Negative Answers To Some Positivity Questions
by
John Lesieutre
A.B., Harvard University (2009)
Submitted to the Department of Mathematics
in partial fulfillment of the requirements for the degree of
MASSAO
ACHUSETTS
U
IffTE
OF TECHNOLOGY
Doctor of Philosophy in Mathematics
JUN 17 2014
at the
MASSACHUSETTS INSTITUTE OF TECHNOLOGY
June 2014
@ John Lesieutre, MMXIV. All rights reserved.
The author hereby grants to MIT permission to reproduce and to
distribute publicly paper and electronic copies of this thesis document
in whole or in part in any medium now known or hereafter created.
Signature redacted
A uthor .................................................
Department of Mathematics
April 24, 2014
----- Signature redacted
Certified b
James McKernan
Professor of Mathematics
Thesis Supervisor
Accepted
by.Signature redacted.......................
William Minicozzi
Chairman, Department Committee on Graduate Students
Negative Answers To Some Positivity Questions
by
John Lesieutre
Submitted to the Department of Mathematics
on April 24, 2014, in partial fulfillment of the
requirements for the degree of
Doctor of Philosophy in Mathematics
Abstract
We construct counterexamples to a number of questions related to positivity properties of line bundles on algebraic varieties. The examples are based on studying the
geometry of varieties that admit pseudoautomorphisms of positive entropy, and in
particular on the action of standard Cremona transformations on blow-ups of projective space at configurations of points.
The main examples include the following: nefness is not an open condition in
families; the diminished base locus of a divisor is not always a closed set; Zariski
decompositions do not necessarily exist in dimension three; asymptotic multiplicity
invariants are not always finite in the relative setting; and the number of FourierMukai partners of a variety can be infinite.
Thesis Supervisor: James MCIKernan
Title: Professor of Mathematics
3
4
Acknowledgments
I am indebted first to my advisor, James McKernan, both for years of patiently
fielding my general questions about algebraic geometry, and for many more specific
suggestions related to this project. Thanks are due too to many others with whom
I have had useful discussions about this work, among them Robert Lazarsfeld, Eric
Bedford, Mircea Mustata, Igor Dolgachev, and John Ottem. The members of my
thesis committee, James McKernan, Bjorn Poonen, and Frangois Charles, had many
useful questions and suggestions both major and minor, which substantially improved
the final product.
I am grateful as well to the many algebraic geometry students in the Boston
area, from whom I have learned a great deal. I am particularly indebted to my
fellow students Tiankai Liu, Roberto Svaldi, and Jennifer Park, who have served
as the first line of defense for my stupid questions for the last five years, and to
Shashank Dwivedi, Yoonsuk Hyun, Kartik Venkatram, and Brian Lehmann for useful
explanations. I also wish to thank my many friends around MIT, whose company
has been an always welcome distraction from this project. Jeff Yelton has my special
thanks for suggesting the present title of this dissertation, after suffering through
more than one version of my talk about it.
At last, I wish to thank some others who have supported me in my mathematical
career: my parents, George and Annie Lesieutre, my brother, Will Lesieutre, and my
fiancee, Ana Enriquez.
This research was supported by an NSF Graduate Research Fellowship under
Grant #1122374.
5
Contents
1 Introduction
1.1 Basic strategy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.2 Overview of results . . . . . . . . . . . . . . . . . . . . . . . . . . . .
9
10
11
2
Preliminaries
2.1 Cones of positive divisors . . . . . . . . . . . . . . . . . . . . . . . . .
2.2 Asymptotic invariants of linear series . . . . . . . . . . . . . . . . . .
2.3 The Cremona action on divisors and configurations of points . . . . .
15
15
17
20
. . . .
23
3 Eigenvectors of the Cremona action
3.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.2 Nefness in families of R-divisors . . . . . . . . . . . . . . . . . . . . .
3.3 A non-closed diminished base locus . . . . . . . . . . . . . . . . . . .
25
26
27
31
4 Multiplicities on CY3s
4.1 Prelim inaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.2 Lifting pseudoautomorphisms . . . . . . . . . . . . . . . . . . . . . .
4.3 Exam ples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
37
37
39
41
5
45
45
48
2.4
(Pseudo)automorphisms of rational and Calabi-Yau threefolds
Zariski decomposition and the MMP
5.1 Zariski decomposition of eigenvectors . . . . . . . . . . . . . . . . . .
5.2 Outcomes of the MMP . . . . . . . . . . . . . . . . . . . . . . . . . .
6 Finiteness of Fourier-Mukai partners
6
55
List of Examples
1.
Nefness is not open in families . . . . . . . . . . . . . . .
2. ... even for big divisors . . . . . . . . . . . . . . . . . . .
3. Jumping of Seshadri constants . . . . . . . . . . . . . . .
4. Bigness is not closed in families . . . . . . . . . . . . . .
5. The diminished base locus is not always closed . . . . . .
6. ... even for big divisors . . . . . . . . . . . . . . . . . . .
7.
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Ec cc(D)[C] need not converge . . . . . . . . . . . . . .
Asymptotic multiplicity invariants are not always finite .
Weak Zariski decompositions do not always exist . . . .
... even for integer divisors . . . . . . . . . . . . . . . . .
Nakayama-Zariski decompositions do not always exist for big divisors
The number of Fourier-Mukai partners can be infinite . .
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8
Chapter 1
Introduction
Suppose that X is a smooth projective variety and that L is a line bundle on X
(we postpone making precise conventions until Chapter 2). In many situations, it
is useful to consider the sections of tensor powers LO' for larger and larger values
of m, and the rational maps #m : X -- + PN defined by these sections. This is a
natural approach, since some power LOm might have non-vanishing sections even if
L itself does not. Assume for now that some power LO' admits a section. Various
complications arise in studying the asymptotic behavior of sections of line bundles as
m increases. For example, the section ring
R(L) =
®
H (X, L
m)
m>O
is not finitely generated in general: for arbitrarily large values of m, the bundle Lom
might have sections which are not generated by sections of smaller tensor powers.
The asymptotic behavior of linear series is straightforward for ample line bundles.
The basic facts concerning ampleness were worked out in the 1950s and 60s by Kodaira, Grothendieck, Kleiman, and many others. In general, few pathologies arise
in this setting. For example, if L is ample, then the section ring R(L) is necessarily
finitely generated. Similarly, the behavior of ampleness under deformation is straightforward: if the variety X and line bundle L are deformed in a family parametrized by
a base S, then L, is ample on X. for all s in a Zariski open subset of S. Similarly, if
L' is another line bundle for which the Chern class ci(L') is sufficiently close to ci(L),
then L' must be ample as well.
More recently, weaker notions of positivity for line bundles have played an increasingly important role in algebraic geometry, especially in connection with birational
geometry and the minimal model program. One such notion is that of nefness: a line
bundle L is said to be nef if D - C > 0 for every curve C C X. This is a weakened
version of the strict positivity possessed by ample divisors. In Chapter 3, we give an
example of a divisor L which is nef on one fiber of a family, but whose deformations
L, are nef only for s in a countable intersection of open sets, which is not itself open,
showing that nefness is less well-behaved than ampleness in this respect.
There are a variety of related invariants, which we will broadly refer to as "asymp9
totic invariants", that encode geometric information about the behavior of the linear
series ImLl for many values of m simultaneously. These might in principle exhibit
a range of pathological behaviors, but in general it is difficult to find examples of
line bundles for which these invariants are both reasonably computable and badly
behaved. The principal aim of this thesis is to give a number of examples in this direction, demonstrating that some conceivable complications related to the asymptotic
theory of linear series do in fact arise in examples.
1.1
Basic strategy
If q5: X -- + Y is a birational map which is an isomorphism in codimension one (i.e.
for which neither # nor 01 contracts any codimension 1 subvariety), and H is a very
ample divisor on Y, then the strict transform of H on X is not ample. Its base locus
coincides with the the indeterminacy locus of #, a set of codimension at least two.
This gives rise to a partial decomposition of the movable cone Mov(X) C N'(X),
which is generated by divisors whose base loci are of codimension at least 2:
U Nef(Xi)
C Mov(X).
Here Xi ranges over the small Q-factorial modifications (SQMs) of X, which include
all smooth varieties Y to which X admits a rational map 0 as above. If X is a
Calabi-Yau variety, the inclusion is in fact an equality: every divisor in the interior of
the movable cone is the strict transform of a nef divisor on some Xi [26]. In general,
however, the inclusion may be strict. (By "Calabi-Yau", we mean only that the
canonical divisor is numerically trivial.)
Mov(Xo)
Nef(Xo)
Suppose now that X is a variety admitting infinitely many SQMs. We will encounter several examples in the coming chapters, including blow-ups of P3 at 8 or
more sufficiently general points, and various Calabi-Yau threefolds. In both cases,
these SQMs exist due to infinite sequences of flops, introduced in Chapter 3. In this
setting, taking the closure of the union on the left is in fact necessary for equality
in many cases: there can exist pseudoeffective divisors on the boundary of Mov(X)
whose strict transforms do not become nef on any SQM of X.
10
Suppose that D is such a divisor, lying on the boundary of Mov(X), and that
X admits a sequence of D-negative small birational maps qi : Xi -- + Xi+,. The
composites 0i = #4 o ... oq 0 define rational maps $i : Xo -- + Xi+1 . If H,, is an ample
divisor on Xn, its strict transform Hn on X 0 is a movable divisor, with base locus
equal to the indeterminacy locus of 4_1. The multiplicity of H along any subvariety
contained in this base locus can be computed by tracking how these multiplicities
change when taking the strict transform along each of the maps #;-1. Moreover, the
classes of the divisors Ht often converge (up to rescaling) to the boundary class D as n
goes to infinity. Computations of the multiplicities of each H then imply something
about properties of the limiting class D, and we will see that these limiting classes D
provide counterexamples to a number of questions.
Broadly speaking, there are two possibilities for the behavior of the indeterminacy
loci of sequences of birational maps for fixed pseudoeffective D:
1. The indeterminacy loci of
/i
: X
--
+
Xi+1 are all contained in some finite
union of curves: for large j, the curves in the indeterminacy locus of
in the indeterminacy loci of #;-1 for some i < j.
2. The number of components of Oi : Xo --
indeterminacy loci of the
#i
+
#j are also
Xi+1 is unbounded with i: the
consist of sequences of distinct subvarieties.
One setting where these examples are particularly transparent is when the small
modifications
#i
all coincide with some fixed pseudoautomorphism
#
: X --
+
X, a
rational map for which neither # nor its inverse contracts any divisor. In this setting,
q induces a linear map #, : N 1 (X) -* N 1 (X). If this map has an eigenvalue greater
than 1, as is often the case, the iterates #'(H) converge (up to rescaling) to the
dominant eigenvector D on the pseudoeffective boundary. These eigenvectors prove
to be an important source of examples.
Owing to the broad range of possible behaviors, our focus throughout will be on
identifying concrete examples in which various pathological behaviors actually arise.
Chapter 3 first deals with an example of Type (1) above, arising from sequences of
flops induced by Cremona transformations on blow-ups of IP3. Chapter 4 then focuses
on analyzing examples of Type (2), particularly in the case of sequences of flops on
a Calabi-Yau threefold. Chapter 5 discusses the implications of both of these constructions for questions about Zariski decompositions in higher-dimensional settings,
and some related questions about the Minimal Model Program. The results of Chapter 6 are largely independent; we observe that the same classical example studied in
Chapter 3 provides a counterexample to a conjecture about the reconstruction of an
algebraic variety from its derived category of coherent sheaves.
1.2
Overview of results
Nefness in families
If 7 : X -+ S is a surjective, proper morphism of varieties, then ampleness is an open
condition on the base: if D is a divisor on X, and there is a point 0 G S for which the
11
restriction Do is ample on X 0 , then in fact D, is ample for all s E S contained in some
open set. This implies that nefness is "almost" an open property: if Do is nef, then
D, is nef for all s in the intersection of countably many Zariski open sets [31], [29].
However, there are few examples known in which nefness is not simply an open
property, and here we will construct the first example on a family of varieties defined
in characteristic 0. The example is quite elementary: the base S is the space of
configurations of 10-tuples of points in P2 , and the fiber X, is the blow-up of P2 at
the points of s. There exists a divisor D that is nef for sufficiently general points
but fails to be so when the 10 points are in certain special kinds of configurations
(three on a line, six on a conic, etc.). This divisor can be realized as the eigenvector
associated with a positive entropy automorphism of the blow-up when the points are
in a certain special configuration (cf. [30]).
Theorem 1.2.1. Let X -+ (P 2 ) 10// PGL(3) \ A be the family of blow-ups of P2 at ten
distinct points. There exists an R-divisor D on X such that De is nef on Xp for very
general p but is not nef for p in countably many codimension-1 subvarieties of the
base.
The diminished base locus
Given a pseudoeffective divisor D, it is sometimes helpful to consider the diminished
base locus B_(D), which is defined by B_(D) = UA ample B(D + A), where B(D)
is the stable base locus nn Bs(nD). This invariant was studied by Nakayama [34]
and Ein, Lazarsfeld, Mustata, Popa, and Nakamaye [16], and has the advantage that
B_(D) depends only on the numerical class of D, unlike the stable base locus B(D).
It is also known as the non-nef locus, for B_(D) is empty if and only if D is nef.
It follows easily from the definition that B_ (D) is at most a countable union of
Zariski closed subsets of X. In Chapter 3 we see an example in which this set is a
bona fide countable union, rather than a Zariski closed set. This is done by finding
a pseudoeffective divisor D on the blow-up of P3 at 9 very general points which has
D -C < 0 for infinitely many curves C, with the union of these curves Zariski dense
on X.
Theorem 1.2.2. Let X be the blow-up of P3 at nine very general points. There
exists a pseudoeffective R-divisor D which has negative intersection with an infinite
sequence of curves Cn, which are Zariski dense on X. In particular,B_(D) is not
Zariski closed.
Asymptotic multiplicity invariants
Suppose X is a smooth projective variety and D is an effective, big divisor on X. For
an irreducible subvariety V C X, let vV(D) denote the order of vanishing along V
of a general member of the complete linear series JD . An asymptotic version of this
invariant is -v(D), defined as the limit
rv (D) = M-+0
lim vv(,mDl)
m
12
where m ranges over the integers for which JmDJ is nonempty. The asymptotic
multiplicity is in some respects better behaved than the usual multiplicity: in contrast
with vV, the invariant av depends only on the numerical class of D, and defines a
continuous function on the big cone of X [34][16]. However, the behavior of UV(D)
as D approaches the pseudoeffective boundary is rather subtle. If A is a fixed ample
divisor on X, then -v can be extended to divisor classes that are only pseudoeffective
by setting
V (D)=
lim o-v
D + IA
.
This limit exists, is finite, and is independent of A, but in general the extension of
ov to the pseudoeffective boundary defines only a lower semicontinuous function:
Nakayama has constructed examples in which it indeed fails to be continuous. We
will study the convergence of -v (D + 1/m A) to the limit in cases where D is the
eigenvector of the map q5, : N1 (X) -+ N 1 (X) induced by a pseudoautomorphism of
positive entropy.
While this invariant can also be defined in relative setting of a proper morphism
:
-r X -+ S and divisor classes in N'(X/S), the proof that it takes on a finite value
relies on the properness of X (via an intersection-theoretic argument), and is not
applicable in the relative setting. Nakayama gave several conditions on D under
which av-(D; X/S) must be finite, but we will see an example of a family over a twodimensional base S for which this invariant is infinite. The definition of Uv in the
relative setting is somewhat more involved, and we postpone the precise statements
to Chapter 4.
Theorem 1.2.3. Let ir : X --+ S be the versal deformation space of a singularfiber
of Kodaira type '2, and let C one of the rational curves in the central fiber. Suppose
that D is a divisor on the boundary of the cone Eff(X/S). Then ac (D; X/S) = 00.
If f : W -+ X is the blow-up along C, then O'E(f*D; W/S) = o.
Zariski decompositions
If X is a smooth projective surface, then any pseudoeffective divisor D can be written
as D = P + N, where P is nef and N is effective (see Theorem 5.1.1 for details). This
decomposition has significant implications for the study of section rings of divisors on
X, and a natural and important problem is to study the extension of this definition
to divisors on varieties of higher dimension. There are various definitions of Zariski
decompositions in higher-dimensional settings, all seeking to maintain the important
properties of the two-dimensional version, while still being defined for as many divisors
D as possible [40].
The usual set-up is to ask for a decomposition f*D = P + N, where f : Y -+ X is
a birational morphism, N is an effective divisor, and P enjoys some sort of positivity
property, typically being movable or nef. We might additionally seek to impose more
conditions, for example that the maps H0 (Y, Oy([mP])) -+ H0 (Y, O(y([mf*DJ)) are
surjective for all m, as is the case for the usual Zariski decomposition in dimension
13
2. If we ask simply that f*D = P + N with P nef and N effective, this is termed a
weak Zariski decomposition by Birkar.
We give two constructions to show that weak Zariski decompositions do not exist
for arbitrary pseudoeffective R-divisors in dimension 3, providing negative answers to
questions of Birkar [7] and Nakayama [34].
Theorem 1.2.4. The R-divisor D\ of Theorem 1.2.2 does not admit a weak Zariski
decomposition, and the divisor D of Theorem 1.2.3 does not admit a relative weak
Zariski decomposition over S.
Finiteness of Fourier-Mukai partners
Although this example is not strictly speaking about positivity, the constructions used
are closely related. If X is a smooth projective variety, the bounded derived category
of coherent sheaves on X contains a great deal of information about X, including its
dimension and Kodaira dimension. It was conjectured by Kawamata that in fact the
derived category of a smooth projective variety X determines the isomorphism type
of X, up to finitely many choices.
This conjecture has been confirmed for several classes of X, including surfaces [22],
abelian varieties [36],[17], varieties with either Kx or -Kx ample [10], and toric
varieties [24]. However, based on the same constructions used in Chapter 3, and
using a basic result of Bondal-Orlov, we will see that the conjecture is not true in
general.
Theorem 1.2.5. There is an infinite set W of configurations of 8 points in P such
that if p and q are distinct elements of W, then Db Coh(Blp(P'3 )) 2 Db Coh(Bl(P 3 )),
but Bl(P
3
) and Blq(P
3)
are not isomorphic.
14
Chapter 2
Preliminaries
For simplicity, we work throughout over the field k = C, though the majority of the
results in the following chapters are valid over an uncountable algebraically closed
field of arbitrary characteristic. By a variety, we mean a separated, integral scheme
of finite type over k. Linear equivalence of divisors on a normal variety is denoted by
~, Q-linear equivalence by -Q, and numerical equivalence by
2.1
Cones of positive divisors
Suppose that X is a normal projective variety. There are several notions of "positivity" for divisors, whose basic properties we will make use of throughout. A basic
reference for these facts is Lazarsfeld's introduction [29]. The primary classes of
interest to us, and their defining properties, are described below.
1. Amp(X). A divisor D is ample if H 0 (X, Ox(mD)) determines an embedding
#1mDI : X - PN for some m > 0. Equivalently, by the Nakai-Moishezon
criterion, a divisor D is ample if Ddimv . V > 0 for any subvariety V of X.
2. Nef(X). A divisor D is nef if D - C > 0 for every curve C.
3. Mov(X). A divisor is called strictly movable if its moves in a linear system
whose base locus is of codimension at least 2. The cone Mov(X) is the closure
of the cone spanned by classes of strictly movable divisors.
4. Big(X). A divisor is big if H 0 (X, Ox(mD)) grows like Cm', where n = dim X.
Equivalently, D is big if it can be written in the form D = A + E, where A is
ample and E is effective (the divisors A and E may be Q-divisors).
5. Eff(X). A divisor a pseudoeffective if it is the limit (in numerical equivalence)
of classes of effective divisors.
15
For each of these classes of divisors, we consider the cone its members generate
inside the finite-dimensional vector space N1 (X) = N1 (X)R. There are inclusions
Amp(X)
C Mov(X)
n
C Big(X)
n
Nef(X)
C Mov(X)
n
c
Eff(X)
The cones in the bottom row are all closed, and the cones Amp(X) and Big(X) are
open. Each cone in the bottom row is the closure of the corresponding cone in the top
row. We note that the terminology in the literature concerning movable divisors is
not entirely uniform; we will say that D is movable if its numerical class is contained
in the closed cone Mov(X), and strictly movable if it is an integer divisor with base
locus of codimension at least 2. These cones of divisors can also be defined in the
generality of a proper family 7r : X -4 S; we postpone these definitions until they
become essential in Chapter 4. Below we collect a few miscellaneous but convenient
lemmas.
Proposition 2.1.1 (e.g. [47], Proposition 2.2.2). Each of the cones Nef(X), Mov(X),
and Eff(X) is strongly convex (i.e. does not contain any line through 0).
Proof. In view of the inclusions Nef(X) C Mov(X) C Eff(X), it suffices to prove this
claim for the pseudoeffective cone. We claim first that if D is a pseudoeffective class
and A 1, ... , An- 1 are ample, then D - A 1 ....
An_ 1 > 0. The proof is by induction on
.
dimension; we first consider the case n = 2. It is clear that D -A1 > 0; suppose that
equality holds, so D -A1 = 0. It follows by the Hodge index theorem that D 2 < 0,
with equality possible only if D = 0. On the other hand, D + nA1 is ample for
sufficiently large n, and so D2 = D - (D + nAi) > 0. It follows that D 2 = 0 and so
D = 0. The claim for X of dimension greater than 2 then follows by cutting X by
general members of very ample linear series.
E
Though we postpone the definition of the pseudoeffective cone Eff(X/S) in the
relative setting to Chapter 4, we stress that the Proposition 2.1.1 does not hold in
this greater generality: it is possible that both D and -D are numerically 0 on a
general fiber, and both are pseudoeffective over S.
Proposition 2.1.2. Suppose that D is a Q-divisor in the interiorof Mov(X). Then
some multiple jmD moves in a linear series with base locus of codimension 2.
Proof. If D is in the interior, then D = E ajDj, with the Di strictly movable and
the ai some non-negative rational numbers. Clearing denominators, the result follows
since each of the Di individually has base locus of codimension 2.
El
Lemma 2.1.3. Suppose that
#
: X --
+
Y is a rational map of normal varieties
that extracts no divisor. Then the strict transform [D]
well-defined linear map 0,, : N 1 (X) -+ N'(Y).
16
H-
[0,,(D)] descends to a
Proof. Fix a resolution
Z
/
y
X - - -*0
and suppose that D and D' are two numerically equivalent divisors on X; we need to
show that q.D and 0,D' are numerically equivalent.
Write f*D = g*D+E and f*D' = g*D'+E'. The divisors E and E' are necessarily
g-exceptional. Then g*(D - D') = E - E', but the negativity lemma implies that the
only g-trivial exceptional divisor is 0 (by [6, Lemma 3.6.2] applied to both the left
D
side and its negative). Hence g*(D - D') = 0 and so D D'.
Lemma 2.1.4. Suppose that f : Y -+ X is a birationalmorphism of normal projective
varieties. Then there is a decomposition
Ni(Y) L f*Nl(X) + ® R [Ei],
Ei
where Ej runs over the set of f -exceptional divisors.
Proof. Write D = f*f*D+E, with E exceptional. If the decomposition is not unique,
then f*D + E ajEj = f*D'+ EZ a'Ej for some D, D' and constants ai and a'. But
then EZ(ai - a')Ej = f*(D - D'), and EZ(ai - a')Ej is an f-trivial exceptional class,
l
which must be 0.
Observe that given a birational morphism f : Y -+ X, we have f~f*D = D for
any D in N1 (X).
2.2
Asymptotic invariants of linear series
Given a divisor D, it is often important to study the rational maps #ImDI defined
by large multiples of D. It is possible that the base locus of some multiple jmDj is
smaller than that of ID1, so the rational map #1ImDl has a larger domain of definition.
For example, a divisor might be ample but not basepoint-free, even though its large
multiples must be. In general, maps #1mDI stabilize to a fixed map, the Iitaka fibration
associated to D.
Theorem 2.2.1 (litaka fibration theorem, [29]). Suppose that X is a normal projective variety, and L is a line bundle on X for which some multiple Lo' admits a
nonzero section. Then the maps Om =
#ImLI
X --
+
Y, are all birationally equivalent
to a fixed map 0, : X, -+ Yx, in the sense that for large, divisible m there exists a
diagram
X
X
Ym "-Y
17
The Iitaka fibration associated to D need not be a morphism unless some multiple
of D is basepoint free. The indeterminacy locus of the maps 0m for large divisible
m is called the stable base locus of D. Unfortunately, the Jitaka fibration associated
to a divisor does not depend solely on its numerical class: two divisors in the same
cohomology class can determine different Iitaka fibrations.
Even the stable base locus of a divisor is not a numerical invariant. For example,
if D is a degree 0 non-torsion divisor on an elliptic curve X, then no multiple of D has
any section and the base locus of D is all of X. On the other hand, if D is trivial then
its base locus is empty. For this reason, it is convenient to consider approximations
to the stable base locus which are in fact numerical invariants.
Definition 2.2.2. Let D be a Q-Cartier divisor on a normal, projective variety X.
The stable base locus of D is the Zariski closed set
B(D)
=
nBs(mD).
Suppose now that D is merely a pseudoeffective R-divisor on a normal projective
variety X. We define two notions of base locus for D.
Definition 2.2.3. The augmented base locus of D is the union
B+(D)=
U
B(D - A).
A ample
D - A Q-Cartier
The diminished base locus (also called the non-nef locus or restricted base locus) is
the union
B_ (D)
U
B(D + A),
A ample
D + A Q-Cartier
where B(D + A) =
nn,
Bs(n(D + A)) is the stable base locus [16].
Both B+(D) and B_(D) depend only on the numerical class of D, since the
ampleness of a class A is also a numerical condition. Note that in general we have
B_(D) C B+(D), and if D is a Q-divisor (so that the stable base locus is defined),
we have inclusions B_(D) C B(D) C B+(D).
The set B+(D) is an intersection of closed sets, hence closed. In contrast, B_(D)
is an a priori infinite union of varieties. In many examples it works out to be a finite
union since B(D + A) stabilizes for small A, in which case the diminished base locus
is Zariski closed. In Chapter 3, we will construct example of an R-divisor for which
B_(D) is not Zariski closed.
We will make use of the following properties of the diminished base locus, which
follow easily from the definition (see [16] for details).
Lemma 2.2.4. Suppose that D is a pseudoeffective R-divisor on a normal variety Y.
1. B_(D) depends only on the numerical class [D] E N 1 (Y).
18
2. B_(D) = 0 if and only if D is nef.
3. If C is a curve with DC
< 0, then Cc B_(D).
4. B_ (D + D') C B_ (D) U B_ (D').
5. Iff : Y' -+ Y is a surjective morphism between smooth varieties, B_(f*D)
f~-'(B_- (D)).
=
6. If {Ai} is a sequence of ample divisors converging to 0 in N 1 (Y), with each
D + Ai a Q-divisor, then B_(D) = U3 B(D + A3 ).
7. If D E Mov(X), then every component of B_(D) has codimension at least 2.
A more refined invariant is Nakayama's asymptotic multiplicity, which roughly
speaking captures the multiplicity with which a subvariety appears in B_(D).
Definition 2.2.5. Suppose that X is a smooth projective variety, and that V is an
irreducible subvariety of X. For a big R-divisor D, define
-v(D) = inf {multv(D') : D' > 0, D' = D}.
Now, given a pseudoeffective R-divisor D, set
-v(D) = lim or (D + cA).
c-+O
Lemma 2.2.6 ([34], Lemma 2.1.2). The limit defining -v(D) exists, is finite, and is
independent of the choice of A.
Proof. For any e, (D + cA) - or(D + eA)F is pseudoeffective, and so
((D + eA) - or(D +eA)F) . A"-
1
> 0.
(D -An- 1 )/(F . An- 1 ) is bounded above independently
It follows that -r(D+ cA)
of e. As -r(D+ cA) is evidently a nondecreasing function of e, the existence and
El
finiteness follows. For independence from A, we refer to [34].
Nakayama proved several properties of the asymptotic multiplicity invariants:
Lemma 2.2.7 ([34], Ch. 2). Suppose that V c X is a subvariety. The function
av : Eff(X) -+ R has the following properties:
1. Suppose that E is a pseudoeffective R-divisor. Then orr(D) = limE-o o-r(D+eE).
2. Suppose that V C X is any subvariety. Then civ(D) = E (f*D), where f
Y -+ X is a the normalized blow-up of the ideal sheaf of V, and E C Y is an
f -exceptional divisor dominating V.
3. If 1 ,...,jm, are prime divisors on X, and 0 < si ur (D) are real numbers,
then D - E siP is pseudoeffective, and or% (D - E si]i) = -r,(D) - si.
19
4. There is an equality B-
(D) =
U {V
: uv(D) > 0}.
The main goal of the following chapters is to give examples illustrating a variety
of pathologies of these invariants.
2.3
The Cremona action on divisors and configurations of points
In what follows, we will often consider the variation of positivity properties of a divisor
on a blow-up of P' at a set of closed points, when the configuration of points blown
up is varied. Although we will generally be concerned with configurations of distinct
points, where the more subtle aspects of the configuration space are not involved, we
briefly recall a GIT construction of a compact parameter space for k-tuples of points,
due to Dolgachev and Ortland [14].
Consider the space (pn)k of ordered k-tuples, with j : (pn)k - P' the jth projection. We construct the space k-tuples up to the diagonal action of G = PGL(n + 1)
via geometric invariant theory, closely following the construction of Dolgachev and
Ortland [14]. It is necessary to describe a G-linearization of an ample line bundle on
_>xj*Opn (f), where f is
(pn)k. Equip this variety with the ample line bundle L = (
chosen so that fk is a multiple of n + 1.
We seek to give a linearization of the ample line bundle L with respect to diagonal
action 6: G x (Pn)k -* (1p)k, i.e. an isomophism *L -+ q*L,where q is the projection
of G x (Pn)k onto (Pn)k. We may regard G as an open subset of pf2+n, whose
complement is a hypersurface of degree n +1; consequently OG (n + 1) is trivial. Then
we see that
k
P*(L)
P
(*
k
7; OP. (f)
3
(7ri 0 6) *(Opn(f)
j=1j1
= p*OG(mf) 0 q*L & p*OG(w(n + 1)) 0 q*L
p*OG
P
0 q*L =~q*L.
The machinery of GIT then gives the existence of the quotient X"s//G. Moreover, on
the stable locus X', the quotient is a geometric quotient.
Theorem 2.3.1 ([14], Ch. 1.2, Theorem 1). A configuration (PI,... , Pk) is semistable
if and only if any i points from among the k span a linear space of dimension at least
i(n + 1)/k - 1. The configuration is stable if the inequality is always strict.
In later chapters, we will mostly be interested in the cases n = 2 and n = 3.
When n = 2, the conclusion means that a point is semistable if no FN/3] points
coincide, and no [2N/3] points are collinear. When n = 3, there may be no more
than FN/4] coincident points, no more than [N/2] collinear ones, and no more than
[3N/4] coplanar ones.
20
The standard Cremona transformation Cr : P" --
+
P" is defined in coordinates
by
[XO; ...; X'] - [XV--; ...; X-'].
The standard Cremona map is toric, and it is easy to give an explicit toric resolution.
First blow up all n + 1 coordinate points, then all the lines between 2 of those points,
then all the planes between 3, etc. In the case n = 3, the polytope description is as
illustrated below.
Let X be the blow-up of P' at the standard coordinate points. An important feature of this resolution is that the induced map Cr : X -- + X is a pseudoisomorphism,
i.e. a rational map which neither extracts nor contracts any divisors. This means that
the pushforward induces an isomorphism q$ : N1 (X) -+N1 (X).
Lemma 2.3.2. The action of the standard Cremona transformation on N1 (X), with
respect to the standardbasis, is given by the matrix
/n
n- I n-i
1
...
-1
---
-1
o
n- 1
Mcr=.
Proof. The strict transform of the exceptional class Ej is a plane through the points
other than pi, i.e.
n
*(Ei) = H + E -
Ek.
k=O
Similarly the strict transform of H that
En
n
1
Ej is the exceptional class E0 . It follows
Ek)
(H + E, -
#*(H) = Eo +
j=1
n
n
k=O
21
= nH - Z(n - 1)E,.
j=0
This gives the matrix claimed.
D
Lemma 2.3.3 ([14], Prop. 1, pg. 86). Suppose that p E E is a configuration of
k distinct points in Ip.
There exists a second configuration q such that there is
a pseudoisomorphism Xp -- + Xq, induced by a standard Cremona transformation
centered at the first n + 1 points of p.
Proof. Applying a linear map, we may assume that the first n + 1 points of p are the
standard coordinate points, and then define the points of q to be the images of the
points of p under a standard Cremona transformation.
The map N 1 (Xp)
N 1 (Xq) is given with respect to the standard bases by the
block matrix (1,r
_0',). The action of the Cremona transformation, along with
permutations of the points, generates the action of a certain Coxeter group on the
space of configurations. We next outline the important facts about this action, and
refer to Dolgachev's surveys [14] and [15] for a fuller account.
Define an inner product on N1 (X) by setting
-4
H.H=n-1,
H.Ei=O,
Ei-Ei=-1,
Ei-Ej=O
Take A C N 1 (X) to be the lattice of elements D with D -Kx = 0 with respect to the
above pairing, i.e. divisor classes dH + EZ miE for which (n+=1)d+
= 0. A
root basis for a lattice is a basis ao, . , k for which a = -2 and ai -a > 0 if i # j.
A root basis for the lattice A is given by
n+1
Ceo = H -
Ej,
ozi = Ez - Ei+1 (I < i < k - 1).
Each ai determines a reflection si : N 1 (X) -+ N 1 (X) via x
+ (x - ai)ai. The
Weyl group associated to this root system is the subgroup of Wn,k c O(N'(X))
generated by the reflections si (0 < i < k - 1). These generating reflections satisfy a
number of relations. First, each si satisfies s? = 1. Moreover, if i > 1, then we have
(ssi+1) 3 = 1, since sisi+1 defines a cyclic permutation of coefficients on Ei, Ei+1 , and
Ei+2. If i and j are both greater than 0 but are not adjacent, then si and sj commute
H->
x
and so (ssj) 2 = 1. At last, it is easy to verify that (sosn+1)3 = 1, while so commutes
with each other reflection. All told, this demonstrates that Wn,k is in fact isomorphic
to a Coxeter group with Dynkin diagram of type
a,
a2
an+1
T2,n+1,k-n-1,
ak-2
as illustrated below.
ak-1
ao
A Coxeter group corresponding to a T,q,r Dynkin diagram is infinite if and only
if 1+ +
1, and rcontains elements acting with eigenvalue greater than 1 if the
r
p q
inequality is strict. For T2,fl+1,knl1, this amounts to 2k < (n + 1) (k - n- 1).- In the
22
case n = 2, this corresponds to k = 9 for parabolic and k = 10 for hyperbolic. For
n = 3, the critical numbers of points are respectively k = 8 and k = 9.
Now, given a configuration of distinct points p E En,k, we can obtain a new config-
uration pi(p) by swapping the ith and (i + I)st points. Similarly, as in Lemma 2.3.3 we
can obtain a new configuration po(p) by making a Cremona transformation centered
at the first four points of p. These maps can easily be checked to satisfy the same
relations as the reflections si acting on N 1 (Xp), and so these maps generate an action
of Wn,k on En,k by birational maps.
In general, given an element w E Wn,k, we will denote by pw : En,k -- + En,k the
corresponding birational map on the configuration space, and by M" : N1 (Xp) N1 (Xpp)) the linear map induced on the numerical groups.
These Coxeter groups have often played a role in the study of the birational
geometry of blow-ups of projective space. Nagata's construction of infinitely many
(-1)-curves on blow-ups of P'2 at 9 very general points makes use of the fact that
the group of type T2 ,3 ,6 is infinite [33], while Mukai's characterization of the blowups of pn which are Mori dream spaces again relies on the finiteness of associated
Coxeter groups [32]. Laface and Ugaglia's study of a higher-dimensional analog of
the Harbourne-Hirschowitz conjecture involves sequences of Cremona transformations
centered at various points [27].
2.4
(Pseudo)automorphisms of rational
and Calabi-Yau threefolds
To implement the strategy outlined in the introduction and find pathologies of asymptotic invariants, we need sources of examples of varieties admitting infinite sequences
of flops. The main two sources will be blow-ups of projective spaces at large numbers
of points, and Calabi-Yau threefolds with non-polyhedral effective cones.
We have seen that given a very general configuration p and a Weyl group element
w E Wn,k, there exists a configuration q and a pseudoisomorphism # : Xp -- + Xq
whose action on N'(Xp) -+ N 1 (Xq) is given by the matrix M, with respect to the
standard bases. It is possible that p and q actually coincide up to automorphism;
that is to say, that p is a fixed point of the action of pw on En,k.
In the case n = 2 this means that XP admits an automorphism whose action
on cohomology is given by M., while in the case n = 3 it means that Xp admits a
pseudoautomorphism acting by M,. If this M, has an eigenvalue greater than 1, then
a theorem of Gromov and Yomdin implies that these maps have positive entropy in
the usual topological sense [18][46]. In this case of n = 2 these automorphisms were
investigated by work of Bedford and Kim [4],[5] and McMullen [30]; indeed, divisor
classes considered in [30] provide examples of the sort in Theorem 3.0.1. In the case
n > 3 the existence of such fixed points was more recently studied by D.Q. Zhang
and F. Perroni [38]. I have learned that recent work of T. Bayraktar also considers
the diminished base loci of a general class of R-divisors constructed as eigenvectors
of pseudoautomorphisms; the example presented here is roughly one in which the
23
inclusion in Theorem 1.1 of [2] is an equality, and the union in question is infinite.
Calabi-Yau threefolds provide another class of varieties which can admit pseudoautomorphisms of positive entropy. The celebrated Kawamata-Morrison conjecture
predicts that every Calabi-Yau for which the effective cone is not a rational polyhedral
cone necessarily admits an infinite group of pseudoautomorphisms.
Conjecture 2.4.1. Suppose that X is a smooth projective Calabi-Yau variety.
1. There exists a rational polyhedral subcone 11 C Nef(X) such that
Nef(X) =
U
g".R,
gEAut(X)
and the interiors of the chambers gFJ are all disjoint.
2. The number of chambers Nef(Xi) is finite up to the action of pseudoautomorphisms.
Any birational map between two Calabi-Yau threefolds may be factored as a
sequence of flops and isomorphisms, so in this case all of these pseudoisomorphisms
may be viewed simply as infinite sequences of flops. This conjecture has been proved
by Kawamata for families with three-dimensional total space and non-trivial base [21],
and by Totaro for dimension 2 in the added generality of log pairs [44]. The conjecture
has also been verified for a variety of examples of Calabi-Yau threefolds of dimension
3. The results in this thesis never explicitly rely on the cone conjecture; instead, the
conjecture motivates us to consider Calabi-Yau varieties as a source of examples in
Chapter 4.
24
Chapter 3
Eigenvectors of the Cremona
action
We now turn to the first set of examples of pathologies of asymptotic invariants of
base loci. Both of these take place on blow-ups of projective space, in dimensions 2
and 3, and make use of the notions and notations of Section 2.3.
Theorem 3.0.1. Let E = (1P2 ) 10// PGL(3) \ A, where A is the locus where two points
coincide, and let 7r : X -+ E be the family whose fiber over p E E is isomorphic to
the blow-up of 1p
2
at the corresponding ten points. There exists an R-divisor C\ on X
such that CA,p is nef for very general p, but there are countably many prime divisors
Vn C E such that Cs,, is not nef if p E V.
The behavior of this example is an instance of the following property of nefness.
Proposition ([29], Proposition 1.4.14). Suppose that X and S are varieties over a
field and r : X - S is a surjective and proper morphism. Let D be an R-Cartier
divisor on X. If Do is nef for some 0 E S, then D, is nef for very general s E S (i.e.
for all s not contained in some countable union of subvarieties).
It seems not to be known whether "very general" may be replaced by "general" in
this statement when the divisor D is in fact Cartier; the example demonstrates that,
at least in the generality of R-divisors, the "very general" of the conclusion is indeed
essential.
A similar method in dimension 3 gives an example of a divisor with non-closed
diminished base locus.
Theorem 3.0.2. Let X be the blow-up of V"at 9 very general points. There exists a
pseudoeffective R-divisor D, on X with the following properties:
1. There is a countable set of curves C, C X with D\ - C, < 0, whose union is
Zariski dense on X.
2. B_(D\) is a countable union of curves.
Further, there exists a big R-divisor D' on Px(Ox e Ox(1)) for which B_(D') is a
countable union of curves, where Ox(l) is any very ample line bundle on X.
25
The next section contains some preliminary lemmas needed for the constructions.
Section 3.2 provides the example of Theorem 3.0.1. Section 3.3 recalls the standard
Cremona transformation Cr : P -- + P', leading to the construction of DA. The
various claims of Theorem 3.0.2 are proved in Section 3.3 as Lemmas 3.3.6, and 3.3.9.
3.1
Preliminaries
We first record a simple observation in the spirit of the Perron-Frobenius theorem,
implying that R-divisors arising as eigenvectors of automorphisms of N1 (X) often
generate extremal rays on the various cones of divisors.
Lemma 3.1.1 (cf. [9]). Suppose that V is a finite dimensional real vector space,
G C V is a closed convex cone with nonempty interior and containing no line, and
T: V -+ V is a linear map with T(G) = G. If T has a real eigenvalue A of algebraic
multiplicity one, with magnitude larger than that of any other eigenvalue, then the
A-eigenvector v\ (with appropriatesign) spans an extremal ray on G.
Proof. Fix a norm I- on V and write V = RvA D W, where W is the direct sum of the
other real Jordan blocks, so that Tjw has all eigenvalues with norm strictly less than
A. Since G has nonempty interior, there exists v G G with nonzero component in the
v-eigenspace. Then
converges to some nonzero multiple of vA. Switching the
sign if needed, we conclude that v, is contained in G.
Suppose that v\ is not extremal, i.e. that there exists a nonzero w E W for which
vA + w and v, - w are both in G. Since its image contains an open set, T is invertible
and T-1 (G) = G. There is a sequence ni for which T-"jw/ IT-njW converges to
a nonzero limit r E V. Since Tjw has eigenvalues less than A, jAnT-nwj grows
without bound as n increases, and VA/ jAnT-nwj converges to 0. It follows that the
two sequences of vectors in G
yT'v
AniT--i (vA
k w)
| AniT -niw|
V'\
|ni T -ni|
W
AniT-ni
Ani T-n W|
converge to kr. The closedness of G implies that both r and -r are contained in G,
contradicting the assumption that G contains no line.
E
Let E = ((Pn)k\A)// PGL(n+1) be the set of k-tuples with all points distinct, and
let 7r : X -+ E be the family whose fiber Xp over p = (P,... , Pk) E E is isomorphic
to the blow-up of Pn at the corresponding k points.
The next lemma shows that if p and q are very general, the movable and pseudoeffective cones of Xp and Xq coincide under the identification 1pq.
Lemma 3.1.2. There is a set U C E, the complement of a countable union of subvarieties, such that if p and q lie in U, then @pq(Eff(Xp)) = Eff(Xq), pq(Eff (Xp)) =
Eff(X,), and 4pq(Mov(Xp)) = Mov(Xq).
Proof. For an integral class D = dH- E mjE, the set of all p for which ho(Xp, Dp) >
0 is a closed subset of E by the semicontinuity theorem. It follows that X, and Xq
26
have the same effective integral classes as long as these two points lie off of the
countably many proper closed subsets that arise in this way, and so the effective and
pseudoeffective cones coincide.
For the movable cone, we again restrict our attention to integral classes. An
integral class DP has base locus of codimension 2 if h0 (XP, Dp) > 1 and Dp has no
fixed part, i.e. there does not exist a nonzero effective class Fp such that h0 (Xp, Dp Fp) = h0 (Xp, Dp). The result follows as above by the fact that for any integral D and
Z
F, each of h0 (Xp, Dp - Fp) and ho(Xp, Dp) is constant for p in some open set.
3.2
Nefness in families of R-divisors
In this section, we adopt the notation of Section 3.1 for blow-ups of P2 at k = 10
points. The proof of Theorem 3.0.1 is contained in Lemmas 3.2.1, 3.2.2, and 3.2.3.
We briefly recall the discussion of Section 2.3 in the special case of P 2 . The standard
Cremona transformation Cr : P 2 _ _ p2 given by [XI; X 2 ; X 3] -+ [X- 1 , X2 , X 1 ] has
a resolution
X
X'
4,I
P
2
47r'
_Cr ,_p2
Here 7r is the blow-up of P 2 at three points, and 7r' contracts the strict transforms of
the lines between any two of those points. We employ the two notations X and X'
to emphasize that the standard bases {h, el, e2 , e3 } and {h', e', e', e' } are different. If
C is any curve on X, then its class on X' in the new basis is given by M([C) where
M=
-1
1
1
2
0
-1
-1
(-1
-1
-1
-1
0
-1
1
-1
0
Suppose now that w E W2,1o is an element of the Weyl group W 2,io described in
the introduction, corresponding to some sequence of Cremona transformations and
permutations of the points. If X is the blow-up of P 2 at a general configuration p E E
of 10 points, then we obtain a map Mfq : N1 (Xp) -+ N 1 (Xq), where q = pw(p).
1
If p is in very general position, then the map -1 : N1 (Xq) -+ N (Xp) is an
o
isomorphism which identifies the effective cones, by Lemma 3.1.2. Let M" =
N1 (Xp) -+ N1 (Xp) be the composition. For very general p, since both Ijand Mpq identify the effective cones, we have M,(Eff(Xp)) = Eff(Xp). Because Mw
preserves the intersection form on N1 (Xp), it satisfies Mw(Nef(Xp)) = Nef(Xp) as
well.
If p is a point for which the effective cone of Xp is larger than that of a very
general configuration, it is not necessarily the case that D- maps effective classes to
effective classes. For example, q might have the first three points collinear, even if p
does not. In fact, the divisor of the example will fail to be nef precisely over certain
27
configurations p with the first three points collinear, the first six on a conic, etc.
These are "nodal relations" among the points, in the terminology of McMullen [30].
Lemma 3.2.1. Suppose that the map M, has a unique eigenvalue A > 1 of magnitude
greater than 1. When the A-eigenvector C\, is written as h rEe=(with the e
permuted to be in descending order), the first three coefficients satisfy r 1 +r 2 +r 3 > 1.
The divisor CAp is nef on X, for very general p.
_
Proof. The inequality on the coefficients follows from [14, Ch. 5, Prop. 4(iv)] with
n = 2; we verify it for an explicit element w in Remark 3.2.4. The claimed nefness
follows from Lemma 3.1.1, with the cone G = Nef(Xp) C N1 (Xp) and with M, for
the linear map T.
Let CA be the corresponding divisor h - E 1 rjej on the total space X. Though
CA,p is nef for very general p, we will see that if p lies on any of countably many
subvarieties V, of E for which XP contains (-2)-curves of certain classes, CA,p is not
nef. Define V to be the set of p E E for which pi, P2, and p 3 are collinear. If po E Vo,
there is a curve 1 C Xpo of class Co = h -e 1 -e 2 -e 3 . Then Cpo = 1-rj-r 2 -r 3 < 0,
and CApo is not nef.
Similarly, pi = pw(po) is a configuration of points with the a curve in the class
M, (Co), then strict transform of that conic on XP,1 has negative intersection with
CApi. Generally, for n > 0 define V+,1 C E to be the strict transform of V" under
Pw.
Lemma 3.2.2. Each V, is a prime divisor not equal to L, and V and V" are distinct
if m -A n. For any point Pn E V, there exists a curve Cn C XP in the class M,(Co).
Proof. To prove these sets are distinct, we will construct a sequence of points pn E
V, \ L such that Xp, contains a curve lying in the class Mn(Co), which is the unique
rational curve of self-intersection less than or equal -2. Let E C P2 be a smooth
elliptic curve. Construct po E V by choosing points on E such that pi, P2, and p3 are
the points of intersection of E with some line i meeting E transversely, and p4 , ... , p1o
have the property that if 3d - E'01 mi = 0, the class dflE mipi is not linearly
equivalent to 0 on E unless m 4
=
...
= mio = 0. This condition will be met if these
points are chosen to be very general. Write 1 and B for the strict transforms of f and
E on XpO.
Suppose that C ~ d7r*h - E
miej is a rational curve with Kxp. -C > 0. Since
KxPO ~ -P, we have . - C < 0, and so P - C = 0. Then 3d 1 mi = 0, and the
hypothesis on the points implies that C ~ h - el - e2 - e3 is the curve 1. It follows that
that under any sequence of Cremona transformations, no three points will become
collinear; indeed, the strict transform of a line containing these three points would
be a (-2)-curve on X. 0 , but f is the only such. We may therefore define a sequence
of points Pn+1 E Vn+ 1 by taking pn+ = p(pn). For each n, the image C, of e is the
unique (-2)-curve on Xp, and lies in the class Mn(Co), where CO = h - el - e2 - e3 .
This implies that the divisors Vn are distinct.
A general point Pn E V is of the form p"(po) for some point po C V. The strict
transform on XpO of a line through the first three points of po has class CO, and this
curve has class Mg(CO) on Xp..
D
28
Lemma 3.2.3. If p E Vn, then
0
\,p is not nef.
Proof For any point p E Vn, there is a curve C, C XP with class M,(Co). Then
C.,p -Cn = (
M"([CAP])
M"([Co]) = I [C,p] - [Go] < 0.
0
Remark 3.2.4. For concreteness, here we give an explicit example of such an element
w E W 2 ,10 , and describe the eigenvector in coordinates. Let M" = WCr o W,, where
- E S 10 is the permutation (8, 9,10,1, 2, 3,4, 5, 6, 7). If p E E is a configuration
with p8, p9, and pio in linear general position, there is a Cremona transformation
CrP : P2 _ _,+ p 2 defined by g- 1 o Cr og, where g an element of Aut(P 2 ) sending
p8, P9, pio to the points [1,0, 0], [0, 1,0], [0,0,1], and Cr is the standard Cremona
transformation. Let p : E -- + E be the rational map given by (pi, ... , pio)
(p8,p9, plo, Crp(pi), ... , Crp(p 7 )). This map is regular off of the set L C E, defined as
the locus of p with some pi on a line through two of p8, pg, and pio. Let p be any
point of E \ L, and set q = p(p).
The action of M, on N'(Xp) is given in the standard basis by the 11 x 11 matrix
'-+
1
MPq : N 1 (Xp)-4 N (Xq)
MUq_(M
0
1
0
S0
17
0
110'
where both of these are 11 x 11 block matrices, but with different block sizes.
In the standard coordinates, the eigenvector of the map M, is approximately
~ (1,
-
0.451, -0.440, -0.408, -0.315, -0.307,
- 0.285, -0.220, -0.215, -0.199, -0.154).
This fails to be nef if the first three points are collinear, if the first six points are on a
conic, if there exists a quartic curve with double points at the first three points and
passing through the next six, ... : each of these is a codimension-1 condition on the
configuration space of points.
Remark 3.2.5. Let 7r : X -+ E be the family of blow-ups of P2 at ten distinct
points, and fix a wr-very ample divisor A. This gives an embedding X -+ PN X E. Let
CX c pN+1 x E be the relative cone over X. Now, let Y be obtained by blowing
up CX along the cone points, with p : Y -4 CX the projection. The variety Y
also admits a map q : Y -+ X, a P-bundle. The fiber over p is isomorphic to the
Pl-bundle IP x(Ox, D Ox, (1)). Write Cn,p for the curve in the class C, in the fiber
of the exceptional divisor of q over p.
Let H be an ample divisor on X with support disjoint from the cone point. Take
CA = p*H + q*C. On the fiber over any p, this is a sum of a big divisor and a
pseudoeffective one, and so is big. For very general p, it is a sum of two nef divisors,
and hence nef. However, if p E V, is such that C\,p is not nef, then we have
CA\,p Cn = 0 +CA - Cn < 0,
29
and so C,
is not nef. Hence nefness is not open in families even for big divisors.
Remark 3.2.6. If D is big and nef in dimension 2, then nefness is indeed open in
families; this is proved by Nakayama [34, above Example 3.2.7]. Hence the threedimensional example here is minimal.
Remark 3.2.7. The example in fact also provides an instance of a more specific type
of the failure of nefness to be an open condition in families: it is an example of an
ample R-divisor on a smooth surface for which the Seshadri constant E(X, L, x) jumps
countably many times.
Recall that if X is a smooth projective surface, L is an ample R-divisor on X, and
x is a closed point of X with T : Y -+ X the blow-up at x, then
E(X,L;x) = max {t: 7r*L - tE is nef}= inf L C
Ccx multX C
It follows from the fact that nefness is a very general condition that the Seshadri
constant c(X, L; x) is constant for very general x, but that it might in principle jump
down over countably many subvarieties. The same example shows that this jumping
phenomenon can in fact occur, at least if we allow A to be R-divisor.
Let 7r : X -+ X' be the blowing down of the 1 0 th exceptional divisor, and let
D = irDA be the pushforward of the divisor DA, so that DA = w*D - r10 E 10 . D has
positive intersection with every curve, since DA is strictly nef, while D 2 > 0, So it
follows from the Nakai criterion that D is ample.
Since DA = *D - r10 E 10 has self-intersection 0 and so lies on the boundary of the
nef cone, for a very general 1 0 th point x the constant E(X', D, x) is simply given by
the coefficient r1 o. On the other hand, if x lies on one of the subvarieties V, then this
class is not nef, and so the Seshadri constant must be strictly less than rio. Hence
the Seshadri constant jumps infinitely many times in this example.
Indeed, if x is in V,, the curve C, computes the Seshadri constant in the sense
that (7r*D - aE1o) - C, = 0, where a = E(X', D, x). This gives
0 = (7*D - aE1o) -C, = (C + (rio - a)E1o) - Cn
rio - a = -
1
CA Co
E1 0 - C.
The numerator of the right-hand side is a negative constant, while the denominator
grows exponentially in n. Hence for large n, the Seshadri constant is very close to
but slightly less the generic value, as expected.
Remark 3.2.8. Lue Pan and Junliang Shen have further analyzed the is example
and computed the volumes of the divisors CA,p as a function of the configuration
p [37]. Their calculation shows that bigness is not, as one might naively expect, a
closed condition in families. The divisor CA is very generally on the boundary of the
pseudoeffective cone and hence not big, but in special fibers in which NE(Xp) jumps,
the divisor lands in the interior of NE(Xp) and so has positive volume.
30
3.3
A non-closed diminished base locus
We now turn to the second example, and will employ the notation of Section 3.1 for
blow-ups of P'. Some notation from Section 3.2 will be reused in the new context. The
example of Theorem 3.0.2 will be constructed as an eigenvector of a map N1 (X) -+
N1 (X) induced on a blow-up of p3 by a certain sequence of Cremona transformations.
We next recall the geometry of the Cremona transformation of PS, working out the
details from Chapter 2 in this case. The standard Cremona transformation of lP3
centered at four non-coplanar points Pi,. . . , p 4 is the birational map Cr : P -- P3
defined by
[X 1 , X 2 , X 3, X 4] - [X i X2 i X3 I Xwhere the coordinates are chosen so the points pi lie at the intersections of the coordinate hyperplanes. The map Cr is toric and is easily seen to have a resolution
Y
p
p
P3 - -
-
XP
p3 and 7r' : X' -+ lp3 are the blow-up of P3 at pi,... ,P4, with
exceptional divisors Ei and El respectively. Let F and F' denote the strict transforms
on X and X' of planes through the three points other than pi, and H and H' the
pullbacks of Cp3 (1). Take lij and I to-be the lines in Ip3 through pi and pj, and
lij and I their strict transforms. The lij are smooth rational curves with normal
bundle Opi (-1) E Opi (-1) and Cr is the flop of these six curves. More precisely, p
is the blow-up of X along the six curves ij, with exceptional divisors isomorphic to
PI1 x 1P, and these are contracted along the other ruling by p'. The strict transform
of F under Cr is the exceptional divisor Ej, while the strict transform of E is FL'.
The indeterminacy locus of Cr: X -- + X' is the union of the six curves lIj; since
this map is an isomorphism in codimension 1, taking strict transforms of divisors
induces an isomorphism M : N' (X) -+ N' (X'), as well as an isomorphism M :
N 1 (X) -s N 1 (X') defined by requiring D - C = MD - RC. This action has been
studied by Laface and Ugaglia in connection with special linear systems of divisors
on P 3 [27],[28]. The following computation is just Lemma 2.3.2 in the case n = 3.
Here both 7r : X
-+
Lemma 3.3.1. The isomorphisms M : N 1 (X) -+ N'(X') and M : N 1 (X)
are given in the standard bases by the matrices
3
M =
-2
-2
-2
-2
1
0
-1
-1
1
0
-1
0
-1
1
-1
-1
0
1 1
-1
-1
-1
0f
2
2
2
2
0
-1
-
-1
0
1
-1
-1
0
-1
-1
-1
-
-1
-1
3
1>
-1
-1
-1
=
,
\
31
-+
0/
N 1 (X')
.
If p E E is a set of k > 4 points with the first four not coplanar, the Cremona
transformation Crp : P' _ _+ P3 induces a small birational map Cr : XP -- + Xq,
where q = (PI,..., P4, Crp(p5 ),... , Crp (pk)).
Corollary 3.3.2. Suppose that p is a k-tuple in P3 with no four points coplanar and
consider the map Cr : XP -- + Xg induced by a standard Cremona transformation
centered at the first four points.
1. If D is any divisor on Xp, then
[Crp (D)] -
0
0 k-4
(D]),
where Crp(D) denotes the strict transform of D.
2. If C is any curve on X, which does not meet the curves lI which make up the
indeterminacy locus of Cr, then
[Crp(C)] =
(
(0
Ik0
Ik_4)
([C]),
D
where Crp(C) denotes the strict transform of C.
Proof. The strict transform of Ei is El for i > 4, so the coefficients on these divisors
are unaffected, and (1) is just Lemma 3.3.1. (2) follows from the fact that if C is
disjoint from the indeterminacy locus of Cr, the intersection of C with a divisor is
unchanged under strict transform, and (
04) is the linear map which preserves
the intersection form.
E
We now focus on the case that k = 9 points are blown up. The blow-up of p3
at k < 7 very general points is a Mori dream space, while the blow-up at k > 8 is
not [32]. S. Cacciola et al. give generators of the movable cone for k < 6 [121, while
A. Prendergast-Smith has studied the movable cone for k = 8 non-general points in
connection with the Kawamata-Morrison cone conjecture [39].
If I is a 4-tuple from among the nine points, there is a birational map Crj
3
P __+ p3 defined as a standard Cremona transformation centered at the first four
points of I, inducing a small rational map Cr : Xp -+ Xq. Given a sequence I =
(Ii, . . . , 1,) of 4-tuples from among the nine points, the composition Cr1 = Crj" o ... o
CrI is not defined in general; four of the points might become coplanar under some
Crj, 1 . However, if p = po is in very general position, arbitrary compositions of
Cremona transformations are defined. When the composition is defined, we write
Cr : Xp,_ 1 -- + Xp, for the induced small rational maps of the blow-ups, and
Cr 1 : Xp, -- + Xp, for their composition.
If 1 C X, is the strict transform of a line through pi and P2, the numerical class
of its strict transform under Cr 1 could be computed using Corollary 3.3.2 if it were
known that the strict transform of 1 under CrIk_1 o .. o CrI is disjoint from the
indeterminacy locus of CrI, for every k < n - 1. Laface and Ugaglia have shown that
32
this is indeed the case for very general blow-ups, and for convenience we provide their
proof.
Lemma 3.3.3 ([28], Lemma 2.6). There exists a configuration p such that the unique
quadric Qo through p1, ... ,p9 is smooth, p1 and P2 lie on a ruling f of Qo, and 1 is
the unique rational curve on Qo with (C - C)QO < -2.
Proof. Fix a smooth quadric Qo C IP 3 , and choose pi and P2 on a ruling f. Let BO
be a smooth elliptic curve of type (2, 2) on Qo through pi and P2, and pick p3, ... , P9
ai = 0, then (biri+ b2r2)IBo - E9= aipi is not linearly
such that if 2(bi + b 2 ) - E
ag = 0; this condition is met if these points
aie is a curve
are very general on BO. Suppose now that C ~ bi*r1 + b2 V)/r 2 -
equivalent to 0 on BO unless a 3 =
.
=
_
with (C - C)O :< -2. Since -K,
B30 is effective, it must be that KQO - C = 0,
aipi ~0. The
so -2(b 1 + b2 ) + Ei9 ai = 0 and Clo ~ (b1O3ri + b2 <$r 2 )LBO - E'
choice of points then guarantees that a3 = - = ag = 0, which in turn implies that
C= 0.
(I1,... , I,,) be a finite sequence of
Theorem 3.3.4 ([28], Proposition 2.7). Let I
3
4-tuples, and let f be the line in 1P between pi and P2, with f its strict transform on
X = XP. There exists an open subset U1 C E such that if p is contained in U1 , the
following hold:
1.
The composition Cr1 : p3
-- +
P3 is well-defined.
2. If f is not contained in the indeterminacy locus of Cr1 , then for each 1 < j n,
the strict transform fj_1 C Xp,_, is disjoint from the indeterminacy locus of
Cr ,.
Proof. The set of points in E for which both conditions are satisfied is open, so it
suffices to a exhibit a single configuration of such points. Let p = po be a configuration
satisfying the conclusions of Lemma 3.3.3, so that in particular f is a ruling of Qo.
Let Q1 be the strict transform of Qo under Crj o ... o Cr1 , and Qj its strict transform
on Xpy.
The following will be proved by induction on j, starting with j = 1:
(i) The map Cr1 , : Xpj,
--
+
Xp, is defined.
(ii) !j_1 is disjoint from the indeterminacy locus of Cr13 .
(iii) ij is the unique rational curve on Qj with (C - C)Q, < -2.
For (i), the transformation Cr j is not defined if the four points at which it is centered
are contained in a plane H c P 3 . Then HJQj_, is a rational curve of type (1, 1) through
these four points, the strict transform of which on Q._1 then has self-intersection less
than or equal to -2, which is impossible by induction. Now, since io was contained
in Qo, its strict transform !j_1 is contained in the quadric Qj_1. No two pa and PA
< -2.
K,
can lie on a ruling r of Q_1, for the strict transform would satisfy (f - f)
Thus the line r between pa and PA meets Q transversely, and f does not meet Qj-1
33
or fj_1, as claimed by (ii). As the indeterminacy locus comprises strict transforms of
lines between two points, we see that Cr,
j_1 : Qj-1 -+ Qj is an isomorphism, and
so ej is again the only rational curve of self-intersection less than or equal -2 on Qj,
proving (iii).
E
We now consider compositions of Cremona transformations centered at judiciously
chosen sequences of quadruples from the among nine points. Let w G W3,9 be any
element of whose action M, : NI(Xp) -+ N 1 (Xp) has an eigenvalue greater than 1,
and let DA be the leading eigenvector.
Lemma 3.3.5. The class D, lies in Mov(X) and spans an extremal ray on Eff(X).
Writing DA = H - ZC 1 riEi, we have r1 + r 2 > 1.
Proof. The first claims follow from Lemma 3.1.1 by taking V = N 1 (X) and T = M,
with G = Mov(X) and G = Eff(X) respectively. The claimed inequality on the
coefficients of the dominant eigenvector is a consequence of [14, Ch. 5, Prop. 4(iv)]
with n = 3.
E
Lemma 3.3.6 (= Theorem 3.0.2, (i), (ii)). If p is very general, there is an infinite
set of curves CG C X = X, such that D, - Cn < 0, and B_(D)
is not closed. The
curves C are Zariski dense on X.
Proof. The strategy is to construct curves C in the classes Mn ( [Co]), where Co is
a line through pi and P2. Let a be the image of w under Wn,k -+ Sn, and set
Ij = (o-(1), . . . , o-J(4)). By Theorem 3.3.4, we can find a sequence of configurations
pj, defined for all integers j, with po = p and such that the maps Cr : Xp 1 -- + Xpj
are defined for all j. We may additionally assume that if f C Xp, is a line not
contained in the indeterminacy locus of Crjj+1 , then for all k > 0 the strict transform
of f on Xp,k is disjoint from the indeterminacy locus of Cr+k+lSuppose that f C X,
is the strict transform of a line between pi and p3 . By
Theorem 3.3.4, as long as pi and pj are not among the base points of Cr 1i, the
composition Crn o ... o Cri is well-defined for all n, and the the strict transforms of
1 are disjoint from the indeterminacy loci of the maps Cr,. Taking f to be the line
between p,.(1) and Pf(2) on X,,, we thus obtain a curve C, C X with class M([CO]),
where [Co] = h - el - e2 is the class of a line through the first two points. Note that
Crj_,,,, : Xp, -- + Xpn,,
is centered at p,,-1(1), ... ,P,-n-1(4). Since a-n(l) = a-n-1( 5 )
and n(2) = n-1( 6 ), f is not among the curves in the indeterminacy locus of Cr_-,.
By Lemma 3.3.5, DA satisfies DA - Co = 1 - (ri + r2 ) < 0, and so for every value
of n,
DA - Cn = (An M,"D.). (MN"Co) = An(D - Co) < 0.
By (3) of Lemma 2.2.4, each curve C, is contained in B_(DA).
However, DA is
movable and so B (DA) contains no divisors by (7) of the same lemma. It follows
that B (DA) is a countable union of curves.
We now show that the curves are Zariski dense. Suppose that S c X is any
surface, and let 4 : S -+ X be the inclusion of a resolution of S. Since DA is
movable, DA does not contain S in its base locus, and thus 0*(DA) is pseudoeffective.
34
If C, C S, then a curve 0, C S mapping finitely to C, has (4O*(DA) - 0,)g =
(DA - Cn)x < 0. However, a pseudoeffective R-divisor on a smooth surface can have
negative intersection with only finitely many curves, namely those in the support of
the negative part of its Zariski decomposition. Thus only finitely many of the curves
0
C, are contained in any surface.
Example 3.3.7. As in Remark 3.2.4, we now work this out explicitly for one element of this group. Let a E S9 be the permutation (6, 7, 8, 9,1, 2, 3, 4, 5), and take
o CrI could equivalently be
I = (u-(1), ... , u-(4)). The composition Cr j o ...
realized by repeatedly making a Cremona transformation centered at P6, ... , p9 and
then cyclically permuting the indices so these points become pi,... ,P4.
Let X = XP be the blow-up at a very general configuration p. Define M,
1
N (X) -+ N 1 (X) and I,: N 1 (X) -+ N 1 (X) by
0
M
MM010Mo =
=
where 1, is the permutation matrix for a-. The class of the strict transform of a
divisor D under Cr 1 n o ... o Cr is (M0 ) Mg([D]). Since D is movable and each
Cr1 , is an isomorphism in codimension 1, this strict transform is a movable divisor as
well. This strict transform is a divisor on a different blow-up Xq (as in Section 3.2),
but if p is very general then by Lemma 3.1.2 this defines a movable class on X as
well, and so M,(Mov(X)) = Mov(X). Thus M,.: N1 (X) -+ N1 (X) is a linear map
which preserves the effective and movable cones. Similarly, if C is a curve with strict
transforms disjoint from the indeterminacy loci of each Crk, its strict transform has
class (a ', )~n AJ,([C]) by Corollary 3.3.2.
The linear transformation Ma has characteristic polynomial p(t) = (t + 1)(t 4
1) t q(t + t- 1 ), where q(t) = t4 - 3t 3 + 4t - 1. M, has four real eigenvalues: 1, -1,
A ~~1.800 and 1/A. To three decimal places, DA is given in components by
DA ~ (1, -0.640, -0.634, -0.615, -0.554, -0.355, -0.352, -0.341, -0.307, -0.197).
As required, the first two coefficients satisfy r, + r 2 > 1.
The first few classes [Cn] = 6h - Epie for the transformation M, are given
below.
5 P6 P7 P8 19
A1p2 P3 1A4
n 6
0
0
0
0
0
0
1
3 1 1 1
2 7 3 2 2 2
3 13 4 4 4 4
4 25 8 8 8 7
5 45 14 14 14 13
1
1
3
4
8
1
1
2
4
8
0
1
2
4
8
0
1
2
4
7
0
1
1
3
4
0
1
1
1
1
0
On a given variety X, the set of divisors for which B_ (D) is not closed has measure
0 in N 1 (X); all such classes are necessarily unstable in the sense of [16]. Nevertheless,
one expects that on "sufficiently complicated" varieties there should exist divisors for
which B_ (D) is not closed. The following gives one result in this direction.
35
Corollary 3.3.8. Suppose that Y is a normal threefold. There exists a finite set of
points q1,... , qj on Y such that if r : Y' -+ Y is the blow-up of the qi, there is an
R-divisor D on Y' for which B_(D) is not closed.
Proof. Fix a separable finite map s : Y -+ P3 , and let p be a very general set of 9
points in P 3 , none of which is contained in the branch locus of s. Take the qi to be
the preimages of these 9 points under s, so there is a map s' : Y' -+ XP . If DA is the
divisor of the previous theorem, then s'*DA is a movable divisor, which has negative
intersections with the preimages of each of the curves C,. As above, it follows that
B-(s'*D\) is a countable union of curves.
E
Though the divisor DA is not big, a standard construction gives a big R-divisor on
a smooth 4-fold with non-closed diminished base locus. Fix an embedding X -+ P'
let CX C PN+1 be the projective cone over X, and take p : Y -+ CX the blow-up
at the cone point. The map p is birational with a unique exceptional divisor E _ X;
write iE : X -+ Y for the inclusion. The variety Y has the structure of a P-bundle
q : Y x PX(O @Ox(1)) -> X.
Lemma 3.3.9. There exists a big R-divisor D' on Y with B_(D') a countable union
of curves.
Proof. Let H be an ample divisor on CX with support disjoint from the cone point,
and set D' = p*H + q*D,. Choosing H sufficiently large, we may assume the base
locus of D' is contained in E. Observe that D' is the sum of a big divisor and a
pseudoeffective one, and thus big.
Properties (2), (4), and (5) of Lemma 2.2.4 imply that B_(D1) c B_(p*H) U
B-(q*D\) = B_(q*D\) = q--1 B(D). Furthermore, the choice of H implies that
B-(D') C E, and so B_(D) C q- 1 B_(DA) n E, which is a countable union of
curves. Moreover, each curve Cj = iE(Cj) has C - D' = q(Cj) - DA < 0, and so
Cj c B_(D'). It follows that B-(D') is a countable union of curves, all contained
in E.
D
Remark 3.3.10. Nakayama has suggested that it might be useful to associate to
a divisor D on a threefold the curve class B(D) = EC ac(D)[C], as a numerical
measure of the size of the base locus of D [34, above Corollary 4.1.4]. Moriwaki has
demonstrated that if D is big and movable, then this sum converges. It is straightforward to compute o-c (DA) for the eigenvector DA; we have ac, (DA) = 1, where
[C.] = r2,(Co). Then
B(D,\)
=
Z cJ(DA)[Ci]
=1
w"(CO).
Ci
The terms in this sum converge to the nonzero class CA, and consequently the sum
does not converge.
36
Chapter 4
Multiplicities on CY3s
We will now describe a strategy for computing the asymptotic invariants of eigenvectors of a different sort of pseudoautomorphism, those of Type (2) in Section 1.1.
Here the indeterminacy locus of on : X -- + X stabilizes to a fixed union of curves
for n > 0, rather than increasing without bound. We will see that this setting gives
rise to asymptotic multiplicity invariants that increase very quickly near the pseudoeffective boundary, and in particular encounter an example in which the relative
version or(D; X/S) does not have a finite value when S is not a point. We describe
the approach in some generality, though our interest for now is admittedly limited
to a single example. This is a Calabi-Yau fiber space of relative dimension one,
first pointed out by Reid and subsequently studied by Kawamata [41, Example 6.8],
[21, Example 3.8(2)]. Our hope is that other examples can be found with certain
properties, which would then provide additional useful examples.
Theorem 4.0.11. Let i : X -+ S be the versal deformation space of a singularfiber
of Kodaira type I2, and let C one of the rational curves in the central fiber. Suppose
that D is a divisor on the boundary of the cone E~f(X/S). Then c-c(D; X/S) = oc.
If f : W --+ X is the blow-up along C, then cE(f *D; W/S) = 0o.
This indicates that Nakayama's divisorial Zariski decomposition does not exist
the relative setting without some additional hypotheses: it is not possible to define
N, (f*D; Y/S) as an R-divisor.
4.1
Preliminaries
Chapter 2 introduced Nakayama's asymptotic multiplicy a- (D) of a divisor D along a
subvariety V C X. This invariant can in fact be defined in the more general setting of
a proper family of varieties ir : X -+ S over a base. These invariants were introduced
in both the relative and absolute settings by Nakayama. Suppose that w : X -* S
is a proper morphism of schemes, and that D is a Cartier divisor on X. The space
N1 (X/S) is defined by divisors on X modulo numerical equivalence over S, so that
D and D' lie in the same class if D - C = D' - C for every curve C contracted by r.
A divisor D is said to be
37
o
7r-nef if D -C > 0 for any curve C contracted by 7r,
" r-movable if the support of ir*7rOx(D) -+ Ox(D) has codimension at least 2
in X,
" ir-effective if ir, Ox(D)
" ir-big if rk(fOx(mD))
#
0,
C -m', where r = dim X - dim S.
In the case that S = Spec k is a point, these definitions coincide with those given in
Section 2.1. Write Nef(X/S) and Eff(X/S) for the corresponding cones of divisors
in N'(X/S). The interiors of these cones comprise the 7r-ample and 7r-big divisors
respectively.
Definition 4.1.1. Suppose that r : X -+ S is a proper morphism, with S an affine
variety, and that V is a subvariety of X. For a 7r-big R-divisor D, define
-v(D;X/S) = inf {multv(D') : D' > 0, D' =, D}.
If D is a yr-big divisor class, then D is 7r-numerically equivalent to an effective divisor
on X, since by assumption the base scheme S is affine. Now, given a r-pseudoeffective
divisor D, fix a ir-ample divisor A and set
av(D; X/S) = lim ar (D + eA; X/S).
This is a nondecreasing function of E, and so the limit in question exists, though
it might be infinite. Nakayama gives a number of settings guaranteeing that the limit
defining uv(D; X/S) must be finite:
Lemma 4.1.2. If any of the following conditions holds, then uv(D; X/S) is finite.
1. S = Spec k is a single point.
2. codim7r(V) = 0 or codim7r(V)
=
1.
3. There exists an effective R-divisor D' on X with D =, D'.
4.
The support of D does not dominate S.
The first of these was proved in Lemma 2.2.6. When the total space is not proper,
however, the intersection-theoretic argument of that lemma is not available, and
different methods are required. We refer to Nakayama's book for discussion of cases
(2)-(4) [34].
Lemma 4.1.3. Suppose that D is ir -nef. Then av (D; X/S) = 0.
Proof. By assumption D+A is 7r-ample for any ample A with D+A Q-Cartier. Since
S is affine, m(D + A) is ample on X and we can find a divisor 7r-linearly equivalent
to rn(D + A) and which does not contain V.
E
38
It follows from Lemma 2.2.7(3) that the number of prime divisors E for which
UE(D; X/S) > 0 is finite. If all these divisors have oE(D; X/S) finite, we define the
divisorial Zariski decomposition D = P,(D; X/S) + N,(D; X/S) by N, (D; X/S) =
UE(D; X/S)E and P,(D; X/S) = D - N,(D; X/S), as in the absolute setting.
EE
4.2
Lifting pseudoautomorphisms
Suppose that
#
: X --
+
X is a pseudoautomorphism.
morphism r : Y -+ X is a small lift of
pseudoautomorphism.
#
We say that a birational
if the induced map
Y-
4
: Y --
+
Y is also a
-Y
f
f
X- - X
If f : Y -+ X is a small lift, then the map
of f.
4' must
permute the exceptional divisors
Example 4.2.1. Suppose that # X -- + X is a pseudoautomorphism and x is a
point not contained in indet #. The blow-up 7r : Blx X -+ X is a small lift of # if
and only if x is a fixed point of q. If x is not a fixed point, then the induced map
4: Y -- + Y contracts the exceptional divisor E, while if x is fixed, then "kE is an
automorphism.
The property of r : Y -+ X being a small lift is quite special. The most interesting
cases should be those in which T : Y -+ X is the blow-up of some components of the
indeterminacy locus of /. In the next section, we describe in detail one such example
for a pseudoautomorphism of a Calabi-Yau threefold over a base.
If 7r : Y -+ X is a birational morphism from a projective variety Y with terminal
singularities, it follows from Lemma 2.1.4 that there is a decomposition NI(Y) =
f*Nl(X) @ E, where E = Di R - [Ei]. If D is a divisor class on X, then pullback
via f does not commute with #,, and 0,,. However, the difference 0. f*D - f*q0"D is
an f-exceptional divisor. Define K : N'(X) -+ E by K = f*#, - O'f*. The next
lemma characterizes the action of the pushforward 0,, : Y -- + Y in terms of this
decomposition.
Lemma 4.2.2. Suppose that f : Y -+ X is a small lift of a pseudoautomorphism
$ : X -- + X. With respect to the decomposition f*Nl(X) e E, 0,, is given in block
form as
The eigenvalues of 0* are the union of those of 0* and those of P, which are roots of
unity. Its eigenvectors are
1. f*vi - (Al - P) 1 Kvi, where vi are the eigenvectors of 0, with eigenvalue A;
39
2. Ej, the exceptional divisors of f, with eigenvalues that are roots of unity.
Proof. For a divisor D on X, V,f*D = f*q,5D - KD, while the exceptional divisors
Ej are simply permuted by 7p; this gives the block form of the map. The eigenvectors
follow from elementary linear algebra.
11
The strategy is now to show that if the divisorial Zariski decomposition P,(f*(D))
is known for some divisor D, the decomposition P,(f*($ D)) can also be computed,
using the strict transform under 4 : Y -- + Y.
Lemma 4.2.3. Suppose q5: X -- + Y is a pseudoisomorphism defined over S. Then
N,(O*D;X/S) = #5N,(D; X/S).
qP,(D;X/S) as well.
If N,(D; X/S) is finite, then P,(#*D; X/S)
=
Proof. It clearly suffices to prove the result for N,(D; X/S). Since # neither contracts nor extracts any divisors, for any prime divisor E we have cE(D; X/S) =
l
a4(E)(q*D; X/S), and the claim follows.
Lemma 4.2.4. Suppose that D is a #-negative divisor on X, with N,(f*D; X/S)
finite. Then P,(f*q5D;X/S) = O*P,( f*D;X/S).
Proof. By assumption, KD is an effective exceptional divisor. By [34, Lemma 3.5.1],
if E is an effective exceptional divisor, we have N, (f*D + E) = N,(f*D) + E. This
means that
Na(f*O*D) = N,(4'*f*D + KD) = N,(O* (f*D + KD)) = O*N, (f*D + KD)
= 4'*N,(f*D) + O*KD = O*N,(f*D) + KD.
We have made use of the fact that E is effective by the negativity hypothesis on D.
It is now simple to compute the positive part of the decomposition:
P,(f*#*D) = f*O*D - No,(f*O*D) = f*#*D - 4'*N,(f*D) - KD
= 4',f*D - N,(V*f*D) = Pu(O'f*D) = 4'P,(f*D).
E
The following observation now makes it possible to compute the divisorial Zariski
decomposition of the eigenvector of 0* on Y, in the absolute setting (where the
negative part of the decomposition is guaranteed to be finite).
Corollary 4.2.5. Let Do be the dominant eigenvector of #* : N 1 (X) -+ N 1 (X), and
Dp be the dominant eigenvector of 4', : N 1 (Y) -+ N 1 (Y). Then P,(f*DO) = Do.
Proof. Let D be any pseudoeffective class on X. Then for every n,
P,(f*(-"#nD)) = A-n4P,(f*D).
Take D = Dp + Dp-,
so that the above reduces to
P,(f*(Do + /\ 2nD,-1)) = Af-4,Pe,(f*D).
40
The left hand side converges to P, (f*DO) by the definition of P, at the pseudoeffective
boundary. Up to appropriate choice of scaling, the right hand side converges to
DO.
4.3
Examples
In this section, we make the computation establishing the example claimed in Theorem 4.0.11. Let 7r : X -+ S be the versal deformation space of a fiber of Kodaira type
This example was introduced by Reid [41] and has been further elucidated in work
of Kawamata [21]. Here dim S = 2 and the fiber over 0 E S is two smooth rational
curves intersecting transversely at two points. Write C1 and C2 for the two rational
curves in the central fiber, C for their union, and pi and P2 for the two singularities
in the fiber at which these curves intersect. There is an exact sequence
'2.
0
>-NI/x
- (NC/x) IC1
> TC2,Pi (@TC2,P2
>0
This has the property that a first-order deformation (determined by a section in
HO(C, Nc/x)) smooths pi if and only if it has nonzero nonzero image in Tc2 ,p, [13, pg.
40]. The sheaf in the middle is the trivial Oc E Oc. In one direction pi is smoothed,
and in another P2 is, so the map sends (1, 0) to (1, 0) and (0, 1) to (0, 1). It follows
that the kernel (i.e. the normal bundle in question) is (c, (-1) T Oc (-1), and that
the flop of C1 or C2 is a standard flop.
There is a map 0 : X --
+
X'/S which flops the curve C1, and an isomorphism
: X' -+ X/S which sends the strict transform C2 on X' to the flopping curve C1
on X, and sends the flopped curve C' to C2. We take # = T 0 0 : X -- + X. The
flop @ is resolved by f : Y -> X which blows up the flopping curve C1. The map
Y -+ X' -+ X contracts the exceptional divisor E1 to C' C X', which is then mapped
to C2 by r. We write (a, b) for the class in N'(X/S) that has intersection a with C1
and b with C2; these two curves span N 1 (X/S).
It is first necessary to compute the map #. : N 1 (X/S) -+ N1 (X/S). Suppose
that D is a divisor on X. Write g*#O(D) = f*D + aE1 . Taking C1 a ruling of E1
contracted by g and C a ruling contracted by f, we obtain 0 = D - C, + aE1 - C1,
whence a = D - C1. Then
T
+ aE1 - C+ = -a
=f*D
#(D) -C+ = g*#(D) #(D) -C2 = g*#(D) C2= f*D -C+ aE - C= D - C2+ 2a,
which gives 0* = (-_).
Since every ir-big divisor becomes effective after some sequence of ir-flops, the
pseudoeffective cone Eff(X/S) is the closure of the union of the images of Amp(X/S)
under the map #, which is readily seen to be the half-space D - (C1 + C2) > 0. The
points (1, -1) and (-1, 1) are not effective. The boundaries between the nef cones of
various models are the rays (n + 1, -n) [23].
The construction of a small lift 7r : W -+ X of # is indicated in the following
41
diagram.
B\t
X
-w
-- X+
-X
The map P permutes E, and E 2 . Since g*#,5(D) = f*D + aEj, the map K sends
(a, b) to aE 2 : the multiplicity if g*#5D along E1 is W is equal to multc+ #5D + a,
and 0 interchanges the two exceptional divisors. It follows that with respect to the
basis C1, C2, El, E 2 , we have
2 1 0 0
-1
0 0 0
*0
1
0
0
0 1
1 0
Fix a divisor Hfn on X with H - C = H - C' = 1. Let fHn be the strict transform
of Hn on X under n applications of q. Using the Jordan decomposition of 0', which
has a 3 x 3 block associated to the eigenvalue 1, we compute
n
0
1
2
3
4
n
multc1 H, multc 2 H
,
0
0
1
-1
1
0
1
3
-3
6
-5
3
10
-7
6
1
3
5
7
9
n(n-1)
2
2n + 1 -2n + 1
n(n2+)
2
Let D be the divisor class on the boundary of Eff(X/S) with D - C1 = 1 and
D. C2 = -1. Since C1 and C2 span Ni(X/S), we see that
Hn =,
(2n)D + HO.
42
Nef(X)
.H
1
It follows that Ha/2n = D, + -HO is a sequence of divisors converging to D",
whose multiplicities along the curves is known. We may now write
1
uci (D; X/S) = lim multc1 (D + -HO)
= lim -
n-+oo 2n
2n
n-4 o
1
multC1 Hn = lim
n-+oO
n-I
4
= oc.
Moreover, by Lemma 2.2.7(2), if f : W -+ X is the blow-up along C1, with exceptional
divisor E, we have UE(f*D; W/S) = oo. This completes the proof of Theorem 4.0.11.
We now make an analogous computation in the case that
#
: X -- + X is a
pseudoautomorphism of a threefold of Picard number 2 that has positive entropy,
and f : Y -+ X is a small lift centered on the indeterminacy locus. Unfortunately I
am not aware of any examples satisying the hypothesis, though there does not seem
to be any obvious reason for it to be impossible. If such an example can be found, an
analogous computation will give a counterexample to another question of Nakayama.
Suppose that # : X -- + X is a pseudoautomorphism of a proper threefold with
p(X) = 2. Assume that there is a finite set union of curves V such that indet(") 9 V
for all n, so that p admits a small lift. We may assume now that A = DA + DA\ is an
ample divisor on X; the hypothesis on p(X) implies that by choosing appropriately
scaled eigenvectors this may be assumed to be the case.
Applying
#,
we obtain
#'A =
A'DO + A-'DO-1. We have
P,(f*OnA) = $,"P(f*A) =
Expressing this in the eigenbasis for
,"f*A.
,
f*A = DV + DV-
-
CEE
E
on f*A =
AnDV) + A-"D,-1 -E
CEE,
E
43
where
CE
=
cE(f *D)
N,(f*$,"A)
+
UE(f *Do-,).
Then
f*$"lA - P, (f*$"A)
= Aff*D + A-"f*D-
-
"D,
A-D
-
1 ,-1
+
Z CEE
E
=AflNo,(f *DOp) + A -nlN.,(f *Do-i) + E
E
Multiplying through by A-n yields
Nu(f*Do + A-2"ff*Do-,) = Nu(f*Do) + A-
E CEE + A-2nN(f*DO-1).
E
Take E = A-2n and we have
Na(f*Do + EA) - N,(f*Do) = \/EZCEE + cN(f*D-1).
E
The right-hand side is much larger than E. Taking m = A2n, we have
A
u-c(DA + -) -uc(D)
It follows that c-c(mDA +A) -u-c(mDA) ~
c
.
/d grows without bound. If an X satisfy-
ing the hypothesis exists, it provides a negative answer to a question of Nakayama [34,
Problem 6.1.9]. This unboundedness is impossible in the interior of the big cone, where
the functions ov(D; X/S) are all locally Lipschitz (and not only in ample directions):
Proposition 4.3.1 ([16]). Suppose that V is a fixed irreducible subvariety of X.
Given any big divisor D, there exists a compact neighborhood K C Big(X) of D and
a constant MK such that the functions ov satisfy
l-v(D 1 - D 2)1 < MK |1D, - D 2 1.
44
Chapter 5
Zariski decomposition and the
MMP
5.1
Zariski decomposition of eigenvectors
The Zariski decomposition for divisors plays an essential role in the study of linear
series on surfaces.
Theorem 5.1.1 (Zariski decomposition theorem, e.g. [40]). Suppose D is a pseudoeffective R-divisor on a smooth projective surface X. There exists an effective divisor
N = Zi aiNi such that P = D-N is nef, (Ni -Nj) is negative definite, and P.Ni = 0.
Many properties of a divisor are determined by the properties of its positive part
P, where they may be easier to get a handle on: for example, D is semiample if and
only if P is.
The situation in dimension three or more is considerably more complicated. There
are several analogues of Zariski decompositions for divisors on higher-dimensional varieties, each imposing conditions which ensure the retention of useful properties of the
two-dimensional version. The more conditions are imposed, however, the smaller is
the class of divisors admitting a decomposition. One notion of Zariski decomposition
which always exists and has proved important is the divisorial Zariski decomposition
of a pseudoeffective R-divisor D, developed by Nakayama.
Definition 5.1.2 ([34]). Suppose that D is an R-divisor. Set N,(D) = EEUE(D).E,
and P,(D) = D - N,(D). This is a finite sum, and P,(D) E Mov(X).
In dimension two, this coincides with the standard Zariski decomposition, but in
higher dimensions P,(D) is only movable and not in general nef. Indeed, it is easy to
see that there is no hope to write every divisor as the sum of a rigid effective divisor
and a nef divisor. Suppose, for example, that X is the blow-up of P3 at two points,
and D is the strict transform of a plane through those two points; this has no rigid
component, yet it is negative on the strict transform of a line through the two points.
To obtain a reasonable notion of a Zariski decomposition, it is necessary to allow
ourselves to blow up the base locus of the linear system, and then write the pullback
to the resolution as the sum of a nef and an effective divisor.
45
This gives the simplest definition of a Zariski decomposition in higher dimensions:
given pseudoeffective R-divisor on a smooth variety X, one might ask for a birational
modification f : Y -+ X and a decomposition f*D = P + N, with P is nef and
N effective. This is termed a weak Zariski decomposition by Birkar. However, this
decomposition lacks any analog of the isomorphism on section rings imposed by a
two-dimensional Zariski decomposition. To achieve a closer approximation of the
surface case, we might additionally require that:
1. Cutkosky-Kawamata-Moriwaki: the maps
H (YOy([mPJ)) -+ Ho(Y,Oy(Lmf*D]))
are all isomorphisms.
2. Fujita: if g : Y' -+ Y is birational, and P' < g*f*D is nef, then P' <g*P.
3. Nakayama: P = P,(f*D) is the positive part of the divisorial Zariski decomposition.
Each of these seeks to extend some property of the usual two-dimensional Zariski
decomposition to the higher-dimensional setting. The survey [40] of Prokhorov introduces the important properties of these and other higher-dimensional versions of
the Zariski decomposition. Nakayama constructed an example of an R-divisor on
a P'-bundle over an abelian surface which admits no Zariski decomposition any of
these three senses [34]. However, the divisor of Nakayama's example is itself big, thus
effective, and trivially admits a weak Zariski decomposition. We will show that DA
of Theorem 3.0.2 does not admit a weak Zariski decomposition, and that D' of the
same theorem does not admit a Zariski decomposition in the sense of Nakayama.
Birkar has shown that the existence of even a weak Zariski decomposition has
significant implications for the minimal model program: if (X, B) is a n-dimensional
lc pair such that Kx + B admits a weak Zariski decomposition, then (X, B) has a
terminal model, assuming the log MMP in dimension n - 1 [8]. However, the nonclosedness of B_ (DA) implies that DA admits no Zariski decomposition in several
standard senses. Recall the form of decomposition in dimension two:
Lemma 5.1.3. Any divisor DA as in Theorem 3.0.2 does not admit a weak Zariski
decomposition.
Proof. Suppose that f*DA = P + N where N is effective. For each n, pick a curve
-+ Cn). Only finitely many of
the C, are contained in Supp N, since these curves are Zariski dense by Lemma 3.3.6.
On the other hand, for any curve C not contained in Supp N, we have C - N > 0,
and so compute dn(DA - C,) = f*DA - Cn = P - C + N - C > 0, a contradiction. E
C, on Y mapping finitely to Cn, and let dn = deg(Cn
Similarly, the divisor D = f*D of Theorem 4.0.11 admits no Zariski decompositions over S in a very strong sense.
46
Corollary 5.1.4. Let W/ S be as in Theorem 4.0.11. There does not exist a birational
morphism g : Y -+ W/S and a decomposition g*D = P+ N, with P a (7r o g)-movable
divisor and N effective.
Y/S) = 0 since P
multE N is finite. Then UE(P +
Proof. Suppose that such a decomposition exists. We have
is movable and codim E = 1, and O-E(N; Y/S)
N; Y/S)
OE(N;
UE(P;
Y/S) must be finite, whereas UE(D; Y/S) is infinite.
0
The divisor of Lemma 3.3.9 is an example of a big divisor on a fourfold which does
not admit a Zariski decomposition in the sense of Nakayama. A similar example is
due to Nakayama, though the example in [34] is somewhat different in character in
that its diminished base locus is closed.
Lemma 5.1.5. The divisor D' C Y does not admit a Zariski decomposition in the
sense of Nakayama.
Proof. This follows from the fact that B (D') is not Zariski closed.
El
We observe that the divisors admitting a weak Zariski decomposition constitute
a cone in N1 (X).
Definition 5.1.6. The b-Nef cone is the union
b-Nef(X)=
U
f*(Nef(Y)).
f:Y-+X
birational
Lemma 5.1.7. The set b-Nef(X) is a (not necessarily closed) convex cone, with
int(Mov(X)) C b-Nef(X) C Mov(X). The set of divisors admitting a weak Zariski
decomposition is the cone b-Nef (X) + Eff (X) C Eff (X).
Proof. That b-Nef(X) is a cone follows from the fact that any two models admit a
common resolution. If f : Y -+ X is a resolution of the base ideal of a Cartier divisor
D with no fixed component, and f*D = P + F the decomposition into basepoint-free
and fixed components, then P is nef with f*P = D. Any Q-divisor in the interior
of Mov(X) has a multiple with base locus of codimension 2 [20, Lemma 2.2], so this
gives the first containment. The second follows from the fact that the pushforward
of a nef divisor is movable.
Suppose that D admits a weak Zariski decomposition f*D = P + N. Then
D = fP + N and D E b-Nef(X) + Eff(X). Conversely, suppose D = f*P + N, so
that f*D = P+E+f*N, with E exceptional. Then -E is f-nef, and by the negativity
lemma E is effective, so f*D = P + (E + f*N) is a weak Zariski decomposition. 0
Remark 5.1.8. If b-Nef(X) is closed, then b-Nef(X) = Mov(X), and every class in
Mov(X) is the pushforward of a nef divisor. In this situation every pseudoeffective
divisor admits a weak Zariski decomposition: the divisorial Zariski decomposition
gives D = P,(D) + N,(D) = fP + N,(D). This is the case, for example, when
dim X = 2 or X is a Mori dream space.
On the other hand, the following conditions are sufficient to guarantee that a
divisor D admits no weak Zariski decomposition.
47
1. [D] spans an extremal ray on Eff(X).
2. D is not numerically equivalent to any effective divisor.
3. There does not exist birational f : Y -+ X and a nef R-divisor P on Y such
that f*P = D.
Each of the divisors DA and D is an example of a class satisfying all of these hypotheses.
5.2
Outcomes of the MMP
Suppose that X is a smooth projective variety of dimension n. The cone theorem
states that the Mori cone NE(X) is generated by curve classes of positive intersection
with Kx together with countably many extremal rays [C] of negative intersection
with Kx.
Theorem 5.2.1 ([26]). Let (X, A) be a dit pair with A effective.
1. There exist countably many rational curves Cj C X with 0 < -(Kx + A) -Cj <
2 dim X, such that
NE(X) = NE(X)(Kx+A) 0 + E
oj±-
2. If H is any ample divisor and e > 0,
NE(X) =
NE(X)(Kx+A+H) o
+
ZR
0 o[Cjl.
finite
Without the extra ample divisor of part (2), the number of extremal rays need not
be finite: there can be infinitely many rays, accumulating on the hyperplane Kx. A
simple example is given by the blowup of p2 at nine or more very general points. This
is a smooth rational surface containing infinitely many (-1)-curves, each generating
an extremal class on NE(X) [29]. However, there are several situations in which the
number of extremal rays is indeed finite.
Proposition 5.2.2. The decomposition
NE(X) = NE(X)(KX+A) !o +
S
~o[0Cj]
finite
holds with only a finite sum on the right hand side if (X, A) is a klt pair satisfying
any of the following conditions:
1. dim X = 2 and ,(X, A) > 0.
2. Kx + A or A is big.
48
3. X is smooth, dim X = 3, A = 0, r(X, A) > 0.
Proof. The first follows from the fact that an effective divisor in Jm(Kx + A) can
only have negative intersection with one of its finitely many components.
The second case follows from (2) of the cone theorem: first assume that (X, A) is
klt, with A big. Since A is big, it is numerically equivalent to A + F with A ample
and F effective. Continuity of the coefficients on a log resolution of (X, F) gives
(X, (1 - e)A + eF) klt for sufficiently small e. Let A' = (1 - e)A + eF be the latter
boundary divisor. Then A = A'+ cA, and so (2) of the cone theorem applied to the
pair (X, A') gives the result. If A is not big, then choose an effective R-divisor F in
the numerical class of Kx + A, and replace (X, A) by (X, A + JF), where J > 0 is
chosen sufficiently small so that the latter pair is nef. This new pair has the same
negative rays as the original one, and the first argument shows the number of rays is
finite.
For the third, recall that there are no flipping contractions on smooth threefolds
for deformation-theoretic reasons. Consequently, every extremal ray determines a
divisorial contraction. The only divisors which could possibly be contracted are the
finitely many in the support of B(Kx), and a given divisor can only be contracted in
E
finitely many ways.
The easiest examples of varieties with infinitely many extremal rays, including the
above rational surface, are all of Kodaira dimension -oc. Kawamata, Matsuda and
Matsuki asked whether this might in fact always be the case:
Conjecture 5.2.1 ([25], Problem 4-2-5). Suppose that (X, A) is a klt pair with
r(X, A) > 0. Then the cone NE(X)(Kx+A)<o spanned by only finitely many extremal
rays.
Uehara observed that some analysis of Namikawa provides a negative answer to
this question in the klt setting: there exists a Calabi-Yau threefold X containing
infinitely many flopping curves C and a divisor A with A - C < 0 for any infinite
number of these curves, yielding infinitely many extremal rays for the klt pair (X, eA)
for sufficiently small e [35]. Uehara suggests that the answer might nevertheless be
affirmative in the case that the boundary A is empty [45]. This conjecture would
have some implications for basic questions about the MMP.
Conjecture 5.2.2. Suppose that Y is a terminal projective threefold. Then the
number of minimal models Xi that can be the outcome of the Ky-MMP is finite.
This conjecture includes cases in which Y has infinitely many minimal models.
Proposition 5.2.3. Conjecture 5.2.1 implies Conjecture 5.2.2 in any dimension
where termination of flips holds.
Proof. Let T be the decision tree for the Ky-MMP: that is, let T be a tree with
a node for each variety Zi that can be encountered in the course of a run of the
Ky-MMP, and an edge between two nodes if the nodes are connected by a flip or
divisorial contraction.
49
Conjecture 5.2.1 implies that each node of T has at most finitely many children,
since at each stage of the MMP there are only finitely many choices of extremal
contraction. Moreover, T does not contain any infinite paths, by termination of flips
in dimension 3. Kdnig's lemma then states that a finitely branching tree with no
infinite paths must in fact be finite, and so there are only finitely many possible end
results of the MMP.
E
It seems to be difficult, however, to control the number of possible outcomes of the
MMP without directly bounding the number of extremal rays at each step and proving
the conjecture directly. The following slightly restricted version of Conjecture 5.2.2
seems more amenable to other approaches, and indeed is closely related to questions
about Zariski decomposition.
Conjecture 5.2.3. Suppose that Y is a terminal projective threefold. Then the
number of minimal models Xi that can be the outcome of the Ky-MMP with scaling
by an ample divisor H on Y is finite.
We briefly recall the definition of the (Kx + A)-MMP with scaling by an ample
divisor H. The most general result guaranteeing the existence of an MMP with scaling
is the following lemma of Birkar and Shokurov, which guarantees the existence of the
divisors required to run an MMP with scaling.
Lemma 5.2.4 ([7], Lemma 3.1). Suppose (X, A + C) be a Q-factorial ic pair with A
and C effective, (X, A) dit, and Kx + B + C is nef. Then either Kx + B is itself
nef, or there exists an extremal ray R with (Kx + B) - R < 0, (Kx + B + AC) - R = 0,
and Kx + B + AC nef.
Granting Lemma 5.2.4, the MMP with scaling proceeds as follows. Fix a klt pair
(X, A) and an ample divisor H such that Kx + A + H nef. Set Xo = X, AO = A,
and HO = H. The initial choice of H now dictates a run of the MMP as follows.
1. Given a klt pair (X, An) with Kx, + A, + snHn nef, set
sn+1 = inf {s > 0: Kxn + An + sHn is nef.}.
By Lemma 5.2.4 there exists an extremal ray R C NE(X) such that (Kx + A)R < 0 and (Kx + A + sn+1Hn) - R = 0. Let cn : Xn -* Y be the contraction
associated to R
2. The contraction cn is either a divisorial contraction, a flipping contraction, or
a Mori fiber space. In the last case, the MMP with scaling is finished. If it is
a divisorial contraction, set Xn+1 = Y, An+ 1 = CnAn. Then Kn+1 + An+ 1 +
An+iHn+ is nef and we return to step 1. If c, is a small contraction, let Xn+1
be the flip of cn : Xn -4 Y. In this case, set An+ 1 = CnAn, Hn+1 = Cn*Hn.
The MMP with scaling terminates either with a Mori fiber space, or on a model
Xn with Kxn + An nef. If Kx + A is pseudoeffective, then at the final step Kx + A
and Kx + A + sn_1 Hn are both nef. In particular Hn is nef over the canonical model
of (X, A).
50
Corollary 5.2.5. Suppose that f : Y -- + X is a terminal variety birational to a
Calabi-Yau threefold X, and that H is an ample divisor on Y. The Ky-MMP with
scaling by H terminates at the model Xi on which the movable divisor f"H is nef.
This corollary gives a link between the cone of pushforwards of nef divisors as in
Remark 5.1.8 and the set of possible outcomes of the MMP with scaling. Indeed,
suppose that X is a Calabi-Yau variety with Picard number 2 and infinitely many
minimal models. Let D be a non-effective class on the pseudoeffective boundary. Then
D admits a weak Zariski decomposition if and only if there is some model f : Y -+ X
for which the Ky-MMP with scaling can terminate at any of the infinitely many
minimal models of X whose nef chambers accumulate at D.
It is worth noting that for some Calabi-Yau threefolds, even with infinitely many
models, it is easy to prove that there is no Y from which the MMP can reach infinitely
many models.
Example 5.2.6. Let 7r : S a P be a rational elliptic surface, obtained by blowing
up the nine points in the base locus of a pencil of plane cubics. If S' is another such
surface, then under mild non-degeneracy hypotheses the fiber product X = S xpi S' is
a smooth Calabi-Yau threefold [43]. The two surfaces S and S' each contain infinitely
many (-1)-curves ei and f'. The products fi xp1 f are then smooth rational curves
with normal bundle Opi (-1) e O, (-1). Thus X contains infinitely many such
rational curves, and admits infinitely many flopping contractions.
However it is not possible to simultaneously flop arbitrary finite sets of these
curves: some of the curves intersect and cannot be simultaneously flopped, and some
configurations lead to non-projective outcomes. Namikawa precisely enumerates the
configurations of disjoint rational curves which may be flopped while remaining in
the projective category.
Theorem 5.2.7 (Namikawa, [35]). There is an infinite set T of finite tuples of disjoint rational curves such that if D is any big and nef divisor on X, then the strict
transform of D becomes nef after flopping the curves corresponding to some element
of T.
In fact Namikawa completely describes the set T of possible flopping configurations, but we will not require such detailed information. This example is somewhat
different in character from the example of Theorem 4.0.11, for there is no pseudoeffective divisor D such that there exists an infinite sequence of D-flops; by the theorem,
any minimal model may be obtained from X by flopping a set of disjoint curves of
the form ei xi m3 . It is impossible to find more than 9 disjoint (-1)-curves on S or
S', so the maximum size of such a set of flipping curves is at most 81; any two models
are connected by a sequence of at most 162 flops.
If D is any big divisor on X, then B_ (D) contains at most 162 curves. Since
B_ (D) = Urn B(D + 1A) is an increasing union, the same holds for any pseudoeffective divisor on X.
Corollary 5.2.8. Let X be the Calabi-Yau of Example 5.2.6, and let f : Y -- + X
be any terminal threefold birational to X. Then f, Nef(Y) meets only finitely many
chambers of Mov(X).
51
Proof. Resolving the indeterminacies of f, we may assume that f is a morphism.
Suppose that H is ample on Y. Then fH can only be negative on the finitely many
curves in f(ex f), and the only models for where f, Nef(Y) hits Amp(Xi) are those
obtained by flopping subsets of f(ex f). By Theorem 5.2.7, there are only finitely
many such. In particular, running a Ky-MMP with scaling by an ample divisor can
only yield finitely many of the minimal models, and Conjecture 5.2.3 has a positive
answer.
We close with one more case in which the original Conjecture 5.2.1 can be proved,
namely when X is the (possibly singular, but terminal) blow-up of a smooth CalabiYau threefold along an ideal sheaf. This condition is somewhat awkward to deal with
insofar as it is not preserved by the operations of the MMP, but it at least rules out
the most obvious approach to the construction of a counterexample. Further, since
it remains possible that Conjecture 5.2.1 is false in general, it seems interesting to
identify classes of varieties for which it actually holds.
Lemma 5.2.9. Suppose that (X, A) is a kit pair with k(X, A) ;> 0. Fix an effective
Q-divisor D = E= 1a2 Di numerically equivalent to Kx + A, and let {Cj} be the
(countable) set of rational curves which are minimal generatorsfor the (Kx + A)negative extremal rays of NE(X) (in the sense that a[Cj1 G N1 (X) is not represented
by a curve for any a < 1). There exists a constant N = N(X, A, D) such that
-N < Di -C, < N for every i and j.
Proof. Let n = dim X, b = mini<i<e{ai} and c = E=1 ai. Fix e > 0 such that
(X, A + cDi) is klt for 1 < i < m. Let C be a minimal generator of any (Kx + A)negative extremal ray. By the theorem on lengths of extremal rays,
-C < 2dimX.
0 < -(Kx+A)
Suppose that Di is any component of D such that Di -C < 0; as Kx + A = D, there
exists at least one such. Apply the cone theorem to the klt pair (X, A + EDi): the
ray [C] is (Kx + A + cDi)-negative and extremal on NE(X), so C is still among the
curves in the decomposition of the cone theorem, whence
0 < -(Kx + A + cDi) -C < 2 dimX.
Taking the differences of these inequalities, we obtain -2n/e < Di - C < 0, a bound
independent of both Di and C. A lower bound on the negative intersections Di -C now
yields an upper bound on the nonnegative ones since C is (Kx+A)-negative. Without
loss of generality, take i = 1. For any extremal C, the assumption (Kx + A) - C < 0
gives (aiD1 - C) + E=k
2 (akDk - ) < 0, whence
0 < D1 -C<
a,D CC akD - C<a 1 k=2
Thus the claim of the lemma holds with N = 2cn/be.
52
2n
( -,
2cn
<b -.
L
Corollary 5.2.10. Let M be a smooth Calabi-Yau threefold and f : X - M a birational morphism from a terminal threefold X. The number of Kx-negative extremal
rays is finite.
Proof. Let {Ei}iie be the exceptional divisors. Since X is terminal, Kx = |E aiE ,
with every Ej appearing with a strictly positive coefficient ai > 0. Recall that
R - [Ei] by Lemma 2.1.4. The image f(ex f) consists
N'(X) = f*Nl(M) E
of a finite set of curves f(Ei) C M.
Suppose that C is a minimal generator of a Kx-negative extremal ray. Since Kx
is exceptional, C is contained in one of the finitely many Ej, which for simplicity
may be assumed to be E1 . For any divisor D, we have f*D - C = 6(D - f(E 1 )), with
6 = deg(f~c).
Applying the first lemma to X (with A = 0), we see that C- Ej E -Zn [-N, N] for
every i, a finite set of possibilities. The values f*D -C depend on the single parameter
6. Since Ej and f*D generate N 1 (X), this data forces the numerical class of C to
lie in one of finitely many 1-dimensional affine subspaces of N1 (X). However, there
are at most two curves C spanning extremal rays lying on a given affine subspace of
NE(X), which can intersect only two extremal rays. It follows that there are only
El
finitely many possibilities for the numerical class of the extremal ray C.
53
54
Chapter 6
Finiteness of Fourier-Mukai
partners
For a smooth projective variety X, let D(X) = Db Coh(X) denote the bounded
derived category of coherent sheaves on X. It is possible that D(X) and D(Y) are
equivalent (as triangulated categories) even if X and Y are not isomorphic. In this
case, X and Y are said to be Fourier-Mukai partners. The first examples of this
phenomenon are due to Mukai, who showed that an abelian variety and its dual have
equivalent derived categories.
The derived category nevertheless contains a great deal of information about the
variety X: if Y is another variety whose derived category is equivalent to that of X,
then X and Y have the same dimension and Kodaira dimension, and, if X is of general
type, they are are birational [22]. Extending a result of Bridgeland and Maciocia [11],
Kawamata proved that if X is a smooth projective surface, there are only finitely
many other smooth projective surfaces Y (up to isomorphism), with D(X) and D(Y)
equivalent as triangulated categories. He asked whether this property might hold in
all dimensions [22, Conjecture 1.5].
We observe here that there are threefolds for which this is not the case. Let p
denote an ordered 8-tuple of distinct points in P 3 , and let X, be the blow-up of P3
at the points of p.
Theorem 6.0.11. There is an infinite set W of configurations of 8 points in P3 such
that if p and q are distinct elements of W, then D(Xp) e D(Xq) but Xp and Xq are
not isomorphic.
The example boils down to three basic observations, made precise in Lemmas 1, 2,
and 3.
1. XP and Xq are isomorphic if and only if p and q coincide, up to permutation
and an automorphism of P3 .
2. If q can be obtained from p by a sequence of standard Cremona transformations
centered at 4-tuples from among the points of p, then X, and Xq are connected
by a sequence of flops of rational curves with normal bundle O(-1) e 0(-1),
and so D(Xp) e D(Xq).
55
3. The orbit of a sufficiently general configuration p of 8 points under standard
Cremona transformations is infinite.
There are several classes of higher-dimensional varieties for which D(X) has been
shown to determine the isomorphism class of X, up to finitely many possibilities.
These include abelian varieties over C [36],[17], toric varieties [24], varieties with Kx
ample, and Fano varieties [10]. It also known that the number of isomorphism classes
of varieties with a given derived category is at most countable [1]. Further discussion
of this problem can be found in [17] and [42].
The example is based on the action of Cremona transformations on configurations
of points in p3, as investigated by A. Coble. Most of the results we will need can
be found in Dolgachev and Ortland's account of Coble's work [14]. We provide selfcontained proofs, with references to the more general theory.
It is worth noting that these examples do not pose any problems for the KawamataMorrison cone conjecture for klt Calabi-Yau pairs [44],[21]. Through 8 general points
in P3 there is a pencil of quadrics, with base locus a degree 4 curve through the
points. If A is the sum of two generic quadrics in this pencil, then (X, A) is a dlt
pair with Kx + A numerically trivial. However, there is no choice of A for which
Kx + A is numerically trivial and the pair is klt, for the divisor obtained by blowing
up the curve in the base locus has log discrepancy 0. If the points are specialized
to the base locus of a two-dimensional net of quadrics, the pair (X, A) can be made
klt, but the configuration is insufficiently general for our construction, and indeed
Prendergast-Smith has demonstrated that the cone conjecture holds for this class of
examples [39]. Note also that although the blow-up of P3 at 7 very general points is
a weak Fano variety, the blow-up at 8 is not.
For the rest of this section, p denotes an ordered 8-tuple of distinct points in P3 ,
with r, : XP -+ P3 the blow-up of the points of p. Write Ei for the exceptional
divisors of 7r, and H for the pullback to XP of the hyperplane class on Ip 3 .
Lemma 6.0.12 ([14], Ch. V.1). The blow-ups Xp and Xq are isomorphic if and only
if p and q coincide, up to an automorphism of Ip 3 and permutation of the points.
Proof. Suppose that q : Xp
Xp \
U Ei
-
Xq -+
p3
*
Xq is an isomorphism. The restriction p3
defines a rational map
: p3 _F_+
p 3,
\
p
with indeterminacy
locus contained in p and hence 0-dimensional. Its inverse is likewise regular outside
a 0-dimensional set, contained in q. But any rational map ) : p3 __+_p3 for which
4 and 4 both have 0-dimensional indeterminacy sets is in fact an automorphism
(e.g. by Theorem 1.1 of [3]). Now, 4 o ?, contracts Ei to a point, and so 7rq must
contract q(Ei) to a point. Consequently V)(p) = q, so the configurations differ by a
D
permutation and automorphism, and q identifies the exceptional divisors.
The basic ingredient in constructing other blow-ups which are derived-equivalent
to a given one is the action of the standard Cremona transformation Cr : p 3 _ _ I 3 ,
defined by [Xo : X, : X 2 : X 3] 4 [X- 1 X - X2 1 : Xg1 ]. A resolution of this
56
rational map is as follows.
Y
p
P 3 _ _ _ Cr
P3
Here 7r blows up the four standard coordinate points. The strict transforms on X of
the six lines f between two of these points are smooth rational curves with normal
bundle 0(-1) E 0(-1). These are flopped by Cr; p blows up these curves to divisors
isomorphic to P' x P, which are then contracted along the other ruling by p'. The
indeterminacy locus of Cr : X -- + X' is the union of these curves. The map 7r' then
blows down the strict transforms of the four planes through three of the four original
points.
Suppose that p is a configuration of 8 points (regarded now as a point on the configuration space (P3 )8 // PGL(4)), and four points are chosen from among p satisfying
the following condition:
(*) No other point of p lies on any plane defined by three of the four chosen points.
Condition (*) implies that the Cremona transformation centered at the four given
points is defined, as these are not coplanar. It also guarantees that no point of p is
on one of the contracted divisors or one of the lines in the indeterminacy locus. We
can define a new configuration q by making a Cremona transformation centered at the
four chosen points, and moving the remaining four points under that transformation:
if pj is one of the chosen points, then qj is defined as the image of the plane through
the other three points (which is contracted by 7r'), while if pj is not a chosen point,
then qj is just the image of pj under the Cremona transformation. The configuration
q is defined up to choice of coordinates, and no two points of q are infinitely near,
since no point of p is on one of the contracted divisors. After blowing up the points
of p and q, there is a rational map Cr: XP -- + Xq which flops six rational curves of
normal bundle 0(-1) E 0(-1).
We will say that p and q are Cremona equivalent if there exists a sequence of
Cremona transformations centered at 4-tuples from among the points which sends
the configuration p to q, and satisfies condition (*) at each step. The Cremona orbit
of p is the set of all configurations that are Cremona equivalent to p. If p is a
very general configuration of points, then any sequence of Cremona transformations
will automatically satisfy (*); however, we prefer not to make any blanket generality
assumption on p at this stage, as it will be useful to consider 8-tuples in slightly
special configurations.
Lemma 6.0.13. If p and q are Cremona equivalent, then D(Xp) 2- D(Xq).
Proof. Assumption (*) on the Cremona transformations implies that there is a sequence of rational maps X, = XpO -- + Xpl -- - - - -- + Xp, = Xq, where each
57
Xpj -- + Xpi flops six curves. The lemma is then a consequence of a fundamental result of Bondal and Orlov [10, Theorem 4.3]: if X and X+ are threefolds, and
# : X -- + X+ is the flop of a rational curve with normal bundle O(-1) G 0(-1),
then D(X) -= D(X+).
E
There is a subtlety here in that each of these rational maps flops six disjoint curves,
while the theorem of Bondal and Orlov is usually stated for the flop of a single curve.
Each map can be factored into a sequence of six disjoint flops, but the intermediate
varieties encountered are no longer projective. However, the needed result is valid
without assuming X and X+ are projective [19, Remark 11.24ii].
Lemma 6.0.14 (cf. [14], Ch. VI). A very general configuration p of 8 points has
infinite Cremona orbit.
Proof. We will define a sequence of configurations in which pn+ is obtained from
p, by a Cremona transformation centered at four points, satisfying condition (*),
and such that the p, are all distinct. Let C = Co be a smooth genus 1 curve in
P3 obtained as the complete intersection of two smooth quadrics. Choose P5, P6, P7,
and ps to be the four points of intersection of some generic hyperplane H with C,
and then choose pi, P2, P3, and p4 to be very general points of C; this guarantees
that if 4d - E8 mi = 0, the class dHIc = mipi c Pico(C) is nonzero unless
2.4]).
= M4= 0 (cf. [28, Lemma
=
Let C denote the strict transform of C on XpO, and say that a prime divisor
D ~ dH - E8= miEi on Xp, is a root divisor if it does not contain C and satisfies
4d - E8 Mi = 0. If D is a root divisor, then D - C = 0 and so D and C are disjoint.
This implies that dHIc i=1 mipi G Pic0 (C) is trivial, and so m, =
= rM..4 = 0.
If M5 , M 6 , M7 , and M 8 are not all equal, then some mi is greater than d, and the
corresponding class cannot be effective. Hence the only root divisor on XPO is the strict
transform of the plane through the last four points, with numerical class H - EjL E .
We now inductively define pn+ by making Cremona transformation centered at
the first four points of pa, and then cyclically permuting the points so that pi comes
last. At each step, we show that the first four points of pn satisfy condition (*).
These transformations induce rational maps Xp, -- + Xpn, 1 ; let C, denote the strict
transform of C on XPn, and C, its image in P3 . Since Co is the intersection of two
quadrics, and Cremona transformations preserve quadrics through the 8 points, each
Cn is also an intersection of two quadrics. In particular, no 0" can be contained in
the indeterminacy locus of Xpn -- + XPn±1.
Suppose that p is a configuration of points and that q is the configuration obtained
by making a standard Cremona transformation centered at the first four points of p.
If D is a divisor in the class dH - E8= miEi on X, then its strict transform on X,
has class d'H' - E8=1 miEj, where d' = 3d - Ei= mi, m' = 2d + mi - E_= mi for
1 < i < 4, and m' = mi for 5 < i < 8 [28]. Write M : NI(Xp) -+ N 1 (Xq) for the
corresponding linear map.
Let M, = PM, where M is as above and P is the permutation matrix which
permutes the exceptional divisors by moving the first one to last. If D is a divisor
58
on Xpi, its strict transform on Xp, has class M,(D). The map M, : N1 (Xp0 ) -+
N1 (Xp,) preserves the effective cones, as well as the property that 4d m = 0.
_ mjEj is a root divisor on Xp, (i.e. prime, not
Suppose that D ~ dH containing C, and with 4d - Z _1 mi = 0). Then the strict transform of D on Xp0 is
a root divisor as well. Since there is a unique such divisor on Xp0 , we conclude that D
has numerical class Mn (H - E_ E2 ) on Xp.. It is straightforward to check that the
classes M"(H i=5 Ei) are all distinct; the argument is indicated in Lemma 6.0.15,
which is postponed until the end of this section.
It follows that condition (*) holds for the configuration pn: if any four points pjl,
4
... , pj 4 of pn were coplanar, then H - E __ Ej would be a root divisor on Xp", which
is possible only if n = 0 and the points in question are P5, P6, P7, and ps. In particular,
the Cremona transformation defining pn+i is well-defined for every n, and this gives
an infinite sequence of configurations connected by Cremona transformations, all
satisfying (*). Since the degrees of the classes M,(H - E8 Ej) grow unboundedly,
there are infinitely many distinct configurations among the pn, even up to permutation
of the points. As the Cremona orbit is infinite for the special configuration po, it is
E
also infinite for very general configurations.
Proof of Theorem. Let p be a very general 8-tuple of points in IP 3 , and let W be
the Cremona orbit of p. By Lemma 6.0.14, W contains infinitely many distinct
configurations of points, even up to permutations. Lemma 6.0.13 then shows that the
blow-ups Xq for q E W are all derived-equivalent, but by Lemma 6.0.12 no two are
isomorphic.
Lemma 6.0.15. For any distinct m and n, the classes M"(H - E8= Ej) and
Mn (H - E8_
E ) are distinct.
Proof. Explicitly, the matrix M, is given with respect to the basis H, Ej as
3
-2
-2
-2
1
1
1
0
-1
-1
-1
-1
-1
-1
1
-1
0 -1
0
-1
0 0 0
0 0
0
0
0 0
M01=
0
0
0 0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1
0 0 0
0
1
0
0
0
1
0
0
0
0 0 0 0 0 0 0
-2
0
-1
-1
-1
0
0
0
1
0
Let M, = SJS 1 be the Jordan decomposition. A computation shows that J has a
3 x 3 Jordan block associated to the eigenvalue 1. One can compute the coefficients
of H - E1_8 Ej in the Jordan basis as the entries of S- 1 (H - E8_= El), and observe
H - E_, EI has nonzero coefficients for two of the generalized eigenvectors in the
nontrivial Jordan block. It follows that the powers M"(H - Ei=
5 Ej) are all distinct.
59
60
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