ARCHMVES
MASSACHU-jETT INSTITUTE
OF IECHNOLOLGY
Log geometry and extremal contractions
JUN 3 0 2015
by
LIBRARIES
Roberto Svaldi
Laurea Specialistica, Universitd degli Studi Roma Tre (2010)
Laurea Triennale, Universitd degli Studi di Pavia (2008)
Submitted to the Department of Mathematics
in partial fulfillment of the requirements for the degree of
Doctor of Philosophy in Mathematics
at the
MASSACHUSETTS INSTITUTE OF TECHNOLOGY
June 2015
@ Roberto Svaldi, MMXV. 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.
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Author....
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Department of Mathematics
March 27, 2015
Signature
redacted
Certified by ........................................................................
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James McKernan, FRS
Professor of Mathematics
University of California, San Diego
Thesis Supervisor
Z-7?
Accepted by
Signature redacted
Chairman,
....................
William P. Minicozzi II
ent Committee on Graduate Theses
2
Log geometry and extremal contractions
by
Roberto Svaldi
Submitted to the Department of Mathematics
on March 27, 2015, in partial fulfillment of the
requirements for the degree of
Doctor of Philosophy in Mathematics
Abstract
The Minimal Model Program (in short, MMP) aims at classifying projective algebraic varieties from a birational point of view. That means that starting from a projective algebraic
variety X, it is allowed to change the variety under scrutiny as long as its field of rational
functions remains the same.
In this thesis we study two problems that are inspired by the techniques developed in
the last 30 years by various mathematicians in an attempt to realize the Minimal Model
Program for varieties of any dimension.
In the first part of the thesis, we prove a result about the existence and distribution
of rational curves in projective algebraic varieties. We consider projective log canonical
pairs (X, A) where the locus Nklt(X, A) of maximal singularities of the pair (X, A) is nonempty. We show that if Kx + A is not nef then there exists an algebraic curves C, whose
normalization is isomorphic to A 1 , contained either in X \ Nklt(X, A) or in certain locally
closed varieties that stratify Nklt(X, A). This result implies a strengthening of the Cone
Theorem for log canonical pairs.
In the second part, we study certain varieties that naturally arise as possible outcomes
of the classification algorithm proposed by the MMP. These are called Mori fibre spaces. A
Mori fibre space is a variety X with log canonical singularities together with a morphism
f : X -+ Y such that the general fiber of f is a positive dimensional Fano variety and the
monodromy of f is as large as possible. We show that being the general fiber of a Mori fiber
space is a very restrictive condition for Fano varieties with terminal Q-factorial singularities.
More specifically, we obtain two criteria (one sufficient and one necessary) for a Q-factorial
Fano variety with terminal singularities to be realized as a fiber of a Mori fiber space. We
apply our criteria to figure out what Fano varieties satisfy this property up to dimension
three and to study the case of certain homogeneous spaces. The smooth toric case is studied
and an interesting connection with K-semistability is also investigated.
Thesis Supervisor: James McKernan, FRS
Title: Professor of Mathematics
University of California, San Diego
3
4
Acknowledgments
First and foremost, I wish to thank my advisor, James McKernan. He was a constant source
of support, encouragement, guidance and dialogue. I wish to thank him in particular for
teaching me so much about mathematics and algebraic geometry and for being so generous
with his ideas and intuitions. My gratitude goes beyond what words can express.
I wish to thank Steven Kleiman and Bjorn Poonen for agreeing to be part of my thesis
committee and providing several useful suggestions that improved this work.
During different stages of the development of this thesis I was hosted by the Department
of Mathematics of University of California San Diego and by the Department of Mathematics
of Princeton University. I wish to thank both institutions for their kind hospitality and for
providing a very enjoyable working environment. In particular, I wish to thank Jinos Kollar,
my host at Princeton, for the valuable conversations that I had with him.
I also greatly benefitted from suggestions and discussions with Morgan Brown, Steven
Lu, John Ottem, Zsolt Patakfalvi, Jorge Vitorio Pereira and Chenyang Xu.
Part of this thesis originated from a project that was started during the summer school
PRAGMATIC 2013 in Catania. I wish to thank the organisers Alfio Ragusa, Francesco Russo
and Giuseppe ZappalA for their kind hospitality, the stimulating research environment and
the financial support during the school. The problem was proposed by Paolo Cascini and
Yoshinori Gongyo. I wish to thank them for their help and support. I am also grateful to
Yujiro Kawamata for his lectures in Catania and his encouragement.
Part of this work was done while the author was supported by NSF DMS #0701101 and
I am especially grateful to the Department of Mathematics at MIT for their
#1200656.
generous support during these years.
Last but not least, come all my friends and colleagues from all these years I spent in
graduate school.
The first thought goes to my mathematical family at MIT: Shashank Dwivedi, John
Lesieutre, Tiankai Liu, Jennifer Park. They have been my safety net throughout these years
inside and outside the office. Thanks to Gabriele di Cerbo for pushing me and sharing with
me many ideas and many attempts at understanding algebraic geometry. And thanks to
Federico Buonerba for endless discussions and mathematical challenges all the way through
America. Thanks also to Calum Spicer for welcoming me in San Diego, for making me feel
at home and now even for teaching me foliations. A special word of thanks goes to my
friends and collaborators Giulio Codogni, Andrea Fanelli and Luca Tasin who put up with
me for the last two years.
During these five years that I spent in graduate school, I have never been alone. Throughout my many lives and my many peregrinations I have had plenty of people on both sides of
the Atlantic Ocean that helped me navigate in this adventure. There is not enough space
to thank you all, but I wish to let you know that you have not been forgotten.
The deepest thank of all goes to my family for making all of this possible since the very
beginning.
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Contents
9
1 Introduction
1.1
1.2
2
3
4
Summary of the results. . . . . . . . . .
1.1.1 Cone Theorem and hyperbolicity
1.1.2 The classification of fibres of Mori
Structure of the thesis . . . . . . . . . .
Notations and preliminaries
2.1 Notation and Conventions . . . . .
2.2 C ones . . . . . . . . . . . . . . . .
2.3 Pairs and their singularities . . . .
2.3.1 The non-klt locus, Ic centers
2.4 The Minimal Model Program . . .
2.4.1 The algorithm of the MMP
2.4.2 MMP with scaling . . . . .
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fibre
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47
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Fano varieties in Mori fibre spaces
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . .
4.2 Preliminary results . . . . . . . . . . . . . . . . . . . . . .
4.2.1 Monodromy Action and Deligne's Theorem . . . .
4.2.2 The monodromy action and the MMP . . . . . . .
4.3 Surfaces, threefolds and other higher dimensional examples
4.3.1 Del Pezzo surfaces . . . . . . . . . . . . . . . . . .
4.3.2 A general procedure to construct Mori fibre spaces
4.3.3 Fano Threefolds . . . . . . . . . . . . . . . . . . . .
4.4 Criteria for Fibre-likeness . . . . . . . . . . . . . . . . . .
4.4.1 General Criteria . . . . . . . . . . . . . . . . . . .
4.4.2 Applications of the Necessary Criterion . . . . . .
4.5 K-stability in the smooth toric case . . . . . . . . . . . . .
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and their stratification
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Hyperbolicity and the minimal model program
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . .
3.2 Dlt modifications . . . . . . . . . . . . . . . . . . . . .
3.3 Subadjunction for higher codimensional lc centers . . .
3.3.1 Canonical bundle formula . . . . . . . . . . . .
3.4 Mori hyperbolicity . . . . . . . . . . . . . . . . . . . .
3.5 Proof of theorem 3.1.3 . . . . . . . . . . . . . . . . . .
3.6 Ampleness and pseudoeffectiveness for Mori hyperbolic
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4.5.1 Preliminaries on toric geometry: primitive
4.5.2 Fibre-likeness implies K-stability . . . . .
4.5.3 MAGMA computations . . . . . . . . . .
Homogeneous spaces . . . . . . . . . . . . . . . .
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69
70
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74
Chapter 1
Introduction
Throughout this thesis by the term variety, we will always mean an integral, separated,
projective scheme over an algebraically closed field k. Unless otherwise stated, it will be
understood that k = C.
When studying the geometry of algebraic varieties, we can consider our object X as an
abstract variety but more often we would like to think of X as a subvariety of projective
space, X c pN. In view of this, one may want to study the geometry X together with line
bundles. In fact, given a line bundle L on X, we can look at its sections, H0 (X, L), and at
sections of its tensor powers, HO(X, L9m), m c N>o. Whenever one of these vector spaces
is non-empty, say H0 (X, LOm) = 0, then we can define a rational map1
OHO(X,L~k):
X
C P(HO(X, Lk)*).
-- + Y
A celebrated theorem of litaka shows that these maps behave well asymptotically.
Theorem 1.0.1.
[53, Thm. 2.1.33] Let X be a normal projective variety and let L be
a line bundle on X. Assume that for some tensor power L m, HO(X, Lo')
' 0. Then
for all sufficiently large and divisible k E N, the rational maps qHO(X,Lgk): X -- + Yk are
birationally equivalent to a fixed projective morphism 0.. : X', -* Y.. of normal varieties.
More precisely, we have the following commutative diagram, again for k E N sufficiently
large and divisible
Xoo-
X
'kHO(X,L~k)
Vk
"
Yoo -"-
Y
k
Y
In view of Iitaka's theorem, we can define a new invariant depending only on X and
L, K(X, L) := dim YO, the Kodaira dimension of L on X. When none of L and its tensor
powers have sections, we say that n(X, L) = -oo. As on a normal variety there is a natural
correspondence between line bundles and Cartier divisors, given a Cartier divisor D, we will
denote by ,(X, D) the Kodaira dimension of the associated line bundle r,(X, Ox(D)).
We can gather all information coming from sections of L, of its multiples and from the
'A rational map is a map of algebraic varieties that may be defined only on an open dense subset of X.
9
corresponding rational maps in a unique algebraic object: the section ring,
R(X, L) :=
e
Ho(X, Lom).
mEN
The problem with the section ring is that most of the time it is not finitely generated; hence
we cannot apply the standard tools from commutative algebra and algebraic geometry.
Nonetheless, the section ring is a very interesting object and one would like to analyze it.
On a smooth variety X there is a very important line bundle, intrinsically defined. It is
the so-called canonical line bundle, the highest exterior power of the cotangent bundle of X
AdimXQ1
We will indicate by Kx a choice of a Cartier divisor whose associated line bundle is the
canonical bundle. The Kodaira dimension of X is defined to be r(X) := n(X, AdimX).
The canonical divisor can be defined also for normal varieties. The problem is that in the
presence of singularities it may not be a Cartier divisor. We refer the reader to Section 2.3
for a discussion of this case.
When r,(X)
0, Iitaka's theorem implies that there exists an intrinsically defined rational map f : X -- + Y. This morphism is often used in algebraic geometry to derive properties
of X in terms of properties of Y and of the fibres of f. For this purpose, it is then important
to understand the geometry of Y and the structure of the fibres of f.
The canonical bundle is very well-behaved as in fact its section ring is finitely generated.
Theorem 1.0.2'.
ring
[11, Cor. 1.1.1] Let X be a smooth projective variety. Then the canonical
R(X) := R(X, Kx) = @ H0 (X, 0(mKx))
mEN
is finitely generated.
The problem is that even though the canonical bundle is intrinsically defined and its
section ring is finitely generated, it may not be easy to describe its sections or those of its
multiples and hence the associated Iitaka fibration.
Thus, one would like to have an explicit way to recover the Iitaka fibration.
In the case of surface this procedure is nothing but a repeated application of Castelnuovo's Contractibility Criterion.
Theorem 1.0.3 (Castelnuovo's Contractibility Criterion). [5, Thm. 2.17] Let X be a
smooth projective surface and let C c X be a smooth curve isomorphic to P1 such that
C2 = -1.
Then there exist a smooth surface Y and a birational morphism contc: X -+ Y
which only contracts C; i.e., the image of C is a point in Y and contc is an isomorphism
outside of C.
Applying Castelnuovo's Theorem repeatedly, starting from a surface X with K(X) > 0,
we get down in a finite number of steps to a birational smooth surface X' which does not
contain any rational curves of self-intersection -1. The adjunction formula implies that Kx,
must be nef on X', i.e. Kx, - C > 0, for every curve C C X'.
Theorem 1.0.4. [57, Chapter 1] Let X be a smooth projective surface. Then there exists a
birational morphism
f: X - X'
10
to a smooth projective surface X' satisfying the following properties:
Case r(X) > 0. Then
1. Kx, is nef and
$|mKxI: X'
-+ Y is actually a morphism which
The variety X' is called a minimal model of X.
general classification:
Moreover, we have the following
2. for sufficiently large m, the map
realizes the Iitaka fibration.
(X )
=
0 The variety X' is either a KS surface, an Enriques surface, an abelian surface or
a bielliptic surface; in particular the first Chern class of the canonical bundle is
0.
,(X)
,(X
1 The variety Y is a curve and the generic fibre of the Iitaka fibration is a curve of
genus 1.
)
=
2 The variety Y is birationalto X' and X and it is obtained from X' by contracting
rational curves of self-intersection -2. The variety Y will be singular, but Ky
will be Cartier and some multiple of Ky is going to be an ample Cartier divisor.
It is actually enough to pick m = 24 for surfaces of Kodaira dimension 0 or 1, as
follows immediately from Kodaira's classification, see [3, and m = 5 for surfaces of
Kodaira dimension 2, [121.
Case r,(X) = -oo.
Then one of the two following properties hold:
1. X' is isomorphic P2 ;
2. there exists a smooth curve C and a P1 -fibrationg: X' -+ C; i.e., there exists a
rank 2 vector bundle E on C s.t. X' - PCc(E).
Definition 1.0.5. A normal projective variety X is called a Fano variety if there exists
n E N such that -nKx is a very ample Cartier divisor, i.e. the morphism 0|-nKxI: X -+
P(HO(X, Ox(-nKx))*) gives an embedding into projective space.
When K(X) = -oc, the morphism X' -+ C and the trivial morpshi
called Fano fibrations as all fibres are Fano varieties.
P2 a X'
pt. are
In the second half of the 20th century, there was a big effort in algebraic geometry to
extend the ideas employed in the surface case to higher dimension. Let us remind the reader
that a Cartier divisor D on a projective scheme S is said to be numerically effective (in
short, nef) if D - C > 0, for every curve C C S.
Conjecture 1.0.6 (Existence of minimal models and Fano fibrations). [Mori, Kawamata,
Kolldr, Reid, Shokurov, ... ] [48, Conjecture 1.29] Given a normal projective variety X of
dimension higher than 2 with mild singularities, there exist analogues of minimal surfaces
and Fano fibrations. That is, there exists a birational map X -- + X' to.a normal projective
variety X' such that some multiple nKx' of Kx' is Cartier and either nKx' is nef or X'
carries a morphism ir: X' -+ Z such that the general fibre of ir is a Fano variety.
11
Continuing with the notation of Conjecture 1.0.6, when KX' is nef we say that X' is a
minimal model for X. For the time being, we call the map 7r a Fano fibration, cf. Definition
1.1.12.
The Minimal Model Program (in short, MMP), initiated in the '80s by Mori and carried
out over the last 30 years by several other authors aims at solving Conjecture 1.0.6 and at
using it to classify algebraic varieties.
The MMP approach to the classification of varieties is divided into two steps that are
very different in principle, but are obtained by similar techniques, e.g., 175], [43], [49], 1521,
[77], [47], [48].
Starting with a normal projective variety X with mild singularities, in the first step,
a solution to Conjecture 1.0.6 would allow the construction of a birational model X' of X
that either is minimal or carries a Fano fibration. This step wuold yield a classification of
algebraic varieties up to birational equivalence.
Once minimal models and Fano fibrations are available, it is possible to decompose
algebraic varieties into three main building blocks:
Fano varieties
Calabi-Yau varieties these are varieties for which the first Chern class of the canonical
bundle is 0;
(log-)canonical models these are varieties for which the canonical divisor is ample (or
the sum of the canonical divisor and some other effective divisor is ample, see next
section).
The classification then proceeds with the construction and analysis of the moduli spaces
for the three building blocks. This is the second fundamental step.
1.1
Summary of the results
In birational geometry, pairs (X, A), where X is a projective variety and A is a sum of prime
divisors on X with coefficients in (0, 1], appear quite naturally. For example, given a quasiprojective variety U, by Hironaka's resolution of singularities, at least in characteristic 0,
there is a smooth compatification X of U with A = X\ U a simple normal crossing boundary
at infinity. Section 2.1 for the definition of simple normal crossing.
Pairs are also often useful in inductive arguments. In fact when applying the strategy
of the MMP in dimension higher than 2, then it is not possible to work just with smooth
projective varieties. The presence of singularities often requires some correction terms when
dealing with the canonical divisor and its restriction to subvarieties, cf. Section 3.2.
All the ideas explained in the previous section regarding minimal models, Fano fibrations
and the classification of algebraic varieties can be naturally extended to the case of pairs
(X, A), if the singularities of the pair are mild, cf. Section 2.1 and Section 2.3 for the precise
definitions. When dealing with pairs, instead of working with the canonical class Kx of X,
we will work with the log divisor Kx + A. Let us give the following definition.
Definition 1.1.1. Let X be a variety. We say that X is Q-factorial if every Weil divisor
on X has a multiple that is Cartier.
12
We will always require from now on that for a pair (X, A) the divisor Kx + A is a linear
combination of Cartier divisors with coefficients in the real numbers.
Definition 1.1.2. Let X be a projective variety and let D be a Cartier divisor on X. We
say that D is effective if HO(X, Ox(D)) 4 0. We say that D is Q-effective if some positive
multiple of D is effective. We say that D is pseudoeffective if for some ample Cartierdivisor
H on X and for any rational e > 0, some positive integral multiple of D + eH is effective.
We say that D is big if the dimension of HO (X, Ox (mD)) grows like mdimX.
In the context of pairs, it is possible to give a very general positive answer to Conjecture
1.0.6. This theorem first appeared in the fundamental work [11].
Theorem 1.1.3 (cf. Section 2.4.1). Let (X, A) be a pair with Kawamata log terminal (in
short, klt) singularities, see Definition 2.3.4. Assume X is Q-factorial. If A is big and
Kx + A is pseudoeffective, or if Kx + A is big, then there is an algorithm which constructs
a birationalmap
w: X -- + X'
to a pair (X', A': =,r,A) such that
o X' is Q-factorial and (X', A') is klt and
o Kx'
+ A' is nef on X' and there is a multiple m(Kx' + A') which is base point free.
In this case we say that X' is a minimal model for Kx + A.
If Kx + A is not pseudoeffective, then there is an algorithm which constructs a birational
map
,: X -- + X'
to a pair (X', A' := rA) and a morphism p: X'
-÷
Z such that
o X' is Q-factorial and (X', A') is klt;
o dim X' > dim Z and
o -(K'
+ A') is ample when restricted to any fibre of p.
That is the analogue of the Fano fibration that we saw in the case of surfaces, cf. Definition
1.1.12
1.1.1
Cone Theorem and hyperbolicity
If a given smooth projective variety X (resp. a pair (X, A)) with log canonical singularities
(see Definition 2.3.4) fails to be a minimal model, that is because there are curves whose
intersection with the first Chern class of the canonical bundle (resp. of Kx + A) is negative.
The curvature of the cotangent bundle (resp. of the line bundle associated to a log canonical
divisor) governs the birational geometry of the variety. We will denote by NE(X) the closure
of the cone generated by classes of curves on X in H2 (X, R).
Theorem 1.1.4 (Cone Theorem). /52, Thm. 1.24] Let X be a smooth variety of dimension
n. There exist countably many rational curves Ci C X such that
o[Ci], 0< -(Kx) -Ci < n +1.
NE(X) = NE(X)KX o!+EZ
i
13
Moreover, for any ample divisor H and any e > 0 there are only finitely many curves
among the Ci above that are (Kx + eH)-negative, i.e.,
NE(X) = NE(X)Kx+,HO0 + E1 Ro[C],
0 < -(Kx + cH) - Cj
finite
An analogous statement is valid for log canonical pairs (X, A), where we substitute Kx
with Kx + A, cf. Theorem 2.4.4.
In general, to get a new variety birational to X which is closer to being a minimal model
for (X, A), one wants to contract the part of X where the curvature of Kx + A is negative.
That part of X is covered by rational curves.
How are the rational curves distributed with respect to A?
For example, let us consider a smooth quasi-projective variety U and a compactifying
pair (X, D), X \ D = U. Assume that Kx + D is not nef. Is it possible to modify the
statement of the Cone Theorem so that it takes into account also U?
This question makes even more sense in view of the following principle.
Iitaka's principle 1.1.5. (See [57, pg. 1121) Whenever we have a theorem about nonsingularvarieties whose statement is dictated by the behavior of the regulardifferentialforms
and the canonical bundle, there should exist a corresponding theorem about logarithmic pairs
(pairs of a non singular variety and boundary divisor with only normal crossings) whose
statement is dictated by the behavior of the logarithmicforms and log canonical bundles, and
vice versa.
Considering as above a quasi-projective variety U as the complement of a pair (X, D),
then Iitaka's principle is just predicting a correspondence between theorems about nonsingular varieties and regular differential forms and theorems about quasi-projective varieties
and their regular differential forms which extend to the boundary of a compactification with
at worst order 1 poles.
The first result that we prove follows this line of thought and shows how on a quasiprojective variety U we may be able to see some algebraic copies of the affine line, when
Kx + D is not nef.
Theorem 1.1.6. [cf. Thm. 3.1.3] Let X be a smooth projective variety and D =
1 Di
a reduced divisor with simple normal crossing singularities on X. Assume that there is
no non-constant morphism f: Al -- (X \ D) and that there is no non-constant morphism
f: Al -+ (Fie1 Di \ 1 Dj), where I C {1,..., k} is a non-empty set of indices.
Then Kx + D is nef.
More generally, let (X, A) be a log canonical pair. Assume that there is no non-constant
morphism f : A' -* (X \ {x E X I (X, A) is not klt at x}) and that there is no non-constant
morphism from Al with values in certain locally closed varieties which stratify the locus of
points where (X, A) is not klt.
Then Kx + A is nef.
Uj
The assumption on the non-existence of copies of Al in the locally closed subvarieties
Di \ Ujer Dj was first introduced by Lu and Zhang in [55] with the name of Mori
hyperbolicity. We generalize their definition to the case of a log pair (X, A) allowing possibly
non log canonical singularities as well, see Definition 3.4.1. Lu and Zhang proved a version
of the above theorem for divisorial log terminal pairs (see Definition 2.3.4), assuming some
niEl
14
factoriality conditions on the components, [55, Thm. 3.1]. Theorem 1.1.6 is a generalization
of their result to the category of log canonical varieties.
Using Theorem 1.1.6, we are able to obtain a more refined version of the Cone Theorem
for log canonical pairs which gives a more precise description of the intersection of (Kx + A)negative curves with the most singular part of (X, A). This may be thought of as an instance
of litaka's principle, i.e., as a partial analogue of the Cone Theorem for an open variety U
together with a compactification (X, D). We give here the statement for log smooth pairs.
The reader can find the more general version in Section 3.5.
Theorem 1.1.7. [cf. Thm. 3.1.5] Let (X, D = E_ Di) be a pair with snc singularities.
Then there exist countably many (Kx + D)-negative rational curves Ci such that
NE(X) = NE(X)KX+D0 +
ERo [Ci].
iEI
Moreover, one of the two following conditions holds:
" Ci
n (X \ D) contains the image of a non-constant morphism f : A1 -+ X or
" there exists a non-empty set of indices I c {1 ... , k} such that Ci n (Fi., Di\U3 1 Dj)
contains the image of a non-constant morphism f : A' -+ (nE1, Di \ UjV 1 Dj).
1.1.2
The classification of fibres of Mori fibre spaces
Definition 1.1.8. Let X be a smooth variety. We say that X is uniruled if there exists a
smooth variety B and a dominant rational map 7r: P1 x B -- + X such that the image of the
generic fibre of the second projection P2: P1 x B -+ B is non-constant.
It is a now classic result, due to Miyaoka and Mori [59], that if through a general point of
a projective variety X there is a curve which intersects KX negatively, then X is uniruled.
In particular, their theorem implies that K(X) = -00.
The fact that also the opposite implication holds is one of the main results of the last
decade in algebraic geometry, due to Boucksom, Demailly, PAun and Peternell.
Theorem 1.1.9. [13, Cor.
equivalent:
0.31 Let X be a smooth projective variety. The following are
1. X is uniruled;
2. Kx is not pseudoeffective and
3. there exists a family of curves passing through the generic point of X and such that
Kx intersects negatively every curve in the family.
Mumford conjectured an even stronger result.
Conjecture 1.1.10 (Mumford). Let X be a smooth projective variety. If r,(X)
X is uniruled.
=
-oo then
We can restate Mumford's conjecture in the following equivalent form.
Conjecture 1.1.11 (Non-vanishing Conjecture). [11, Conjecture 2.11 Let X be a smooth
projective variety. Assume that Kx is pseudoeffective. Then Kx is effective, i.e., /-i(X) ;> 0.
15
In Theorem 1.1.3 we saw that when Kx is not pseudoeffective the MMP gives us a way
of constructing a birational map X -- + X' together with a morphism X' -+ Z such that the
general fibre is a positive dimensional Fano variety. By the Miyaoka-Mori theorem, X' and
consequently X are uniruled.
The precise definition for the kind of morphism that we want to allow in the case of
varieties with non-pseudoeffective canonical divisor is the following. The reader will find the
definition of numerical equivalence in Section 2.1.
Definition 1.1.12. Let f : X -+ Y be a dominant projective morphism of normal varieties.
Then f is called a Mori fibre space (in short, MFS) if the following conditions are satisfied:
1.
f
has connected fibres, with dim Y < dim X;
2. X is Q-factorial with at most log canonical singularities;
3. all curves on X contained in the fibres of
f
are numerically equivalent and
4. -Kx is ample along every fibre of f.
Mori fibre spaces arise as final outcomes of a run of the MMP. For this reason they
have been widely studied for the last thirty years in the context of classification of higher
dimensional varieties. There is a natural question which does not appear to have been
considered in the literature.
Question 1.1.13. Which Fano varieties (within a suitable class of singularities) can be
realized as general fibres of a Mori fibre space?
The question is easily answered for every Fano variety with log canonical singularities
and Picard number one: the constant morphism to a point trivially gives such variety the
structure of a Mori fibre space.
This part of the thesis aims at giving evidence that for Fano varieties of Picard rank
strictly greater than 1 being the general fibre of a Mori fibre space is a very restrictive
condition.
The notion of "general fibre" will be clarified later in Section 4.2: the idea is to determine
an open dense subset of the base on which the fibres are "good enough", see Definition 4.2.16.
In our work, we will also assume the existence of a dense open set over which the fibres
of the MFS are Q-factorial. This is due to some technical reasons connected to the structure
of birational data in families of varieties. For the same reason, we shall restrict to the class
of Fano varieties with terminal singularities, see Definition 2.3.4 for the notion of terminal
singularities.
Given a terminal Q-factorial Fano variety F, we will denote by Mon(F) the maximal
subgroup of GL(H 2 (F, Z)/ Torsion) which preserves the birational data of F (cf. Definition
4.2.11).
Theorem 1.1.14.
9 Sufficient criterion: A terminal Q-factorial Fano variety F can be realised as a
general fibre of a Mori fibre space if the invariant subspace H 2 (F, Q)Aut(F) for the
action of the automorphism group of F on H2(F, Q) is equal to the line spanned by
the first Chern class of KF.
16
"
Necessary criterion: A terminal Q-factorial Fano variety F cannot be realised as
a general fibre of a MFS if the invariant subspace H2 (F, Q)Mon(F) for the action of
Mon(F) on H2 (F, Q) has dimension at least 2.
" Characterisation for rigid varieties: Assume that H1 (F, (Q)*)
sufficient criterion turns into a characterisation.
=
0.
Then the
Our results rely upon a careful study of the monodromy action on Mori fibre spaces. On
a suitable open subset of the base of a Mori fibre space we know the morphism is nothing
but a family of Fano varieties: the monodromy of the family, acting on first Chern classes
of line bundles, can only fix the multiples of the class of the canonical bundle. Hence many
invariants of the fibres that are naturally defined in terms of their groups of divisors modulo
numerical equivalence must be highly symmetrical, cf. Theorem 4.2.9.
The case of 2-dimensional Fano varieties that appear as general fibres of Mori fibre
spaces had been worked out already in [60, Theorem 3.5] when the dimension of the total
space of the Mori fibre space is 3. In Section 4.3 we give an alternative proof using our
criteria, allowing total spaces of arbitrary dimension. Moreover, we are able to obtain a full
classification also for threefolds in terms of deformation types.
Theorem 1.1.15.
" Surfaces: a smooth del Pezzo surface can be realised as the general fibre of a MFS if
and only if it is not isomorphic to the blow-up of P 2 in one or two points.
" Threefolds: the deformation type of a smooth Fano threefold F with p(F) > 1 can be
realised as a general fibre of a MFS if and only if it is one of the 8 classes appearing
in Table 4.3.3.
In particular, the property of being a general fibre of a Mori fibre space is invariant under
smooth deformation up to dimension 3; moreover, in these cases, the necessary criterion of
Theorem 4.1.2 is actually a characterisation.
In the case of smooth toric Fano varieties, we completely classify in low dimension those
varieties that can be realised as the general fibre of a MFS (see Table 4.5.3). Moreover,
we are able to draw the following interesting connection to the existence of Kihler-Einstein
metrics, see Sections 4.1 and 4.5 for definitions and more speculations on this matter.
Theorem 1.1.16 (cf. Thm 4.5.7). Let F be an n-dimensional toric Fano variety and let
E C Zn be the fan associated to it. Let A be the polytope in Z whose vertices are the integral
generators of the 1-dimensional cones of E.
If F can be realised as the general fibre of a MFS then the the sum of all vectors corresponding to vertices of A is 0. In particularthis implies that F is K-stable or, equivalently,
that there exists a Kdhler-Einstein metric on F.
1.2
Structure of the thesis
In Chapter 2 we introduce the basic notations and survey known results in birational geometry, particularly, those regarding the singularities of varieties and pairs and the MMP. The
results presented in the chapter are foundational for the rest of the thesis.
17
In Chapter 3, we introduce the notion of Mori hyperbolicity and discuss the relations
with the theory of adjunction along higher codimensional strata. We prove theorems 1.1.6
1.1.7 and also provide an ampleness criterion for dlt pairs which are Mori hyperbolic.
In Chapter 4, we discuss the notion of being a general fibre of a MFS and its topological
and algebro-geometric implications. We prove some necessary and sufficient conditions for
a Q-factorial terminal Fano variety to be a general fibre of a MFS. We work out some
constructions which allow us to prove some affermative results for certain classes of Fano
varieties. Moreover, we discuss several examples: the low dimensional case, toric varieties
with a specific focus on K-stability and certain types of homogeneous spaces.
The results presented in this thesis are based on the papers [71], [21]. The results in
Chapter 4, based on [21], are joint work with Giulio Codogni, Andrea Fanelli and Luca
Tasin.
18
Chapter 2
Notations and preliminaries
This chapter will serve as a collection of notations and preliminary facts that will be freely
used during the whole exposition in the next chapter.
2.1
Notation and Conventions
- By the term variety, we will always mean an integral, separated, projective scheme
over an algebraically closed field k. Unless otherwise stated, it will be understood that
k=C.
- Given a scheme S, when we talk of a point of
unless otherwise stated.
- Let S be a scheme and let
S, we always mean a closed point of S,
f:
X -+ S and 9: Y -+ S be two schemes over S. We say
that a morphism of schemes h: X -+ Y is a morphism over S if the following diagram
commutes
X
Y.
h
f
/g
S
A similar definition holds in the case of a rational map h': X -- + Y of S-schemes.
- A K-Weil (K = Z, Q, R) divisor is a K-linear combination of Weil divisors and the
analogous definition is valid in the K-Cartier case. Unless, otherwise specified, we will
often indicate K-Weil divisors just by referring to K-divisors. When K is not specified
it should be understood that K = Z. When we want to indicate that the divisors are
also K-Cartier we will do so explicitly. Given a Cartier divisor L, we will denote by
ILI the complete linear system of L, [53, page 121.
- For d E R a real number the round down (resp. the round up) of d, [d] (resp. Fdl)
denotes the largest (resp. the least) integer which is at most d (resp. at least d). If
D = E diDi is an R-divisor on a normal variety X, then the round down (resp. round
up) of D is [DJ = EiLdijDi (resp. [D1 = EiFdi]Di), The fractional part of D is
{D} = D - [Dj.
- If D =
E diDi is an R-divisor on a normal variety X, where the Di are the distinct
prime components of D, then we define D*C :=
19
Ei
diDi, c E
R, where * is any of
,> I<I > <
= Eic1 djDj (where di # 0) is the union of the prime divisors
appearing in the formal sum, Supp(A) = Uicr Di.
- The support of A
- Two K-divisors D1 and D2 are K-linearly equivalent if their difference is a K-linear
combination of principal divisors on X.
In this case, we write Di ~K D2,
- Two K-divisors Di and D2 are K-numerically equivalent if their difference is a K-linear
combination Ei aiEj of Cartier divisors such that degc(jEj aEj) is 0 on every curve
C on X.
In this case, we write D 1 =K D2,
- A 1-cycle with coefficients in K, on X is a formal finite K-linear combination of integral
curves on X.
- Two 1-cycles C1, C2 are numerically equivalent if (D -CI) = (D - C2) for every Cartier
divisor D.
- Given a map f: X -+ S, two K-divisors D 1, D2 on X are K-linearly equivalent over
S, denoted D -f,K D 2 , if D1 - D2 ~K f*(A) for some K-Cartier divisor A on S. A
similar definition holds for numerical equivalence over S.
- Given a map f: X -+ Y, a 1-cycle with coefficients on K relative to f on X is a
formal finite K-linear combination of integral curves on X that lie in the fibers of f,
C = Ei aiCi, aj E K, Ci c X, fCi = 0.
- A log pair (X, A) consists of a normal variety X and a R-Weil divisor A > 0 such that
Kx + A is R-Cartier.
- A log pair (X, D =-Eic ajDj) is simple normal crossing (snc) if X and every component of D are smooth and for every p E X one can choose a neighborhood U - p in
the 6tale or analytic topology and local coordinates xj around p such that for every
component Di of D containing p there is index c(i) for which Di n U = (xc(j) = 0).
- If (X, A) is snc, a stratum of (X, A) is either X or an irreducible component of the
intersection njEJDj, where J is a non-empty subset of the prime divisors appearing
in A with coefficient 1. Given a (closed) stratum, W, the corresponding open stratum
is obtained from W by removing all the strata strictly contained in W.
v = ordD for a prime divisor D on some
variety X' birational to X. A geometric valuation v of k(X) is called exceptional if it
is not realized by a divisor on X.
- A valuation v of k(X) is called geometric if
- Given a normal variety X, a K-b-divisor is a (possibly infinite) sum of geometric
valuations vi of k(X) with coefficients in K,
ID =bivi,
bi E K, Vi E I,
iEI
such that for every normal variety X' birational to X, only a finite number of the vi
can be realized by divisors on X'.
20
- Let D = Jgir bivi be a K-b-divisor on X, and let X' be a normal variety birational
to X. Let I' be the set of those i E I such that vi is associated to a divisor Di on X'.
The trace of D on X' is defined by
Dx, =
biDi.
iCI'
- A projective morphism f : X -+ Y between normal varieties is said to be a contraction
if f*Ox = Oy. By Zariski's Main Theorem, [40, Cor. 11.41, this is equivalent to the
fibers of f being connected.
- A resolution of a normal variety X is a proper birational morphism ir: Y -+ X such
that Y is smooth and 7r is an isomorphism outside the singular locus of X. In characteristic 0, resolutions always exist by Hironaka's algorithm.
- For a birational morphism f : X -+ Y the exceptional set Exc(f) C X is the set of
points of X where f is not biregular, that is, f- 1 is not a morphism at f(x).
- A subset C of a real vector space V is called a cone if C is a sub-semigroup of the
additive group of V that is closed under multiplication by positive real numbers.
21
2.2
Cones
Let X be a projective variety. Then, we are especially interested in studying the features of
codimension 1 varieties on X.
Definition 2.2.1. A Weil divisor D on X is called Q-Cartierif a positive integer multiple
of D is a Cartierdivisor.
For K = Z, Q, R, we define N (X)K to be the K-module of K-Cartier divisors modulo
numerical equivalence
N 1 (X)K
{K-linear combinations of Cartier divisors}/
=K
It is a finitely generated K-module. For this and other basic facts, the reader can consult
[53, Chap. 11.
Dually, we can define a similar object for curves. NI(X)K will be the K-module of
1-cycles with coefficients in K modulo numerical equivalence
N(X)K
=
{K-linear combinations of integral curves}/
=K
When we do not specify the coefficients of N1 (X) and NI(X), it should be understood
that we mean N1 (X)R and Ni(X)R. By the definitions it is clear that there is a natural
perfect bilinear pairing
N1 (X)K
XK
Ni(X)K
-*
K
(2.1)
given by taking the intersection of Cartier divisors with 1-cycles.
In N 1 (X)R and NI(X)R there are several cones that are important to birational geometry.
Amp(X) The ample cone is the cone generated by classes of ample divisors. A Cartier
divisor D is ample if for some sufficiently large natural number m, the linear system
ImD gives an embedding of X in projective space.
Nef(X) The nef cone is the cone of classes [D] that have non-negative intersection D -C > 0
with any integral curve C on X. Nef(X) is closed as it is the intersection of countably
many closed subsets.
Mov(X) The movable cone is the closure of the cone generated by classes of divisors D
such that for some positive integer m the base locus of the linear system ImD has
codimension at least 2 in X.
Big(X) The big cone is the cone generated by classes of big divisors. A divisor D is said to
be big, if
limsup dim HO(X, Ox(mD)) >0, n=dimX
m-+oo
m
Eff(X) The pseudoeffective cone is the closure of the cone generated by classes of effective
divisors on X in N1 (X)R. Its interior corresponds to Big(X), [53, Thm. 2.2.24].
NE(X) The cone of effective curves is the closure of the cone generated by classes of integral
curves on X in N1(X)R.
22
In the course of our exposition, if D is an R-Cartier divisor, we often abuse notation by
writing D also for its class. We will indicate by p(X) the dimension of N1 (X), which is
called the Picard number of X.
All the cones defined above are strongly rational cones (i.e., they do not contain lines)
and have non-empty interior, see, e.g., [83, Prop. 2.2.2].
We have the following basic result relating Nef(X) and NE(X).
Theorem 2.2.2. [53, Thm. 1.4.26] Let X be a projective variety. Then the dual of Nef(X),
under the duality established in 2.1, is given by NE(X). Moreover, a divisor D is ample if
and only if its intersection with any nonzero class contained in NE(X) is strictly positive.
Hence, the interiorof Nef(X) is equal to Amp(X).
We have analogous definitions in the relative case. Let us consider a projective morphism
of varieties 7r: X -÷ S. Unless otherwise specified, all curves and divisors will be on X. We
have the following definitions:
N1 (X/S)K
{K -linear combinations of Cartier divisors}/ -K
N 1 (X/S)K = {K -linear combinations of integral curves contracted by w}/
=
When we do not specify the coefficients of N 1 (X/S) and N 1 (X/S), it should be understood that we mean N1 (X/S)R and N1(X/S)R. Again, by definition, there is a perfect
pairing analogous to the one in 2.1. Here, too, we are interested in certain cones.
Amp(X/S) The relative ample cone is the cone generated by classes of divisors D such that
for any fiber of r the restriction of D is ample.
Nef(X/S) The relative nef cone is the cone of classes [D] that have non-negative intersection
D - C > 0 with any integral curve C contained in the fibers of 7. Nef(X/S) is closed
and its interior corresponds to Amp(X/S).
Big(X/S) The relative big cone is the cone generated by classes of relatively big divisors. A
divisor D is relatively big, if the rank of 7r,(Ox(mD)) at the generic point of S grows
like mdimX-dimS, i.e. Ox(D) is big along the generic fiber of 7r.
Eff(X/S) The relative effective cone is the closure of the cone generated by classes of divisors
D such that 7rr(Ox(D)) $ 0 on S. Its interior corresponds to Big(X/S).
NE(X/S) The relative cone of effective curves is the closure of the cone generated by classes
of integral curves on X in N1 (X/S)R.
We will indicate by p(X/S), the so-called relative Picard number, the rank of N1 (X/S).
In the relative case, it is not true anymore that all cones above are strongly rational. This
is immediately seen by considering a projective birational morphism of varieties 7r: X -4 S.
In this case, every divisor D on X is big as the generic fiber is just a point. Nonetheless,
Nef(X/S) is still the dual of NE(X/S) as the following generalization of Theorem 2.2.2
implies.
Theorem 2.2.3. [52, Thm. 1.44] Let ir: X -+ S be a projective morphism of algebraic
varieties. Then the dual of Nef(X/S) is given by N E(X/S). Moreover, a divisor D is ample
over S if and only if its intersectionwith any nonzero class contained in NE(X/S) is strictly
positive. Hence, the interior of Nef(X/S) is equal to Amp(X/S).
23
2.3
Pairs and their singularities
Let be X a normal quasi-projective variety. As X is normal, the smooth locus X" is an
open set whose complement has codimension at least 2. Then the canonical divisor Ksrnm
on the smooth locus of X can be naturally extended to a Weil divisor on X which we will
denote by Kx
Definition 2.3.1. We say that X is Gorenstein (resp. Q-Gorenstein) if it is CohenMacaulay and the canonical class is Cartier (resp. Q-Cartier). X is said to be Q-factorial
if every Weil divisor is Q-Cartier.
A log resolution for a log pair (X, A) is a projective birational morphism w: X'which the following three conditions hold:
X for
1. the exceptional locus of 7r is a divisor E;
2. E supports a ir-ample divisor and
3. Supp(E +
7r; 'A) is a simple normal crossing divisor.
Given a log resolution of (X, A), we can write
Kx' +
;1A +
biEi = 7r*(Kx + A),
(2.2)
where the Ei are the irreducible components of E.
Definition 2.3.2. The log discrepancy of Ei with respect to A is a(Ei; X, A) := 1 - bi.
The definition of log discrepancy can be extended to geometric valuations of k(X). For
an exceptional valuration v, there exists a log resolution 7r: X' -+ X on which v is realized
as the valuation associated to a prime Cartier divisor D C X', see [52, Lemma 2.45].
The log discrepancy of v, a(v; X, A), is defined as the log discrepancy of D. The center
of v on X, cx(v), is the subvariety 7r(D) C X.
Definition 2.3.3. The discrepancy of a pair (X, A) is
logdiscrep(X, A) := inf{a( v; X, A)
v geometric valuation exceptional over X}.
For Z C X an integral subvariety and qz its generic point, we define
a(rz; X, A)
=
inf
,cx(V)=z
a(V; X, A).
The log discrepancy of divisorial valuations is the central object in the study of singularities of log pairs. It is a well known fact (cf. [52, Cor. 2.31]) that
0 < logdiscrep(X, A) < 2
or
logdiscrep(X, A) = -oo.
The Minimal Model Program focuses on those pairs whose log discrepancy is nonnegative.
Definition 2.3.4. A log pair (X, A) is Kawamata log terminal (in short, klt) if logdiscrep(X, A)
0 and [AJ = 0.
A log pair (X, A) is log canonical (lc) if logdiscrep(X, A) > 0
24
>
A log pair (X, A) is divisorial log terminal (dit) if the coefficients of A are in [0, 1] and
there exists a log resolution 7r: X' -+ X such that all exceptional divisors have log discrepancy
< 1.
We say that the pair is terminal if logdiscrep(X, A) > 1 and canonicalif logdiscrep(X, A)
1.
2.3.1
The non-klt locus, ic centers and their stratification
Definition 2.3.5. Let (X, A) be a log pair and Z C X an integral subvariety. Then Z is a
non Kawamata log terminal center (in short, a non-klt center) if a(qz; X, A) < 0.
The non-klt locus, Nklt(X, A), of the pair (X, A) is the union of all the non-klt centers of
X,
U
Nklt(X, A)
Z.
{Zla(7z;X,A) O}
The non lc locus, Nlc(X, A), of the pair (X, A) is the union of all the centers of strictly
negative log discrepancy, i.e.,
U
Nlc(X, A):=
Z.
{Zla(7z;X,A)<0}
A non-klt center Z C X is called a log canonical center (lc center) if a(rlz; X, A)
and Z is not contained in Nlc(X, A).
=
0
We will often write Nklt(A) (resp. Nlc(A)) instead of Nklt(X, A) (resp. Nlc(X, A)).
If we pass to a log resolution of (X, A), 7r: X' -+ X and write as in (2.2)
Kxi + Ax, = Kx + Z biA' = Kx, +r-A +
bi Ei = r*( Kx + A),
then Nklt(A) = 7r(Supp(Zi~bi>1 A')) and Nlc(A) = 7r(Supp(Eisbi>1 A )).
The complement in X of Nklt(A) is the biggest open set on which A has just klt singularities and, analogously, the complement of Nlc(A) is the biggest open set of X on which
A has lc singularities.
It is easy to see (cf. [52, Lemma 2.29]) that all valuations of log discrepancy 0 with
respect to A that are not contained in Nlc(A) are given either by the components of A=4
or by blowing up the strata of A-4 and repeating the same procedure. Hence, the lc centers
are nothing but the closures of the lc centers for the pair (X \ Nlc(A), AIX\Nlc(A)).
The union of the lc centers of (X, A) is a subvariety of X, but it carries a richer structure.
It is in fact a subvariety stratified by the lc centers and it will be important for us to keep
track of the strata.
Definition 2.3.6. Let (X, A) be a log pair. Given an lc center W for (X, A), the total
space of the stratificationassociated to (X, A) on W is given by
J
Strat(W, A)
wI;w
W' Ic center
the union of the log canonical centers contained in W.
25
W,
An important result about the structure of the non-klt locus, that we will need in the next
sections of the paper, is the following connectedness theorem, originally due to Shokurov
and Kolldr.
Theorem 2.3.7. [49, Theorem 17.41 Let (X, A) be an lc pair and let 7r: X -+ Y be a
contraction of normal projective varieties. Assume that -(Kx + A) is relatively nef and
relatively big over Y. Then every fiber of 7 has a neighborhood (in the classical topology) in
which Nklt(A) is connected.
2.4
The Minimal Model Program
As we explained in the introduction, the Minimal Model Program aims at giving a birational
classification of algebraic varieties. In this section, we will give an overview of the technical
aspects of the realization of the program, providing the reader with the necessary results
that will be used in the next chapters. Our main sources in terms of notations and results
will be [11] and [52].
We start by considering a log pair (X, A) with lc singularities and a projective morphism
f : X -+ S. As usual the non-relative case will just follow by taking f to be the constant
morphism to a point. Let us introduce the following definitions.
Definition 2.4.1. A proper birational map f: Z -- + Y between normal quasi-projective
varieties is said to be a birational contraction if the restriction of f-1 to every prime divisor
D C Y is a birationalmap.
Definition 2.4.2. Let f : Z -- + Y be a birationalcontraction between normal quasi-projective
varieties and let D be an R-Cartier divisor on Z such that D' := fD is also R-Cartier.
We say that f is D-non-positive (resp. D-negative) if there exists a common resolution
p: W -* Z, q: W -+ Y such that
p*D = q*DI + E
where E > 0 is q-exceptional (resp. E is q-exceptional and it contains the strict transform
of the f -exceptional divisor).
Definition 2.4.3. Let (X, A) be a lc pair and f : X -+ S a projective morphism of quasiprojective varieties. A minimal model of Kx + A over S (or (Kx + A)-minimal model over
S) is a log pair (Y, Ay), where g: Y -+ S is a scheme over S together with a birational
contraction ir: X - - + Y over S such that
1. Ay = *A;
2. Ky + Ay is nef over S and
3. the birational contraction i is (Kx + A) -non-positive.
If Kx + A is not nef over S, then the Cone Theorem describes the structure of the
closure of the relative cone of effective curves in terms of intersection with Kx + A.
Theorem 2.4.4 (Cone Theorem). 131, Thm. 3.3 and Rmk. 3.41 Let (X, A) be a dlt pair
and f : X -+ S a projective contraction of normal varieties. Then there exist countably many
26
rational curves Ci
c X contained in the fibres of f such that
NE(X/S) = NE(X/S)Kx+AO +
Ro[Ci], 0 < -(Kx +A)
- Ci < 2n.
Moreover, for any H f-ample divisor and any e > 0, there are only finitely many curves
among the Ci above that are (Kx + A + eH) -negative, i.e.
NE(X/S)
=
NE(X/S)Kx+A+,H>o +
R>O Ao]
finite
In particular, the (Kx + A)-negative extremal rays are discrete and the boundary of the
negative part of NE(X/S) is locally polyhedral away from the hyperplane (Kx + A) -C = 0.
Every extremal ray, thanks to their local discreteness, corresponds dually to a codimension one face of Nef(X/S). Let D be a class in the relative interior of a facet Y corresponding
to the intersection Nef(X/S) n [Ci]', for [Ci] the class of some (Kx + A)-negative extremal
curve as in the Cone Theorem. As D lies in the relative interior of F, for simplicity we can
assume that it is a Q-Cartier divisor. Then, [D]I n NE(X/S) = R>o [Ci]. In particular,
taking 0 < e < 1, it is immediate to see, using for example a compact slice of NE(X), that
D - e(Kx + A) is ample over S.
Theorem 2.4.5 (Base Point Free Theorem). [31, Thm. 4.1] Let (X, A) be a dlt pair and
f : X -+ S a projective morphism. Let D be a relatively nef Q- Cartier divisor, such that for
some a > 0, aD - (Kx + A) is ample over S. Then there exists k s.t. JkD| is base point
free over S, i.e., there exists a commutative diagram of projective morphisms
X
T
h
f
/g
S
and kD ~f h*H, for some relatively ample divisor H on T.
The Base Point Free Theorem implies that the codimension one faces that arise as above
all correspond to the pullback of Nef(Y) under some morphism f : X -+ Y.
Theorem 2.4.6 (Contraction Theorem). [31, Thm. 3.3] Let F C NE(X/S) be a (Kx+A)negative face of dimension d. Then, there exists a unique contraction morphism contF: X -+
Y over S to a projective variety Y such that for every effective curve C on X contF,wC = 0
if and only if [C] E F. Moreover, we have the following exact sequence
0 -+ N'(Y/S) -+ N'(X/S) -+ Zd
_
0
where the first map is simply given by pullback via cont* and the second map is given by the
intersection with classes in the linear span of F inside NE(X/S).
In the case where F is just an extremal ray Rho[C] as in the Cone Theorem, we obtain a
map such that the relative Picard number p(X/Y) is equal to 1 as in fact all curves contained
in the fibers are numerically equivalent to some multiple of C. Moreover, in this case we
can give a first classification of the types of morphism that appear.
27
Divisorial contraction COntR>0 [C]: X -+ Y is birational, the exceptional locus is an irreducible divisor. Moreover, if X is Q-factorial then so Y is.
Flipping contraction contR, [C]: X -+ Y is birational, the exceptional locus is of codimension at least 2. In this case, Y is never Q-factorial regardless of X, [52, Cor.
2.63].
Mori fibre contraction (or Mori fibre space) contR;>o[C]: X -+ Y and dim X > dim Y.
If X is Q-factorial then so Y is.
2.4.1
The algorithm of the MMP
We are now ready to describe the algorithm behind the Minimal Model Program. We start
with a pair (X, A) with dlt singularities and a projective contraction of normal varieties
f: X -+ S. We moreover assume that X is Q-factorial.
Our goal is to construct a sequence of birational models of (X, A, f) over S
-+ (Xi, Ai,
fA)
--
+
(Xi+1, Ai+ 1, fi+1) --
+
--
-
-
ending with a projective contraction f.: X* -+ S having nice geometric features.
Below, we describe an algorithm that aims at achieving such goal. An instance of this
algorithm, will be called a run of the (Kx + A)-MMP.
Step 0 We start by defining (Xo, A 0 , fo) = (X, A, f) and we define the triples (Xi, Ai, fi)
inductively following the steps below. Each triple (Xi, Ai, fi) should have the following
properties:
" Xi is Q-factorial;
" (Xi, Ai) is a dlt pair;
"
fi:
Xi -+ S is a projective morphism.
Step 1 Case 1: KXi
+ Ai is nef over S: then we define
(X*,I A* , f.') :=(Xi, Ai, fi)
and the algorithm stops.
Step 2: Kxi + Ai is not nef over S: by the Cone Theorem and the Contraction
Theorem, we know that there exists a (Kx, + Ai)-negative extremal ray Ri and
a contraction morphism
contR.
Xi
/9Y1
S
over S which only contracts curves whose classes are in the ray R. We then have
three possibilities:
1. If contR : Xi
-+
Y is a divisorial contraction, then we
just define
(Xi+l, Ai+l, fi+1) :=::: (Yi, cont Ri,* Ai 9i)
28
and we repeat the procedure. Notice that p(Xi+1/S) = p(Xi/S) - 1. In
particular there can only be finitely many divisorial contractions in our algorithm, as the relative Picard number is always non-negative, for a projective
morphism.
2. If contR : Xi -+ Y is a small contraction, then Y will not be Q-factorial
anymore. To overcome this problem, the strategy is to find a new variety
X , birational to Xi and Q-factorial such that the diagram
-
Xi - - -
- >. Xt
h
cont\
Yi
commutes and Kxt + 7ri,,Ai is relatively ample over hi. The horizontal map
is not a morphism, but it will be an isomorphism in codimension 1, i.e., the
locus outside which it is an isomorphism has codimension at least 2. This
map is usually called the flip of (Xi, Ai). It follows from the definition of 7ri
that if Xt exists, then it should be defined as
Xt = ProjcY
(@ fi,* Oxi([LKxi + Ail))
neN
If the flip of Xi exists, then we define
(Xi+l,Ai+1,fi+):= (X+,
A
i o hi)
and we go back to Step 1.
3. If contRi is a Mori fibre space, then we define
(X*,I A* , f*) := (Xi, Ai, fi)
and the algorithm stops.
In order to obtain a fully functional algorithm, we need to check
1. that at each step the conditions in Step 0 are satisfied;
2. that flips exist and
3. that the algorithm terminates in finitely many steps.
The answer to the first problem above has been known for a long time and is affirmative,
in the case of dlt singularities. In fact, it is not hard to see that when the map Xi -- + Xi+1
defined above is birational then it is actually a (Kx + A)-negative birational contraction in
the sense of Definition 2.4.2, see [52, Lemma 3.38].
As a consequence, one can prove the following theorem.
Theorem 2.4.7. [52, Chap. 3J Let (X, A) be a dit pair and f : X -+ S a projective morphism. Then at each step of a run of the (Kx + A)-MMP, (Xi, Ai) is Q-factorial with dit
singularities and fi: X -+ S is projective.
29
As for the existence of flips, that is one of the greatest achievements of the last decade
in algebraic geometry. The answer is affirmative once again.
Theorem 2.4.8. ([11, Cor. 1.3.1][10, Cor. 1.2], [39, Cor. 1.8]) Let Ir : (Y, F) -* T be a
kit pair relative over T and assume that either F is relatively big and Ky + r is relatively
pseudoeffective or Ky + F is relatively big. Then the ring
ProjoT
(Q 7r*Oy([Ky
+ IF))
nEN
is finitely generated.
In particular, kit flips exist. More generally, flips exist for dit and even ic pairs.
Thus, we have just seen that the finiteness of the algorithm only depends on the finiteness
of sequences of flips. This is conjectured to hold true.
Conjecture 2.4.9. [8, Conjecture 1.1] Let (X, A) be a log canonical pair. Then every
sequence of (Kx + A) -negative flips terminates.
2.4.2
MMP with scaling
We describe now a slightly different algorithm which in some cases allows us to get some
positive answers to Conjecture 2.4.9.
We will allow perturbation of the pair (X, A) by effective divisors C such that the
singularities of (X, A + C) are bounded and Kx + A + AC is nef over S for some positive
real number A. The advantage that we gain from allowing such perturbation is that we are
going to have better control on our choice of the extremal rays along which we perform the
contractions. We call this procedure a Minimal Model Program for Kx + A with scaling of
C.
Lemma 2.4.10. [9, Lemma 3.1] Suppose (X, A+ C) be a Q-factorial lc pair relative over a
scheme S with A > 0,C > 0, (X,A) dit and Kx + A+C nef over S. Then either Kx + A
itself is nef over S, or there exists an extremal ray in NE(X/S) such that (Kx + A) - R <
0, (Kx + A + AC) . R = 0 and Kx + A + AC is nef over S.
We are now ready to describe the MMP with scaling.
Let us start by fixing a dlt pair (X, A) and an ample divisor H such that Kx + A + H
is lc and nef over S. Set Xo = X, AO = A, HO = H. We can then define a run of the
(Kx + A)-MMP with scaling of H in the following way.
Step 1 Case 1: Kx, + Ai is nef over S: then we define
(X*,I A* , f.) :=(Xi, Ai, fi)
and the algorithm stops.
Step 2: Kx, + Ai is not nef over S: there exists si such that Kxi + Ai + siHi is
nef over S. We define
si+1 = inf{s > 0 1Kx, + Ai + sHi is nef over S}.
Using Lemma 2.4.10, we can find a Kxi + Ai-negative extremal ray R1
as in the previous section we obtain a contraction morphism contci: Xi
associated to the facet in Nef(Xi) orthogonal to R,.
30
-÷
and
Y
Then we proceed in the same way as above.
1. If contR : Xi
-*
Y is a divisorial contraction, then we just define
(Xi+,, Ai+1, fi+1, Hi+1) := (Yi, contai,*Ai, gi, contai,,,Hi)
and we go back to Step 1.
2. If contR;: Xi -+ Y is a small contraction, then let Xt be the flip:
Xi - - - - - - 1 X i
1
contR
Yi
Then we define
(Xi+ 1, Ai+ 1, fi+1, Hi+1) :=(Xi+, 7rz-, *At, gi o hi, 7ri, Hi)
and we go back to Step 1.
3. If contR%: Xi
-+
Y is a Mori fibre space, then we define
(X*,I A* , f*) :=(Xi, Ai, fi)
and the algorithm stops.
As in the case of a run of the (Kx + A)-MMP over S, we way that a run of (Kx + A)MMP over S with scaling of H terminates if the algorithm just described terminates in
finitely many steps. The good news is that the MMP with scaling is known to terminate in
certain cases.
Definition 2.4.11. A pair (Y, Ay) is a good minimal model for the pair (X, A) over S (or
(Kx + A)-good minimal model over S) if (Y, Ay) is a (Kx + A)-minimal model over S and
moreover Kx + Ay is semiample over S.
Theorem 2.4.12. [11, Cor. 1.3.3, Cor. 1.4.2J Fix a pair (X, A) with Q-factorial klt
singularities. If A is big and Kx + A is pseudoeffective, or if Kx + A is big then there is a
run of the (Kx + A)-MMP with scaling of an ample divisor
(Xo, Ao) --
+
- -. --
+
(Xi, Ai) -- + - - - --
+
(Xk, Ak) = (X*, A*)
which terminates with a (Kx + A)-good minimal model, (X*, A*).
If Kx + A is not pseudoeffective then there is a run of the (Kx + A)-MMP with scaling
of an ample divisor
(Xo,A o)
+
-.
-- +
(Xi,
Ai)
+
-.
-- +
(Xk, Ak) = (X*, A*)
Z.
which terminates with a Mori fiber space p: X,
The latter part is true even when (X, A) has dlt singularities.
-
Hence the termination of sequences of flips remains still open in quite a few cases.
Let us just finally mention that it is possible to formulate an even stronger conjecture
that predicts the existence of good minimal models.
31
Conjecture 2.4.13 (Abundance conjecture). /11, Conjecture 2.3J Let (X, A) be a log canonical pair with Q-factorialsingularities. Assume that Kx + A is pseudoeffective.
Then any run of the (Kx + A)-MMP terminates with a (Kx + A)-good minimal model.
32
Chapter 3
Hyperbolicity and the minimal model
program
3.1
Introduction
Rational curves on algebraic varieties have been carefully studied since the early days of
algebraic geometry. In the last century, many authors turned their attention to the study of
the existence/absence of rational curves and their distribution on a given variety, providing
some interesting discoveries and conjectures. The interested reader can consult [27] for a
survey of classical and more recent results.
Looking back at the Cone Theorem, one immediately understands how restrictive hyperbolicity is in terms of log-divisors and their positivity.
Theorem 3.1.1 (Cone theorem, weak version). Let (X, A) be a projective pair with log
canonical singularities. Assume that X does not contain any rational curve. Then the
divisor Kx + A is nef.
In fact, if Kx + A is not nef, the Cone Theorem implies that there must be a (Kx + A)negative rational curve in X.
In view of this, the following question seems quite natural.
Question 3.1.2. What kind of geometric properties should the pair (X, A) have in order
for Kx + A to be nef?
The purpose of this chapter is to prove the following theorem connecting positivity of
log pairs and hyperbolicity properties of a certain stratification induced by the log pair on
the ambient variety.
k Di be a reduced
Theorem 3.1.3. Let X be a smooth projective variety and let D =
simple normal crossing divisor on X. Assume that there is no non-constant morphism
f : A' -+ X \ D. Assume also that there is no non-constant morphism f : A'-+
1 Di \
Uj 1 Dj, where I C {1, ... , k} is a non-empty set of indices.
Then Kx + D is nef.
More generally, let (X, A) be a log canonical pair. Assume that there is no non-constant
morphism f : A' -* X \ {x E X I A is not klt at x} and the same holds for all the open
strata of the non-klt locus.
Then Kx + A is nef.
f.,
33
We can thus think of Theorem 3.1.3 as a way to answer the previous question.
The assumption on the non-existence of copies of A 1 in the open stratification on X
induced by a simple normal crossing divisor was first introduced by Lu and Zhang in [55]
with the name of Mori hyperbolicity. We generalize their definition to the case of a log pair
(X, A), allowing possibly non log canonical singularities as well, see Definition 3.4.1. Lu
and Zhang proved a version of the above theorem for dlt pairs, assuming some factoriality
conditions on the components, [55, Thm. 3.1]. We generalize their result to the category of
log canonical varieties.
The notion of Mori hyperbolicity for a log pair (X, A) has an inherently inductive nature.
Hence, it is fair to expect that some sort of inductive approach could possibly lead to the
above theorem. Indeed, this is the strategy that we adopt in the course of the proof. A
fundamental step is represented by the following result which makes clear the connection
between the positivity of a pair and the geometry of the non-klt locus of (X, A).
Theorem 3.1.4. [cf. Cor. 3.4.3] Let (X, A), be a log pair. Assume that (X, A) is Mori
hyperbolic. Then Kx + A is nef if it is nef when restricted to its Nklt(X, A).
The proof of Theorem 3.1.3 is then carried out by conducting a careful analysis of
adjunction along lc centers of codimension greater than 1, by means of the canonical bundle
formula. This way, we also obtain a weak formulation of subadjunction for log canonical
pairs (cf. Theorem 3.5.8) and the following strengthening of the Cone Theorem.
Theorem 3.1.5. [cf. Thm. 3.5.9] Let (X, A) be a log canonical pair. Then there exist
countably many (Kx + A)-negative rational curves Ci such that
NE(X) = NE(X)KX+A
O+
ZR> [Ci].
iEI
Moreover, at least one of the two following conditions hold:
" Ci
n (X \ Nklt(A)) contains the image of a non-constant morphism f : A1 -+ X or
" there exists an open stratum W of Nklt(A) such that Ci
non-constant morphism f: A 1 -+ W.
n W contains the image of a
We moreover describe a criterion for the ampleness of Mori hyperbolic dlt pairs.
A fundamental result in algebraic geometry, cf. Theorem 3.6.1, shows that ampleness
of R-Cartier divisors on a projective scheme X can be tested simply looking at its restrictions over all subvarieties of X, via self-intersection numbers and intersection numbers with
integral subvarieties of X.
For a Mori hyperbolic pair (X, A), we prove that this criterion can be restated in a much
simpler form: in fact, it is enough to test ampleness only along the lc centers of A.
Theorem 3.1.6. [cf. Cor. 3.6.51 Let (X, A) be a dlt pair. Assume that (X, A) is Mori
hyperbolic.
Then the following are equivalent:
1. Kx + A is ample;
2. (Kx + A)dim X > 0 and
(Kx +
A)dimw
.
W > 0, for every log canonical center W c X for the pair (X, A).
34
The chapter is structured as follows: in the first two sections we recall some classical
ideas from birational geometry together with the basic definitions and theorems regarding
adjunction along higher codimension le centers. In Section 3.4, we define Mori hyperbolicity
and describe some of its properties. Section 3.5 is devoted to the proof of Theorem 3.1.3,
while in Section 3.6 we prove Theorem 3.1.6.
3.2
Dit modifications
When dealing with a pair (X, A) that is not log smooth easy examples show that the
adjunction formula might need the introduction of a correction term. That is, given a
component D of A of coefficient 1, it could happen that in the adjunction formula
(Kx + D)ID
#
KD.
For more details on this, see [49, 16].
However, when (X, A) is dlt, it is possible to modify the theory and obtain something
analogous to the classical adjunction setting, that furthermore behaves well when restricting
to higher codimension lc centers.
Theorem 3.2.1. Let (X, A) be a dit pair and W C X a
naturally defined R-divisor Diff*A > 0 such that
(Kx + A)|w
lc center. There exists on W a
Q Kw + Diff * A
and the pair (W, Diff * A) has dit singularities. Moreover, the non-kit locus of (W, Diff* A)
is equal to the union of the lc centers of A strictly contained in W.
The divisor Diff *A can be defined inductively starting as in [49, Sec.16] from the case
in which W = D is a divisor. Then
(Kx + D + (A - D))D -Q KD + Diff* A.
Working inductively, Diff*wA is constructed analogously whenever W is an irreducible component of a complete intersection of divisors in [AJ. In the case of dlt singularities, every
lc center is of this form.
Definition 3.2.2. The divisor Diff * A from Theorem 3.2.1 is called the different of A on
W.
An important fact, that will be needed multiple times in the following sections is that,
starting with an lc pair, there always exists a crepant resolution giving a dlt pair.
Theorem 3.2.3. Let (X, A = EZ bjDj) be a log pair, 0 < bi < 1. Then there exists a
Q-factorial pair (Y, Ay = E> bjAj 0) and a birationalmap 7r: Y -+ X with the following
properties:
1. Ky + Ay = 7r*(Kx + A);
2. the pair (Y A' := Eij.<j bjAj + Eijbj>1 Ai) is dlt;
3. every divisorial component of Exc(7r) appears in A' with coefficient 1;
35
4. -7r-1 (Nklt(A)) = Nklt(Ay) = Nklt(A'y).
Proof. For the proof of (1), (2), (3) one can refer to [50, 3.10]. Let 7rz: (Z, Az) -+ X be a
modification satisfying these properties. Then
Az
=
A<, + A
=
E
biDi + E
ijbj<1
biDi
(3.1)
ijbi>1
and (Z, A<') is a klt pair. Moreover, as Kz + Az = 7r* (Kx
+ A),
Kz + A< 1
(3.2)
Therefore, we can run a relative (Kz + A<1 )-MMP over X, 4: (Z, A<')
0',Af ) and reach a model Z' on which the following conditions hold true:
--
+ (Z', A
a) (Z', A<1) is a Q-factorial, klt pair;
b) Kz' + A!f + A jI = 7r*,(Kx + A), where A:
structural map;
:= O*A$ 1 and 7rz': Z' -+ X is the
c) Kz' + A </ is 7rz'-nef and by (3.2) the same holds for -Ar>/l.
Properties a) and b) imply that Nklt(A' + Azf) = Supp(Ai:). In fact, the inclusion
Nklt(AZl + A!l) 2 Supp(Aj:) follows form Definition 2.3.5. To prove the other inclusion, let W be a non-klt center not contained in Supp(A jf ). There exists a log resolution
r: (S, AS) -+ (Z', Ajl + Aj1) and a component F1 of As whose coefficient is > 1 and
cz'(F) = W. As cz'(F1) g A !', it follows that a(Fi; Z', A </) < 0 as well, which is
impossible as (Z', A V) is kit.
Finally, take another dlt modification
(Y, AY) --+ (Z', Azf +
F)
FjcSupp(A ,)
with properties (1), (2), (3) from the statement of the theorem. The divisr -k*(Aj)
will be a 7r-nef divisor, where 7 = 7rz' o 4. The support of -0*(A') contains all and only
those components of Ay of coefficient > 1. By negativity, [52, Lemma 3.39], 7r: Y -- X
satisfies condition (4) of the theorem.
3.3
3.3.1
Subadjunction for higher codimensional ic centers
Canonical bundle formula
Definition 3.3.1. [34] An ic-trivial fibration is the datum of a contraction of normal varieties 7r: Y -+ Z and a pair (Y, Ay) s.t.
1. (Y, Ay) has subic singularities over the generic point of Y, i.e., Nlc(Ay) does not
dominate Z and Ay could possibly contain components of negative coefficient;
36
2. rank 409*([A*(YA)1) = 1, where ir = 7r o 1 and 1: Y -+ Y is a log resolution
of (Y, Ay). A*(Y, A) is the b-divisor whose trace on Y is defined by the following
equality
K; = *(Ky + Ay) +
aiDi+A*(YA)ip.
S
3. there exist r E N, a rationalfunction 0 E k(Y) and a Q-Cartierdivisor D on Y s.t.
Ky + Ay +
1
r
-(0)
= wr*D,
i.e. Ky + Ay ~,Q 0.
At times, we will denote an lc-trivial structure by
r: (Y, Ay)
-+
(3.3)
Z.
Definition 3.3.2. An integral subvariety W C Z is an lc center of an lc-trivial fibration
ir: Y -+ Z, if it is the image of an lc center Wy C Y for (Y, Ay).
Remark 3.3.3. A sufficient condition for (2) in definition 3.3.1 to hold is that Ay is log
canonical, in which case,
=FKj - 7r*(Ky + Ay) +
[A*(Y, Ay)p
E]
a(E,Y,Ay)=1
is always exceptional over Y.
Let us notice that under this hypothesis, an
structure in the sense of [46, Def. 2].
lc-trivial fibration is also a crepant log
Example 3.3.4. One of the main reasons to study lc-trivial fibrations comes from resolutions and adjunction. Let (X, A) be an lc pair and W C X an lc center. In the purely
lc case, when (X, A) is not dlt, the structure of Nklt(A) is not as easily determined as in
Theorem 3.2.1. Nonetheless, Theorem 3.2.3 shows that it is always possible to pass to a dlt
pair crepant to the original one. Let 7r: X' -+ X be a dlt modification as in the Theorem
3.2.3, with
Kxl + Ax' = ,r*(Kx + Ax).
Let S be a log canonical center of Ax', i.e., an irreducible component of intersections
of components of coefficient 1. Let W be its image on X. Taking the contraction in the
Stein factorization of 7rls: S -+ W and considering the pair (S, Diff * Ax/) yields an lc-trivial
fibration.
Starting with an lc center S minimal among those dominating W the singularities of
(S, Diff *Ax,) are actually of klt type over the generic point of W.
Definition 3.3.5. Given an lc-trivial fibration -x: (Y, Ay) -+ Z as above, let T C Z be a
prime divisor in Z. The log canonical threshold of r* (T) with respect to the pair (X, A) is
aT
= sup{t E RI (Y, Ay + t7r*(T)) is lc over T}.
We define the discriminant of
7r:
(Y, Ay) -+ Z to be the divisor
Bz
:= ET (1
- aT)T.
(3.4)
It is easy to verify that the above sum is finite: a necessary condition for a prime divisor
to have non-zero coefficient is to be dominated by some component of Bz of non-zero
coefficient. There finitely many such components on Y. Hence, Bz is an R-Weil divisor.
37
Definition 3.3.6. Let 7r: (Y, Ay) -÷ Z be an ic-trivialfibration. With the same notation as
in equation (3.3), fix # e k(Y) for which Ky + Ay + (0) = ,r*D. Then there is a unique
divisor MZ for which the following equality holds
1
r
Ky + Ay + -(#)
=
r*(Kz + Bz + Mz).
(3.5)
The Q- Weil divisor Mz is called the moduli part.
When dealing with an ic-trivial fibration, 7r: (Y, Ay) -+ Z, one can pass to a higher
birational model of Z, Z', take a higher birational model Y' of the normalization of the main
component of the fibre product Y xz Z' and form the corresponding cartesian diagram,
Y
Y'
Z <cr
Z'.
(3.6)
By base change, we get a new pair, (Y', Ay,), from the formula
Kyl + Ayt = r*,(Ky + Ay ).
It follows from the definition that, under these hypotheses, 7r': (Y', Ay,) -+ Z' will be
an ic-trivial fibration as well, allowing to compute Bz' and Mzi.
The discriminant and the moduli divisor have a birational nature: they are b-divisors,
as their definition immediately implies that
r*Bz'
=
Bz, and r*Mz' = Mz.
As they are b-divisors, we will denote them using the symbols B and M, respectively.
Fujino and Gongyo proved, generalizing results of Ambro, that these divisors have interesting features.
Theorem 3.3.7. ([34], [2]) Let vr: (Y, Ay) --+ Z be an ic-trivialfibration. There exists a
birational model Z' of Z on which the following properties are satisfied:
(i) Kzl + Bzt is Q-Cartier, and p*(Kz, + Bz')
I: Z" -+ Z'.
Kz" + Bz" for every higher model
(ii) Mzt is nef and Q- Cartier. Moreover, /* (Mz') Mz" for every higher model p: Z" -+
Z'. More precisely, it is b-nef and good, i.e., there is a contraction h: Z' -+ T and
Mz' = h*H, for some H big and nef on Z'.
When the model Z' satisfies both conditions in the theorem, we say that B and M
descend to Z'.
3.4
Mori hyperbolicity
Definition 3.4.1. Let (X, A = E> biDi), 0 < bi < 1 be a log pair. We say that (X, A) is a
Mori hyperbolic pair if
1. there is no non-constant morphism f : A'
38
-+
X \ Nklt (A);
2. for any W C X
lc center, there is no non-constant morphism
f: A'
-
W
\ {(W n Nlc(A))
U Strat(W, A)}.
The following result is already implicitly contained in [55, 4]. We restate it here for the
reader's convenience since it does not appear there in this generality. The following proposition is the starting point of our approach to the proof of Theorem 3.1.3.
Proposition 3.4.2. Let (X, A =
such that (X, A' = Zjlbi<1 biDi +
>i biDi
Zibi
;> 0) be a normal, projective, Q-factorial log pair
1 Di) is dlt.
Suppose that Kx + A is nef when restricted to Supp(A l). Then
" either Kx
+ A is nef or
" there exists a non-constant morphism
f:
A1 -+ (X \ Nklt(A)).
Proof. Suppose Kx + A is not nef. Then there exists a (Kx + A)-negative extremal ray,
R in the cone of effective curves, NE(X). Since Kx + A is nef on Nklt(A), R is both a
(Kx + A')-negative and a (Kx + AiC)-negative extremal ray. In particular, there exists an
extremal contraction p: X -+ S associated to R.
As R does not contain classes of curves laying in Nklt(A), p induces a finite morphism when
restricted to Nklt(A). Thus, the Q-factoriality of X implies that we are in either of these
three cases:
1) p is a Mori fibre space and all the fibres are one dimensional;
2) IL is birational and the exceptional locus does not intersect Nklt(A);
3) p is birational and the exceptional locus intersects Nklt(A).
As p is a (Kx + A<)-negative fibration and Kx + All is klt, then all of its fibres are
rational chain connected, by [37, Corollary 1.51. Moreover,
Rltu.Ow = 0,
(3.7)
by relative Kawamata-Viehweg vanishing [54, page 1501. Thus, Theorem 2.3.7 implies that
Nklt(A') = Nklt(A) is connected in a neighborhood of every fibre.
In case 1), the generic fibre of p is a smooth projective rational curve. Theorem 2.3.7 implies
that the generic fibre intersects Nklt(A) in at most one point. This conludes the proof in
case 1).
In case 2), as the fibres of p are rationally chain connected, there exists a rational projective
curve contained in X \ Nklt(A). This conludes the proof in case 2).
In case 3), the positive dimensional fibres are chains of rational curves and by the vanishing
in 3.7 above, the generic fibre has to be a tree of smooth rational curves. By Theorem 2.3.7,
Nklt(A) intersects this chain in at most one point. In particular, there exists a complete
rational curve C such that C n (X \ [AJ) = f(A 1 ), where f is a non-constant morphism.
This conludes the proof in case 3).
F1
In the case of a general log pair, using dlt modifications we get the following criterion, which
will be fundamental in the proof of Theorem 3.1.3.
39
.
Corollary 3.4.3. Let (X, A = E> biDi ;> 0), 0 < bi <; 1 be a log pair. Assume that there is
no non-constant morphism f: A' -+ X \ Nklt(A),
Then Kx + A is nef if and only if Kx + A is nef when restricted to Nklt(A).
Proof. Nefness of Kx + A immediately implies nefness of its restriction to every subscheme
of X. Hence, we just have to prove the converse implication.
Let 7r: (X', Ax,) -+ (X, A) be a dlt modification for (X, A) as in Theorem 3.2.3. We can
reduce to proving nefness for Kx' + Ax,. As 7r(Nklt(Ax,)) = Nklt(A), Kx' + Ax, is nef
when restricted to Nklt(Ax,).
Suppose Kx' + A' is not nef. By the proposition, there exists a non-constant morphism
f : A' -+ (X' \ Nklt(A')). This contradicts the assumption in the statement of the corollary,
as the properties of dlt modifications imply that the image of 7r o f lies in X \ Nklt (A). E
Let us notice that in the above corollary, we did not impose any condition on the singularities
of A, besides the coefficients being in [0, 1].
3.5
Proof of theorem 3.1.3
We will work inductively on the strata of Nklt(A). Namely, we will prove that Kx + A is
nef when restricted to every stratum of Nklt(A). As the union of all the strata is the non-klt
locus itself, the theorem will follow from Corollary 3.4.3.
Step 1. Start of the induction: the case of minimal Ic centers.
When W is a minimal lc center, then nefness of (Kx + A) w follows from the following
classical result in the MMP.
([33], /2] or /421). Let (X, A) be a log canonical pair and W a minimal lc center. Then there exists an effective divisor Aw on W s.t.
(W, Aw) is klt and
(Kx + A)|w ~R Kw + Aw.
Theorem 3.5.1 (Kawamata subadjunction).
Since (Kx + A)lw -R KW + Aw and by definition of Mori hyperbolicity W does not
contain rational curves, it follows that Kw + Aw must be nef by the Cone theorem.
Step 2. Moving the computation to the spring of W.
We assume now that W is no longer minimal and that Kx + A is nef when restricted
to any other stratum W' strictly contained in W. Recall the following notation
U
Strat(W,A)=
W'
w'w,
W' Ic center
to indicate the union of all substrata contained in W.
Let us fix a dlt modification of (X, A), 7r: (X', A') -+ (X, A). We also fix a non-klt
center W c X and let S C X' be an ic center, minimal among those dominating W. Let us
consider the Stein factorization
7
S
7rls: S -* Ws E
W.
The variety WS is normal, projective and is naturally equipped with the R-divisor
L: = spr*w(Kx + A).
40
The morphism 7rs: S -+ Ws is an lc-trivial fibration with respect to As = Diff *Ax, on S,
as we saw in Example 3.3.4 and it is also a dlt log crepant structure. The following theorem,
due to Kollir, shows that the contraction irs: S -+ Ws already contains all the relevant
data in terms of the geometry of the non-klt locus.
Theorem 3.5.2. [46, Cor. 11] Let 7r: (Y, A) -+ Z be a dit crepant log structure and S C Y
be an ic center, with ir(S) = W. Consider the Stein factorization
7rw : S -1-
Ws
(3.8)
W
s2
and let AS: = Diff *Ay be the different of Ay on S. Then:
1. ?rs: (S, AS) -+ Ws is a dlt, crepant log structure;
2. Given an lc center Zs C Ws for ?rs, sprw(Zw) c W is an lc center for ,r: (Y,Ay)A
Z. Every minimal lc center of (S, As) dominating Zs is also a minimal lc center of
(Y, Ay) and dominates ir(Zw).
3. For Z c W an lc center of 7risI: (S, As) -+ W, every irreduciblecomponent of spr- I (Z)
is an lc center of irs: (S, As) -+ Ws. We denote the total space of this stratification
by
Strat(Ws,As):=
V
U
w'
w'-4w,
Ic center
U
V irreducible
component of
spr 1 (W)
Remark 3.5.3. Kolidr proved that the isomorphism class of the variety Ws over W in
Theorem 3.5.2 is independent of the choice of S. He also proved that for any two pairs
(Si, AsI), (S 2 , As 2 ) such that the Si are minimal among the Ic centers dominating W the
varieties S1 and S 2 are birational and there exists a common resolution pi: W -+ Si, i = 1, 2
such that
p*(Ks 1 + AsI) = pi(Ks 2 + As 2 ),
see [46, Thm. 1].
Definition 3.5.4. [46, Def. 18 and page 10] With the notation of Theorem 3.5.2, let S be an
lc center of (Y, A) minimal with respect to inclusion among the lc centers T with ir(T)
W.
We call the pair (S, As = Diff * Ay) a source of W.
The normal variety Ws appearingin the Stein factorizationof the morphism irisI: S -+ W
in 3.8 is called the spring of W.
Proving nefness of (Kx + A)I W is equivalent to proving nefness of L and we can assume
that L is nef on Strat(Ws, As) since
Strat(Ws, As) = sprw
1
(Strat(W, A)),
by 3. in Theorem 3.5.2.
Hence, without loss of generality, we could substitute the triple (W, (Kx+A) Iw, Strat(W, A))
with the triple (Ws, L, Strat(Ws, As)). In fact, if L is not nef, then we will show that there
exists a non-constant morphism f : A' -+ Ws \ Strat(Ws, As). By Theorem 3.5.2, it follows
that there exists a non-constant morphism f': A' -+ W \ Strat(W, A), violating the Mori
hyperbolicity assumption for W.
41
To ease the notation, in the following we will denote Ws simply by W and Strat(Ws, As)
by Strat(W, As).
Step 3. Constructing a good approximation for L on W.
By the results of Section 3.3, there exist sufficiently high birational models S' of S and
W' of W together with a commutative diagram
S <
S'
7rS I
(3.9)
Is'
W <rW'
having the following properties:
1. r*(L) = Kw' + Bw' + Mw,;
2. (W', Bw') is log smooth and sublc, i.e., Bw' is not necessarily effective;
3. Kw, + Bw' descends to W' and Mw' is nef and abundant.
4. (S', As') is a sublc pair, where K' + As' = rs,(Ks + As);
In this context, we compare singularities of (W', Bw') with those of the original pair
(W, A).Lemma 3.5.5.
With the above notation and hypotheses, we have that r(Nklt(Bw'))
Strat(W, As).
Proof. We know that rs'(Nklt(As')) = Nklt(As) and 7rs(Nklt(As)) = Strat(W, A). As
the diagram in (3.9) commutes, we need to prove that 7rs'(Nklt(As)) = Nklt(Bw'). The
definition of Bw' implies that every stratum of Nklt(Bw') c W' is dominated by a stratum
of Nklt(As'), hence Nklt(Bw') c 7rs'(Nklt(As')). The opposite inclusion is also true, as
given a stratum of Nklt(As
5 ), up to going to higher models of W' and S', we can suppose
that D is a divisor whose image D' on W' is a divisor, too. In this case, by the definition of
Bw' and since it descends to W', D' c Nklt(Bw'). Thus, Nklt(Bw') D 7rs,(Nklt(As,)).
0
As proving that L is nef is equivalent to proving that, for any given ample Cartier divisor
A on W and any given e > 0, L + EA is nef, we focus on the divisor
r*(L + EA) = Kw' + Bw' + Mw' + r*(eA).
(3.10)
By construction, we can assume that there exists an effective divisor E supported on
the exceptional locus of r and -E is relatively ample over W. Hence, there exists a positive
number 0 , < c, such that for any 0 < 6 < 0, Mw' + r*(eA) - 6E is an ample divisor on W'.
Lemma 3.5.6. For every c > 0, there is a suitable choice of 6 and of an effective R-divisor
QE ~R Mw' + r*(EA) - 6E for which the following equalities hold
Nklt(Bw' + 6E + Qe) = Nklt(Bw' + 6E) = Nklt(Bw').
With this notation,
r*(L + EA)
RKw' + Bw' + 6E + Qr.
42
(3.11)
Proof. The first equality is a consequence of [54, Proposition 9.2.261, once we choose 6 small
enough so that QE is ample. The second equality follows immediately from the fact that we
can choose 6 to be arbitrarily small, since (W', Bw') is log smooth and sublc.
E
Step 4. End of the proof.
Using Lemma 3.5.6, we define a new divisor on W
FE := r*(Bw' + 6E + Qe).
The pair (w, FE) is a log pair and its coefficients are real numbers in [0, 1]. By construction,
those coefficients in Bw' + 6JE + Q, that are strictly larger than 1 were those of components
that are exceptional over W. Also, L + eA -R KW + F and we are reduced to proving
nefness for Kw +FE, for e < 1. The pair (W, F) fails to be lc but Nklt(FE) = Strat(W, As),
by Lemma 3.5.5 and Lemma 3.5.6. Moreover, Kw + F is nef, more precisely ample, when
restricted to its non-klt locus. Hence, it is nef on W by Corollary 3.4.3. Since this holds
for arbitrary choice of e > 0, it follows that L is nef on W, terminating the proof of the
inductive step and of the theorem.
0
Remark 3.5.7. In Section 3.5, we proved the following (very) weak version of (quasi log
canonical) subadjunction. Surely, this is not the most desirable version of subadjunction
that is expected to hold, as we explain below.
Theorem 3.5.8. Let (Y, A) be a log canonicalpair and 7r: Y -+ Z be an lc trivial fibration.
Let A be an ample divisor on Z.
Then for all e, 6 > 0, there exists an effective divisor FE,6, with coefficients in [0, 1]
satisfying the linear equivalence relation
Kz + Bz+ Mz + cA ~R Kz + FEs.
The pair (Z, FE,6) is not log canonical, but there exists a log resolution 7r: Z' that the log discrepancy of the 7r-exceptional divisors is bounded below by -6, i.e.
Z such
a(E; Z, FE,) > -6, for every E C Z' prime divisor exceptional over Z.
A much stronger result should hold under the hypotheses of Theorem 3.5.8. The moduli
b-divisor, M, is expected to be semi-ample on a sufficiently high birational model of Z. That
would easily imply that, for a certain choice of Mz, (Z, Bz + Mz) is log canonical. If that
were to be true, the proof of Theorem 3.1.3 could be considerably simplified. In fact, L
would be linearly equivalent to the lc divisor Kz + Bz + Mz and
Nk1t(Bz + Mz) = Nk1t(Bz) = Strat(W, A).
In the proof of the Theorem 3.1.3 we showed that if Kx + A is not nef, there is a nonconstant morphism f: A' -+ X whose image is contained in an lc center W c X and it
does not intersect the Ic centers strictly contained in W. In particular, from the inductive
procedure used in the proof, we see that it is possible to select W to be a minimal lc center
among those on which the restriction of Kx + A is not nef.
43
Theorem 3.5.9. Let (X, A) be a log canonical pair. Then there exist countably many
(Kx + A)-negative rational curves Ci such that
NE(X) = NE(X)Kx+A O +
R o[Ci].
iEI
Moreover, one of the two following conditions hold:
n (X \ Nklt(A))
f : A' 1 X \ Nklt(A);
" Ci
contains the image of a non-constant morphism
" there exists an lc center W C X such that Ci n (W \ Strat(W, A)) contains the image
of a non-constant morphism f : A' -+ (W \ Strat(W, A)).
In an attempt to expand the above results to arbitrary singularities, the following questions appear quite natural.
Question 3.5.10. Let (X, A =
biDi > 0),0 < bi
1, be a Mori hyperbolic log pair.
Assume Kx + A is nef when restricted to Nlc(X, A). Is Kx + A nef? Is it possible to drop
the assumption 0 < bi < 1?
Most of the proof of Theorem 3.1.3 applies to the case of varieties with worse singularities
than log canonical, through the language and techniques of quasi log varieties introduced in
[1]. It seems that, in order to finish the proof, one would have to prove a stronger version
of the Bend and Break Lemma. Unfortunately, we are not able to prove such a result at
this time, hence the above question remains still open. Some results in this direction were
recently proved by McQuillan and Pacienza in [58], for quotient singularities.
To address Question 3.5.10, one could mimic the same proof as for Theorem 3.1.3.
Namely, starting with a log pair (X, A) such that the coefficients of A are in [0, 1], no
matter what the singularities of A are, it is sufficient to prove that Kx + A is nef on
Nklt(A), by Corollary 3.4.3. As there is very little control on the non lc locus of A (cf. [1,
Theorem 0.2]), it seems inevitable to assume the nefness for the restriction of Kx + A. In
this setting, the formalism of the canonical bundle formula is not available anymore, but in
order to study adjunction or just the restriction of Kx + A to lc centers of A, the formalism
of log varieties can be used (cf. [1] and [32]). Again, working by induction, one can restrict
to a given stratum, W, and assume that nefness is known for the smaller strata and the
intersection with the non-lc locus. Assuming by contradiction that (Kx + A) Iw is not nef,
then we can find a contraction morphism 7r: W -+ S which contracts curves with (Kx + A)negative class in a given extremal ray contained in NE(X). It is not hard to prove that
the fibres of 7r will contain rational curves. The hard part is to prove that it is possible to
deform one of these curves to a rational curve whose normalization supports the pull-back
of A at most one point. The classical tool to deform curves is surely the Bend and Break
Lemma, although in this case, we need not only to be able to deform a curve, but also we
would like to be able to control its intersection with the components of A. Hence, ideally,
one would like to prove a stronger version of the Bend and Break Lemma that makes the
above construction possible.
44
3.6
Ampleness and pseudoeffectiveness
pairs
for Mori hyperbolic
In the dlt case, one can go further and describe conditions that imply ampleness of Kx + A
as described in Theorem 3.1.6 in the introduction. Let us first recall the following classic
result.
Theorem 3.6.1 (Kleiman). Let X be a proper variety and let D be a Cartierdivisor on X.
Then D is ample on X if and only if for every proper subvariety variety Y 9 X
jDdimY
>0.
We will also need the following definition.
Definition 3.6.2. Let (X, A) a log canonical pair. An R-divisor D is log big (with respect
to (X, A)) if D is big and DIw is big for any lc center W of A.
Proposition 3.6.3. Let (X, A) be a log canonical pair. Then the following are equivalent:
(1) the divisor Kx + A is ample;
(2) the divisor Kx + A is big, its restriction to Nklt(A) is ample and Kx + A has strictly
positive degree on every rational curve intersectingX \ Nklt(A).
If (X, A) is dlt, then the above conditions are also equivalent to:
(3) the divisor Kx +A is nef and log big and it has strictly positive degree on every rational
curve.
Remark 3.6.4. The assumption on the bigness of Kx + A in the proposition is necessary
as the following example shows.
Let E be a curve of genus 0. Then KE ~ 0 and the pair (E, 0) is terminal (hence, log
canonical) with empy non-klt locus. The curve E clearly does not contain rational curves,
nonetheless KE is not ample.
Proof of Proposition 3.6.3. Clearly condition (1) implies conditions (2) and (3).
Condition (2) implies that Kx + A is nef. In fact, by the Cone Theorem, an extremal
ray contained in NE(X) on which Kx + A is negative is spanned by the class of a rational
curve C C X. As Kx + A is ample along Nklt(A), C must intersect X \ Nklt(A), which
gives a contradiction.
Thus, Kx + A is big and nef and it is ample along Nklt(A). It follows that Kx + A is
semiample, by [32, Thm. 4.1]. The corresponding morphism is either an isomorphism or it
has to contract some rational curves intersecting X \ Nklt(A) as implied by [37, Thm. 1.21.
But this also gives a contradiction, as the intersection of Kx + A with such curves must be
strictly positive. Then (2) implies (1).
Let us prove that (3) implies (2). Since Kx + A is nef and log big, it is also semiample.
By induction on the dimension and using Theorem 3.2.1, it follows that Kx + A is ample
along [AJ, which concludes the proof.
Theorem 3.6.5. Let (X, A) be a Mori hyperbolic log canonical pair.
Then the following are equivalent:
45
(1) Kx + A is ample;
(2) Kx + A is big and its restriction to LA] is ample.
If (X, A) is dlt, then the above conditions are also equivalent to:
(3) Kx + A is log big.
Remark 3.6.6. As (X, A) being Mori hyperbolic implies that Kx + A is nef, condition 2)
in the corollary is equivalent to the condition stated in Theorem 3.1.6:
(Kx+ A)dimx > 0 and (Kx+ A)dimw
.
W > 0, for any ic center W.
Remark 3.6.7. The assumption on the bigness of Kx + A in the theorem is necessary.
In fact, the pair (P', {0} + {oo}) is log canonical and its lc centers are the points 0 and
oc. The divisor Kp + {0} + {oo} is clearly ample along the two lc centers, yet the divisor
is linearly equivalent to 0.
Proof of Theorem 3.6.5. Again, (1) implies (2) and (3). Moreover, as (X, A) is Mori hyperbolic, it is nef.
Let us prove that (2) implies (1). As Kx + A is big and ample along LAJ, to prove
its ampleness on X, it suffices to prove that Kx + A intersects all rational curves on X
with strictly positive degree. Let us assume there exists a rational curve C such that
(Kx + A) - C = 0. We can assume that Kx + (A - cLAJ) - C < 0, for any e > 0. Let us
notice that Kx + (A - e LA]) is ample along [A] for 0 < c < 1. Passing to a dlt modification
as in Theorem 3.2.3, we can assume that X is Q-factorial and the proof is the same as that
of Proposition 3.4.2.
If (3) holds, then by induction on dim X it follows immediately that Kx + A is ample
along [A]. Moreover, the definition of log bigness implies that Kx + A is also big, which
terminates the proof.
El
46
Chapter 4
Fano varieties in Mori fibre spaces
4.1
Introduction
In this chapter we focus on a natural question which arises in the context of the classification
of complex algebraic varieties and in the Minimal Model Program. The purpose is to clarify
the geography of Mori fibre spaces.
Question 4.1.1. Which Q-factorialFano varieties can be realised as generalfibres of a Mori
fibre space?
Although every Fano variety of Picard number one is a Mori fibre space over a point,
this work gives evidence about the restrictiveness of this condition for varieties of higher
Picard rank.
The notion of "general fibre" will be clarified later in Section 4.2: the idea is to determine
an open dense subset of the base, on which the fibres are "good enough" (cf. Definition
4.2.16).
Fano varieties play an essential role in the birational classification of projective varieties
with negative Kodaira dimension. Their importance was already highlighted in low dimension in [61]. The seminal work [11] shows that every Q-factorial variety with klt singularities
and non-pseudoeffective canonical divisor is birational to a Mori fibre space (or simply MFS),
i.e., to a contraction morphism with positive dimensional Fano fibres and relative Picard
number one. In this work, we will also assume the existence of a dense open set of the base
over which the fibres are Q-factorial.
Since Mori fibre spaces arise as final outcomes of a run of the MMP, they have been
widely studied for the last thirty years in the context of classification of higher dimensional
varieties.
It is important to underline that distinct Mori fibre spaces can belong to the same
birational class, as shown already in dimension 2 by elementary transformations between
ruled surfaces. Relations between Mori fibre spaces within a birational class (the so-called
Sarkisov program) were investigated by [22] in low dimension. The same picture has been
proved to endure in higher dimension in [38]: two Mori fibre spaces within the same birational
class can be related via a sequence of very easy birational maps, called Sarkisov links.
Another interesting notion for Mori fibre spaces which appears in the literature is birational
rigidity (cf. [15]). Although so many properties have been investigated, the geometric
structure of Mori fibre spaces remains quite mysterious and very few explicit examples are
known.
47
In this work we focus on the classification of the fibres of Mori fibre spaces rather than
the total space.
The main results of this paper are the following criteria (cf. Theorem 4.4.1, Theorem 4.4.4
and Theorem 4.4.5). We will denote by Mon(F) the maximal subgroup of GL(N 1 (F), Z)
which preserves the birational data of F (cf. Definition 4.2.11) and by Aut(F) the image of
the natural homomorphism of the automorphism group of F with value in GL(N 1 (F), Z).
Moreover, for a given group G acting on N' (F)Q, we will denote by N1 (F)G the G-invariant
subspace, i.e. the subspace of classes v E N1 (F)Q such that gv = v, Vg E G.
Theorem 4.1.2.
" Sufficient criterion: A terminal Q-factorial Fano variety F can be realised as the
general fibre of a MFS if
N1(F) ut(F) = QKF.
" Necessary criterion: A terminal Q-factorial Fano variety F such that
dim N1(F) Mon(F) > I
cannot be realised as the general fibre of a MFS.
" Characterisation for rigid varieties: Assume that H 1 (F, TF)
Then the sufficient criterion turns into a characterisation.
=
0, i.e. F is rigid.
Our results rely upon a careful study of the monodromy action on Mori fibre spaces.
Remark 4.1.3. The notion of being realised as the general fibre of a MFS (or fibre-likeness,
see Definition 4.2.16) is quite subtle and it is important to remark that the necessary criterion
holds true also for Mori fibre spaces which are not isotrivial.
We can use our criteria to prove the following theorem (cf. Theorems 4.3.1 and 4.3.8,
Corollaries 4.3.2 and 4.3.13).
Theorem 4.1.4.
* Surfaces: A smooth del Pezzo surface can be realised as the general fibre of a MFS if
and only if it is not isomorphic to the blow-up of P2 in one or two points.
* Threefolds: The deformation type of a smooth Fano threefold F with p(F) > 1 can be
realised as the general fibre of a MFS if and only if it is one of the 8 classes appearing
in Table 4.3.3.
In particularfibre-likeness is invariant under smooth deformations for Fano varieties of
dimension up to three; moreover, in these cases the necessary criterion of Theorem 4.1.2 is
actually a characterisation.
Let us point out that cubic surfaces in P 3 provide examples of varieties that do not
satisfy the sufficient criterion in Theorem 4.1.4 but can be realised as the general fibre of
a MFS. To deal with Fano threefolds in Section 4.3, we give some ad hoc versions of the
necessary criterion explicitly in terms of the birational geometry of F. We do not know if
there are higher dimensional examples of smooth Fano varieties which are not fibre-like but
still verify the necessary criterion of Theorem 4.1.2.
48
Remark 4.1.5. The 2-dimensional case of Theorem 4.1.4 has been worked out in [60,
Theorem 3.51 when the dimension of the total space of the Mori fibre space is three. In
Section 4.3 we give an alternative proof using our criteria, allowing total spaces of arbitrary
dimension.
Mori and Mukai classified all smooth Fano threefolds with Picard number bigger than
one up to deformation into 88 classes in [62] and [63]: Theorem 4.1.4 shows how restrictive
the fibre-likeness condition is.
The second part of Theorem 4.1.4 can be deduced by combining our criteria with the
classification result of Prokhorov, see [68, Theorem 1.2]. We prove the theorem without
using Prokhorov's work, but looking directly at Nef(F).
On the positive side, we can give the following examples of varieties of higher dimension
and large Picard number which can be realised as the general fibre of a MFS.
Definition 4.1.6. Let Z be a variety and L 1 ,..., Lk Cartierdivisors on Z.
Let I be the subvariety of Z x|L1|x ..I|Lk| such that for any divisors D1 c IL1|,..., Dk E
|LkI, the fiber of the projection morphism I -+ |L x ... LkI above (D 1,..., Dk) equals the
complete intersection D1 n ... n Dk in Z.
We call I the incidence variety.
Theorem 4.1.7 (= Theorem 4.3.5). Let F be a smooth Fano variety. Let Z be a smooth
projective variety. Assume that Li, 1 > i > k are basepoint-free effective Cartier divisors
on Z and that F is the complete intersection of the divisors Li in the ambient variety Z.
Let I the incidence variety in Z x |L1| x ... Lk I-. Suppose that there is a finite subgroup
G of Aut(Z) which is fixed point free in codimension one and whose action can be lifted to
I. Assume that G does not preserve F. If
dim N 1 (Z)G = 1,
then F is fibre-like.
We will give a few concrete applications of this result; let us point out one of them.
Corollary 4.1.8 (= Corollary 4.3.6). Take positive integers r, k, d, and n > 2 such that
kd < n + 1. Let F be a smooth complete intersection of k divisors of degree (d,. .. ,d) in
(Pfl)r. Then F is fibre-like.
In the case of smooth toric Fano varieties, we obtain the following necessary condition.
Theorem 4.1.9 (= Thm 4.5.7). Let F be a toric Fano variety and let E c N be the fan
associated to it. Let A be the polytope whose vertices are the integral generators of the
1-dimensional cones of E.
If F can be realised as the general fibre of a MFS then the barycentre of A is in the
origin.
Remark 4.1.10. Smooth fibre-like toric varieties have been completely classified in low
dimension: as Table 4.5.3 shows, it is a rather restrictive condition.
We do not know if the fibre-likeness condition (i.e., the property of being realised as
the general fibre of a MFS) is open in families. Our necessary criterion in Theorem 4.1.2
is invariant under flat deformation, while the sufficient criterion is closed in families, but it
produces just a special kind of Mori fibre spaces: the isotrivial ones.
49
Corollary 4.1.11 (cf. Cor. 4.3.3 and Thm. 4.5.7).
.
In the toric case we find an interesting relation between fibre-likeness and K-stability,
see [66, Def. 2.6] for the definition of K-stable and K-semistable. We recall that, by [19],
K-stability for a Fano variety is equivalent to the existence of a Kdhler-Einstein metric.
" A smooth del Pezzo surface is K-stable if and only if it can be realised as the general
fibre of a MFS;
" if a smooth toric Fano variety F appears as the general fibre of a Mori fibre space,
then it is K-stable.
Furthermore, we suspect that fibre-like smooth Fano threefolds are K-stable. Inspired
by [66], we propose the following question, which is the relative version of a conjecture by
Odaka and Okada.
Question 4.1.12. Is it true that every smooth Fano variety which can be realised as the
general fibre of a MFS is K-semistable?
The chapter is structured as follows: in Section 4.2 we introduce the necessary definitions
and preliminary results and we start discussing the structure of Mori fibre spaces from the
topological and the algebro-geometric perspective. In Section 4.3, we recall the case of
surfaces and we fully classify what smooth Fano threefolds are fibre-like. In Section 4.4 we
show some sufficient and necessary conditions for being fibre-like. Section 4.5 deals with
the smooth toric case and discusses some relations to K-stability. In section 4.6 we classify
certain fibre-like homogeneous spaces.
4.2
4.2.1
Preliminary results
Monodromy Action and Deligne's Theorem
We will discuss some facts about monodromy of fibrations in Fano varieties. Let us recall a
basic definition.
Definition 4.2.1. A normal projective variety F is said to be Fano if its anti-canonical
divisor -KF is Q-Cartier and ample, i.e., there exists a sufficiently high multiple -nKF
which is Cartier and the invertible sheaf OF(-nKF) is very ample.
In this subsection, we deal with projective morphisms between normal varieties
f: X -+ Y
such that fOx = Oy, i.e., the fibres are connected. Moreover we assume that X is Qfactorial with rational singularities and -Kx is relatively ample over Y.
The assumption on the ampleness of -Kx implies that the general fibre is a Fano variety.
For the time being, we will try to keep our treatment at the widest possible level of generality.
We consider two sheaves on Y: the first one is R 2 fQ, whose fibre at t is H 2 (F,Q);
the second one is RlfG 0 Q, whose fibre at t is Pic(Ft)Q. The first Chern class defines a
morphism
(4.1)
ci: Rlf*Gm 9 Q -+ R 2 f*Q.
50
When X is mildly singular, for example when X has Kawamata log terminal singularities,
the above assumptions imply that the map cl is an isomorphism (the relative version of
Kawamata-Viehweg vanishing applies).
When both X and Y are smooth, by Sard's Theorem there is an open dense subset U
of Y where f is a submersion. Using Ehresmann's Theorem (cf. [78, Proposition 9.3]), it
follows that f is a locally trivial fibration of topological spaces over U. The existence of a
non-empty Zariski open subset UtoP of Y where f is a locally trivial topological fibration
f
holds also in the singular case by a delicate argument due to Verdier.
Theorem 4.2.2. [76, Corollary 5.11 Let X and Y be separated algebraic spaces of finite type
over the complex numbers and f : X -+ Y a morphism. Then there exists a Zariski open set
U C Y over which the restriction of f is a topologically trivial fibration (in the Euclidean
topology).
In our exposition, we will denote the open set from the theorem by UtoP. Unfortunately,
the characterisation of UtOP is not easily obtained and leads to the concept of equisingularity
(cf. [72]). On UtoP, the sheaf R 2f'Q is a local system. Thus, there is a monodromy action
of 7ri(UtOP,t) on the fibre (R2 f*Q)t. In this set-up, we can obtain a more refined result.
Theorem 4.2.3 ([51], [25]). Let U =Uf be the open subset, possibly empty, of Y where
1. Y is smooth;
2.
f
is flat;
3. the fibres of f are Q-factorial with terminal singularities.
Then the sheaf RlfF,, 0
Q
is a local system on U with finite monodromy.
Proof. Since the fibres of f over U are terminal with Q-factorial singularities they satisfy
[51, Conditions 12.2.1] as explained in [51, Remark 12.2.1.4]. Hence we can define the sheaf
GAP 1 (X/U) as in [51, Def.12.2.4] and show that it is a local system with finite monodromy
isomorphic to R 1 f.G 0 Q at a very general point of U. The following step is taken from
[25, Prop.6.5]. The authors first show that gj\/1 (X/U) is isomorphic to R 1 fGm 09Q at the
general point of U; that essentially follows from Verdier's result and the isomorphism (4.1).
Then they show that actually the isomorphism holds at every point of Uf.
We remark that in [25] the base is a smooth curve and the Fano varieties involved have
terminal singularities. Nonetheless, the same argument applies verbatim to smooth bases of
arbitrary dimension.
Remark 4.2.4. Assumption 3. in the statement of the theorem is needed to define the sheaf
GAf(X/U). By the discussion in [51, Remark 12.2.1.4] this hypothesis may be weakened to
the requirement of fibres having Q-factorial singularities and being smooth in codimension
2.
In general, the above set Uf may be empty. Nonetheless, the reader should keep in
mind at this stage that our goal is to classify fibres of a Mori fibration. In particular, in
the following sections we will often be able to assume that either the singularities of X are
terminal, hence the general fibre of f will also be terminal, or by construction the fibres of
f will have terminal singularities.
Finally, let us comment on the Q-factoriality of the fibres. In [51, Thm.12.1.10], the
authors show that in a flat family T -+ S of varieties with rational singularities and smooth
51
in codimension 2, there is an open subscheme W c S parametrizing Q-factorial fibres.
These hypotheses are clearly satisfied in the case of a family of Fano varieties with terminal
singularities or with klt singularities and smooth in codimension 2. The problem is that
even assuming that T is Q-factorial, then the set W may be empty as the following example
shows.
Example 4.2.5. Let C be a projective curve of genus g > 1 and let C' be a degree 2 6tale
cover of C and call i the associated involution. Let Q be a quadric in P 4 of rank 3, i.e.,
the projective cone of P1 x Pl for the Segre embedding. The action of Z/2Z switching the
factors on P1 x P1 lifts to an automorphism of Q, which we denote by g. Define T to be
the quotient of Q x C' by the involution (g, i). Then T maps to C and the fibres are all
isomorphic to Q. The class group of Q x C' has rank 3 as it is isomorphic to the direct sum
of the class groups of Q and C'. On T, the monodromy induced by the quotient reduces
the rank of the class group to 2. The Picard group of T instead has rank 2 as the Picard
group of Q x C'. We conclude that the morphism T -+ C is an isotrivial fibration of relative
Picard number 1 such that T is Q-factorial but none of the fibres is Q-factorial.
On UtOP, the monodromy action on R 2 fQ can be explicitly described as follows. Take
the class of a loop -y in 7r,(U, t). Pull back X to the interval I = [0, 1] via -y. Trivialise
the family X1 . The identification between the fibre over 0, F0 , and the fibre over 1, F
is a homeomorphism: this homeomorphism gives the monodromy action. If we change the
representative of [-y] we may change the automorphism, but not its action on H2 (Fe, Q). See
[78, Section 9.2.1, 15.1.1 and 15.1.2] for more details.
In general, we will consider the monodromy action of the fundamental group of different
open subsets of Y. However, if we have a normal variety W with a closed subvariety Z, the
natural morphism
7ri(W \ Z) -+ ri(W)
is always surjective ([35, 0.7 (B)j). As we will be investigating Mori fibre spaces, restricting
to open subsets of UtoP is not going to change the monodromy action very much and in
particular it will not affect the dimension of the invariant part as that is of dimension 1.
Restricting a cohomology class to each fibre we have a morphism
2
(y
n
, TTO(TT y?
2
-
4
By evaluating at t we get an isomorphism
H0 (U, R 2 fQ)
-+ H 2 (FQ)"1(At.
Composing the above maps we obtain a morphism
p: H 2 (X,Q)
-+
H 2 (Ft,Q)
When X and Y are smooth, a result due to Deligne (cf. [78, Theorem 16.24]) states that
p is surjective for every t E U, providing a method to easily identify the invariant part
of cohomology. Deligne's result is rather general and it concerns all cohomology groups
and cohomology classes which are not, in general, algebraic. However, in our case, the
cohomology classes we are interested in are just the classes of divisors, so we can generalise
Deligne's result to the singular case for NI.
52
Theorem 4.2.6. Let
f: X -+ Y
be a dominant morphism of projective normal varieties, where X is Q-factorialwith rational
singularities. Assume that -Kx is f-ample. Take U = Uf as in Theorem 4.2.3 and assume
it contains a point t. Then the restriction map
p: N1 (X)Q -4 N(Ft)"1'(Ut)
is surjective for every t in U.
Proof. By Theorem 4.2.3 and [51, Corollary 12.2.9], g9 1 (f-1(Uf)/Uf) is a local system
on Uf whose global sections are Nl(f- 1 (Uf)/Uf)Q. By general properties of local systems
and the exponential sequence, cf. [78, Lemme 16.17], this vector space can be identified
with H2 (Ft, Q) 71(Uft) which is isomorphic to N1 (Ft)7"('t), since Ft is Fano with terminal
singularities and by [35, 0.7 (B)]. Hence,
Nl f-'(Uf)/Uf)Q
N'(Ft)7q'Ut
where the isomorphism is given by the restriction to Ft. By definition of N
there is a surjection
N(f-1 (Uf))Q - N(Ft)1'(U't)
(f-1 (Uf)/Uf)Q,
again given by restricting to Ft.
The Q-factoriality of X implies that the surjectivity of the restriction map extends also to
N1 (X)Q, proving the statement of the theorem.
E
4.2.2
The monodromy action and the MMP
In this section we show that the monodromy action preserves some information about the
birational geometry of terminal Q-factorial Fano varieties (a general reference on this topic
is [26]). As in the previous section we think that Ft is a Fano variety which appears as
a fiber in a given morphism f: X -+ Y with -Kx relatively ample over the open set Uf
defined in Theorem 4.2.3.
First of all, the monodromy action preserves the intersection pairing. Indeed, it can be
seen as an action on the cohomology algebra H* (Ft, Z). The class of the canonical divisor
of Ft is, by adjunction, the restriction Kx IF,; hence it is preserved by the monodromy. Call
n the dimension of Ft. The Q-valued bilinear form
b(A, B) := (KF)n-2 - A. B,E
on N'(Ft)Q is also preserved.
What else is preserved?
When all fibres are smooth Fano varieties of the same dimension, Wisniewski proved
in [811 and [82] that the nef cone is locally constant. In the terminal Q-factorial case, [25,
Theorem 6.8] shows that the movable and pseudoeffective cones are preserved.
As Ft is Fano, it follows that it is also a Mori dream space, [11, Cor. 1.3.2]. In particular,
the movable cone Mov(Ft) admits a finite decomposition into polyhedral cones, called Mori
chambers decomposition, [41, Prop. 1.11]. The cone Nef(F) is one of the chambers in the
decomposition.
53
Definition 4.2.7. A log pair (X, A) is said to be 1-canonical if logdiscrep(X, A) > 2.
We say that a normal Q-Gorenstein variety X is 1-canonical if the log pair (X, 0) is
canonical.
1-
In [25, Theorem 6.9], De Fernex and Hacon show that the Mori chambers are locally
constant in the following cases:
" families of 3-dimensional Fano varieties;
* families 4-dimensional and 1-canonical Fano varieties;
* families of toric Fano varieties.
In these cases the nef cone of fibres is locally constant in the family. In [74], Totaro has
shown that under weaker assumptions than the one just illustrated the Mori chambers are
not preserved by the monodromy action.
We can preove the following theorem.
Theorem 4.2.8. Keep notation as in Theorem 4.2.3. Up to shrinking Uf there exists an
open set Uf c Uf C Y on which the monodromy action preserves Nef(Ft).
Before we prove Theorem 4.2.8 let us introduce a useful tool that we will use to complete
the proof. Given a morphism f: X -+ Y a projective morphism of schemes and a class
3 E N1 (X/Y)z, we will denote by
Mor (P1, X/ Y,
the subscheme of the relative Hilbert scheme of X over Y parametrizing morphisms g: P1 9
X such that (f og)(P) = {pt.} and the class of [g,,P 1 ] = 3. This is a quasi-projective scheme
of finite type that comes equipped with a proper map 7r: Mor(P1 , X/Y,#3) -+ Y, [45, Chap.
1].
Proof. By Theorem 4.2.3, there is a finite 6tale cover p: V -+ Uf trivialising the monodromy
action.
For every fibre Ft of fv, the restriction map Ni(Xv/V)Q -_ N1(Ft)Q is an isomorphism.
Since -Kx is f-ample, by the cone theorem we know that NE(XV/V) - the cone generated by effective classes of curves - is rational polyhedral: in particular it is generated by a
finite number of rays R 1,... , Rk and each ray is generated by the class of a curve Ci. Any
Ci is represented by an integral lattice point in Ni(Xv/V)Q.
By Kawamata's rationality theorem, [52, Thm. 3.5] the primitive generators of NE(Xv/V)
are integral lattice points that lie between the hyperplanes HO = {v E Ni(X/Y) I Kx . v =
0} and H2 , = {v E Ni(X/Y)z IKx - v = 2n}, where n = dimX. The number of integral
points contained in NE(Xy/V) lying between HO and H2n is of course finite. Let us denote
the set of such points by C.
Now, to C we can associate the variety
M :=
U
Mor(P,XV/Vi3).
OEC
Let N be the union of those irreducible components of M that do not dominate V via 7r,
and let T = 7r(N). Let us remark that T is a Zariski closed set of V, as M is a variety of
finite type proper over V.
54
Define W : V \ T and Xw = f- 1 (W). The claim is that p(W) is the Zariski open set
we are looking for. In fact, consider the rational polyhedral cone NE(Xw/W). As above,
the extremal rays are finite and their generators have bounded degree with respect to -Kx.
Since every component of M that does not belong to N dominates V, it follows that the
classes that generate the extremal rays of NE(Xw/W) move over W. But this means that
the restriction Nl(XW/W)Q -+ N1(Ft)Q identifies NE(Xw/W) and NE(Ft). The inclusion
NE(F) C NE(Xw/W) follows in fact directly from the definition, while the opposite one is
E
a consequence of the last observation.
Our result does not provide any effective method to characterize the open subset where
the monodromy preserves the nef cone.
To ease the notation, from now on we will denote by U the open subset constructed in
Theorem 4.2.8.
Take an element g of 7r,(U, t). Assume it exchanges two maximal faces 9 i and 92 of
Nef(Ft). These faces give contractions
7ri: Ft -+ Gi,
which correspond to the first step in a run of the KFt-MMP. The pull-back via 7i identifies the
Nef(Gi) with the facet 9j. The information whether the map 7ri is a divisorial contraction
or a flipping contraction or a Mori fibration is encoded in the cohomology ring. As a
consequence, each of the above types is preserved under monodromy.
Theorem 4.2.9. Same notation as in Section , let us consider the flat family of terminal
Fano varieties over the open set Uf defined in Theorem 4.2.3. Let t E Uf be a point and let
Ft be the fibre over t. Assume that the monodromy action identifies two maximal faces 9 1
and 92 of the nef cone of Ft. Then the two maps
i1: Ft
G1
Ft
-+ G2
72:
correspond to the same kind of step in the MMP. In the case of the divisorial contraction,
the monodromy action exchanges the exceptional divisors.
Moreover, the varieties G 1 and G 2 (and the morphisms 7n 1 and 72) are deformation
equivalent.
Proof. To ease the notation throughout the proof we will indicate Ft simply by F.
To prove the first part of the theorem, we do a case-by-case analysis.
Divisorial contraction As 7ri: Ft -+ Gi is birational, given a divisor H in the relative
interior of rir Nef(Gi), we have (Hdim Ft) > 0. The exceptional locus is an irreducible
divisor, call it Di. It is clear that Di is the only effective divisor on F such that
(Hdim Ft-1 - D) = 0, for every H in the relative interior of the corresponding face.
Moreover, we can characterise the dimension of 7i(Di) as the maximal integer k such
that (Hk - D2 ) 7 0, i.e., the numerical dimension of the restriction of H to Di.
Flipping contraction as 7j: Ft -+ Gi is birational, given a divisor H in the relative interior
of ir Nef(Gi), we have (HdimFt) # 0. The smallness of 7ri is equivalent to the fact that
for every effective divisor E E 9i we have (HdimFt-1 - E) > 0. Both these conditions
are preserved by the monodromy action (as the effective cone is preserved).
55
Mori fibre contraction as dim Ft > dim Gi, given a divisor H in the relative interior
of 7r Nef(Gi), we have (HdimF,) = 0 and dim Gi is the maximum integer such that
(Hk) # 0. Hence, in this case, we also know that the dimension of the base of the
fibration is preserved by monodromy.
Let us prove that there exists a flat deformation from G, to G2 . The monodromy action
is finite, so after a finite 6tale cover p: V -4 U, we obtain a family
fp:
Xy
-+
V
with trivial monodromy action on the fibres. This means that the restriction morphism
N1 (Xv)Q -+ N1 (F)Q is surjective for every fibre of fp. As 9 1 and 92 are two faces identified
under the monodromy action on the family XU -4 U, we can find two point t1 , t 2 on V and
a Cartier divisor H on Xy whose restrictions to Ft, (resp. Ft2 ) lie in the relative interior of
g1 (resp.
92).
We want to construct the variety X
relative to V
Projov ( neN fp.Oxv (nH)), getting a morphism
g
Xy
fp
7r
V
The fibre of 7r over t2 is Gi. The restriction of g over t is the contraction given by
the face 9j. We are going to prove that X and 7r exist and they are flat ove V. Since
H is Cartier and f, is flat the sheaves Ox, (nH) are flat over V. Moreover, as any fibre
F, is terminal H2 (F,OF,(nHIF)) = 0, i > 0, n > 0. So by the classical theory of Cohomology and Base Change, [64, Cor. 2], the sheaves 7r,,Oxv(nH) are locally free sheaves.
Hence EnEN fp.Oxv (nH) is a flat sheaf of algebras. The finite generation now follows from
Castelnuovo-Mumford regularity, up to passing to a sufficiently large multiple of H, [53,
Example 1.8.241.
Hence 7r is flat and hence G 1 is deformation equivalent to G 2 , i.e., we have an actual flat
deformation of the contraction given by gi to the contraction given by 92.
E
Let us notice that in the proof of the deformation equivalence of the Gi, we have not used
at all the fact that 91 and G2 are maximal faces of the Nef cone. Hence, we immediately get
the following corollary.
Corollary 4.2.10. Using the same notations as above. Let us consider the flat family of
terminal Fano varieties over the open set Uf defined in Theorem 4.2.3. Let t E Uf be a point
and let Ft be the fibre over t. Assume that the monodromy action identifies two faces 91 and
92 of the nef cone of Ft and let us denote by
7r,:
Ft
7r2: Ft
-
G1
-+ G 2
the two corresponding contraction morphisms.
Then the varieties G1 and G 2 (and the morphisms 7r1 and 7r 2 ) are deformation equivalent.
The previous discussion motivates the following definition.
56
.
Definition 4.2.11 (The groups HMon and Mon). Let F be a normal n-dimensional Fano
variety with terminal singularities. We denote by Aut(F)0 the largest subgroup of Aut(F)
which acts trivially on the Niron-Severi group with Q-coefficients. Then the group HMon(F)
is defined as
HMon(F): = Aut(F)/ Aut(F)0
The group Mon(F) is defined as the maximal subgroup of GL(N 1 (F), Z) which preserves:
" the line spanned by the class of KF;
" the Q-valued bilinearform b(A, B): = (KF)n
2
.
A -B;
" the nef cone;
" the type of step (divisorial, flipping, fibre type) of the MMP associated to the facets of
the nef cone and the exceptional divisor;
" the deformation type of the images of the maps defined by the faces of the nef cone.
Remark 4.2.12. The group HMon(F) is the subgroup defined as the image of the natural
homomorphism
Aut(F) --+ Mon(F) 9 GL(N'(F), Z)
Example 4.2.13. As an example, let us discuss the case of a Fano manifold F of Picard
number 2.
In this case there are only two possibilities: either Mon(F) is trivial or it has order 2. In
fact, as Nef(F) is invariant, if the action of Mon(F) is not the trivial one, then it permutes
the two primitive vectors v1, v2 generating the extremal rays of the cone.
When this case occurs, the canonical divisor lies on the line spanned by v 1 + v2 and
Theorem 4.2.9 shows that the extremal rays of Nef(F) correspond to the same type of
contraction in the KF-MMP and the images of F under the two contractions are deformation
equivalent.
Another easy example is for del Pezzo surfaces (cf. Section 4.3): the generic del Pezzo
surface of degree 3 has no automorphisms (HMon is then trivial in this case), but Mon is
certainly not trivial, as we will show in Section 4.4.
The group Mon(F) is invariant under flat deformations which preserve the nef cone,
whereas the group HMon(F) can jump when we deform F. Let us state the definitive form
of our result for further references.
Definition 4.2.14 (Isotrivial fibration). A morphism
f: X -+ Y
is isotrivial if there exists an open dense subset U of Y such that all the geometric fibres
over Y are isomorphic to a fixed variety.
Equivalently, every point t in U has an Euclidean neighbourhood over which f is holomorphically trivial; moreover, there exists a finite 6tale cover U' -+ U such that XU, a Xu x u U'
is a trivial family (cf. [70, Proposition 2.6.10]).
When we are dealing with isotrivial fibrations, the identification between the fibres over
two distinct points is given by a holomorphic automorphism of Ft. This does not give us a
57
homomorphism from 7ri(U, t) to Aut (F), but it allows us to assume that the action of [Y]
is induced by a (non-unique) element of Aut(Ft). We remark that isotriviality is a special
condition. The only case when it is granted for free is when Ft is rigid. To summarize, in
these two sections we proved the following result.
Theorem 4.2.15. Let
f: X -+ Y
be a dominant morphism of projective normal varieties, where X is Q-factorial with rational
singularities and on an open dense subset of Y the fibres of f are terminal and Q-factorial.
Assume that -Kx is f-ample. Then there exists a maximal open dense subset U = U, of
Y such that
" the morphism
f : XU
-*
U is a flat family of Q-factorial Fano varieties with terminal
singularities;
" for every t in U, the monodromy action of 7r1(U, t) on N1 (Ft)Q factors through the
finite group Mon(Ft) defined above. If the fibration is isotrivial, the monodromy factors
through HMon(Ft).
Moreover, the restrictionmap
p: N 1 (X)Q -+ N'(Ft)7"(U't)
is surjective for every t in U.
We can finally introduce the key notion for our purposes.
Definition 4.2.16 (Fibre-like). A normal Fano variety F with terminal Q-factorial singularities is said to be fibre-like if it can be realised as a fibre of a Mori Fibre Space f : X -+ Y
over U', where U' is as in Theorem 4.2.15.
4t.3U
kSurfCesZ., U11hrCefvUlds and
tER11 11ighLE dimens1blinalex
-
In all the examples we produce, the total space X will have quotient singularities, which
are well known to be klt.
ples
4.3.1
Del Pezzo surfaces
The classification of fibre-like surfaces was carried out in 160, Theorem 3.5, Addendum to
item 3.5.2] when the total space X has dimension three. We generalize this result to higher
dimensional total spaces. Let us denote by Sd the blow up of P2 at 9 - d general points.
P2
,
Theorem 4.3.1. A del Pezzo surface S is fibre-like if and only if it is isomorphic to
PI x P' or Sd, with d < 6.
Proof. To show that S 7 and S8 are not fibre-like we can apply either Theorem 4.4.4 or
Theorem 4.4.5. To show that P1 x PI 1 and Sd, with d < 6 and d 7 3, are fibre-like we
can apply Theorem 4.4.1 and the classical analysis of the automorphism group of del Pezzo
surfaces (cf. [44] and [29]). Now, let F be S3 , a smooth cubic in p3. A generic F does not
have automorphisms (cf.
[69]) so we can not apply the sufficient criterion 4.4.1; however,
58
we can argue as follows. Let X be the incidence variety in P3 x PHO(P 3 , O( 3 ))v. This is
a smooth ample divisor. Hence by Lefschetz hyperplane theorem it has Picard number 2.
The projection
X -+ PH (P3, 0 ( 3 ))v
is a Mori fibre space which contains all cubic surfaces as smooth fibres, so S3 is fibre-like.
E
We remark that we can handle in a similar way also P1 x P1 and Sd with d < 5.
The cubic surface is an example of a fibre-like variety where the sufficient Criterion 4.4.1
does not apply. A consequence of our case-by-case proof is the following.
Corollary 4.3.2. When F is a surface, the necessary criterion 4.4.4 is actually a characterization of fibre-likeness. Moreover, fibre-likeness is preserved by smooth deformations.
By comparing our result with the classification of K-stable smooth del Pezzo surfaces
([73, Theorem 1.4]), we also obtain the following corollary.
Corollary 4.3.3. A smooth del Pezzo surface is K-stable if and only if it is fibre-like.
4.3.2
A general procedure to construct Mori fibre spaces
The abstract notation will be heavy, so we start with an example. Let F be a smooth
divisor of bidegree (2,2) in P 2 x p 2 . Let a be the involution of P 2 x p 2 and pN := PH0 (P 2 X
P2 , 0(2, 2 ))V. Consider the incidence variety I in pN X p 2 X p 2 ; it is a smooth divisor of
degree (1, 2, 2). We can apply Lefschetz's theorem to show that the Picard number of I is
3. We have a fibration
7r: I -* pN
whose relative Picard number is 2. The involution a acts on this fibration. Let X := I/o
and Y - PN/a. In this way, we obtain a fibration
f : X -+ Y,
with relative Picard number 1. The singularities are finite quotient singularities, they are
klt and Q-factorial by [52, Proposition 5.15 and Corollary 5.21]. We conclude that f is a
Mori fibre space. By moving F by a generic element of PGL(3) x PGL(3), we can always
assume that it is not preserved by a; hence the action of a is free on a neighbourhood of F
in Z. This means that F is a smooth fibre of f. The previous argument shows that every
smooth divisor of bidegree (2, 2) in P 2 x p 2 is fibre-like.
We now generalize this construction. Let F be a smooth Fano variety. Suppose that F
is a complete intersection of divisors L 1 ,... , Lk in a smooth ambient variety Z, where the
line bundles OZ(Li) are basepoint-free. Call I the incidence variety in Z x ILI x ... x JLkk
see the introduction to this chapter for the definition of incidence variety. The variety I is
smooth because the line bundles are basepoint-free.
Lemma 4.3.4. The restriction morphism
p: Pic(Z x ILii x
...
is surjective.
59
x ILkI)
-4
Pic(I)
Proof. Consider the projection
7r: I -+ Z.
The fibres are divisors of multi-degree (1, . . ., 1) in ILiI x
equidimensional, so 7r is flat. Let H be a fibre of 7r.
Below, we will show that the sequence
Pic(Z)
-÷
. x ILkI. In particular, they are
Pic(J) -+ Pic(H)
is exact. Since the subgroup p(Pic(Z x ILii x ...x ILki)) c Pic(I) contains the image
of Pic(Z) and surjects to Pic(H) we obtain the surjectivity of p.
To complete the proof of the exactness of the sequence in 4.3.2 we argue as in the last
part of the proof of Theorem 4.4.1. Let L be a line bundle on I whose restriction to H is
trivial. Since R 2 rQ is locally constant on Z, we have that c1(L) is trivial on every fibre.
The fibres are Fano, so L itself is trivial on every fibre. The map 7r is flat, so, by the seesaw
principle, cf. [64, Corollary 6, p. 54] or [51, Proposition 12.1.4]), a line bundle which is
l
trivial on each fibre is the pull-back of a line bundle from the base.
Suppose that there is a finite subgroup G of Aut(Z) which is fixed point free in codimension one and whose action can be lifted to I; fix such a lifting. Assume that G does
not preserve F.
Theorem 4.3.5. Keep notation as in Lemma
4.3.4. If
dim N (Z)S
=
1,
then F is fibre-like.
Proof. We construct explicitly a Mori fibre space which will have F as a smooth fibre. Let
X := I/G and Y = (LiL x ... x ILk )/G. We claim that the projection
f: X -+ Y
ILiI are projective spaces, we have
x ILI) - Pic(Z)
Pir(z X ILI I x ..-.
Lemma 4.3.4 and the hypothesis dim N1 (Z)
dim N' (X)Q
=
dim N 1 (I)G
x
Pic(ILil1)
x
-P;( -, -hi(|
)
has relative Picard number one. Since the
= 1 imply that
<
= dim N' (ILI I x ...x |Lk I)G + 1
N1 (X x ILi I x ...x ILk I)G
=dim N' (Y)Q + 1.
The variety F is a smooth fibre of f because it is not fixed by G. Since the action of G on Z
is fixed point free in co-dimension one, the singularities of X and Y are klt and Q-factorial
by [52, Proposition 5.15 and Corollary 5.21].
E
We remark that it should not be easy to check if the singularities are terminal, as
explained in the remark after [52, Corollary 5.21]. Let us apply our result. Denote by (P")r
the cartesian product of r copies of P'.
Corollary 4.3.6. Take positive integers r, k, d, and n > 2 such that kd < n + 1. Let F be a
smooth complete intersection of k divisors of degree (d,... , d) in (P,)r. Then F is fibre-like.
60
Proof. The condition kd < n + 1 ensures that F is Fano. Let G be the symmetric group on
r elements. It acts on (Pf)r permuting the factors. By acting by a general automorphism
of (Pn)r we can arrange that G does not fix F. We can now apply Theorem 4.3.5.
El
The Mori fibre space will in general depend on a choice of the lifting of G to a subgroup
of Aut(I). For instance, if G acts trivially on the linear systems, the base Y will be a
product of projective spaces. If the lifting is nontrivial, the base will be a singular variety
with smaller Picard number. Moreover, we could have chosen a smaller G. It is enough that
the action of G is transitive on the copies of Pn and fixed point free in co-dimension one on
()pn).
Corollary 4.3.7. Fix two integers r and n > 2 greater than 1. Denote by Li be the line
bundle 0(1,..., 0,..., 1) on (Pn)y, where the 0 appears only at the i-th position. A smooth
complete intersection F of multi-degre (Li,... , Lr ) in (Pnl)r is fibre-like.
Clearly, there are many variants of these corollaries. In the next section we will give a
few more specific examples.
4.3.3
Fano Threefolds
The results described in the previous sections show that there are quite a few restrictions
on the geometry of a Fano variety F in order for it to be fibre-like. We are interested in
understanding how strong these restrictions are. As vague as this question may appear,
drawing on the classification of smooth Fano threefolds due to Mori and Mukai (cf. [621 and
[63]), we are able to show that in this context most threefolds do not satisfy these restrictive
conditions.
We will refer to [62, Tables 2, 3, 4, 5] where a full description of the deformation types
of Fano threefolds is given.
Theorem 4.3.8. Let F be a smooth Fano threefold with Picard number greater than 1. Then
F is fibre-like if and only if its deformation type appears in Table 4.3.3.
Remark 4.3.9. In the second column of Table 4.3.3 we use the numbering adopted in [62].
Exactly the same numbering will be used throughout our proof. We remark that entry la
and lb have the same deformation type. Alternative descriptions of these manifolds, which
we will use, can be found in [681.
Remark 4.3.10. The last column in each table presented in [62] enumerates all the possible
ways a Fano threefold can be obtained from another Fano threefold by blowing up a curve.
Alternatively, in the language of this section, they describe all the facets of the nef cone
corresponding to a divisorial contraction in which the image of the exceptional divisor is
a curve. We remark that the contractions are listed without multiplicity; this means that
there could be more than one face giving the same contraction.
Proof of Theorem 4.3.8. For the reader's convenience, we will divide our analysis based on
the Picard number of the Fano threefolds that we examine.
Fano varieties of Picard number 2
The nef cone of a Fano variety F of Picard number 2 is a rational polyhedral cone of
the form R;>oD 1 + R;>oD 2 , for D 1 , D 2 two nef, semiample (integral) Cartier divisors on F.
In this representation, we always assume that the classes of the Di are primitive in the
N6ron-Severi group.
61
-K
12
12
[62]
(6a)
(6b)
p(F)
2
2
2
(12)
2
20
3
(28)
2
28
4
5
(32)
2
(1)
3
48
12
6
(13)
3
30
7
8
(27)
3
4
48
24
IDeformation
type of F
D
F is a divisor of bidegree (2, 2) in P 2 x p2.
F is a 2 : 1 cover of a smooth divisor W of bidegree
(1, 1) in P 2 X p 2 branched along a member of I-Kw L.
F is the blow-up of P 3 with center a curve of degree
6 and genus 3 which is an intersection of cubics.
Alternatively, F is the intersection of three divisors
of bidegree (1, 1) in P3 x p3.
F is the blow-up of Q C P 4 with center a twisted
quartic, a smooth rational curve of degree 4 which
spans P 4
F is a divisor of bidegree (1, 1) in P 2 x p2
F is a double cover of P1 x P1 x P1 whose branch
locus is a divisor of tridegree (2, 2, 2).
F is the blowup of a smooth divisor of bidegree
(1, 1) in ]P 2 X p 2 with center a curve C of bidegree
(2, 2) on it, such that C " W " p 2 x p 2 + p 2 is an
p2
embedding for both both projections P 2 x p 2
1
F = P x P' x P'.
F is a smooth divisor of multi degree (1, 1,1, 1) in
P1 X P, x P1 x P 1
.
.
No
la
lb
.
(1)
Table 4.1: Deformation types of Fano varieties in Theorem 4.3.8
.
Remark 4.3.11. As the nef cone is Mon(F)-invariant, dim Nef(F) = 2 and the only invariant subspace for the action of Mon(F) is the span of the anticanonical class, it follows that
the sum of the primitive generators of Nef(F) must be a multiple of the canonical class, i.e.
there exists A < 0 such that
AKF ~ Dl + D2
+
This is another useful condition: e.g., a Fano variety F isomorphic to a smooth divisor of
type (1, 2) contained in P 2 x p 2 cannot be fibre-like. Let i: F " P 2 x p2 be the inclusion of
F in P2 x p 2 . We denote by P1, P2 the projections of p 2 x p 2 onto the its two factors. By
Lefschetz hyperplane theorem, Nef(F) = i* Nef (P 2 x P 2 ). Then Nef(F) = Ryoi*p*(Op2)
Ryoi*p2*(Op2) and the two classes are the primitive generators of the cone. The adjunction
formula implies that
KF =
(Kp2Xp2 +
F)IF = (Kp 2 xp2 + Op2Xp2(1, 2))IF-
It is immediately clear that [KF is not contained on the line spanned by the sum of the two
primitive generators of Nef(F).
Using Corollary 4.4.9, we can immediately exclude the families corresponding to the
following entries of Table 2 of 162]:
(1 - 5), (7 - 11), (13 - 20), (22), (23), (25 - 31), (33 - 36).
Entry number (12), the intersection of three divisors of bidegree (1, 1) in P3 x p3 , is fibre-like
62
because of Theorem 4.3.5. Using Remark 4.3.11, we can also exclude entry (24).
The variety corresponding to entry (6a) is a divisor of degree (2, 2) in P 2 X p 2 and the
variety corresponding to entryto entry (32) is a divisor of type (1, 1) in P2 X<p 2 . They are
fibre-like because of Corollary 4.3.6. Entry (6b) is a 2 : 1 cover F of a smooth divisor W of
bidegree (1, 1) in P 2 x p2 branched along a member of I - Kw 1. We can construct inside
I - KwI x I - KwI the universal family Z for F (cf. [3, Chapter 1.17]). The variety Z is
smooth and projective and it has Picard number 3. We remark that W has an involution
o-; its action can be lifted to both I- KwI and Z. Letting Y := I- KwI/- and X := Z/cwe obtain a Mori fibre space which contains F as a smooth fibre.
Entry (28) can be described as a smooth complete intersection of L:
f*H and L 2
f*2H - E, where f : Z -+ P5 is the blow-up of the Veronese surface V, E is the exceptional
divisor and H is an hyperplane in P5 . We want to apply Theorem 4.3.5. To this end we
construct an order 2 automorphism C of Z such that C*Li = L 2 . The automorphism C is a
special Cremona transformation. The Veronese surface is the intersection of 6 quadrics, so
we have a Cremona transformation of P5 whose indeterminacy locus is V. By blowing up V,
we get a regular map from Z to P5 which contracts the secant variety of V, so this new map
is again a blow-up. We conclude that C lifts to a regular automorphism of Z. One checks
that it acts nontrivially on the Picard group. A general reference for this kind of Cremona
map is [30].
Fano varieties of Picard number 3
Entry (1) is a double cover of P1 x P1 x P1 whose branch locus is a divisor of tridegree
(2, 2, 2). We will use the same notation as [20, Section 54]: this Fano variety can be realised
as a member of the linear system 12L + 2M + 2NI in the toric variety Z with weight data
XO
X1
Yo
1
0
0
1
0
0
0
1
0
Y1
0
1
0
ZO
Zi
0
0
1
0
0
1
W
1
1
1
L
M
N
+
Also for this variety Theorem 4.3.5 applies, since Z carries a natural action of the symmetric
group S3 which exchanges the divisors L, M and N and so lifts to the linear system 12L
2M + 2NI. This shows that entry (1) is fibre-like.
Entry (13) can be alternatively described as a smooth complete intersection of 3 divisors
of multi-degree (0, 1, 1), (1, 0, 1) and (1, 1, 0) in P 2 x p 2 x p 2 . It is fibre-like because of
Corollary 4.3.7.
Using Table 3 of [62] and Corollary 4.4.9, we can immediately exclude the families
corresponding to the following entries of the table:
(2 - 8), (11), (12), (14 - 16), (18), (20 - 24), (26), (28 - 31).
Remark 4.3.12. Let F be a Fano variety of Picard number 3. Suppose that the nef
cone contains two facets for which the images of the corresponding contraction morphisms
are deformation equivalent. Then these may be identified by the action of Mon(F). In
particular, the primitive generators of the two facets are exchanged and their sum is then
invariant. Hence it has to belong to the span of the canonical class, if F is fibre-like.
When the two facets correspond to divisorial contractions the same holds true for the
sum of the two exceptional divisors Ei, with i = 1, 2. In particular, E1 + E 2 has to be ample.
63
Using the previous remark, the following entries can be shown not to be of fibre-like
type:
(3), (9), (10), (17), (19), (25).
Fano varieties of Picard number 4
In Table 4 of [62] entry (1), a divisor of multidegree (1, 1,1,1) in P1 x 1 x
x P 1, is
fibre-like because of Corollary 4.3.6. Using Corollary 4.4.9, it is immediate to see that we
can exclude the families corresponding to the following entries of the table:
(3 - 6), (8 - 11), (13).
Using the natural generalisation to Picard number 4 of Remark 4.3.12, we can exclude
the following entries, too:
(2), (7), (12).
Fano varieties of Picard number 5
In this case the only Fano threefolds are the following.
" Let Y be the blow up of a quadric Q C P3 along a smooth conic contained in it. The
Fano variety F is the blow-up of Y with center three distinct exceptional lines of the
blow-up Y -+ Q; then the sum of the three exceptional divisors over the lines must be
a (negative) multiple of Kx and it is ample. That is clearly false true, as one can see
by taking an exceptional line for the map Y -+ Q other than those already blown up.
* F is the blow-up of Y = P(OpixP 1 (1,0) D Opi x Pi (0,1)) with center two exceptional
lines 11, 12 of the blow-up 4: Y -+ P3 such that 11 and 12 lie on the same irreducible
component of the exceptional set of q; such F is not of fibre-type by Proposition 4.4.7.
" Products
P' x Sd, d < 6,
where Sd is a del Pezzo of degree d. A quick analysis shows immediately that the
projection onto the second factor must be Mon(Sd)-invariant as Nef(P 1 x Sd) =
Nef(Pl) x Nef(Sd).
Proof. As a consequence of our case-by-case proof we have the following corollary.
Corollary 4.3.13. When F is a threefold, the necessary criterion 4.4.4 is actually a characterization of fibre-likeness. Moreover, fibre-likeness is preserved by smooth deformations.
4.4
4.4.1
Criteria for Fibre-likeness
General Criteria
In this section we present two criteria, one sufficient and one necessary, which detect the
fibre-likeness in a rather general setting. The necessary criterion is based on Theorem 4.2.15.
When the Fano variety is rigid, we obtain a characterisation. Here by rigid we simply mean
that H 1(F, TF) = 0.
64
Theorem 4.4.1 (Sufficient Criterion). A Fano variety F with terminal Q-factorial singularities and such that
N1(F) ut(F) = Q[KF]
is fibre-like.
Moreover, there exists a Mori fibre space f : X -+ Y such that the base Y is a curve and
the fibration is isotrivial.
Remark 4.4.2. Before giving the proof, let us remark that KF is always fixed by Aut(F).
In particular, if we fix m E N s.t. -mKF is very ample, then the action of Aut(F) lifts
faithfully to a linear action on I - mKF. In other words, the hypothesis of the theorem is
that the subspace of N1 (F)Q fixed by Aut(F) is minimal.
Proof of Theorem 4.4. 1. We know that HMon(F) is finite since the nef cone Nef(F) of F is
rational polyhedral and HMon(F) permutes its faces.
Pick a set of generators [fi],..., [fe] of HMon(F). Call G the sub-group of Aut(F)
generated by fi, ... , fg. Take a genus g curve C and denote by ai and bi the generators of
its fundamental group. There is a unique relation between the ai and bi and it is the one on
the product of commutators:
albial bil a2 b2 a2- bs1 -- agba-1 b- 1
=
1.
We define a surjective morphism
p:
iri(C, t)
-+
ai
-
bi
Let
O
G.
fi
fi- 1
be the universal cover of C. We define
X := F x
0/ri(C, t),
where ri (C, t) acts on F via p. The action of 7i, (C, t) is free and properly discontinuous;
hence X is an analytic space with terminal singularities, i.e., the same type of singularities
of F. The natural projection
f: X -+ C
is an isotrivial fibration with fibre isomorphic to F.
The variety X is projective.In fact, indicating the embedding induced by the linear
system of I- mKFI by 1-nmKpi: F -+ pN, N = dim - mKFI , we know from Remark 4.4.2
that the action of 7ri(C, t) extends also to pN and the map Ol-mKFI is equivariant for the
action.
65
Hence we have the following commutative diagram
Fx
0
X = F x 0C/ r(C, t)
I
C
pNX(42
d
1mF
> pN X 0/7ri(C, t) = Z
idI
C.
As above, the singularities of Z are the same as PN and so Z is smooth. Moreover,
as the action of Aut(F) is contravariant for <ImKFIxida, Z maps to C and every fibre is
isomorphic to pN. In particular, the anticanonical sheaf Oz(-Kz) is relatively ample over
C. Since C is itself projective, it follows that Z is projective.
The variety X is Q-factorial. This is simply a consequence of [51, Cor. 12.1.9], as in
view of the hypotheses of the theorem, F is Cohen-Macaulay, in particular S3 , [52, Thm.
5.22] and the codimension of the singular locus of F is at least 3.
To finish the proof, we need to show that p(X/C) = 1. We fix a point t on C and
consider the fibre Ft of f over t. That is isomorphic to F via the map q defined in 4.2. We
will denote the inclusion of Ft in X by t: F -+ X. As p(C) = 1 it suffices to show that
p(X) = 2. We consider the sequence
0
-
N'(C)Q 1+ N'(X)Q
+ N(F )
-+
0.
(4.3)
If this sequence is exact, then p(X) = 2.
The injectivity of f* follows by the connectedness of the fibres and the projection formula.
The vector space N 1 (Ft)G is generated by -KF,. By adjunction t*Kx = KFe, so C* is
surjective. We have im f* C ker C*. We need to show the opposite inclusion, but this follows
by the same reasoning as in the proof of Lemma 4.3.4 from the seesaw principle.
E
Remark 4.4.3. The same proof works verbatim in the case where X is a Fano variety with
rational Q-factorial singularities of dimension > 3 and F is smooth in codimension 2, i.e.
the singular locus has codimension at least 3. In fact the assumption on the terminality of
X has been used only to allow us to use [51, Cor. 12.1.9]. But all the hypotheses in the
corollary are verified with the weaker assumptions just explained.
We state now a necessary criterion for F to be fibre-like.
Theorem 4.4.4 (Necessary Criterion). A Fano variety with Q-factorial terminal singularities for which
dim N(F) Mon(F) > 1
is not fibre-like.
Proof. We argue by contradiction. Let
f: X
be a Mori fibre space.
Y
By the definition of fibre-like, there exists an open dense subset
66
U = Uf of Y such that the map
p: N1 (X)Q
-+ N1(Ft)'
(Ut)
for a given fibre Ft isomorphic to F. Let us show that N 1 (F)7 (U 't) is one dimensional. We
first prove that sequence
0
-÷
N1 (Y)Q -
N1 (X)Q
4
N(F)
'(Ut)
-+
0.
is exact. Since F is connected, the map f* is injective on N 1 (Y)Q. The inclusion im f* C
ker p holds because the composition Ft -+ X -+ Y factors through a point. The map p is
not the zero map. Now, we use the fact that we are dealing with a Mori fibre space. The
relative Picard number is one, so:
dim N1 (X)Q = dim N 1 (Y)Q + 1.
Thus, the sequence is exact and
dim N1(Ft)"'<Ut) = 1
By Theorem 4.2.15, the monodromy action factors through Mon(F), so
N1(F)
Mon(F)
-
QKF-
This contradicts our hypothesis.
In the next section we will introduce a more handy version of this criterion. Let us finish
this section by considering the rigid case.
Theorem 4.4.5 (Characterisation - Rigid case). A rigid Fano variety F with Q-factorial
terminal singularitiesis fibre-like if and only if
N1(F)
ut(F) =
QKF
In this case, F is a fibre of an isotrivial Mori fibre space over a curve.
Proof. The "if" part is Theorem 4.4.1. The "only if" part follows from Theorem 4.4.4: just
remark that if F is rigid the monodromy action factors through HMon(F) (cf. Theorem
4.2.15).
E
If F is not rigid this characterisation is false. A counterexample is the del Pezzo surface
of degree 3 (see Section 4.3).
4.4.2
Applications of the Necessary Criterion
The group Mon(F), defined in 4.2.11, is in general difficult to describe. Roughly speaking, it
can be thought of as the group of symmetries of the nef cone preserving some other features
coming from the birational geometry of the underlying variety. Taking this point of view,
we can rephrase this criterion in terms of the birational geometry of F. The idea is that,
if the faces of the nef cone are different from the view point of birational geometry, then
67
N1 (F)Mon(F) must be big. Let us give an easy example. Assume that F has Picard number
2. The nef cone has two faces, g1 and 92. Each face gives a contraction
7i: F -÷ G.
Corollary 4.4.6. Keep notations as above. If
dim G1 7 dim G2
then F can not be fibre-like.
Proof. The group Mon(F) can not exchange g 1 and G2, so it is trivial.
E
Case by case, one can cook up more refined versions of this corollary. Let us give more
examples.
Corollary 4.4.7. Let F be a Fano variety obtained as the blowup of another Fano variety
G and assume there are no other facets of Nef(F) whose associated contraction is divisorial
with image a variety deformation equivalent to G. Then, F cannot be a fibre-like Fano.
Proof of Corollary 4.4 7. The face of Nef(F) corresponding to the pullback of Nef(G) is
invariant by Mon(F). In fact, the type of an extremal contraction and the deformation type
of its image are preserved under the action of Mon(F). Hence, the uniqueness implies that
the map must be preserved by such action.
It is enough to show that on such face there is a fixed point and, consequently, a fixed
one-dimensional subspace. As, in order to be a fibre-like Fano, the only subspace preserved
by Mon(F) could be the span of KF and this does not lay on the pullback of Nef(G), the
required contradiction is immediate. As we explained above, if Nef(G) is stable by Mon(F),
then the class of the exceptional divisor is fixed as well.
D
The above criterion can be generalised quite easily. Let us explain how, by means of
some examples.
Example 4.4.8. Let F be a Fano manifold that possesses a unique Mori fibre contraction
to a variety G, with dim G = k > 0. Then the face of the nef cone of F corresponding to
the nef cone of G is stable under Mon(F). In particular, the primitive generators of the
extremal rays (in the lattice N1 (F) C N'(F)) of such a face are going to be permuted by
Mon(F). In particular their sum will be Mon(F)-invariant. Hence, F cannot be fibre-like.
This is the case, for example, for the projectivisation F of the vector bundle associated to
the sheaf Opi xi G Opi pi (1, 1). Recall that F is isomorphic to the blow-up of the cone over
a smooth quadric in P3 with center the vertex. p(F) = 3 and the facets of Nef(F) are given
by the Mori fibre contraction F -+ Pi x Pl and the two small contractions F -+ F, i = 1, 2,
given by contracting the two rulings of the exceptional copy of P1 x P1.
The above analysis can be formalised into the following statement.
Corollary 4.4.9. Let F be a Fano variety and assume that the nef cone of F contains a
facet G corresponding to a certain variety G. Assume that for any other facet 'W of the nef
cone, the corresponding variety H is not deformation equivalent to G. Then, F cannot be a
fibre-like Fano.
68
I__ I- -- I-
. .
I
So far we have dealt with the case of a facet globally fixed by Mon(F). What happens
to facets that are translated around the nef cone?
Let F be a Fano variety and F be a facet of Nef(F) and let L be the sum of the primitive
generators of the extremal rays spanning F. Let F1, ... , Fk be the facets corresponding to
translates of F under Mon(F). Again, for each of the facets _ 1 , .. ., Fk, let L 1 , . . . , Lk be the
sum of the primitive generators of the extremal rays spanning the facet. The Li constitute
the orbit of L under the action of Mon(F). Hence, L, + - -- + Lk is Mon(F)-invariant. In
order for F to be fibre-like, it has to be a negative multiple of KF, in particular it has to be
ample.
When F corresponds to a divisorial contraction, then the same reasoning applies to show
that the sum E + El + -- - + Ek of the exceptional divisors relative to the different facets
must be a multiple of -KF; otherwise F will not be fibre-like.
Example 4.4.10. Let
Qa
smooth quadric Q C p3. Let p: R -+ Q be the projective space
bundle P(OQ E OQ(1, 1)).
The map p has two sections Eo, E(1,1) corresponding to the two projections of OQ E Og(1, 1)
on its factors.
Let F be the Fano variety obtained as the blow-up of R along an elliptic curve C contained
in E0 . We will denote by 7r: F -+ Q the given map.
The generic fibre of 7r is P1, but over p(C) C Q the fibres are chains of two copies of
P 1 intersecting at a point. The variety F has exactly two different divisorial contractions
4j: F -+ PQ(OQ D OQ(1, 1)), i = 1, 2 given by contracting the two components of the
fibres of r over p(C), respectively. Hence, for F to be fibre-like, the sum of the exceptional
divisors for the Oi must be ample. But this is not possible as such sum has intersection 0
with the generic fibre of 7r.
4.5
K-stability in the smooth toric case
In this section we prove that any smooth fibre-like toric Fano variety is K-stable. Let us
point out that there are smooth toric varieties that are K-stable but not fibre-like, such as
pl X p2
4.5.1
Preliminaries on toric geometry: primitive collections
We start recalling some notation and basic facts. For more details, see [241, [4] and [17].
Let F be an n-dimensional smooth toric Fano variety. Up to isomorphism, there is a
unique pair (N, E) consisting of a free abelian group N of rank n and a fan E c NQ whose
support is all of NQ. Let M = Hom(N, Z).
Let A be the polytope whose vertices are the integral generators of the 1-dimensional
cones contained in E. We denote the set of all vertices of A by V(A).
Let N (F) be the group of 1-cycles on F modulo numerical equivalence and set N (F)Q =
N1 (F) 0 Q. Inside Ni(F)Q we consider the Kleiman-Mori cone NE(F) generated by the
effective 1-cycles. There is the following basic exact sequence:
0 4N1(F) -+ Zv(A) --+ N - 0
(4.4)
M -+ Zv(A) - N'(F) - 0.
(4.5)
and dually
0
-
69
In this subsection, we also need some notation and results about primitive collections.
Definition 4.5.1. A subset P C V(A) is called a primitive collection if the cone generated
by P is not in E and for each x E 7 the cone generated by P \ {x} is in E.
For a primitive collection P = {x1,... , xk} denote by o(P) the minimal cone in E such
y be generators of a-(P). By smoothness of F, there
that x 1 +. . . + Xk C 9-(P) . Let Yi, .
exist positive integers bi such that
X1+
+...+
xk
= bly+...+ byh;
let r(P) be this relation.
Definition 4.5.2. The linear relation r(P) is called the primitive relation of P and the
cone o-(P) is called the focus of P. The integer k is called the length of r(P) and the degree
of P is defined as degP = k - E bi.
Using the exact sequence (4.4) we have the following identification between N (F) and
the group generated by relations among the vertices of A:
N1 (F) 2
(bx)XEV(A) E ZV(A)
E
bxx =
0}.
Remark 4.5.3. By abuse of notation we denote with r(P) also the cycle associated, via
the previous isomorphism, to the relation
X1+...+
Xk - (biy1+...+bhyh))=0.
It is not hard to see that deg P = -(KF - r(P)), [18, page 1271. Since F is a Fano variety
and deg P = -(KF - r(P)), any primitive relation has strictly positive degree.
We will need the following result (cf. [4, Proposition 3.2]).
Proposition 4.5.4 (Batyrev). Let F be a smooth toric Fano variety. Then there exists a
primitive collection P such that o-(P) = 0.
Remark 4.5.5. Let
aix + -+
akk = blyi + -+
bhyh
be a relation among vertices of A with all {ai} and {bj} positive integers. Assume that
E ai ;> E b. Then by Lemma 1.4 in [17], (X1 . -- ,xk) X E.
4.5.2
Fibre-likeness implies K-stability
It is known that the symmetry of the polytope A is related to the K-stability, which is
known to be equivalent to the existence of a Kdhler-Einstein metric (cf. 180] and [7]) of the
associated Fano variety. Mabuchi proved in [56] the first result relating the K-stability with
the triviality of the barycentre of A. This result was generalised in the singular setting in
[6].
Theorem 4.5.6 ([6, Cor. 1.2]). Let F be a Gorenstein toric Fano variety.
K-stable if and only if the barycentre of A is the origin.
70
Then F is
The proof of the previous result is analytic and passes through the existence of KdhlerEinstein metrics. Applying the theorem above, we can see that there are Gorenstein terminal
Q-factorial toric varieties which are fibre-like, but not K-stable, e.g., the weighted projective
space P(1, 1, 1, 1, 2).
In this context our main result is the following.
Theorem 4.5.7. Every smooth toric fibre-like Fano variety F is K-stable.
Before proving the theorem, we need to recall some convex geometry.
Remark 4.5.8. A basic fact is the following: the intersection of a convex polytope with an
affine space is again a convex polytope.
Lemma 4.5.9. Let P be an n-dimensional convex polytope in an affine space W
Rn and
let H be a k-dimensional affine subspace intersecting the interior of P. Set P' := P n H
and consider a facet F' of P'. Then there exists a unique face F of P of dimension at least
k - 1 such that
o F'=FnH;
o H intersects F in its relative interior.
Proof. The polytope P is defined by a collection of inequalities {x E W I ai - x < bi, i
1, ... t}, with ai, bi E Qd, t > n + 1 and H is defined by a collection of equations {X E
W I cj , x = dj, j = 1,..., d - k}, with cj, dj E Qd. The facet F' is then defined, up to
reordering the indices i by {x E W I aix = bi, i = 1,... l, aix
bi, i = l+ 1, ... t, cj x =
x
d3., j =1, . . . , d - k}, with 1 < n - k + 1. The set {r E W I aix = bi, i = 1, .... 1, aiz
bi, i = 1 + 1, ... t} defines the unique face F of P of dimension at least k - 1 with the
required properties.
D
If we now consider the action of a subgroup of automorphisms of the polytope on the
vertices V(P), we obtain the following lemma.
Lemma 4.5.10. Let P be an n-dimensional polytope in an affine space W ~ Rn and let
G be a finite subgroup of GL(W, Q). Assume that P is invariantfor the action of G on W
and that WG is k-dimensional and intersects the interiorof P. Then the action of G on the
vertices of P has at least k + 1 orbits.
Proof. We prove this lemma by induction on k. In the case k = 0, there is nothing to prove.
the fixed locus WG is a line, which meets two distinct (possibly not maximal) faces F1
and F2 of P. We immediately obtain two invariant sets:
V(Fi) \ V(F2 ),
and V(F 2 ) \ V(F1 ).
These give at least two orbits.
We now prove the inductive step. Since the intersection of a convex polytope with an
affine space is again a convex polytope, we can consider the intersection polytope P' :=
P n WG. Let F' be one of its (k - 1)-dimensional facets. Using Lemma 4.5.9, we can find a
(unique) face F of P, cut in its interior by WG in a k -1 affine space such that F' = FnWG.
Let H be the smallest affine subspace containing F; it is preserved by the action of G. By
induction, we obtain at least k orbits of vertices contained in F. The extra orbit is obtained
by the vertices of P not contained in F.
71
Proof of Theorem 4.5.7. Any smooth Fano toric variety is rigid (cf. [26, Corollary 4.6]), so
we can apply Theorem 4.4.5: F is fibre-like if and only if dim N 1 (F) ut(F) = 1
After tensoring by Q the exact sequence (4.5), we obtain
0 -÷ MQ
QV) -+ N'(F)Q
0,
-÷
(4.6)
where MQ is the n-dimensional Q-vector space containing the dual polytope of A. There
is a natural action of Aut(A) on MQ and QV(A) and a natural homomorphism Aut(A) -÷
Aut(F), which make the sequence above equivariant for Aut(A). Moreover by [23, Corollary
- N1(F)Aut(F). Let us denote by t be the number of orbits of
the action of Aut(A) on V(A).
It is easy to see that if we take the Aut(A)-invariant part of the exact sequence in 4.6,
we obtain again an exact sequence. Moreover, in this case it follows immediately that
4.7]) we have N1(F)Aut(A)
(QV(A))Aut(A)
-
QV(A)/Aut(A)
-
Qt
Let now G be Aut(A) and t be the number of orbits of the action of G on V(A). Set
k := dim MG; the observation from the last paragraph and the sequence (4.6) imply that F
is fibre-like if and only if t - k = 1. Therefore, we want to prove that if G has exactly k + 1
orbits on V(A), then the barycentre of A is the origin.
Since we are working with Q-vector spaces, M and N are isomorphic as G-modules; in
particular dim NG = dim MG = k.
Let A' be the intersection polytope NG n A. For every facet F' of A', one can apply
Lemma 4.5.9 to find a unique face F of A cut by NG in its interior such that T' - F n NG.
Lemma 4.5.10 says that V(F) splits in at least k orbits. Since F is fibre-like, V(F) splits
in exactly k orbits: another orbit is given by the set of vertices V(A)\V(.F).
Let now TFl and F two distinct facets of A', which correspond to two faces F and F2
of A and determine two sets Si and S2 of k orbits each in V(A). We claim that Si # S2:
otherwise
V(F1) =
U S1
=
U 2
2),
which would imply F1 = F2.
Since there are exactly k + 1 ways to choose k elements in a set of cardinality k + 1 and
A' has at least k + 1 facets (actually exactly k + 1 by the above argument), we conclude
that any collection of k orbits is supported on a face F of A.
Let P be a primitive collection with trivial focus, whose existence is guaranteed by
Proposition 4.5.4. Since any set of k orbits must be contained in a face, P must involve
at least one vertex from every orbit. Acting with G on P we obtain a family of primitive
collections {Pi}l<i<. such that u(Pi) = 0 and UPi = V(A). Assume that Pi n Pj # 0 for
some i,j, i.e., Pi = {x 1 ,.. .,Xk} and P = {x1,...,Xh,yh+1,- -,yk} with y, # x for any
s, t. Then
Xh+1 +... +Xk
yh+1 ...yk,
which is impossible by Remark 4.5.5, because Xh+1, - - -, Xk generate a cone in E.
This implies that all the Pi are disjoint and, as a consequence, that the sum of all vertices
of A equal the origin, i.e., the barycentre of A is trivial. The theorem is proved.
D
The previous corollary seems to be the relative version in the toric case of the following
very general conjecture by Odaka and Okada.
72
__
DImension
2
2
2
3
3
4
4
4
4
4
# Vertices
6
4
3
6
4
10
12
8
5
5
6
6
14
12
18
12
9
8
7
14
8
18
15
20
24
12
16
12
10
9
6
6
6
6
6
6
7
7
8
8
8
8
8
8
8
8
8
ID
V2
2
4
5
21
23
63
100
142
146
147
1003
1013
1930
5817
P1 X P1
2
P
(PT)3
P3
V4
V2 x V2
(PY)4
p 2 X p2
6
5
10
Description
4
P
(PT)5
P5
V6
Wi
(V2 )
(pl)
3
7568
6
8611
8631
8634
8635
80835
2 3
(P )
3 2
(P )
P6
(P1)7
P7
80891
V8
W2
106303
277415
442179
2
(V4 )
4
(V2 )
W3
8
2) 4
(P)
(P
4 2
(P 8)
P
_
.W
...
..
....
"._'
790981
830429
830635
830767
830782
830783
Table 4.2: Smooth toric Fano varieties of dimension at most 8 that are fibre-like.
Conjecture 4.5.11 ([66, Conj. 5.1]). Any smooth Fano manifold X of Picard rank 1 is
K-semistable.
4.5.3
MAGMA computations
.
Theorem 4.5.7 and its proof show that the fibre-like condition is rather restrictive.
The table on the next page collects the smooth Fano toric varieties (up to dimension 8)
which are fibre-like. It has been obtained using the software MAGMA together with the
Graded Ring Database [16] (for further details on the classification, cf. [65]).
In Table 4.5.3, the IDs of the Fano polytopes are the ones introduced in [16].
The varieties Vd are known as Del Pezzo varieties (see [79] for more details). There is no
classical description for the varieties W 1 , W2 and W3
We would like to finish off by stating the following speculation.
73
Conjecture 4.5.12. Let A be a smooth fibre-like polytope of dimension d. Assume that d
is an odd prime number. Then either X(A) = Pd or X(A) = (Pl)d.
4.6
Homogeneous spaces
In this section we classify certain fibre-like homogeneous spaces; the upshot is that most of
them are not fibre-like. Let us fix the notation.
Definition 4.6.1. A complex homogeneous space is a complex manifold X endowed with a
transitive action of a group G of holomorphic transformations of X.
In this section we will consider homogeneous spaces F that are projective and endowed
with a transitive action of a semi-simple algebraic group G. Alternatively, F can be defined
as a quotient of G by a parabolic subgroup P. The varieties F that are part of this class of
projective homogeneous spaces with semi-simple algebraic action of a group G are always
rational Fano varieties (general references are [14] and [28]). For the rest of the section,
whenever we use the expression homogeneous space, we refer to this particular class of
spaces.
The isomorphism class of F is determined by the conjugacy class of P; conjugacy classes
of parabolic subgroups are in bijective correspondence with subsets of the nodes of the
Dynkin diagram of G. We picture them by marking the corresponding nodes of the diagram;
the resulting decorated diagram is the Dynkin diagram of P. We denote by Ep the group
of symmetry of the Dynkin diagram preserving the marked nodes; it is a finite group and
it is isomorphic to HMon(F) (see the proof of Corollary 4.6.3). We are going to use the
following classical result (cf. [28, Theorems 1 and 2]). The group Ep is denoted by E, in
[28].
Theorem 4.6.2 (Demazure). Let F = GP
be an homogeneous space of Picardnumber at
least 2, with G simple. Then F is rigid; that is, h1 (F, TF) = 0. Moreover, the automorphism
group of F is isomorphic to the semi-direct product of G and Ep.
The homogeneous spaces which are called exceptional in [28] have Picard number one,
so we can ignore them as they trivially have the structure of Mori fibre spaces.
The outer automorphism group of G, which is denoted by E in [28], is known to be
equal to the symmetries of the Dynkin diagram, see e.g., [67, Section 10.6.10]. We will use
Theorem 4.1.2 and 4.6.2 to prove the following.
Corollary 4.6.3. Let F = GP
be a homogeneous space of Picard rank at least 2. Then F
is fibre-like if and only if is it isomorphic to one of the following varieties:
1. F = F(n, k) parametrises pairs of subspaces (L, V) in Cn such that dim L = k,
dimV = n - k and L c V;
2. F = F(n) parametrises n-dimensional isotropic subspaces of (C 2 n, Q), where
non-degenrate symmetric form;
Q
is a
3. F = FT parametrises pairs of isotropic subspaces (L,V) in (C 8 , Q), where Q is a
nondegenrate symmetric form, L is a line, V is four-dimensional and L C V (the
upper-scriptT stands for triality);
74
T
F = FT is a contraction of a facet of Nef(F ); more explicitly, either F[ = F(4) or
V is forced to belong to one of the two connected component of the grassmanniansof
isotropic 4-dimensional subspaces of C 8
.
4.
5. F = G/P, where G is the exceptional group E6 and V is associated to one of the two
pairs of roots conjugated by the automorphism of the Dynkin diagram.
Before giving the proof, let us give some details about these varieties. Case 1 is realised
as an homogeneous space with G = SLn; it has Picard number two and the faces of the
nef cone are given by the projection onto grassmannians. If we fix a quadratic form Q, we
get an automorphism of F given by OQ(L, V) = (V', L') which exchanges the faces of the
nef cone. In Case 2, G = SO 2n. The linear subspaces of an even dimensional quadric are
divided into two families, which correspond to the even and odd spin representations (cf.
[67, Section 11.7.2] or [36, Section 6.1]). This variety has Picard number two; the action
of an improper orthogonal transformation exchanges the faces of the nef cone. The third
variety is homogeneous for G = SO 8 ; it has Picard number 3. The relevant automorphisms
are realised via triality (e.g., [67, Section 11.7.3]).
Proof. Since F is rigid, because of Theorem 4.1.2 we have just to study the action of
Aut(F) = G x Ep on N'(F)Q. The group G acts trivially in cohomology, so we are left
with the action of the finite group Ep. The group N'(F)Q is spanned by the line bundles
associated to the simple roots of P (cf. [141), so we can identify a basis of N'(F)Q with
the set of the marked nodes of the Dynkin diagram of P. This identification is equivariant
with respect to the group of symmetry Ep; in particular HMon(F) equals to Ep. In other
words, dim N1 (F)Aut(F) = 1 if and only if the group of symmetry Ep acts transitively on
the set of marked nodes. Dynkin diagram and their symmetries are classified (e.g., [67,
Section 10.6.10]); by direct inspection, we conclude that the only F which are fibre-like are
the ones listed above. More explicitly, the Dynkin diagrams B", Cn, E7, E8, F4 and G2
have no symmetries, so the homogeneous spaces for the respective groups are fibre-like if
and only if the Picard number is one. An has just an order two symmetry, so for each pair
of conjugated nodes one gets a fibre-like homogeneous space of Picard number 2; this is case
1. The Dynkin diagram Dn, for n > 4, has just an order-two symmetry which fixes all nodes
except the two nodes associated to the Spin representations, this gives case 2. D4 has the
symmetric group on three elements as group of symmetries, this is the so-called triality and
gives cases 3 and 4. The Dynkin diagram E6 has an order-two symmetry which gives case
5.
E
75
76
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