fixed points, or only involving ODPs.
\begin{quote}
{\em Simple constructions for surfaces involving quotients by a group
action often have $3$-fold analogs having\index{quotient singularity}
quotient singularities, for example, the\index{terminal!quotient sing.\
$\frac{1}{r}(1,a,r-a)$} Veronese cone singularity.}
\end{quote}
The first 3-fold quotient singularity to come to my attention was the
Veronese cone. As everyone knows, the Veronese surface $V_4\subset\PP^5$
is the image of $\PP^2$ under the embedding given by all quadratic
monomials. To see it by equations, we take coordinates $x_1,x_2,x_3$ on
$\PP^2$ and write down
\[
M = \begin{pmatrix}
u_{11} & u_{12} & u_{13} \\
u_{12} & u_{22} & u_{23} \\
u_{13} & u_{23} & u_{33}
\end{pmatrix} \quad\text{where}\quad u_{ij}=x_ix_j.
\]
Here $M$ is the generic symmetric $3\times3$ matrix. The Veronese surface
is defined by $\rank M\le1$. On the other hand, it's obvious that the
projective coordinate ring of $V_4$ is $k[V_4]=k[x_1^2,x_1x_2,\dots,
x_3^2]$. This is, of course, the ring of even polynomials in $x_1,x_2,x_3$,
or equivalently, the ring of invariants of $\{\pm1\}$ acting by
$(x_1,x_2,x_3)\mapsto (-x_1,-x_2,-x_3)$. Its Spec, the affine cone $X$ over
$V_4$, is the\index{quotient singularity} quotient singularity
$\half(1,1,1)$ in modern terms.\index{terminal!quotient sing.\
$\frac{1}{r}(1,a,r-a)$} If you resolve it by a blowup $f\colon Y\to X$,
you obtain a 3-fold $Y$ containing an exceptional surface $E\iso\PP^2$
with normal bundle $\Oh_E(E)\iso\Oh_{\PP^2}(-2)$; by the adjunction
formula, $K_Y$ restricted to $E$ is $\Oh_{\PP^2}(-1)$.
Since $K_Y$ has negative degree on $E$, all sections of $K_Y$ in a
neighbourhood of $E$ vanish along $E$; it is an easy exercise in the same
style to see that also all sections of $nK_Y$ vanish along $E$ with
multiplicity $\ge n/2$. Thus
\[
f_*\Oh_Y(nK_Y)=f_*\Oh_Y\Bigl(nK_Y-\frac{n}{2}E\Bigr) \quad\text{and}\quad
2K_Y=f^*(2K_X)+E.
\]
If we start from an Abelian surface $A$ and pass to the quotient surface
$\Sbar=A/\{\pm1\}$ by the action of $\{\pm1\}$, then $\Sbar$ has 16 ODPs
at the fixed points of the group action, that is, the 2-torsion points of
$A$. Resolving these leads to $-2$-curves on the minimal nonsingular model
$S\to\Sbar$, which is then a K3 surface, the {\em Kummer surface} of $A$.
In contrast, the Kummer variety $\Xbar=A/\{\pm1\}$ of an Abelian 3-fold $A$
has 64 Veronese cone singularities at the 2-torsion points of $A$, and
resolving these by the blowup $X\to\Xbar$ is not necessarily such a good
thing to do: you achieve nonsingularity at the expense of spoiling the
property $2K_{\Xbar}=0$, replacing it by $2K_X=\sum E_i$ where $E_i$ are
the exceptional $\PP^2$s over the singularities.
The Veronese cone\index{terminal!quotient sing.\ $\frac{1}{r}(1,a,r-a)$}
appears in many similar contexts: for example, a popular construction of
the Enriques surface (possibly originally due to Burniat) is to divide a
complete intersection of 3 quadrics $S=Q_1\cap Q_2\cap Q_3\subset\PP^5$ by
the free action of $\Z/2$ acting by $\diag(1,1,1,-1,-1,-1)$ (think of
diagonal quadrics for clarity). We can make $S$ avoid the two fixed planes
$\PP^2_\pm$ because we have three quadrics $Q_i$ to spare. If we try to
generalise this construction to a 3-fold complete intersection of 3
quadrics in $\PP^6$, one of the fixed components $\Pi$ must be $\PP^3$ (at
least), and we can't avoid having the 8 points $(Q_1\cap Q_2\cap
Q_3)\cap\Pi$ as fixed points. This is one construction for Enriques--Fano
3-folds, that typically have 8 Veronese cone singularities.
\paragraph{History}
I shared an office for a few weeks with Bernard Saint-Donat on arriving at
the IHES in 1970, and we both visited the Warwick Symposium ran by David
Mumford in 1970--1971; I gave a Cambridge Part~III course around 1974
based on his paper on projective models of K3 surfaces \cite{S}. This
contains the example of the K3 surface $S_5\subset\PP(1^3,2)\subset\PP^6$
embedded in the Veronese cone; its general hyperplane section is a
canonical curve of genus 6 with a $g^2_5$, that is, the Veronese embedding
of a plane quintic, and so it is an exceptional case for which the
projective image is not an intersection of quadrics (see \cite{S},
Theorem~7.2, (ii)). It has\index{Fano!3-fold} the $\Q$-Fano extension
$X_5\subset\PP(1^4,2)$ having the singularity $\frac{1}{2}(1,1,1)$, and
when following Iskovskikh's work, I was at first surprised to find it
excluded from the company of nonsingular Fanos (\cite{I}, Part~II,
Prop.~1.7 and \S2).
As another example, in the context of Fano's study of anticanonical
models\index{anticanonical!model} of 3-folds $V_{2g-2}\subset\PP^{g+1}$,
Iskovskikh asked for an example of a nonsingular $V_{2g-2}$ containing a
linearly embedded plane $\PP^2$. If $E$ is such a plane then, by the
adjunction formula, its normal bundle is given by
$\Oh_E(-E)=\Oh_{\PP^2}(2)$, so that $E$ should contract to a Veronese cone
point. Examples such as $\PP(1^3,2)$, $X_3\subset\PP(1^4,2)$ and
$X_4\subset\PP(1^3,2,2)$ show that these really occur. (I wrote a letter
to Iskovskikh in August 1978 attempting to prove that this cannot happen;
but I found the examples soon after this.)
\paragraph{The 1970--71 Warwick symposium} I followed Deligne to Warwick
in the summer of 1971, and met Artin, Bombieri, Mumford, C.P.~Ramanujam,
Seshadri and many others; for example, I learnt more about G.I.T. from
standing in a lunch queue with Seshadri than from several weeks' study of
the book earlier that year. I took part in a seminar on etale cohomology
organised by Saint-Donat and Anders Thorup, and had the embarrassing task
of giving half-prepared lectures on the etale base change theorem with
Mike Artin in the audience.
When Elaine Greaves threw me out of the old typewriter room (two
clapped-out old mechanical typewriters, reserved for the use of graduate
students), neither of us suspected that she would be my trusted henchman
throughout the planning and organisation of the two subsequent Warwick
algebraic geometry symposiums in 1982--83 and 1995--96. Several letters
and rough drafts for \cite{C3-f} were written on exactly the same
typewriters when I arrived at Warwick in 1978. (In the meantime, our
equipment budget has improved somewhat.)
\subsection{Canonical rings and the first canonical 3-folds}
\label{sec!can}\index{canonical!3-fold}
There are many places in Enriques' work where he constructs a canonical
surface\index{canonical!surface} by what seems to be an extraordinarily
ingenious argument based on the geometry of the canonical or bicanonical
map, but that becomes much more transparent when viewed in terms of graded
rings.
\begin{quote}
{\em Graded rings such as the canonical ring\index{canonical!ring} of a
variety of general type contain more information than individual linear
systems (for example, $m$-canonical maps).}
\end{quote}
The simplest illustration for my purposes is Enriques' construction of a
surface $S$ with $p_g(S)=4,q(S)=0,K_S^2=6$ as the normalisation
$S\to\Sbar_6$ of a sextic surface $\Sbar_6\subset\PP^3$ having an ordinary
double curve along a plane cubic curve $C\subset\PP^3$. My interpretation
is as follows: the canonical ring\index{canonical!ring} of $S$ needs 4
generators $x_i\in H^0(K_S)$ in degree 1, then (at least) 1 more generator
$y\in H^0(2K_S)$ in degree 2, simply because quadratic monomials $x_ix_j$
only provide 10 elements of $H^0(2K_S)$, whereas by Riemann--Roch
\[
h^0(2K_S)=\chi(\Oh_S)+K^2=11.
\]
Now\index{plurigenus formula} we easily find that we need 2 relations in
degree 3 and 4 between these generators, giving $S$ as the complete
intersection $S_{3,4}\subset\PP(1^4,2)$. No ingenuity here, this is just
the general complete intersection; we recover Enriques' model by assuming
that the two relations are $x_0y=a_3$ and $y^2=b_4$, and eliminating $y$,
giving the equation of the sextic as $\Sbar_6:(a_3^2=x_0^2b_4)$.
One of Enriques' most remarkable example of this kind is his construction
of a surface of general type with $p_g(S)=2,K_S^2=1$; in this case it is
easy to see that the canonical system $|K_S|$ is a pencil of curves of
genus 2 with a single transverse base point $P$, which is a Weierstrass
point on each $C\in|K_S|$. Then $\Oh_C(2K_S)=K_C$, so that $|2K_S|$ maps
each $C$ as a double cover of $\PP^1$. Since the image $\PP^1$s all
contain the image of $P$, the image is the quadric cone
$\fie_{2K_S}(S)=Q\subset\PP^3$, and $S$ is the double cover of $Q$
branched in the intersection of $C$ with a quintic and over the vertex.
The modern construction is the weighted hypersurface
$S_{10}\subset\PP(1,1,2,5)$.\index{weighted!hypersurfaces and c.i.}
Putting together the idea of canonical ring\index{canonical!ring} derived
from Enriques' constructions and the experience of the Veronese cone
described in Section~\ref{sec!Ver}, it was natural to look for examples
such as $X_{14}\subset\PP^4(1,1,2,2,7)$. This example can perfectly well be
treated from the point of view of classical geometers: it has a canonical
pencil $|K_X|$ consisting of surfaces $S_{14}\subset\PP(1,2,2,7)$. Each
such surface has the equation $z^2=f_7(x^2,y_1,y_2)$. If you substitute
$X=x^2$ and $Z=xz$, and divide all the degrees by 2, this becomes the
hypersurface $S_8\subset\PP(1,1,1,4)$ (that is, $\PP(2,2,2,8)$) defined by
$Z^2=Xf_7(X,y_1,y_2)$. In other words, my 3-fold has a pencil of surfaces
with $p_g=3$, $K^2=2$, each a double cover of $\PP^2$ whose octic branch
curve happens to split up as a septic plus a line, meeting of course in 7
nodes.
With what we know today, it is immediate that $X_{14}$ has 7 Veronese cone
singularities along the $(y_1,y_2)$-axis. The striking thing here is that
$K_X^3=\frac{1}{2}$; in other words, the plurigenera $P_n(X)$ grow like
$\frac{1}{6}n^3\times\frac{1}{2}$. On the other hand, a 3-fold of general
type that has a nonsingular model with $K_X$ nef\index{nef} has $K_X^3$ a
positive and even integer, so $K_X^3\ge2$. The plurigenera are manifestly
birational invariants, and on a minimal model, we can use vanishing to
deduce that $P_n$ grows like $\frac{1}{6}n^3\times K_X^3$. Thus my example
has no nonsingular model with $K_X$ nef. (Compare also Ueno \cite{U}.)
It might be a fun exercise to find a treatment in the old Italian style of
examples such as $V_{6,10}\subset\PP(1,2^3,3,5)$, which has $p_g=1$,
$K^3=1/2$, and 15 Veronese cone points. The main point, however, is that
canonical rings\index{canonical!ring} already give a simple construction of
dozens of examples, with features that would be hard to study in other
ways.
\paragraph{History}
The inevitability of working with the canonical ring\index{canonical!ring}
comes out clearly in the study of algebraic surfaces such as Godeaux
surfaces, which I studied around 1974, following Bombieri's paper on
canonical models\index{canonical!model} and his lectures at the IHES and
the Warwick symposium in 1970. For me, the pleasure of seeing how simple
constructions with graded rings generate effortlessly many classes of
algebraic surfaces was the main motivation for trying to do the same in
dimension~3. It is clear that there is still a whole lot more mileage to
be got out of these ideas.
After two years of my research fellowship at Christ's College, Cambridge,
I applied to the British Council for a second year on their Soviet
exchange, from September 1975. The Soviet ministry didn't come up with my
placement by the due starting date for their own reasons, and when they
did, it turned out to be at Minsk. It's not plausible that they excluded
me from Moscow because of the bad company I kept during my first year, and
much more likely that this was a tit-for-tat because a Cambridge
department had refused to take some party stooge.
In October 1975, after finishing my paper on elliptic Gorenstein surface
singularities \cite{ellGor}, and while hanging around Cambridge, waiting
for my placement, I got interested in 3-fold examples of the above type,
and even wrote a primitive computer program (in Fortran!)\ to search for
examples of canonical 3-folds\index{canonical!3-fold} as hypersurfaces
$X_d\subset\PP^4(a_0,\dots,a_4)$ with all $a_i \mid d$ and $\sum a_i=d+1$.
Unfortunately, the only singularity I really believed in and was confident
of being able to resolve was the Veronese cone, so that although my
program found a few examples such as \cite{Fl}, 15.1, No.~14:
$X_{18}\subset \PP(1,2,2,3,9)$, which\index{terminal!quotient sing.\
$\frac{1}{r}(1,a,r-a)$} has $2\times\frac13(1,1,2)$ quotient
singularities\index{quotient singularity} along the $z,t$ axis, I didn't
recognise them as valid examples, and missed out on discovering terminal
singularities of higher index. What a fool I was, Watson!
\enlargethispage{\baselineskip}
Notes written during my first month at Warwick in 1978 contain 21 cases of
canonical complete intersections with Veronese cone points as their only
singularities, including the example $V_{6,6,6}\subset\PP(2^4,3^3)$ with
$p_g=0$ discussed in \cite{YPG}, Example~2.9. The same notes and letters
from the period conjectured falsely and repeatedly that the index of
3-fold canonical singularities\index{canonical!singularities} is always
$\le2$. In a letter around 1977, Ueno suggested I look at
$\frac{1}{3}(1,1,1)$ for an index 3 singularity. Of course this has index
1, and I ignored the suggestion. But it is possible that what Ueno wrote
was a misprint for
$\frac{1}{3}(1,1,2)$; in any case, I would have stumbled on this somehow
if I had interpreted his suggestion more sympathetically. Nick
Shepherd-Barron discovered the singularities $\frac1r(1,1,r-1)$ of index
$r$ in early 1979 in response to my false conjecture, and Dave Morrison
provided\index{terminal!quotient sing.\ $\frac{1}{r}(1,a,r-a)$} the
general case of terminal\index{quotient singularity|)} quotient
singularities \hbox{$\frac1r(1,a,r-a)$} in a letter of April 1980. I
originally proposed to Nick that we should write
\cite{C3-f} as a joint paper, but he didn't want to, although he
contributed several of the key ideas to it, in particular the first
conjectural statement that the hyperplane section of a Gorenstein
canonical singularity\index{canonical!singularities} is either rational or
elliptic. I realised at the Angers conference in July 1979 that we had a
simple proof of this conjecture using nothing much more than the
adjunction formula.
In the year after Angers I spent a couple of weeks in Bonn at the
invitation of Van de Ven and Hirzebruch, where I discovered that I knew
how to write out formally the\index{famous@``famous 95''} ``famous 95''
list of K3 hypersurfaces.
\section{Towards the definition of minimal model}
\index{minimal model|(}
The logic behind minimal models is much less clear-cut than for canonical
models,\index{canonical!model} and we arrived at the definition by
interpolating between the canonical model and birational nonsingular
models, with Mori and me working from opposite ends. (Compare \cite{M3},
Section~9 for a complementary historical discussion.) Whereas the
preceding chapter argued the inevitability of canonical
models,\index{canonical!model} this chapter has the more modest task of
putting together some considerations that make the definition of minimal
model look at least reasonable. The material here does not use the Mori
cone,\index{Mori!cone} and is mostly simpler than the eventual solution
of the minimal model problem by Kawamata, Koll\'ar, Mori, Shokurov and
others (see Chapter~\ref{ch!Mori}).
\enlargethispage{\baselineskip}
Section~\ref{sec!nef} discusses\index{Zariski decomposition|(} Zariski
decomposition of a divisor on a surface to explain the\index{nef|(} nef
condition on a divisor. Section~\ref{sec!flop} goes back to the material
of Section~\ref{sec!A1} to discuss\index{small!resolution} small
resolutions of ODPs and the classic flop, and Section~\ref{sec!min} plays
with examples of the construction of \cite{C3-f} and \cite{Pagoda}
constructing\index{resolution!of singularities} partial resolutions of
canonical singularities,\index{canonical!singularities} in order to give
some feeling for the condition that $K$ is nef. Section~\ref{sec!deg}
discusses the special case of relative minimal models of surface
fibrations.
\subsection{Zariski's paper and nef divisors on surfaces} \index{nef}
\label{sec!nef}\index{nef}
Zariski's paper \cite{Z} on the asymptotic form of Riemann--Roch for a
divisor on a surface forms a crucial bridge between the Italian tradition
of surfaces and modern work on 3-folds. My attention was drawn to it by
Mumford's appendix, referred to in Bombieri \cite{B} for the proof of the
finite generation of the canonical ring\index{canonical!ring} of a surface
of general type. The paper stresses the notion of a divisor $D$ being {\em
eventually free\/} (that is, the linear systems $|nD|$ is free for some
$n>0$), and its consequence {\em numerical eventually free\/}
or\index{nef} {\em nef\/} (that is, $D\Ga\ge0$ for every curve $\Ga\subset
S$).\index{nef} Zariski introduces $\Q$-divisors, the idea of Zariski
decomposition, and\index{Zariski decomposition} the related
characterisation\index{quasieffective} of quasieffective in terms of the
existence of Zariski decomposition.
The degree of a divisor on a curve $C$ is an integer, so the notion of
positive degree is not hard to find, and when $D$ has large degree, RR
takes the simple form $h^0(C,D)=1-g+\deg D$. Zariski's aim is to study the
Riemann--Roch space $H^0(S,nD)$ for a divisor on a surface, especially for
$n\gg0$; there is not much problem when $D$ is ample, so the whole point
is to deal with negative features of $D$.
Assume that some multiple of $D$ is\index{effective!divisor} effective,
say $mD=\sum d_i\Ga_i$ for some $m>0$ with $d_i>0$. If $D\Ga<0$ for some
curve on $S$ then $\Ga$ must be one of the $\Ga_i$, say $\Ga_1$, and
necessarily $\Ga_1^2<0$. Then the fixed part of the linear system $|nD|$
contains $\Ga_1$ with multiplicity $\ge na_1$, where
$a_1=(-D\Ga_1)/(-\Ga_1^2)$. In other words, decreasing $D$ down to
$D_1:=D-a_1\Ga_1$, where $a_1$ is chosen so that $D_1\Ga_1=0$, leaves all
the $H^0$ unchanged: $H^0(nD)=H^0(nD_1)$ for all $n>0$. We thus find
ourselves playing games with the quadratic intersection form on the
components of $\Ga_i$, with the flavour of orthogonal complement or
Gram--Schmidt orthogonalisation. The theory proceeds systematically via
the\index{Zariski decomposition} {\em Zariski decomposition} of $D$,
defined to be an expression $D=P+N$ where
\begin{enumerate}
\renewcommand{\labelenumi}{(\roman{enumi})}
\item $P$ is a nef $\Q$-divisor;\index{nef|)}
\item $N$ is effective, negative definite and orthogonal to $P$.
\end{enumerate}
The point is that the base part of the linear system $|nD|$ must be at
least $nN$ for any $n>0$, so that $h^0(nD)=h^0(nP)$. For any divisor $D$,
the existence of a Zariski decomposition is an important dichotomy: it
exists if and only if $D$ is\index{quasieffective} quasieffective, and
then $D\Ga<0$ happens for at most finitely many curves having a negative
definite intersection pairing.
It seems to be a common misapprehension that just defining a Zariski
decomposition in terms of orthogonal complement somehow makes it a one-off
process in bilinear algebra, so I emphasise the following point:
\begin{quote}
{\em Calculating the Zariski decomposition of a divisor on a surface is an
inductive process, and is a kind of minimal model program.}
\end{quote}
Continuing the argument from above, if $D_1\Ga_2<0$ we have to subtract off
a multiple of $\Ga_1$ and $\Ga_2$ to make $D_2$ orthogonal to both $\Ga_1$
and $\Ga_2$, and so on by induction as long as $D_j$ is not nef. We don't
know whether or not $\Ga_2$ will fall in the negative part just by looking
at $D$ and $\Ga_2$ without running the inductive process. (I stress the
point, at the risk of repetition: whether a rational curve $\Ga\subset S$
on a surface with self-intersection $\Ga^2\le-2$ is contracted on passing
to the minimal model is not determined just by a neighbourhood of $\Ga$,
but by how many contracted $-1$-curves it meets in the course of $S\to
S_1\to\cdots$.)
If $S$ is a surface with $p_g>0$ and $D=K_S$, then calculating the Zariski
decomposition of $D$ is exactly the same thing as running a minimal model
program on $S$. The first $\Ga_1$ is a $-1$-curve because $K_S\Ga_1<0$ and
$\Ga_1^2<0$. We subtract off a multiple of $\Ga_1$, and from that point on
always work in the orthogonal complement to $\Ga_1$, which is equivalent
to working only with the pullback of divisors from the\index{contraction}
contraction $S\to S_1$ of $\Ga_1$. The negative part of the Zariski
decomposition is the\index{discrepancy} {\em discrepancy}
$K_S-K_{S_{\mathrm{min}}}$ of the morphism to the minimal model $S\to
S_{\mathrm{min}}$. Kawamata's work on\index{log!category} log minimal
models of log surfaces (or ``noncomplete'' surfaces) is a straightforward
application of Zariski decomposition of log canonical divisors $K+D$.
\index{minimal model}\index{Zariski decomposition|)}
\paragraph{History} I first visited Moscow for one year 1972--73 under the
British Council exchange. By that time, Soviet citizens who talked to
foreigners were no longer imprisoned or shot (or at least, not very much),
but it was still not easy for a fairly ignorant and frivolous foreigner to
get into serious conversation with people (except taxi drivers). Although
I spent a lot of time with Moscow mathematicians, it was not until I got
home that I realised that several of my colleagues had distinguished
dissident connections. During my second visit in 1975--76 things were
different: I was less ignorant, having read Solzhenitsyn and Robert
Conquest on the Stalin terror and the even more terrible collectivisation
holocaust. But more, I believe that most Russians were well informed and
cynical about their political system, and keen to criticise it and to hear
about the West. Even people who were party members would crack jokes such
as ``Is socialism scientific? No -- they would have tried it out on
animals first''.
Minsk in 1975 was different; I was more or less the first capitalist seen
in the town, and there were various confused reactions: some of my
neighbours in the students' hostel were ordered by their komsomol
organisers (communist youth league) not to consort with me. The foreign
office in the university was splendidly friendly and incompetent, and
cheerfully signed orders to the visa office to give me permission to
travel to Moscow, where I stayed for several months, illegally as it
turned out. The student hostel on Oktyabr'skaya Ulitsa, next to the vodka
factory, was a real master\-piece of Mr Brezhnev's 5 year plans -- the
doors and windows didn't fit, and the elevator had to be turned off after
6 pm (in a 13 storey building) because of the inconvenience of calling out
the repair man at night when it broke down. Over many months in Minsk, I
searched everywhere to buy a teapot, but in vain; while making myself into
a \hbox{3-folder} by studying\index{weighted!projective space} Zariski's
paper, weighted projective spaces, the Riemann--Roch formula for 3-folds
and\index{simultaneous resolution} simultaneous resolution of Du Val
singularities, I made tea in a jam jar, and tried to pour it into the
glass without scalding my hands. The jam jar was the big kind you got 5
kopecks back on; at the time vodka was 4 roubles and 12 kopecks a
half-litre, with 12 kopecks back on the bottle; these prices were stable
over several decades.
\paragraph{1976--77 my first visit to Japan} I was aware of young Japanese
algebraic geometers at the AMS summer institute at Arcata in 1974, and I
wrote to Kodaira to invite myself to Tokyo. He put me in touch with
Miyaoka, who was also doing Godeaux surfaces. I was very lucky with my
contacts, and made many friends at once. As my host at Tokyo University,
Iitaka looked after me with great kindness; he introduced me to his young
graduate student Kawamata, then working on\index{log!category} log
varieties. Miyaoka had worked on my notes on Bogomolov's inequality
$c_1^2\le4c_2$, and lectured on his proof of the famous inequality
$c_1^2\le3c_2$ during my first couple of weeks in Tokyo. Among many other
kindnesses, Iitaka advised me to go to the March 1977 spring meeting of
the Japan Math.\ Soc.\ at Kyoto ``to meet new people''; this was when I
first met my wife Nayo.
\subsection{Simultaneous resolution and the classic flop}\label{sec!flop}
As mentioned in Section~\ref{sec!A1}, an easy topological argument shows
that the Milnor fibre $M$ and the minimal resolution $Y$ of the surface
ordinary double point $xy+z^2=0$ are diffeomorphic. In the 1950s, Atiyah
\cite{A} discovered a much more convincing reason for the diffeomorphism
$M\iso Y$, with an argument that was to have far-reaching consequences for
3-fold geometry.
For this, think of the family as $\sX:(xy+z^2=t)\subset\C^3\times\C$,
with $\sX\to\C$ given by $t$. Replace the deformation parameter $t$ by
$\sqrt t=\tau$, thus replacing $\sX$ by
$\sX':(xy+z^2=\tau^2)\subset\C^3\times T$. The fibres of $\sX'\to T$ are
the same fibres as before, but now each fibre $X_t$ with nonzero $t$
occurs twice as $X'_{\pm\tau}$. The new total space $\sX'$ is the 3-fold
ODP $xy+z^2-\tau^2=0$, or $xy=zt$ after the trivial change of coordinates
to $\tau\pm z$. Now the graph of the rational function $x/z=t/y$ provides
a resolution of\index{resolution!of singularities} singularities $\sY\to
\sX'$, and moreover, one in which $X_\tau$ moves in a smooth 1-parameter
family together with the minimal resolution $Y_0\to X'_0$. (It is an easy
exercise to see that the blowup of a line through a surface ODP has the
same effect as the blowup of the point itself.) This is called a {\em
simultaneous resolution}\index{simultaneous resolution} of the surface
singularity.
Now consider the 3-fold ODP $X:(xy=zt)$, forgetting the family of surfaces
for the moment. Let $Y\subset X\times\PP^1$ be the graph of the rational
function $x/z=t/y$. Then $f\colon Y\to X$ is a birational morphism with
fibre $L\iso\PP^1$ over $0$. It is easy to check that $Y$ is nonsingular,
and that the normal bundle to $L$ in $Y$ is
$\Oh_{\PP^1}(-1)\oplus\Oh_{\PP^1}(-1)$. This is an example of a {\em small
resolution}:\index{small!resolution} the exceptional locus contains no
divisors.\index{factorial@$\Q$-factorial}\footnote{The small morphism
$f\colon Y\to X$ is closely related to the class group: $\Oh_X$ is not a
UFD because $xy=zt$; the function $x/z=t/y$ can be written with two
essentially different denominators, which was the point of the reference
to \cite{red} in Section~\ref{sec!A1}. \endgraf
An $n$-fold point $P\in X$ is {\em $\Q$-factorial\/} if every codimension
1 irreducible subvariety of $X$ through $P$ has a multiple that is a
principal divisor. Although the $\Q$-factorial property may seems
somewhat arcane at first sight, it is very simple, and plays a central
role in\index{Mori!theory} Mori theory. To understand it, please do the
following elementary exercise: if $P\in X$ is $\Q$-factorial, then for
every partial resolution $f\colon Y\to X$ every component of the
exceptional locus of $f$ has pure dimension $n-1$. (This is called {\em
van der Waerden purity}.)}
We could have paired the factors $x,y$ and $z,t$ in the other order, and
considered instead the rational function $x/t=z/y$. This gives a second
resolution $f'\colon Y'\to X$ with exactly the same properties as $f$.
It might seem at first sight that $Y=Y'$, but this is not so. Indeed,
since $Y\subset X\times\PP^1$ and the ratio $(x:z)$ is the coordinate on
$Y$, the divisors of zeros and poles of $x/z=t/y$ are two disjoint
surfaces cutting $L=\PP^1$ transversally at $0$
\begin{figure}[ht]
\centerline{\epsfbox{flop.ps}}
\caption{The flop $Y\broken Y'$. }
\label{fig!flop}
\end{figure}
and $\infty$; these are the birational transform on $Y$ of the planes
$x=t=0$ and $z=y=0$ of $X$ (see Figure~\ref{fig!flop}). On $Y'$ however,
the same rational function $x/z=t/y$ has divisor of zeros and poles that
meet transversally along $L'$, marking the two factors of the normal bundle
$\Oh_{\PP^1}(-1)\oplus\Oh_{\PP^1}(-1)$. The ``same surfaces'' (more
precisely, their birational transforms) have intersection number $+1$ with
$L$ and $-1$ with $L'$.
The birational map $\fie\colon Y\broken Y'$ is the {\em classic flop}. To
think of it as a correspondence, consider its closed graph $W\subset
Y\times Y'$. Then $W$ fits into the traditional picture
\[
\begin{matrix}
\renewcommand{\arraycolsep}{0em}
& W \\
\kern1em \swarrow \kern-0.8em && \kern-0.8em \searrow \kern1em \\[2pt]
Y \kern1em &\broken& \kern1em Y' \\[2pt]
\kern1em \searrow \kern-0.8em && \kern-0.8em \swarrow \kern1em \\
& X
\end{matrix}
\]
with $W\to X$ the blowup of the ordinary double point having exceptional
surface the quadric $Q=\PP^1\times\PP^1$ with normal bundle $\Oh(-1,-1)$.
The two sides $W\to Y$ and $W\to Y'$ are the respective blowups of
$L\subset Y$ and $L'\subset Y'$ or the contractions of $Q$ along its two
rulings.
Flops relate closely to the\index{nef} nef condition: let $\fie\colon
Y\broken Y'$ be a classic flop as above. Because $\fie$ is an isomorphism
outside a set of codimension $\ge2$, a divisor $D$ on $Y$ has a birational
transform $D'$ on $Y'$. In other words, it is natural to identify the
divisors on $Y$ and $Y'$. On the other hand, as we saw in
Figure~\ref{fig!flop}, if $DL>0$ then $D'L'<0$.
On a surface, Zariski had proved (\cite{Z}, Theorem~6.1) that if $|D|$ is
a\index{mobile linear system} {\em mobile} linear system (that is, $|D|$
has no fixed part) then some multiple $|nD|$ is {\em free} (that is,
$|nD|$ has no base points). I tried to prove the same thing for 3-folds
until Iskovskikh and Bogomolov told me it was obvious nonsense, because of
flops. The conclusion from this is that if you're hoping for Zariski
decomposition for a divisor $D$ on a 3-fold $X$, you can't expect it to
take place on the 3-fold itself. If it makes sense at all, you first
subtract off some ``negative part'' $N$ so that
$(D-N)\Ga<0$ holds for only finitely many curves $\Ga_i$, and then hope to
``flip'' these\index{flip} by a birational modification $Y\broken
Y'$ so that the $\Ga_i$ are replace by $\Ga_i'$ with $(D-N)\Ga_i'>0$.
The classic flop is also central to the questions of the Weil divisor class
group and projectivity of 3-folds. For example, let $\Pi$ be a plane in
$\PP^4$, and $X_d$ a general 3-fold hypersurface of degree $d\ge3$
containing $\Pi$. The Weil divisor class group of $X_d$ is generated by
$\Pi$ and the hyperplane class. Then $X_d$ can be written $xA+yB$ where
$\deg A,B=d-1$, and in general has $(d-1)^2$ ODPs at the points
$x=y=A=B=0$. Taking the graph of the rational map $x/y=-B/A$ amounts to
blowing up $\PP^4$ along $\Pi$, and is a\index{small!resolution}
projective small resolution of $X_d$ that introduces a flopping curve $L$
above each node. In terms of local coordinates, each curve $L$ can be
flopped, but carrying out a flop of any strict subset of the $(d-1)^2$
lines $L$ results in a nonprojective variety.
It is known by work of Brieskorn \cite{B} and Tyurina \cite{Tyu} in the
late 1960s that\index{simultaneous resolution} simultaneous resolution
applies to any family of surfaces with at worst Du Val singularities
(after passing to a suitable cover of the base, like taking the root
$\tau=\sqrt{t}$ above). For example, a 1-parameter family of surfaces
having nonsingular fibres when $t\ne0$ and a Du Val singularity of type
$A_{n-1}$ above $t=0$ pulls back on taking the $n$th root $\tau=\sqrt[n]t$
to
\[
xy=z^n-\tau^n=\prod_{i=1}^n z_i, \quad\text{where}\quad z_i=z-\ep_i\tau
\]
(and $\ep_i$ runs through the roots of unity). The minimal resolution
$Y_0\to X_0$ of the central fibre has a chain on $n-1$ copies of $\PP^1$,
and the family has $n!$ different\index{simultaneous resolution}
simultaneous resolutions corresponding to ordering the factors $z_i$. You
pass between them by flops, corresponding to transpositions of the $z_i$.
The idea of simultaneous resolution means that you can allow ODP in
families of surfaces (K3 surfaces, surfaces of general type, del Pezzo
surfaces, etc.), in the knowledge that they can be resolved in families.
The moral for 3-folds, and an important reason underlying the success of
my program in \cite{C3-f} and \cite{Pagoda} is that by taking cyclic
covers, a canonical model\index{canonical!model} is always only a finite
distance away from a nonsingular minimal model.
\paragraph{History} Local Torelli says that moduli of nonsingular K3s is
locally isomorphic to the period space, but the moduli problem is not
locally separated near surfaces with extra $-2$-curves. This phenomenon is
related to the need to make a cover to kill the monodromy before obtaining
a simultaneous resolution, and to the ambiguity in the choice of the
simultaneous resolution\index{simultaneous resolution} that gives flops:
on the Hodge theory side of Torelli, you need the extra
information\index{effective!cone} of the effective cone (in other words,
the lattice spanned by $-2$-curves has a distinguished Weyl chamber
consisting of effective curves). If you're really careful with the
definitions, you get a Torelli isomorphism between the moduli and period
stacks; it is likely that Deligne had this aspect of the Torelli problem
in mind for me in my first year. The problem was solved soon after by
Burns and Rapoport \cite{BR}, and their solution was an important
background item in my work on \cite{Pagoda}. Meiki Rapoport also studied
with Deligne, and was my office mate for 2 years at the IHES
I always referred to the classic flop as {\em Atiyah's flop} during the
1970s and 1980s, until Gavin Brown pointed out to me that it occurred in
Zariski papers in the 1930s. As soon as you start looking back into the
past, you see that it occurs in factoring the standard Cremona
transformation
\[
T_{\mathrm{tet}}:\PP^3\to\PP^3 \quad\text{given by} \quad
(x_1,\dots,x_4)\mapsto\Bigl(\frac{1}{x_1},\dots,\frac{1}{x_4}\Bigr),
\]
which flops the 6 edges of the coordinate tetrahedron; it must have been
well understood in this context by Cremona and Noether around 1870.
(According to Hilda Hudson's bibliography, $T_{\mathrm{tet}}$ was studied
by Magnus in 1837 and by Beltrami in 1863.)
\subsection{From canonical models to minimal models}\label{sec!min}
\index{minimal model}
As discussed above, there is no choice in \cite{C3-f} about the
definition of canonical models\index{canonical!model} and canonical
singularities.\index{canonical!singularities} You must simply take whatever
comes as the Proj of the canonical ring\index{canonical!ring} of a variety
of general type (assumed finitely generated). The point I want to make here
is this:
\begin{quote}
{\em A little knowledge on canonical singularities shows that they admit a
restricted class of blowup\index{crepant} called\/ {\em crepant partial
resolution}. Applying\index{resolution!of singularities} this to a
canonical model leads to a minimal model. This procedure is a close analog
of the minimal resolution of Du Val surface singularities.}
\index{minimal model}
\end{quote}
The ordinary double point, the $A_2$ double point and the Veronese cone
point described in Sections~\ref{sec!A1}--\ref{sec!Ver} are among the
simpler kind of canonical singularities\index{canonical!singularities}
called {\em terminal singularities}.\index{terminal!singularities} Any
blowup of them leads to exceptional divisors that are {\em
discrepant}.\footnote{A linguistic point to help the non-English
reader:\index{discrepancy} {\em discrepancy} means a difference or
contradiction, as when the honest cop finds a discrepancy between the
politician's small declared income and large detected expenditure. Here it
means the difference $K_Y-f^*K_X$, that is,
$1/r$ times the divisor of zeros on $Y$ of a local basis of $rK_X$. The
word {\em crepant}\index{crepant} is an English pun meaning not
discrepant; it seems to have somewhat unpleasant reverberations in several
European languages.} For example, as we saw in Section~\ref{sec!Ver}, the
blowup $Y\to X$ resolving the Veronese cone point has
$K_Y=K_X+\frac{1}{2}E$; the discrepancy\index{discrepancy} divisor
$\frac{1}{2}E$ prevents $K_Y$ from being\index{nef} nef. See \cite{YPG}
for more details.
A simple example of a more typical canonical
singularity\index{canonical!singularities} would be a triple point such as
$X:(x^3+y^3+z^3+t^k=0)\subset\C^4$ with $k\ge3$. If $k=3$, this is the
cone over a nonsingular cubic surface $E$, a del Pezzo surface with
$K_E=\Oh_E(-1)$. The blowup $X_1\to X$ is nonsingular, and has $E$ as a
divisor with normal bundle $\Oh_E(E)=\Oh_E(-1)=\Oh_E(K_E)$, so that
$K_{X_1}=K_X$; we say that $X_1\to X$ is\index{crepant} a {\em crepant}
resolution, and $E$ a crepant exceptional divisor. For higher $k$, the
exceptional divisor $E$ is a cubic cone, and $X_1$ has in turn a
singularity $X:(x^3+y^3+z^3+t^{k-3}=0)$, and we can repeat.
Consider again the\index{crepant} crepant resolution $Y\to X$ of the cone
over a non\-singular cubic surface $S\subset\PP^3$, and write $E\iso S$
for the exceptional surface. Any of the 27 lines $L$ of $E$ is a $(-1,-1)$
curve on $Y$, and has a flop $Y \broken Y_1$ that takes $E\to E_1$ by
contracting the $-1$-curve $L$ and introducing a flopped curve $L_1$. Then
$Y_1$ is an alternative resolution of $X$, and its exceptional locus
consists of a del Pezzo surface $E_1$ of degree 4 together with a line
$L_1$ that meets $E_1$ transversally. This illustrates the fact that flops
occur all over the place, and that you almost never expect to get
uniqueness of minimal models of \hbox{3-folds}. In the present case,
crepant resolutions $Y'\to X$ (that is,\index{minimal model} minimal
models of $X$) correspond one-to-one with faces of the nef\index{nef} cone
of the surface $E$, of which there are several hundred. For example, since
there are 72 different contraction morphisms $E\to\PP^2$ that contract 6
disjoint lines on $E$, there are 72 different\index{crepant} crepant
resolutions $Y\to X$ whose exceptional locus consists of the del Pezzo
surface of degree 9 ($\PP^2$ with normal bundle $\Oh(-3)$) and 6 flopping
lines sticking out of it.
\paragraph{Problem} We now know that flops account for the nonuniqueness
of\index{minimal model} minimal models of \hbox{3-folds}. Thus any two
birational minimal models have the same Betti numbers, the same
singularities, etc. All very tame stuff. But in dimension $\ge4$, who
knows?
\paragraph{History} The general result highlighted above is the main
content of my two papers \cite{C3-f} and \cite{Pagoda}. In \cite{C3-f}, I
showed that a suitable chain of blowup can be used to extract all the
crepant\index{crepant} divisors living above a canonical but not terminal
point. In \cite{Pagoda}, which was one of the main outcomes of my 1981
visit to Kyoto, I used the Brieskorn--Tyurina theory of simultaneous
resolution\index{simultaneous resolution} to resolve nonisolated cDV
points.
The flop $Y\broken Y_1$ is clearly related to the\index{projection}
projection of the del Pezzo surface $S_4$ of degree 4 to a cubic surface
$S_3$, and should be viewed as a kind of affine cone over it: $X$ is the
cone over $S_3$, and $Y_1$ contracts to a 3-fold $X_1$ having the cone
over $S_4$ as its singularity, but with a projective line sticking through
it. This picture is the original motivation for\index{flip} Francia's
flip: consider the Veronese surface $V\subset\PP^5$, and its projection
from a point to the scroll $\FF(1,2)\subset\PP^4$. Write $X$ for the
affine cone over $\FF(1,2)$; then $X$ is the first flipping singularity.
It has a ruling by planes that define a\index{small!resolution} small
resolution $X^+\to X$ with exceptional locus a line $L^+$ having normal
bundle $\Oh(-1)\oplus\Oh(-2)$, so that $K_{X^+}\cdot L^+=1$. On the other
hand, it has a small partial resolution $X^-$ corresponding to the
projection $V\broken\FF(1,2)$ where $X^-$ has a Veronese cone point with a
line $L^-$ sticking through it, and $K_{X^-}\cdot L^-=-\frac12\,$.
\subsection{Minimal models via semistable degenerations of
surfaces}\label{sec!deg}\index{semistable!degeneration|(}
Degeneration of surfaces is a special case of the\index{minimal model}
problem of minimal models and classification of 3-folds that has attracted
a lot of attention. Instead of just a 3-fold, we consider a 3-fold $X$
with a fibre space structure $X\to B$ over a base curve $B$. This has a
number of advantages:
\begin{enumerate}
\item For most purposes we can treat the generic fibre $X_{\mathrm{gen}}$
first, and assume that it is a minimal model, typically a K3 surface or a
minimal surface of general type.
\item The minimal model problem then reduces to a neighbourhood of each
degenerate fibre.
\item After making a base change by a ramified cover $B'\to B$ (replacing
$X$ by a model birational to $X'=X\times_BB'$ as in
Section~\ref{sec!flop}), we can assume that $X\to B$ satisfies the
conclusions of Mumford's\index{semistable!degeneration} semistable
degeneration theorem.
\end{enumerate}
For reasons of space, I will not say very much about this big and
technically difficult topic. Mumford raised the problem in the early 1970s
of modifying a\index{semistable!degeneration} semistable degeneration of
K3s to one of several normal forms corresponding to the possible unipotent
filtrations of mono\-dromy; in other words, if we believe that Torelli for
K3s extends to infinity, we can use the compactification of the period
space as a model for how to compactify the moduli problem.
Mumford's problem was solved around 1976 by Viktor Kulikov \cite{Ku}, who
proved the existence of a minimal model for the 3-fold total space of a K3
fibre space with\index{semistable!degeneration} semistable degeneration;
this was the first time that 3-fold\index{minimal model} minimal models
were proved to exist in substantial cases. Kulikov works in the
nonsingular category, without projective assumptions. His main aim is to
get rid of any surface component of the degeneration that appears in the
canonical class with positive multiplicity, first using 3-fold flops to
reduce it to a minimal ruled surface and then contracting it out. The main
difficulty is that using a flop to contract a $-1$-curve on a component of
the degeneration has the effect of introducing another $-1$-curves
somewhere else, possibly even on a different fibre of the same ruled
surface, like a serpent that bites its own tail. (Think of Hironaka's
nonprojective examples. This kind of thing would be excluded by projective
assumptions.) Kulikov has to work hard with graphs and combinatorics and
so on to exclude this kind of loop. His paper contained major innovations,
together with some minor difficulties of exposition and inaccuracies, and
figuring it out led to a flurry of activity in the USA around 1980
(\cite{PP}, \cite{deg}).
I heard Kulikov's lectures in Moscow in 1976 while his program was in the
unfinished state, and still involved formidable technicalities. At this
time my ideas on canonical and minimal models\index{canonical!model} were
just forming. His systematic use of flops predate my published work by
several years, and must have influenced me. However, since my own approach
is always to run away from difficult things wherever possible, and I had my
own agenda involving canonical\index{canonical!model} and\index{minimal
model} minimal models, I did not get involved in the details of his proof,
and my main conclusion was that I really preferred to wait until the
general minimal model program could be sorted out.
Several years later, after Kawamata's\index{log!category} log surfaces,
canonical models and\index{Mori!theory} Mori theory were all on-line,
Tsunoda proposed a more general program to construct\index{minimal model}
minimal models of semistable\index{semistable!degeneration|)} degenerations
of surfaces with $K$ nef,\index{nef} by applying Mori theory
to\index{Mori!theory} Kawamata's log surfaces. The experts seem to believe
that Tsunoda's work over several years contains a more or less complete
proof of semistable minimal models. However, in the intervening time
Shokurov, Kawamata, Mori and most recently Corti have given more
convincing versions of semistable\index{flip} flips\index{semistable!flip}
and the semistable minimal model program. It remains a strong possibility
that the future will see the easiest way to general\index{Mori!flip} Mori
flips through semistable minimal models. (Compare Corti's proof given in
\cite{KM}, Chapter~7.)\index{minimal model|)}
\paragraph{The December 1981 Kinosaki meeting}
Japanese algebraic geometers run a regular workshop at Kinosaki, a famous
spa town in the north of Kyoto prefecture. The tradition apparently
started because of a family connection with the owners of one of the
ryokan (traditional Japanese inn), for whom December is the off-season. At
Kinosaki, you're supposed to visit all the other baths, walking around the
streets in the evenings in yukata (evening kimono) and wooden clogs, and
holding traditional greased paper umbrellas if it's snowing.
Mathematicians gather in the hotel lobby in the evenings to discuss math
and other matters, and to sample generous quantities of the finest sake
and Japanese whisky. Up to the late 1980s, the lectures were still held in
the intimate environment of the ryokan's dining room.
Tsunoda announced his solution to the minimal model problem for
semi\-stable degenerations of surfaces at the Kinosaki meeting in December
1981. After the workshop, Tsunoda, Maehara, Kurke and I spent an afternoon
visiting the famous beauty spot Ama-no-hashidate. The Buddhist temple
Nari-ai-ji at the top of the hill contained a Boddhisatva that is a Yuigon
Bosatsu, that is, has the unusual power of granting any request, even if
only prayed for a single time (the normal practice is to require 10,000
repetitions, a bit like trying to get your kids off the phone). On hearing
of this, Tsunoda and I both pressed our palms together, bowed our heads,
and solemnly prayed for 3-fold canonical rings\index{canonical!ring} to be
finitely generated. We forgot to ask help in solving the problem for
ourselves.
\section{The Mori cone and Mori minimal models}\label{ch!Mori}
\index{Mori!cone}
Castelnuovo and Enriques, together with every geometer up to Zariski and
Mumford, expressed the classification of surfaces in terms of birational
invariants, especially the plurigenera and irregularity. In these terms,
two of the main aims of the theory are Castelnuovo and Enriques' criteria
\begin{align*}
P_2(S)=q(S)=0& \ \implies \ \text{$S$ is rational; and} \\
P_{12}=0& \ \implies \ \text{$S$ is ruled or rational.}
\end{align*}
Here the assumptions and the conclusions are birational, and for this
reason, the classification of surfaces is often described as {\em
birational classification}. This is one of several traditional
preconceived notions you must unlearn if you wish to become a 3-folder.
The systematic approach to higher dimensional classification initiated by
Iitaka and Ueno follows the same logic. This approach gives the right
results (such as the Iitaka fibration) in cases when the plurigenera grow,
or you have other information such as a nontrivial Albanese map. But if
the plurigenera and irregularity are zero, it doesn't really give anything
other than optimistic conjectures that the variety should be covered by
rational curves, conjectures that are somewhat cheap because there is no
plan for proving them. My approach described in the preceding chapter,
although it set itself biregular aims, and was natural enough in view of
my main interest in varieties of general type and their canonical
rings,\index{canonical!ring} depended on even stronger assumptions than
Iitaka's program: general type and finitely generated canonical ring.
Mori's interest seems to have started out from\index{Fano!variety} Fano
varieties (varieties with $-K$ ample that classical geometers describe as
``close to rational'') and the existence of rational curves on them. The
springboard for his first paper in minimal models \cite{M1} was his
earlier solution of the Hartshorne conjecture characterising $\PP^n$ by
ampleness of the tangent bundle. (I believe that this conjecture
originated with a question of Deligne after Hartshorne's lecture on ample
vector bundles at the 1970--71 Warwick symposium. I hope Mumford and
Zeeman remembered to get it into the final SRC report.) A classical
dimension count predicts that the space of rational curves is positive
dimensional at any point, but we know that this type of argument cannot
prove it is nonempty; Mori's key breakthrough is his extraordinary method
of {\em bending-and-breaking} and {\em reduction to characteristic $p$} to
prove\index{bending-and-breaking} the existence of rational curves.
The Kleiman--Mori cone described in Section~\ref{sec!Klei} gives a
strategy for minimal models based\index{extremal!contraction}
on\index{extremal!ray} contracting extremal rays
(Section~\ref{sec!extrr}). In contrast to Iitaka's and my methods, these
ideas work best of all for\index{Fano!variety} Fano varieties, whose Mori
cone\index{Mori!cone} is a finite rational polyhedron with rays spanned by
extremal rational curves. The contraction part can be done very generally
by a circle of ideas that Kawamata's school call the {\em X method}, and
modifications of its proof lead to the general cone theorem
(Section~\ref{sec!X}). Flips were the last foundational problem to be
solved by Mori in the late 1980s, and still lack a simple and wholly
convincing argument.
\subsection{Kleiman criterion and the Kleiman--Mori cone}\label{sec!Klei}
The Riemann--Roch formula $\chi(\Oh_S(D))=\frac{1}{2}D(D-K_S)+\chi(\Oh_S)$
for a divisor $D$ on a nonsingular surface $S$ is quadratic in $D$. It
follows that for $D^2>0$, either $h^0(nD)$ or $h^0(-nD)$ grows
quadratically with $n$. Grothendieck's proof of the algebraic Hodge index
theorem follows easily from this: the index of the intersection form on
the algebraic part of $H^2(S,\R)$ is $(+1,-(\rho-1))$, and effective
divisors\index{effective!divisor} with $D^2>0$ form a half-cone in
$H^2(S,\R)$ that we draw as a future light cone.
The Kleiman criterion \cite{Kl} starts with the question of the cone of
nef divisor classes $\Nef S \subset H^2(S,\R)$. The class of an ample
divisor is both in the interior of $\Nef S$, and of the $D^2>0$ cone; thus
ample divisors form a subcone of the future light cone. Conversely, it
turns out that a nef divisor is in the {\em closure} of the ample cone in
$H^2(S,\R)$: if $D$ is nef and $H$ is ample then the index polynomial
$p(\la)=(D+\la H)^2$ takes positive values for $\la\gg0$, while $D+\la H$
is nef\index{nef} for any $\la\ge0$. But $p(\la)$ cannot go from being
positive to being negative while $D+\la H$ is nef, and it follows that
$D^2\ge0$ and $D+\ep H$ is an ample $\Q$-divisor for any $\ep\in\Q$,
$\ep>0$.
Thinking about effective and nef divisors on surfaces tends to obscure a
key point that comes out naturally for a higher dimensional variety $V$:
\begin{quote}
{\em Rather than $H^2(S,\R)$ and its {\em quadratic} intersection form,
it is better to think of the {\em bilinear} pairing between the two dual
vector spaces $H^2(V,\R)$ and $H_2(V,\R)$.}
\end{quote}
The first of these contains the classes of $\Q$-Cartier divisors, and the
second the cone of\index{effective!curve} effective curves $\NE V$ and
its closure $\NEbar V\subset H_2(V,\R)$, the\index{Mori!cone} {\em
Kleiman--Mori cone}.
The Kleiman criterion says that a Cartier divisor $L$ is ample on $V$ if
and only if its divisor class in $H^2(V,\R)$ is strictly positive on
$\NEbar V\setminus\{0\}$. This condition is slightly stronger than saying
$L\Ga>0$ for every curve $\Ga$ in $V$; it requires equivalently that
$L-\ep H$ is still\index{nef} nef for any ample $H$ and $\ep>0$, or that
$Lz>0$ for $z$ a {\em limit} of effective curves.
The Kleiman--Mori cone $\NEbar X\subset H^2(X,\R)$ is a closed convex cone
by definition. You don't expect to be able to calculate it in general, but
enough cases are known to suggest that both $\NEbar X$ and the relation
between $\NE X$ and its closure $\NEbar X$ can be pretty well arbitrary in
the half-space $K_Xz\ge0$. Mori's discovery concerns the half-space
$K_Xz<0$: on a nonsingular $n$-fold $\NEbar X\cap\bigl((K_X+\ep
H)z\le0\bigr)$ is finite rational polyhedral for any ample $H$ and any
$\ep>0$. The next two sections sketch two approaches to proving this:
Mori's argument by\index{bending-and-breaking} bending-and-breaking, and
the rational threshold argument related to Kodaira vanishing.
\paragraph{History} I learned the idea of the positive cone implicitly
from Zariski \cite{Z}, from working with Bogomolov on asymptotic RR, and
more especially from Mumford's comments on my notes on Bogomolov.
The use of cones such as the positive cone, $\Nef$ and $\NE$ can
reasonably be traced back to Nagata's papers on rational surfaces
\cite{N}. If you look for Kleiman's paper \cite{Kl} in the library of the
Kyoto University math department, you will find that Vol.~84 of Ann.\ of
Math.\ is particularly dirty and dog-eared, as if held over the
photocopier dozens of times. I presume that Nagata has set Kleiman's paper
as background reading for successive generations of students. (Both
Hironaka and Mori were Nagata's M.Sc.\ students.)
\subsection{Mori cone theorem and Mori contractions}\label{sec!extrr}
\index{Mori!cone}
Mori first introduced\index{bending-and-breaking} bending-and-breaking
under very strong assumptions in the context of the Hartshorne conjecture,
where he must prove the existence of a rational curve in a variety
$\Ga\subset V$ (that is to be a straight line $\Ga=\PP^1\subset\PP^n$, so
that $-K_V\Ga=n+1$). He extended the method in the proof of his original
cone theorem, and later Koll\'ar and Miyaoka generalised it further in
several directions. I sketch Koll\'ar's ideal form of the theorem.
\begin{quote}
{\em Let\/ $C_0\subset X$ be a curve with\/ $-K_XC_0>0$ on a nonsingular\/
$n$-fold\/ $X$. Then there exists a rational curve\/ $\Ga\subset X$ passing
through any point\/ $Q\in C_0$ with\/ $0<-K_X\Ga\le n+1$.}
\end{quote}
\newcommand{\scpa}[1]{\medskip\noindent{\sc#1}\enspace}
\scpa{Step~1:} Replace the embedded curve $C_0\subset X$ by the map
$f\colon C\to C_0\subset X$ from a nonsingular curve, with $P\in C$ such
that $f(P)=Q$.
\scpa{Step~2:} The deformation of the map $f$ is controlled by the
cohomology of the pullback $f^*T_X$ of the tangent bundle of $X$, a rank
$n$ vector bundle on $C$ of degree $-K_XC_0>0$. Suppose that this degree is
not just positive, but very large compared to $n$ and the genus of $C$.
Then
\begin{description}
\item{{\bf bending:}} we can deform $f$ as a morphism $F_0\colon C\times
T_0\to X$ over a parameter curve $T_0$ so that the image curve $f_t(C)$
sweeps out a surface, while fixing the value $f(P)=Q$.
\item{{\bf breaking:}} let $T$ be the projective completion of $T_0$, so
that $F\colon C\times T\broken X$ is a rational map; then $F$ is {\em
certainly not a morphism} in a neighbourhood of the section $\{P\}\times
T$. Because this section has self-intersection 0, but $F$ contracts it to a
point in $X$.
\end{description}
We get a\index{bending-and-breaking} rational curve on $X$ through $P$ by
resolving the indeterminacy of $F$ by blowups and taking the image of a
suitable exceptional curve, and we can arrange for it to have bounded
degree and $0<-K_X\Ga\le n+1$ by rerunning an appropriate version of the
same argument if necessary.
\scpa{Step~3:} Now comes the really clever bit. If we are in characteristic
$p$, we can replace $f$ by its composite $f^{n}$ with a suitable iteration
of the Frobenius map. This allows us to pump up $\deg f^*T_X$ to be as
large as we like, while fixing the genus. Thus the required rational curves
exist in almost all characteristic $p$; an argument on finiteness of the
Hilbert scheme implies that they also exist in characteristic 0.\quad
Q.E.D. \medskip
It follows from this result that any cycle in $z\in\NEbar$ with $(K+\ep
H)z<0$ necessarily splits off a rational curve of bounded degree. The Mori
cone\index{Mori!cone} theorem (for a nonsingular variety) is a rather
formal consequence of this.
The kind of contraction\index{contraction} morphisms $\fie\colon X\to Y$
we are interested in are determined by certain faces of $\NEbar X$; namely
if $Y$ is projective and $H$ ample on it, then $L=\fie^*H$ is\index{nef}
nef, and $L^\perp\cap\NEbar X$ is the cone\index{effective!curve} of
effective curves contracted by $\fie$, that is, the relative cone
$\NEbar(X/Y)$. A face of $\NEbar$ is a relative cone for a contraction if
and only if it has a rational supporting hyperplane corresponding to an
eventually free divisor $L$, and the morphism $\fie_{nL}\colon X\to Y$ for
$n\gg0$ (or rather, its Stein factorisation) is uniquely determined up to
isomorphism by the face.
In \cite{M2}, Mori only contracted\index{extremal!ray} extremal rays on
nonsingular 3-folds in the $Kz<0$ half of the cone, and obtained the
famous classification that I'm sure you know about.
\paragraph{History} I want to stress that the\index{Mori!cone} Mori
cone came as a huge surprise to everyone, and that even now his leap of
imagination seems quite incredible. I first heard that Mori was working on
3-fold contractions in a letter from Mumford around 1980 on practicalities
of a visit to Warwick. Just in passing, he scribbled a note, that I
paraphrase from memory: Mori had given a Harvard seminar, proving that if
$K_X$ is not nef on a smooth 3-fold, there exists a contractible surface,
either a scroll, or $\PP^2$ with normal bundle $\Oh(-1)$ or $\Oh(-2)$, or
a quadric surface. P.S. (clearly added as an afterthought) That is, unless
we are in the uniruled case.
In other words, even after Mori had described his work at a seminar, none
of us had really grasped the significance of his cone theorem, or the extra
precision that his methods\index{Mori!fibre space} give us in the uniruled
case. For about 3~years after this, I regularly wrote to Mori to bother
him for more details, both for my own sake and for my Moscow
correspondents. The research announcement \cite{M1} only aroused our
curiosity, and I clearly remember getting the preprint of \cite{M2} and
wondering what all this clever-clever stuff about cones had to do with
contracting surfaces on 3-folds. My next reaction was to look among the
varieties I knew for counter\-examples to his conclusions
for\index{extremal!contraction} extremal contractions that were conic
bundles\index{conic bundle} or\index{divisorial contraction} divisorial
contractions to ODPs.
I learned\index{Mori!theory} Mori theory, and many of the details of
Mori's paper \cite{M2} from a series of lectures by Miyanishi at Ueno's
seminar at Kyoto University in 1981, and its applications to Fanos from
talking to Mukai who was at that time working with Mori \cite{MM} on the
classification of nonsingular\index{Fano!3-fold} Fano 3-folds with
$B_2\ge2$. The 2-ray game\index{2-ray game} plays an important role in
their study (without the name). In 1982--83 I organised my first Warwick
symposium. During the first half of this year, I got together with several
long term symposium visitors (Beltrametti, Dolgachev, Palleschi and
Tsunoda) to run a seminar on Mori's paper and subsequent developments.
\subsection{The X method, contractions and thresholds}\label{sec!X}
Two related issues presented themselves around 1981:
\begin{enumerate}
\renewcommand{\labelenumi}{(\roman{enumi})}
\item If $X$ is a 3-fold minimal model of general type (that is, at worst
terminal singularities,\index{terminal!singularities} $K_X$ nef\index{nef}
and $K_X^3>0$), can we prove that $K_X$ is eventually free? This is
equivalent to the finite generation of the canonical
ring\index{canonical!ring} of a minimal model $X$.
\item If $F\subset\NEbar X$ is a face of the\index{Mori!cone} Mori cone
contained in the half-space $K_Xz<0$, can we prove that there exist a
morphism $X\to Y$ contracting exactly the curves in $F$? Mori proved the
contraction theorem for\index{contraction!theorem} an\index{extremal!ray}
extremal ray on a nonsingular 3-fold $X$, but his proof seems impossible
to generalise, since it involves a detailed knowledge of $X$ in a
neighbourhood of the contracted divisor.
\end{enumerate}
Both these questions were answered at the same time by Kawamata's theorem
\cite{Ka}: if $D$ is a\index{nef} nef Cartier divisor and $D-\ep K_X$ is
ample for some $0<\ep\ll1$ then $D$ is eventually free. This statement
only has content if $D$ is a supporting hyperplane of $\NEbar X$ defining a
nontrivial face $D^\perp \cap\NEbar X$ in the half-space $K_Xz<0$.
The method of proof is a complicated game that has subsequently been
refined by many authors, notably Kawamata himself and Shokurov. Kawamata's
graduate students, who have had to suffer this method more than most, have
with characteristic Japanese linguistic inventiveness christened it the
{\em X method\/}; the instruction manual is \cite{KMM}. The method is
powerful, we know how to use it pretty well in practice, and it's been
around for 20 years without anyone thinking of a better one, but (in my
opinion) it is still messy and not optimal, and we don't really understand
it.
Kodaira vanishing is the theorem that if $X$ is a nonsingular projective
$n$-fold over a field of characteristic 0 and $D$ and ample divisor, then
\[
H^i(X,D+K_X)=0 \quad\text{for all $i>0$.}
\]
Many generalisations are known, for example Kawamata--Viehweg vanishing:
we can allow $D$ to be an ample $\Q$-divisor with fractional part
supported on a normal crossing divisor, and get the conclusion
$H^i(X,\ulcorner D\urcorner+K_X)=0$. When using the X method, we blow up
the $n$-fold $X$ together with any divisors we can see on it to a
nonsingular variety $V$ with a normal crossings divisor
$D$, then manoeuver into position to apply vanishing at the level of the
different strata of $V,D$, thus obtaining surjective restriction maps.
\begin{quote}
{\em Kodaira vanishing and the X method give a reason of principle why
life is simple around the $K$ negative boundary of\/ $\NEbar$. These ideas
apply to give the\index{Mori!cone} Mori cone theorem and
the\index{contraction!theorem} contraction theorem for very general class
of singular varieties $X$.}
\end{quote}
Very roughly, the argument is as follows: suppose $K_X$ is not
nef,\index{nef} and let $D$ be an ample Cartier divisor. Consider $D+tK_X$
as $t$ increases; then there is some critical value\index{nef!threshold} or
{\em nef threshold\/} at which $D+tK_X$ stops being ample
\[
t_0=\sup\bigl\{t\bigm| \text{$D+tK_X$ is nef}\bigr\}.
\]
just before we reached $t_0$, the divisor $D+tK_X$ is ample. But if we add
$K_X$, then $D+(t+1)K_X$ is not nef; however, Kodaira vanishing applies to
it, because $D+tK_X$ is ample. This implies that something very rational
is going on, and, following \cite{R}, Kawamata and Koll\'ar proved that
$t_0$ is a rational number whose denominator can be bounded in terms of the
dimension $n$ and the index of $X$.
\paragraph{History}
During my 1981 stay at Kyoto University, I made extensive attempts to
prove the contraction theorem\index{contraction!theorem} for 3-folds of
general type with $K_X$ nef.\index{nef!threshold} I was hoping to prove,
for example, that
$|10K|$ is free. I tried to use ideas of the following type: if $D$ is a
divisor on $X$ passing through a point $P$ with multiplicity $\ge3$, and
$\si\colon X_1\to X$ the blowup of $P$ with exceptional locus $E$, then
$D^{(3)}=\si^*D-3E$ is an\index{effective!divisor} effective divisor, and
by exactly the same argument as in the Bombieri--Kodaira--Ramanujam method
for surfaces,
\[
P\in\Bs|D+K_X| \iff H^1(\Oh_{D^{(2)}})\to H^1(\Oh_{D^{(3)}}) \text{ is
not surjective}.
\]
For example, if $D$ is a reduced surface with an ordinary triple point at
$P$, then $P$ is a base point of $D+K_X$ if and only if the blowup of the
elliptic singularity $P$ increases the irregularity of $D$, rather than
decrease the geometric genus (so that $P$ ``disconnects $D$'', by analogy
with a node disconnecting a curve in Bombieri's method). If $D$ is a little
bit ample, then this should be impossible by some kind of Lefschetz
hyperplane section argument in topology. I'm not sure why this idea still
can't be made to work.
At the July 1981 conference in Tokyo, in connection with my \cite{Pagoda}
lecture, I discussed these kind of problems with Kawamata. This was just
before he went to Berkeley to take up his Miller fellowship. I received
several of his preprints (including \cite{Ka}) during the Warwick
symposium, and lectured on them and started developing my own version of
them \cite{R}. A particular aim was to massage the statement of Kawamata's
theorem on eventually freedom to imply\index{contraction} contractions of
faces of the\index{Mori!cone} Mori cone.
Over many months during the Warwick symposium, Tsunoda came to discuss
attempts to prove the Mori cone by deforming curves. I considered these to
be doomed to failure because of the Francia flip:\index{flip} in that
case, there is just a single curve on which $K$ is negative, and of course
it doesn't deform. (I was wrong: a few years later, Koll\'ar \cite{Ko}
found an ingenious argument involving Mori's idea of quotient stacks to
extend the deformation argument to some singular $n$-folds, in particular
those with hypersurface\index{quotient singularity} or quotient
singularities.) I stumbled on the threshold proof of the\index{Mori!cone}
Mori cone theorem in preparing lectures on \cite{R}, and in trying to find
another way around the cone theorem to answer Tsunoda's questions.
Shokurov, who discovered a version of the same argument independently of
me \cite{Sh}, wrote in the summer of 1983 that he could do the
nonvanishing trick that made these methods work for general
\hbox{$n$-folds}. The cone and\index{contraction!theorem} contraction
theorems are now proved whenever it makes sense.
In the report to SRC on the 1982--83 year I wrote:
\begin{quote} {\em
\dots the framework for the study of 3-folds and higher dimensional
varieties is just in the process of construction; we have just got to the
stage when conventional wisdom was that `there is not satisfactory
extension to higher dimensions of the theory of minimal models of
surfaces, or of the details of the classification' to the point when
everyone now believes that `there is a good theory provided that we work
in the right class of singular varieties'. It is quite clear that the
Warwick year 1982/83 will go down as a key period when substantial cases
of the general theory of higher dimensional varieties were first proved.}
\end{quote}
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\bigskip
\noindent
Miles Reid,\\
Math Inst., Univ. of Warwick,\\
Coventry CV4 7AL, England\\
e-mail: miles@maths.warwick.ac.uk \\
web: www.maths.warwick.ac.uk/$\sim$miles
\end{document}
editing scrap
\subsection{} Mori's and my approaches were opposites in at least 3
respects: (i) My main interest at the start was varieties of general type,
whereas his were $\PP^n$ and Fanos. (ii) He worked down towards a minimal
model from a general nonsingular variety by contractions, whereas I worked
up from the canonical model towards a minimal model by blowing up. Whereas
I was mainly thinking in terms of linear systems of divisors, as in the
Italian tradition, he invented the dual point of view of 1-cycles.
(iii) The experience of working with algebraic curves using divisors and
linear systems suggests two possible approaches to the study of a higher
dimensional variety $X$. The Italian approach in terms of divisors and
linear systems on $X$, and Mori's innovative use of the set of all curves.
If our purpose is to prove very ampleness via vanishing theorems, we may
well try something like this: blow up a point, take the pullback of the
divisor and subtract off a small multiple of the exceptional divisor, and
ask whether the resulting combination is still ample. We run at once into
curves in the guise of Seshadri constants.
(It all goes to show that opposites can work creatively together.)
We attacked the problem from opposite ends also in a quite different
sense: I worked primarily with divisors and linear systems in the Italian
tradition, and thus suffered from the traditional preconceived notion that
what we were supposed to contract was a divisor; whereas Mori's unique
insight was to work with rational curves and the Mori cone, and his
contractions are driven by its extremal rays I discuss the background to
the early subject in Sections~\ref{sec!nef}--\ref{sec!flop}, and the Mori
cone and the minimal models program in
Section~\ref{sec!extrr}--~\ref{sec!min}.