Geometric invariants and Geometric

consistency of Manin’s conjecture

Akash Kumar Sengupta

A Dissertation

Presented to the Faculty

of Princeton University

in Candidacy for the Degree

of Doctor of Philosophy

Recommended for Acceptance

by the Department of

Mathematics

Adviser: Professor Janos Kollar

June 2019

c© Copyright by Akash Kumar Sengupta, 2019.

All rights reserved.

Abstract

Manin’s conjeture states that the asymptotic growth of the number of rational points

on a Fano variety over a number field is governed by certain geometric invariants (a

and b-constants). In this thesis we study the behaviour of these geometric invariants

and show that Manin’s conjecture is geometrically consistent. In the first part, we

study the behaviour of the b-constant in families and show that the b-constant is

constant on very general fibers of a family of algebraic varieties. If the fibers of the

family are uniruled, then we show that the b-constant is constant on general fibers.

In the second part, we study the behaviour of the a-constant (Fujita invariant) under

pull-back to generically finite covers and prove a conjecture of Lehmann-Tanimoto

about finiteness of covers. In the last part, based on joint work with B. Lehmann

and S. Tanimoto, we prove geometric consistency of Manin’s conjecture by showing

that the rational points contributed by subvarieties or covers with larger geometric

invariants are contained in a thin set.

iii

Acknowledgements

I am deeply grateful to my advisor Professor Janos Kollar for his guidance, constant

support and encouragement throughout the years. I also would like to thank him for

teaching me so much about mathematics.

I am very thankful to Professor Brian Lehmann and Professor Sho Tanimoto for

their ideas in the joint work [LST18], the results of which constitute a part of my

thesis. I learnt a lot of mathematics during the collaboration with them.

I would like to thank Professors Gabriele Di Cerbo, Mihai Fulger and Zsolt Patak-

falvi for many mathematical discussions. I am thankful to Professor Gabriele di Cerbo

and Professor Nick Katz for agreeing to be in my committee.

I would like to thank my friends from the math department: Amitesh Datta, Lena

Ji, Yuchen Liu, Charlie Stibitz, David Villalobos Paz, Ziquan Zhuang. They made

my life in Fine Hall much more enjoyable.

I am very thankful to my friends Naman Agarwal, Shoumitro Chatterjee, Sumegha

Garg, Sravya Jangareddy, Divyarthi Mohan, Niranjani Prasad, Nikunj Saunshi and

Karan Singh, who have been a constant support and made my time in Princeton very

memorable. I am also thankful to my friend Yajnaseni Dutta for many mathematical

conversations over the years.

Lastly, I would like to thank my parents who have been a constant source of

inspiration for me.

iv

Contents

Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iii

Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iv

1 Introduction 1

1.1 Manin’s b-constant in families . . . . . . . . . . . . . . . . . . . . . . 5

1.2 Fujita invariant of finite covers . . . . . . . . . . . . . . . . . . . . . . 7

1.3 Geometric consistency of Manin’s conjecture . . . . . . . . . . . . . . 8

2 Preliminaries 11

2.1 Neron-Severi group. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2.2 Minimal and Canonical models. . . . . . . . . . . . . . . . . . . . . . 13

2.3 Boundedness of singular Fano varieties . . . . . . . . . . . . . . . . . 20

2.4 Geometric invariants. . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

2.4.1 Fujita invariant or a-constant . . . . . . . . . . . . . . . . . . 22

2.4.2 the b-constant . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

2.5 Manin’s conjecture and the thin exceptional set . . . . . . . . . . . . 32

2.6 Twists . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

3 The b-constant in families 45

3.1 Global invariant cycles. . . . . . . . . . . . . . . . . . . . . . . . . . . 45

3.2 Constancy on very general fibers . . . . . . . . . . . . . . . . . . . . . 48

3.3 Family of uniruled varieties . . . . . . . . . . . . . . . . . . . . . . . 50

v

4 Fujita invariant of finite covers 54

4.1 Boundedness of degree of covers . . . . . . . . . . . . . . . . . . . . . 55

4.2 Finiteness of adjoint-rigid covers . . . . . . . . . . . . . . . . . . . . . 56

5 Geometric consistency of Manin’s conjecture 61

5.1 Boundedness of breaking thin maps . . . . . . . . . . . . . . . . . . . 61

5.1.1 Boundedness of subvarieties . . . . . . . . . . . . . . . . . . . 62

5.1.2 Finiteness of covers . . . . . . . . . . . . . . . . . . . . . . . . 65

5.2 Thinness of rational points from breaking thin maps . . . . . . . . . . 80

Bibliography 98

vi

Chapter 1

Introduction

A motivating problem in the field of algebraic geometry is to classify all algebraic

varieties (classically, algebraic varieties are geometric objects which arise as a zero set

of multivariable polynomials). Birational geometry is the study of algebraic varieties

upto birational equivalence and the minimal model program proposes a solution to

the birational classification problem [KM98, BCHM10]. On the other hand, in the

field of number theory, we are interested in rational points on algebraic varieties

(e.g. rational numbers which solve polynomial equations with rational coefficients).

In the minimal model program, one of the building blocks of algebraic varieties

are Fano varieties, i.e. varieties with ample anti-canonical bundle (for example,

projective space Pn, hypersurfaces of degree at most n in Pn and so on). Also, from

the arithmetic point of view, Fano varieties tend to have a lot of rational points.

One of the guiding problems in number theory is Manin’s conjecture about rational

points on Fano varieties defined over a number field [FMT89, BM90]. The conjecture

states that the asymptotic growth of the number of rational points on a Fano variety

is governed by certain geometric constants (called the a and b-constants). Thus

Manin’s conjecture provides a bridge between birational geometry and number theory.

1

Manin’s conjecture:[FMT89, BM90, Pey03] Let X be a Fano variety defined

over a number field F and L a big and nef line bundle on X with an associated height

function HL. Then, after possibly replacing F by a finite extension, there exists a

thin set Z ⊂ X(F ) such that

#{x ∈ X(F ) \ Z|HL(x) ≤ B} ∼ c(F,X(F ) \ Z,L)Ba(X,L)(logB)b(X,L)−1

as B →∞.

In the conjectural formula above, c(F,X(F ) \ Z,L) is Peyre’s constant as in

[Pey95, BT98b]. The exponent a(X,L) is the Fujita invariant or the a-constant

[Fuj85] and b(X,L) is called the b-constant of X and L [FMT89]. The Fujita invariant

and the b-constant depend on the geometry of X, whereas Peyre’s constant is more

arithmetic in nature. The geometric invariants are defined as follows:

Let X be a smooth projective variety over a field of k of characteristic 0 and L a

big Q-divisor on X. Let Eff1(X) ⊂ NS(X)R be the cone of pseudo-effective divisors.

The Fujita invariant or the a-constant is defined as follows (cf. [Fuj85])

a(X,L) = min{t ∈ R|[KX ] + t[L] ∈ Eff1(X)}

The b-constant is defined as follows (cf. [FMT89, BM90])

b(X,L) = codim of minimal supported face of Eff1(X)

containing the class of KX + a(X,L)L.

2

For a singular variety X, the a,b-constants of L are defined to be the a,b-constants

of π∗L on a resolution π : X −→ X.

Manin’s conjecture is known to be true in various cases, for example when X is a

projective space [Sch79, Pey95], smooth quadric hypersurface [FMT89], toric variety

[BT96a, Sal98], generalized flag variety [FMT89] and so on. However it is not yet

known in complete generality.

A motivating problem that arises from Manin’s conjecture is to understand the

thin set Z, whose existence is predicted in Manin’s conjecture. The notion of thinness

arises naturally in the context of Hilbert Irreducibilty theorem due to Serre [Ser92,

Proposition 3.3.1].

Definition 1.0.1 A morphism f : Y −→ X is thin if it is generically finite onto its

image and admits no rational section (e.g. inclusion of a subvariety, a finite cover of

degree ≥ 2). A subset Z ⊂ X(F ) is a thin set if Z = ∪ifi(Yi(F )) for finitely many

thin maps fi : Yi −→ X.

Initially, Manin’s conjecture was stated with Z being contained in a closed subset

[FMT89, BM90]. For example, if X is a cubic surface and Y ⊂ X is a line defined

over the base field, then Y has higher a and b-constants than those of X, and as a

result Y contains more rational points than the predicted number for X. Therefore

it is necessary to remove Y while counting rational points (i.e. Y (F ) ⊂ Z). In

[BT96b, LR14], counterexamples to the closed set version of Manin’s conjecture were

provided. These counterexamples arise from the existence of subvarieties and finite

covers with higher a and b-constants. In [Pey03] a modified statement of Manin’s

conjecture was presented that predicts the existence of a thin set Z (instead of a

3

closed set) such that the asymptotic formula holds outside of Z.

Let f : Y → X be a thin map such that

(a(Y, f ∗L), b(Y, f ∗L)) > (a(X,L), b(X,L)). (1.1)

If Manin’s conjecture holds for (X,L) as well as for (Y, f ∗L), then the asymptotic

growth rate of the number of rational points on Y would dominate the growth of

rational points on X. Therefore if Manin’s conjecture if geometrically self-consistent

(cf. Section 2.5), then the thin exceptional set Z must contain the image f(Y (F ))

for any breaking thin map f : Y −→ X, i.e. a thin map satisfying inequality 1.1.

Therefore, from the perspective of Manin’s conjecture, we would like to study the

behaviour of the a and b-constants under pullback by thin maps f : Y −→ X. Note

that, if κ(KY + a(Y, f ∗L)f ∗L) > 0, then after possibly replacing Y by a birational

model we may assume that we have morphism to the (KY + a(Y, f ∗L)f ∗L)-canonical

model π : Y −→ T . Therefore, in order to study rational points contributed by Y , we

may study the rational points contributed by fibers of the family π : Y −→ T . Hence

it is important to study the behavior of a and b-constants in families of varieties as

well.

In this thesis we study the behaviour of the geometric invariants (a and b-

constants) under pull-back by thin maps and in families. We also prove that the

rational points contributed by breaking thin maps are indeed contained in a thin set,

proving that Manin’s conjecture is geometrically consistent. In the following sections

we describe the main results of this thesis.

4

1.1 Manin’s b-constant in families

In this section we study the behaviour of the geometric invariants in families. The

behaviour of the a-constant in families is well-understood. Let f : X −→ T be a

family of projective varieties and L an f -big and f -nef Q-Cartier Q-divisor. By

semi-continuity the a-constant of the fibers a(Xt, L|Xt) is constant on very general

fiber (cf. [LT17, Theorem 4.3]). It follows from invariance of log plurigenera [HMX13]

that if the fibers are uniruled then the a-constant is constant on general fibers.

In Chapter 3 of the thesis, we investigate the behaviour of the b-constant in

families and prove the following theorems that answer the questions posed in [LT17].

The following results are from the paper [Sen17a].

Theorem 1.1.1 Let f : X −→ T be a projective morphism of irreducible varieties

over an algebraically closed field k of characteristic 0, such that the generic fiber is

geometrically integral. Let L be an f -big Q-Cartier Q-divisor. Then there exists a

countable union of proper closed subvarieties Z = ∪iZi ( T , such that

b(Xt, L|Xt) = b(Xη, L|Xη)

for all t ∈ T \ Z, where η ∈ T is the generic point. In particular, the b-constant is

constant on very general fibers.

If the fibers of the family are uniruled, then we have the following:

Theorem 1.1.2 Let f : X −→ T be a projective morphism of irreducible varieties

over an algebraically closed field k of characteristic 0, such that the generic fiber is

geometrically integral. Let L be an f -big and f -nef Q-Cartier Q-divisor. Suppose

5

a general fiber Xt is uniruled. Then there exists a proper closed subscheme W ( T

such that

b(Xt, L|Xt) = b(Xη, L|Xη)

for t ∈ T \W and η ∈ T is the generic point. In particular, the b-constant is constant

on general fibers in a family of uniruled varieties.

The result above was proved in [LT17, Proposition 4.4] under the technical

assumption that κ(KXt + a(Xt +Lt)Lt) = 0 and used to prove geometric consistency

of Manin’s conjecture in a special case [LT17, Theorem 1.5]. In the result above we

get rid of the technical assumption on the Iitaka dimension and prove the statement

for families of uniruled varieties in general.

The ideas in proving our results are as follows. To prove Theorem 1.1.1, we ana-

lyze the behaviour of the b-constant under specialization and combine this with the

constancy of the Picard rank and the a-constant in very general fibers to obtain the

desired conclusion. The key step for Theorem 1.1.2 is to prove constancy on closed

points when k = C. We run a (KX + aL)-MMP over the base T , to obtain a relative

minimal model X 99K X ′ where a = a(Xt, L|Xt). We pass to a relative canonical

model φ : X 99K Z over T and base change to t ∈ T , to obtain φt : Xt 99K Zt as the

canonical model for (Xt, aLXt). Using a version of the global invariant cycles theorem

(see Lemma 3.1.1), we observe that b(Xt, Lt) is same as the rank of the monodromy

invariant subspace of H2(Y ′Z ,R), where Y ′z is a general fiber of X ′t −→ Zt. Then using

topological local triviality of algebraic morphisms we conclude that the monodromy

invariant subspace has constant rank.

6

1.2 Fujita invariant of finite covers

In this section of the thesis we study the behaviour of the a-constant (Fujita invari-

ant) under pullback by generically finite morphisms. The result in this section is

from the paper [Sen17b].

Fujita introduced the invariant κε(X,L) = −a(X,L) as Kodaira energy and stud-

ied its properties in [Fuj85, Fuj92, Fuj97]. By [BDPP13], we know that a(X,L) > 0

if and only if X is uniruled. The a-constant was introduced in the context of Manin’s

conjecture in [FMT89, BM90]. In [LTT18], motivated by Manin’s conjecture, the

authors studied the behaviour of the a-constant under restriction to subvarieties. We

know (by [HJ17, LTT18]) that if X is uniruled and L is a big and nef line bundle

then there exists a proper closed subscheme V ⊂ X such that any subvariety Z ⊂ X

satisfying a(Z,L|Z) > a(X,L) is contained in V . In [LT17], a similar finiteness

statement was conjectured about the behaviour of the a-constant under pull-back to

generically finite covers.

In Chapter 4, we prove the statement conjectured by Lehmann-Tanimoto [LT17].

In particular we prove the following:

Theorem 1.2.1 (see [LT17, Conjecture 1.7] Let X be a smooth projective uniruled

variety and L a big and nef Q-divisor. Then, upto birational equivalence, there are

only finitely many generically finite covers f : Y −→ X such that a(Y, f ∗L) = a(X,L)

and κ(KY + a(Y, f ∗L)f ∗L) = 0.

The conjecture was proved in the case of dim(X) = 2 in [LT17]. The authors

also showed that the conjecture holds for Fano threefolds X and L = −KX if

index(X) ≥ 2 or if ρ(X) = 1, index(X) = 1 and X is general in its moduli. Their

idea was to reduce the statement to the finiteness of the etale fundamental groups of

7

log Fano varieties. We take a different approach to prove the conjecture in general. It

follows from the boundedness results in [Bir16a, Bir16b] that the degree of morphisms

f : Y −→ X satisfying the hypothesis of Theorem 1.2.1 is bounded. Therefore it is

enough to show that the branch divisors of all such morphisms are contained in a

fixed proper closed subscheme V ( X. We show that if B ⊂ X is component of the

branch divisor and B 6⊂ B+(L), then a(B,L|B) > a(X,L). Here B+(L) is a closed

subset of X such that L|B is big for any subvariety B 6⊂ B+(L) ([Laz04]). Therefore,

by [HJ17], there is a fixed (depending only on X and L) closed subscheme V ( X

such that B ⊂ V ∪B+(L).

1.3 Geometric consistency of Manin’s conjecture

The main result in this section proves that Manin’s conjecture is geometrically con-

sistent, i.e. the rational points contributed by breaking thin maps are contained in a

thin set. The results in this section are from the joint work [LST18] with B. Lehmann

and S. Tanimoto.

In Chapter 5, we prove the following:

Theorem 1.3.1 Let X be a smooth geometrically uniruled geometrically integral

projective variety defined over a number field F . Let L be a big and nef Q-divisor

on X. As we vary over all breaking thin maps f : Y −→ X with Y geometrically

integral, the set of rational points

⋃f

f(Y (F ))

is contained in a thin subset of X(F ).

8

This theorem generalizes earlier partial results in [BT98b], [HTT15], [LTT16],

[LT17a], [Sen17b].

The idea for proving the above theorem is as follows. First we prove a finiteness

statement over the algebraic closure F .

Theorem 1.3.2 Let X be a smooth projective uniruled variety over an algebraically

closed field of characteristic 0 that admits an embedding into C. Let L be a big and

nef Q-divisor on X. There exists a finite set of breaking thin maps fi : Yi −→ X such

that for any breaking thin map f : Y −→ X there is an Iitaka base change Y of Y

with a morphism f : Y −→ X such that f factors rationally through one of the fi’s.

We refer to Definition 2.4.3 for the precise definition of Iitaka base change. In

principle, it should be thought of as a base change of the map π : Y 99K W to the

(KY + a(Y, f ∗L)f ∗L)-canonical model. The ingredients in proving the above result

are the geometric behaviour of the a and b-constants, the minimal model program

and the boundedness of singular Fano varieties. The result above is a generalization

of previous finiteness results in [HJ17, LT17, Sen17b].

The second step is to derive thinness over F . Note that a finiteness statement

can not hold over F due to the presence of twists. A twist fσ : Y σ −→ X is a

morphism over F such that it becomes isomorphic to f : Y −→ X after base change

to the algebraic closure F . The twists are parametrized by the Galois cohomology of

the automorphism group Aut(Y /X) (see Section 2.6). Using Hilbert’s Irreducibility

Theorem [Ser92, Proposition 3.3.1] we show that the rational points contributed by

twists of a fixed map are contained in a thin set.

Theorem 1.3.3 Let X be a geometrically uniruled normal projective variety over a

number filed F . Suppose f : Y −→ X is a generically finite morphism from a normal

projective variety. As we vary over all σ ∈ H1(Gal(F/F ),Aut(Y /X)) such that the

9

corresponding twist Y σ is irreducible and fσ : Y σ −→ X is a breaking thin map, the

set of rational points ⋃σ

fσ(Y σ(F ))

is contained in a thin subset of X(F ).

10

Chapter 2

Preliminaries

In this chapter we discuss the preliminaries and build the necessary background. We

work in characteristic 0 throughout this thesis.

2.1 Neron-Severi group.

Let k be a field of characteristic 0. A variety defined over k is an integral separated

scheme of finite type over k. Let X be a smooth proper variety over a field k. The

Neron-Severi group NS(X) is defined as the quotient of the group of Weil divisors,

Cl(X), modulo algebraic equivalence. We denote N1(X) = Div(X)/ ≡, the quotient

of Cartier divisors by numerical equivalence. We denote NS(X)R = NS(X) ⊗ R

and similarly N1(X)R. By [Ner52], NS(X)R is a finite-dimensional vector space and

its rank ρ(X) is called the Picard rank. If X is a smooth projective variety, then

NS(X)R ∼= N1(X)R. From now on, we will often drop the subscript R and write

N1(X) for the space of R-1-cycles modulo numerical equivalence, when it is clear

from the context that we are dealing with the R-vector space.

Remark 2.1.1 Let X be a smooth variety over an algebraically closed field k. If

k ⊂ k′ is an extension of algebraically closed fields, then the natural homomorphism

11

NS(X) −→ NS(Xk′) is an isomorphism. So the Picard rank is unchanged under base

extension of algebraically closed fields.

Let X −→ T be a smooth proper morphism of irreducible varieties. Suppose

s,t ∈ T such that s is a specialization of t, i.e. s is in the closure of {t}. Let Xt

denote the base change to the algebraic closure of the residue field k(t).

Proposition 2.1.2 ([MP, Proposition 3.6]) In the situation as above, it is possible

to choose a specialization homomorphism

spt,s : NS(Xt) −→ NS(Xs)

such that

(a) spt,s is injective. In particular ρ(Xs) ≥ ρ(Xt).

(b) If spt,s maps a class [L] to an ample class, then L is ample.

If ρ(Xs) = ρ(Xt), then the homomorphism NS(Xt)R −→ NS(Xs)R is an isomor-

phism.

Let X −→ T be a smooth projective morphism of irreducible varieties over C. In

Section 12 of [KM92], the local system GN 1(X/T ) was introduced. This is a sheaf in

the analytic topology defined as:

GN 1(X/T )(U) = {sections of N 1(X/T ) over U with open support}

for analytic open U ⊂ T , and the functor N 1(X/T ) is defined as N1(X ×T T ′) for

any T ′ −→ T . It was shown in [KM92, 12.2] that GN 1(X/T ) is a local system with

finite monodromy and GN 1(X/T )|t = N1(Xt) for very general t ∈ T . We can base

change to a finite etale cover of T ′ −→ T so that GN 1(X ′/T ′) has trivial monodromy.

12

Then we have a natural identification of the fibers of GN 1(X ′/T ′) and N1(X ′/T ′).

Therefore, for t′ ∈ T ′ very general, the natural map N1(X ′/T ′) −→ N1(X ′t′) is an

isomorphism. One can prove the same results over any algebraically closed field of

characteristic 0, by using the Lefschetz principle.

2.2 Minimal and Canonical models.

A pair (X,∆) is a normal variety X with an R-divisor ∆ over a field k. We refer to

[KM98] for the standard definitions of singularities of pairs (e.g. terminal, canonical,

klt etc).

Definition 2.2.1 Let X be a smooth projective variety and D be a Q-divisor on X.

We define

κ(X,D) = lim supm→∞logh0(X,OX(bmDc))

logm

If X is a normal projective variety and D an Q-Cartier divisor, then we define

κ(X,D) = κ(X, π∗D)

for a resolution of singularities π : X −→ X. It is easy to see that the definition is

independent of the choice of the resolution.

Let (X,∆) be a klt pair, with ∆ a R-divisor and KX + ∆ is R-Cartier. Let

f : X −→ T be a projective morphism. A pair (X ′,∆′) sitting in a diagram

X X ′

T

φ

f

f ′

13

is called a Q-factorial minimal model of (X,∆) over T if

(1) X ′ is Q-factorial,

(2) f ′ is projective,

(3) φ is a birational contraction,

(4) ∆′ = φ∗∆

(5) KX′ + ∆′ is f ′-nef,

(6) a(E,X,∆) < a(E,X ′,∆′) for all φ-exceptional divisors E ⊂ X. Equivalently,

if for a common resolution p : W −→ X and q : W −→ X ′, we may write

p∗(KX + ∆) = q∗(KX′ + ∆′) + E

where E ≥ 0 is q-exceptional and the support of E contains the strict transform

of the φ-exceptional divisors.

A canonical model over T is defined to be a projective morphism g : Z −→ T

with a surjective morphism π : X ′ −→ Z with connected geometric fibers from a

minimal model such that KX′ + ∆′ = π∗H for an R-Cartier divisor H on Z which is

ample over T .

Over an algebraically closed field k, if KX + ∆ is f -pseudo-effective and ∆ is

f -big, then by [BCHM10], we may run a (KX + ∆)-MMP with scaling to obtain a

Q-factorial minimal model (X ′,∆′) over T . It follows that (X ′,∆′) is also klt. Then

the basepoint freeness theorem implies that (KX′+∆′) is f ′-semi-ample. Hence there

exists a relative canonical model g : Z −→ T . In particular, if ∆ is a Q-divisor, the

OT -algebra

R(X ′,∆′) =⊕m

f ′∗OX′(bm(KX′ + ∆′)c)

14

is finitely generated. Let X ′ −→ Z −→ ProjT (R(X ′,∆′)) be the Stein factorization

of the natural morphism. Then Z is the relative canonical model over T .

In particular, let X be a smooth projective variety over an algebraically closed

field k. Let ∆ be a big Q-divisor such that KX + ∆ is pseudo-effective and (X,∆) is

klt. If we take T = Spec(k) in the above discussion, we can conclude that the section

ring

R(X,∆) =⊕m

H0(X,OX(m(KX + ∆))

is finitely generated. Let W = Proj(R(X,∆)). Note that dim(W ) = κ(X,KX + ∆).

The natural map X 99K W is the rational map to canonical model for (X,∆). Note

that if k is not algebraically closed, the section ring R(X,∆) is still finitely generated,

since it is the direct sum of section rings of the geometric components of Xk. From

now on, even while working over a non-algebraically closed field, the rational map

X 99K W is called the canonical model for (X,∆).

For the rest of this section we work over an algebraically closed field.

The following result relates the relative MMP over a base to the MMP of the

fibers (see [dFH11, Theorem 4.1] and [KM92, 12.3] for related statements).

Lemma 2.2.2 Let f : X −→ T be a flat projective morphism of normal varieties.

Suppose X is Q-factorial and D be an effective R-divisor such that (X,D) is klt. Let

ψ : X −→ Z be the contraction of a KX + D-negative extremal ray of NE(X/T ).

Suppose for t ∈ T very general, the restriction map N1(X/T ) −→ N1(Xt) is surjec-

tive and Xt is Q-factorial.

15

Let t ∈ T be very general. If ψt : Xt −→ Zt is not an isomorphism, then it is a

contraction of a KXt +Dt-negative extremal ray, and :

(a) If ψ is of fiber type, so is ψt.

(b) If ψ is a divisorial contraction of a divisor G, then ψt is a divisorial contraction

of Gt and N1(Z/T ) −→ N1(Zt) is surjective.

(c) If ψ is a flipping contraction and ψ+ : X+ −→ Z is the flip, then ψt is a flipping

contraction and X+t is the flip of ψt : Xt −→ Zt. Also, N1(X+/T ) −→ N1(X+

t )

is surjective.

Proof. Since the natural restriction map N1(X/T ) −→ N1(Xt) is surjective for very

general t ∈ T , any curve in Xt that spans a KX + D-negative extremal ray R of

NE(X/T ), also spans a KXt + Dt negative extremal ray Rt of NE(Xt). For t ∈ T

general, the base change Zt is normal and the morphism Xt −→ Zt has connected

fibers, hence ψt∗OXt = OZt . Hence ψt is the contraction of the ray Rt for very general

t ∈ T .

If ψ is of fiber type, then so is ψt for general t ∈ T . Let us assume that ψ is

birational.

Suppose ψ is a divisorial contraction of a divisor G. Then all components of Gt are

contracted. By the injectivity of N1(Xt) −→ N1(X/T ), we see that ψt is an extremal

divisorial contraction of Gt (and Gt is irreducible). Since Xt is Q-factorial, we have

the surjectivity of N1(Z/T ) −→ N1(Zt).

Suppose ψ is a flipping contraction and φ : X 99K X+ is the flip. For very general

t ∈ T , Xt −→ Zt is a small birational contraction of the ray Rt. Also, X+t −→ Zt

is also small birational and KX+t

+ (φ∗D)t is ψ+-ample for t ∈ T general. Therefore

φt : Xt 99K X+t is the flip. The surjectivity of N1(X+/T ) −→ N1(X+

t ) follows from

ψt being an isomorphism in codimension one.

16

The next Proposition allows us to compare minimal and canonical models over a

base to those of a general fiber.

Proposition 2.2.3 Let f : X −→ T be a smooth morphism. Suppose X is smooth

and ∆ is an f -big and f -nef R-divisor such that (X,∆) is is klt. Suppose the local

system GN 1(X/T ) has trivial monodromy. Let φ : X 99K X ′ be the relative minimal

model obtained by running a (KX + ∆)-MMP over T and π : X ′ −→ Z be the

morphism to the canonical model over T . Then for a general t ∈ T ,

(1) The base change φt : Xt 99K X ′t is a Q-factorial minimal model of (Xt,∆t),

(2) Also, πt : X ′t −→ Zt is the canonical model of (Xt,∆t).

Proof. (1) Since GN 1(X/T ) has trivial monodromy, the natural restriction morphism

N1(X/T )∼−→ N1(Xt) is an isomorphism for t ∈ T very general. Then Lemma 2.2.2

implies that, for very general t ∈ T , the base change φt : Xt 99K X ′t is a composition

of steps of the (KXt + ∆t)-MMP. In particular, X ′t is Q-factorial for a very general

t ∈ T . The fibers X ′t have terminal singularities, by [LTT18, Lemma 2.4]. Hence

[KM92, 12.1.10] implies that there is a non-empty open U ⊂ T such that X ′t is

Q-factorial for t ∈ U . For a general t ∈ T , the conditions (2)-(6) in the definition of

a minimal model follows easily. Therefore, (X ′t,∆′t) is a Q-factorial minimal model

of (Xt,∆t) for general t ∈ T .

(2) Let g : Z −→ T be the relative canonical model. Now Z is normal. Therefore,

for a general t ∈ T , the base change Zt is normal and X ′t −→ Zt has geometrically

connected fibers. Also, KX′ + ∆ = g∗H where H is a π-ample R-Cartier divisor on

Z. By adjunction, KX′t+ ∆′t is pull-back of an ample R-Cartier divisor on Zt. Hence,

17

X ′t −→ Zt is the canonical model for general t ∈ T .

Definition 2.2.4 Let ψ : X 99K X ′ be a proper birational contraction (i.e. ψ−1 does

not contract any divisors) of normal quasi-projective varieties. Let D be a R-Cartier

divisor such that D′ = ψ∗D is also R-Cartier. We say that ψ is D-negative if for some

common resolution p : W −→ X and q : W −→ Y , we may write

p∗D = q∗D′ + E

where E ≥ 0 is q-exceptional and the support of E contains the strict transform of

the ψ-exceptional divisors.

Recall that if ψ : X 99K X ′ is a KX+∆-minimal model, then it is KX+∆-negative

by definition. Further, if (X,∆) is terminal and p : W −→ X, q : W −→ X ′ is a

common resolution, then q : W −→ X ′ is KW + p∗∆-negative.

In general, if we have a KX + ∆-negative contraction ψ : X 99K X ′ of a terminal

pair (X,∆), then the pushforward (X ′, ψ∗∆) might not be terminal since ∆ might

contain the ψ-exceptional divisors as components. If ∆ is big and nef, then the

following lemma shows that we can achieve the desired conclusion by passing to a

resolution. This result will be used in Chapter 4 for proving the boundedness of

degrees of adjoint-rigid covers preserving the a-constant.

Lemma 2.2.5 Let X be a normal variety with terminal singularities and D a big

and nef R-Cartier divisor. Let ψ : X 99K X ′ be a KX +D-negative contraction. Then

we may choose a common resolution p : W −→ X and q : W −→ X ′ with ∆ ∼R p∗D

such that (W, ∆) and (X ′,∆′ = µ∗∆) are both terminal.

Proof. Since D is big and nef, we can find ∆ ∼R D such that (X,∆) is terminal

(see the proof of [LTT18, Theorem 2.3]). Let p : W −→ X and q : W −→ X ′ be a

18

common log resolution

W

X X ′

p q

ψ

Let Ej be the ψ-exceptional divisors and Fi the p-exceptional divisors. Note that

the q-exceptional divisors are Fi and the strict transforms Ej of the ψ-exceptional

divisors. We have

KW + p−1∗ ∆ = p∗(KX + ∆) +

∑i aiFi

where ai > 0. We may add p-exceptional divisors to obtain

KW + p∗∆ = p∗(KX + ∆) +∑

i biFi

with bi > 0. Since ψ is KX + ∆-negative, we may write

p∗(KX + ∆) = q∗(KX′ + ∆′) +∑

j cjEj +∑

i diFi

where ∆′ = ψ∗∆ and cj > 0, di ≥ 0. Therefore we have

KW + p∗∆ = q∗(KX′ + ∆′) +∑

j αjEj +∑

i βiFi

with αj,βi > 0. As p∗∆ is big and nef, we may choose ∆ ∼R p∗∆ with the coefficients

of q-exceptional divisors in ∆ sufficiently small such that (W, ∆) is a simple normal

crossing terminal pair and

KW + q−1∗ q∗∆ = q∗(KX′ + q∗∆) +

∑j α′jEj +

∑i β′iFi

with α′j,β′i > 0. Therefore (X ′, q∗∆) is also terminal.

19

2.3 Boundedness of singular Fano varieties

Let X be a normal projective variety of dimension n and D an R-divisor. The volume

of D is defined by

vol(X,D) = limm→∞

n!h0(X,OX(bmDc))mn

If D is nef then vol(X,D) = Dn. Also D is big iff vol(X,D) > 0. The volume

depends only on the numerical class [D] ∈ N1(X).

We recall the following well-known result.

Lemma 2.3.1 [HMX16, Lemma 1.5.1] Let f : Y −→ X and g : X −→ W be a

birational morphism of normal projective varieties and D an R-divisor on X.

(1) vol(W, g∗D) ≥ vol(X,D),

(2) If D is R-Cartier and Ei are f -exceptional, then

vol(f ∗D +∑

i aiEi) = vol(X,D)

for ai > 0.

We recall the definitions and results related to the BAB-conjecture.

Definition 2.3.2 Let X be a normal projective variety and ∆ an effective boundary

R-divisor such that KX + ∆ is Q-Cartier. We say the (X,∆) is ε-log canonical (resp.

ε-klt) if the coefficients of ∆ are at most (resp. strictly smaller than) 1 − ε and

for a resolution of singularities π : X −→ X with exceptional divisors Ei, we have

a(Ei, X,∆) ≥ −1+ ε (resp. a(Ei, X,∆) > −1+ ε) where the dicrepancies a(Ei, X,∆)

20

are defined by the equation

KX + π−1∗ ∆ = π∗(KX + ∆) + a(Ei, X,∆)Ei.

The following is the BAB-conjecture proved by Birkar.

Theorem 2.3.3 [Bir16b, , Theorem 1.1] Let n be a natural number and ε > 0 a real

number. Then the set of projective varieties X such that

(1) X is of dimension n with a boundary divsior ∆ such that (X,∆) is ε-log canon-

ical

(2) −(KX + ∆) is big and nef,

form a bounded family.

A consequence of the above theorem is the boundedness of anticanonical volumes.

Corollary 2.3.4 (Weak BAB-conjecture) Let n be a natural number and ε > 0 a

real number. There exists a constant M(n, ε) such that, for any normal projective

variety X satisfying

(1) X is of dimension n such that there is a boundary divisor ∆ with (X,∆) is ε-klt

and KX is Q-Cartier.

(2) −(KX + ∆) is ample,

we have

vol(−KX) < M(n, ε).

2.4 Geometric invariants.

Let X be a smooth projective variety defined over a field k. The pseudo-effective

cone Eff1(X) is the closure of the cone of effective divisor classes in NS(X)R. The

21

interior of Eff1(X) is the cone of big divisors Big1(X)R. The cone of nef divisors

is denoted by Nef1(X). Let N1(X) be the cone of R-1-cycles modulo numerical

equivalence. The intersection pairing induces a duality between N1(X) and N1(X).

The cone of pseudo-effective cone of curves Eff1(X) is the dual to Nef1(X) and the

cone of nef curves Nef1(X) is dual to Eff1(X) under the intersection pairing.

Let L be a pseudo-effective Q-Cartier divisor on X. The stable base locus of L

[Laz04] is defined as

B(L) :=⋂m∈N

Bs(mL)

where the intersection is of the base loci of mL such that mL is Cartier. The aug-

mented base locus of L is defined as

B+(L) :=⋂A

B(L− A)

where the intersection is over all ample Q-Cartier divisors A. It is known that B+(L)

is a closed subset of X [ELM+06]. By [Nak00, Theorem 0.3], if L is big, then L|Z is

big for any subvariety Z 6⊂ B+(L).

2.4.1 Fujita invariant or a-constant

Definition 2.4.1 Let L be a big Q-Cartier Q divisor on X. The Fujita invariant or

a-constant is defined as

a(X,L) = min{t ∈ R|KX + tL ∈ Eff1(X)}.

If L is nef but not big, we define a(X,L) = ∞. For a singular projective variety

we define a(X,L) := a(X, π∗L) where π : X −→ X is a resolution of X. It is

invariant under pull-back by a birational morphism of smooth varieties and hence

22

independent of the choice of the resolution [HTT15, Proposition 2.7]. By [BDPP13]

we know that a(X,L) > 0 if and only if X is geometrically uniruled.

It was shown in [BCHM10, Corollary 1.1.7] that, if X is uniruled with klt

singularities and L is ample, then a(X,L) is a rational number. If L is big and nef,

ratioanlity of the a-constant still holds due to [HTT15, Theorem 2.16]. If L is big

and not nef, then a(X,L) can be irrational [HTT15, Example 2.6]. For a smooth

projective variety X, the function a(X, ) : Big1(X)R −→ R is a continuous function

[LTT18, Lemma 3.2].

Remark 2.4.2 [LT19, Section 3.2] We have the following formula for the a-constant

when a(X,L) > 0 ,

a(X,L) = min{rs∈ Q|dim(H0(X, sKX + rL)) > 0}.

We make the following definition for convenience.

Definition 2.4.3 Let X be a smooth projective variety over a field of characteristic

0 and let L be a big and nef Q-divisor on X. Let π : X 99K W be the canonical

model associated to KX + a(X,L)L and let U be the maximal open subset where

π is defined. Suppose that T → W is a dominant morphism of normal projective

varieties. Then there is a unique component of T ×W U mapping dominantly to T .

We call such an X an Iitaka base change of X; it is naturally equipped with maps

X → T and X → X.

The behaviour of the a-constant under pull-back to a generically finite cover is

depicted in the following inequality.

Lemma 2.4.4 Let f : Y −→ X be a generically finite surjective morphism of varieties

and L a big Q-Cartier Q-divisor on X. Then

23

a(Y, f ∗L) ≤ a(X,L).

Proof. Since the a-constant is computed on a resolution of singularities. We may

assume X and Y are smooth. As f : Y −→ X is generically finite, we may write

KY = f ∗KX +R

for some effective divisor R. Let a(X,L) = a. Then we have

KY + af ∗L = f ∗(KX + aL) +R.

Since R ≥ 0 and KX + aL is pseudo-effective, we see that KY + af ∗L is also pseudo-

effective. Hence a(Y, f ∗L) ≤ a(X,L) = a.

Note that the a-constant is independent of base change to another field.

Proposition 2.4.5 [LST18] Let X be a smooth projective variety over a field k and

L a big and nef Q-divisor. Let k′/k be a field extension of k. Then for any irreducible

component X ′ of Xk′ , we have a(X ′, L|X′) = a(X,L) and κ(X ′, KX′+a(X ′, L′)L|X′) =

κ(X,KX + a(X,L)L).

Proof. Let {Xi} be the irreducible components of Xk′ . We may replace k′ by a finite

extension of k such that all the Xi’s are defined over it. For any two irreducible com-

ponents Xi and Xj, there is an element of σ ∈ Gal(k′/k) that induces an isomorphism

σ : Xi −→ Xj. Since σ∗KXj = KXi , we see that

H0(Xi,m(KXi + aL|Xi)) ' H0(Xj,m(KXj + aL|Xj))

24

for any m ∈ Z and a ∈ Q such that m(KX + aL) is Cartier. Also, note that we have

H0(X,m(KX + aL)) =⊕i

H0(Xi,m(KXi + aL|Xi)).

Therefore, by Remark 2.4.2, we have a(Xi, L) = a(X,L) and κ(Xi, KXi+a(Xi, L)L) =

κ(X,KX + a(X,L)L).

We recall the following result about the behavior of the a-constant in families.

Theorem 2.4.6 ([LT17, HMX13]). Let f : X −→ T be a smooth projective

morphism of varieties over an algebraically closed field k such that the fibers are

uniruled. Let L be an f -big and f -nef Q-Cartier divisor on X. Then there exists a

nonempty subset U ⊂ T such that a(Xt, L|Xt) is constant for t ∈ U and the Iitaka

dimension κ(KXt + a(Xt, L|Xt)L|Xt) is constant for t ∈ U .

The following result states that the conclusion of the above theorem holds even if

the base field is not algebraically closed. Note that we do not assume that the generic

fiber is geometrically integral.

Theorem 2.4.7 ([LST18]) Let f : X −→ T be a smooth projective morphism over a

field k of characteristic 0 such that every fiber is integral and geometrically uniruled.

Let L be an f -big and f -nef Q-Cartier divisor on X. Then there exists a nonempty

Zariski open subset U ⊂ T such that a(Xt, L|Xt) is constant for t ∈ U and the Iitaka

dimension κ(KXt + a(Xt, L|Xt)L|Xt) is constant for t ∈ U .

Proof. Consider the base change fk : Xk −→ Tk to the algebraic closure of k. Let

X ′ be an irreducible component of Xk. Then fk(X′) is contained in some irreducible

component T ′ of Tk. By applying Theorem 2.4.6 to the restriction fk : X ′ −→ T ′,

25

we see that there is an open subset U ′ ⊂ T ′ such that the a-constants and the Iitaka

dimension of the fibers are constant. Now for t ∈ T , if t ∈ T ′ maps to t, then by

Proposition 2.4.5 we have an equality of the a-constant and the Iitaka dimension for

the fibers Xt and X ′t. Therefore, by taking Galois conjugates of T ′ \ U ′, we obtain a

closed subset Z ⊂ T such that the a-constant and the Iitaka dimension of the fibers

is constant on U = T \ Z.

We have the following result about the behaviour of the a-constant under restric-

tion to subvarieties.

Theorem 2.4.8 (see [HJ17, Theorem 1.1], [LTT18, Theorem 4.8] ) Let X be a

smooth uniruled projective variety and L a big and nef Q-divisor over a field of

characteristic 0. Then there is a proper closed subset V ( X such that any subvariety

Y satisfying a(Y, L|Y ) > a(X,L) is contained in V .

Note that it is enough to prove the statement when k is algebraically closed. Over

an algebraically closed field, the above result was proved in [HJ17] when L is big

and semi-ample. In [LTT18], it was proved assuming the weak BAB-conjecture. By

[Bir16b], now we know that the BAB-conjecture holds (see Section 2.3). Therefore

the above result works for L big and nef.

Lemma 2.4.9 Let X be a smooth projective variety and L a big and nef Q-divisor.

(1) Let π : U → W be a family of subvarieties of X such that the evaluation map s :

U → X is dominant. Then for a general fiber Uw, we have a(Uw, L) ≤ a(X,L).

(2) If U = X and π : X → W is the morphism to the canonical model for

(X, a(X,L)L), then for a general fiber Xw we have a(Xw, L) = a(X,L).

Proof. Part (1) was proved in [LTT18, Proposition 4.1] and part (2) was proved in

[LTT18, Theorem 4.5] over an algebraically closed field. However the same argument

goes through over an arbitrary base field.

26

2.4.2 the b-constant

Let X be a smooth projective variety over a field k of characteristic 0. Let L be a big

and nef Q-divisor on X. Let FX be the minimal supported face of Eff1(X) containing

KX + a(X,L)L. Let FX ⊂ Nef1(X) denote the dual face FX = F∨X . Note that FX is

the face consisting of classes with vanishing intersection with KX + a(X,L)L.

Definition 2.4.10 The b-constant (or b-invariant) of (X,L) is defined as

b(k,X, L) = codimFX .

Note that we have b(k,X, L) = dimFX . The b-cosntant is invariant under

pullback by a birational morphism of smooth varieties [HTT15, Proposition 2.10].

For a singular variety X we define b(X,L) := b(X, π∗L), by pulling back to a

resolution. By Remark 2.1.1, if we have an extension k ⊂ k′ of algebraically closed

fields, the pull back map N1(X) −→ N1(Xk′) is an isomorphism and the pseudo-

effective cones are isomorphic by flat base change. Also, KX + a(X,L)L maps to

KXk′+a(Xk′ , Lk′)Lk′ under this isomorphism. Therefore the b-constant is unchanged,

i.e. b(k′, Xk′ , Lk′) = b(k,X, L). From now on, when our base field is algebraically

closed we write b(X,L) instead of b(k,X, L) However, the b-constant might change

if the ground field is not algebraically closed, but it can only increase.

Proposition 2.4.11 [LST18] Let X be a smooth geometrically integral projective

variety defined over a field k and L a big and nef Q-divisor. Let k′/k be a finite

extension. Then

b(k,X, L) ≤ b(k′, Xk′ , L|Xk′ ).

Proof. The action of Gal(k/k) on N1(Xk) factors through a finite group and we

have Nef1(Xk′) = Nef1(Xk)Gal(k/k′) for any finite extension k′/k. Since, Gal(k/k′) ⊂

27

Gal(k/k), we have Nef1(Xk)Gal(k/k) ⊂ Nef1(Xk)

Gal(k/k′). Hence we conclude that

dimFX ≤ dimFX′k by intersecting with the hyperplane defined by KX + a(X,L)L.

We not the following lemma about the birational compatibility of the b-constant

over non-algebraically closed fields.

Lemma 2.4.12 [LST18] Let f : X ′ −→ X be a generically finite morphism of smooth

projective varieties such that f maps every geometric component of X ′ birationally

to a geometric component of X. Let L be a big and nef Q-divisor. Then the push-

forward map f∗ : N1(X ′)→ N1(X) induces an isomorphism f∗ : FX′ ' FX .

Proof. We have N1(X ′) = f ∗N1(X)⊕⊕

iREi, where Ei are the f -exceptional divi-

sors. By Proposition 2.4.5 and the birational invariance of the a-constant, we have

a(X ′, f ∗L) = a(X,L). We may write

KX′ + a(X ′, f ∗L)f ∗L = f ∗(KX + a(X,L)L) + E

where E is f -exceptional and the contains all the Ei’s with positive coefficient. There-

fore if α ∈ Nef1(X ′) such that (KX′ + a(X ′, f ∗L)f ∗L) ·α = 0 then (KX + a(X,L)L) ·

f∗α = 0 and Ei · α = 0. Hence f∗ maps FX′ to FX .

Using the direct sum decomposition of N1(X ′), we see that if f∗α = 0 for α ∈ FX′ ,

then α · D = 0 for any D ∈ N1(X ′), i.e. α = 0. Therefore f∗ is injective. Now for

any β ∈ FX , we have (KX′ + a(X ′, f ∗L)f ∗L) · f ∗β = (KX + a(X,L)L) · β = 0. Hence

f ∗β ∈ FX′ and f∗ is surjective.

Let X be a smooth uniruled projective variety over an algebraically closed field

k and L a big and nef Q-divisor on X. The following result (contained in [LTT18])

gives a geometric interpretation of the b-constant.

Proposition 2.4.13 Let φ : X 99K X ′ be a KX + a(X,L)L-minimal model. Then

28

(1) b(X,L) = b(X ′, φ∗L).

(2) If κ(KX + a(X,L)L) = 0 then b(X,L) = rkN1(X ′)R.

(3) If κ(KX + a(X,L)L) > 0 and π : X ′ −→ Z is the morphism to the canonical

model and Y ′ is a general fiber of π. Then

b(X,L) = rkN1(X ′)R − rkN1π(X ′)R = rk(im(N1(X ′)R −→ N1(Y ′)R))

where N1π(X ′)R is the span of the π-vertical divisors and N1(X ′)R −→ N1(Y ′)R

is the restriction map.

Proof. Part (1) is the statement of Lemma 3.5 in [LTT18]. Part (2) follows from part

(1). By abundance, KX + a(X,L)φ∗L is semi-ample. Then κ(KX + a(X,L)L) = 0

implies that KX + a(X,L)φ∗L ≡ 0. Hence, b(X,L) = b(X ′, φ∗L) = rkN1(X ′)R. Part

(3) follows from the proof of Theorem 4.5 in [LTT18].

Remark 2.4.14 Let X be a smooth uniruled projective variety over an algebraically

closed field k. Let L be a big and nef Q-divisor. Let E be an irreducible effective

divisor such that KX + a(X,L)L− cE ∈ Eff1(X) for some c > 0. Then, for any nef

curve class α ∈ Nef1(X) such that (KX + a(X,L)L) · α = 0, we have E · α = 0.

Therefore E ∈ FX . We claim that Span(FX) is spanned by finitely many irreducible

effective divisors Ei such that KX + a(X,L)L− cEi ∈ Eff1(X) for some ci > 0.

Since L is big and nef, we may write a(X,L)L ∼Q A + ∆, where A is ample and

(X,A + ∆) is terminal. Also, we may choose a basis {Ei} of Span(FX) where Ei ∈

Eff1(X) are pseudo-effective divisor classes sufficiently close to KX + A + ∆. Then

we may write Ei ≡ KX + Ai + ∆ for Ai ample such that (X,Ai + ∆) is terminal.

By [BCHM10, Theorem D], we may assume Ei are effective. Therefore Span(FX) is

29

spanned by irreducible effective divisor classes. Note that KX + a(X,L)L − ciEi ∈

Eff1(X) for some ci > 0.

The following result helps us compute the b-invariant over an arbitrary base field.

Lemma 2.4.15 [LST18] Let X be a smooth geometrically uniruled geometrically

integral variety over a field k of characteristic 0. Let L be a big and nef Q-divisor.

Let π : X 99K W be the canonical model for (X, a(X,L)L). Suppose there is a

non-empty open subset U ⊂ W such that (i) U is smooth, (ii) π is well-defined and

smooth over U . Let w ∈ U be a a closed point and w a geometric point over w. (Note

that N1(Xw) naturally embeds into N1(Xw)). Then,

(1) we have

b(k,X, L) = dim (N1(Xw) ∩N1(Xw)πet1 (U,w))/Span({Ei}ri=1)

where Ei are the irreducible divisors which dominate W such that KX +

a(X,L)L− ciEi ∈ Eff1(X) for some ci > 0.

(2) Let i : Xw ↪→ X denote the inclusion. Then i∗(FXw) ⊂ FX and the induced

map i∗ : Span(FXw)→ Span(FX) is a surjection.

(3) If Xw is a general fiber of π then i∗ : FXw → FX is a surjection.

Proof. (1) We may assume that π is a morphism by replacing X by a resolution.

Let η be the generic point of U . Note that Gal(k(η)/k(η)) ' πet1 (U, η), where k(η)

denotes the function field of U . Hence N1(Xη)πet

1 (U,η) is the Galois invariant part of

N1(Xη), i.e.

N1(Xη) ' N1(Xη)πet

1 (U,η) ⊂ N1(Xη).

Since we have a surjection N1(X) � N1(Xη) we obtain that the restriction map

N1(X)→ N1(Xη) ' N1(Xη)πet

1 (U,η) is surjective.

30

By [MP, Theorem 1.1] we know that there exists s ∈ U such that the special-

ization morphism N1(Xη) → N1(Xs) is an isomorphism. Since the generic fiber

of π is rationally connected, the Picard rank of the geometric fibers are constant.

Hence N1(Xη)∼−→ N1(Xw). Since πet

1 (U, η) ' πet1 (U,w) and the monodromy actions

commute with the isomorphism given by the specialization morphism, we have

N1(Xη)πet

1 (U,η) ∼−→ N1(Xw)πet1 (U,w).

As the specialization morphism commutes with restriction from N1(X) we conclude

that we have a surjection N1(X) � N1(Xw)πet1 (U,w) for w. Since the action of

Gal(k/k) on N1(X) and N1(Xw) factors through a finite group, by taking invari-

ants we obtain a surjection

N1(X) ' N1(X)Gal(F/F ) � N1(Xw) ∩N1(Xw)πet1 (W ◦,w)

Now, by Remark 2.4.14 we know that Span(FX) is spanned by irreducible effective

divisors E ∈ N1(X) such that KX + a(X,L)L − cE ∈ Eff1(X) for some c > 0.

Note that the restriction of E to N1(Xw) is zero iff π(E) ( W . Therefore by taking

quotients by the span of Ei’s, we obtain the claim.

(2) First we show that i∗ maps FXw to FX . Note that since Xw has trivial normal

bundle in X, the argument of [Pet12, Arxiv version, Theorem 6.8] shows that if

D ∈ Eff1(X) is a pseudo-effective divisor the D|Xw is pseudo-effective. Therefore, by

duality, i∗(Nef1(Xw)) ⊂ Nef1(X). By Lemma 2.4.9 we know that a(X,L) = a(Xw, L).

Hence, for any α ∈ Nef(Xw) since (KX + a(X,L)L) · α = KXw + a(Xw, L)L · α.

Therefore we have i∗FXw ⊂ FX . By part (1), we have an injection,

N1(X)/Span(FX) ∼= N1(Xw)∩N1(Xw)πet1 (W ◦,w)/Span(i∗FX) ↪→ N1(Xw)/Span(FXw).

31

By duality, we conclude that the map i∗ : Span(FXw)→ Span(FX) is a surjection.

(3)Note that it is enough to prove the statement over an algebraically closed base

field, by taking Galois invariants. Therefore we may assume that k is algebraically

closed. We may write a(X,L)L ∼Q A+ ∆ such that (X,A+ ∆) is terminal where A

is ample. Hence (X, 12A+ ∆) is terminal and FX is a (KX + 1

2A+ ∆)-negative face.

Therefore, by [Leh12, Theorem 1.3], the face FX contains finitely many extremal rays

generated by movable curve classes, say αi. Let g : C → T be a covering family such

that [Ct] = αi for some i. Since (KX+a(X,L)L)·αi = 0, we know that f∗(Ct) = 0. Let

s : C → X be the morphism to X. Let w ∈ U be a general point and Cw = s−1(Xw).

Since every curve Ct is contained in some fiber of π, we see that Cw → g(Cw) is a

covering family for Xw. Therefore, Cw ∈ Nef1(Xw) and i∗([Cw,t]) = [Ct] and hence i∗

is surjective on faces.

2.5 Manin’s conjecture and the thin exceptional

set

Let X be a smooth projective variety over a number field F . Let L = (L, || · ||) be a

big and nef line bundle on X with an adelic metric. We have corresponding height

function HL : X(F )→ R≥0 associated to L (see [Tsc09, Section 4.8] for the definition

of adelic metric and height function). If L is ample, we know that the set of rational

points of bounded height are finite by Northcott’s finiteness theorem. If L is big and

nef, then the height function still satisfies a Northcott property for rational points

outside of the proper closed subset B+(L) ⊂ X. Here B+(L) denotes the augmented

base locus defined in Section 2.4. In particular, for B > 0, there are finitely many

rational points in x ∈ (X \B+(L))(F ) such that HL(x) ≤ B. We denote the number

32

of rational points of bounded height contained in a subset Q ⊂ X by

N(Q,L,B) = #{x ∈ Q(F )|HL(x) ≤ B}.

The following conjecture predicts an asymptotic formula for the number of rational

points with bounded height when X is geometrically rationally connected geometri-

cally integral.

Conjecture 2.5.1 (Manin’s Conjecture.) Let X be a smooth geometrically ratio-

nally connected and geometrically integral variety defined over a number field F . Let

L a big and nef line bundle on X with an associated height function HL. Then, after

possibly replacing F by a finite extension, there exists a thin set Z ⊂ X(F ) such that

N(X(F ) \ Z,L,B) ∼ c(F,Z, L)Ba(X,L)(logB)b(F,X,L)−1

as B → ∞, where a(X,L) and b(F,X,L) are the geometric invariants defined in

Section 2.4, the constant c(F,Z, L) is Peyre’s constant (see [BT98b, Pey95].

We note that Manin’s conjecture is proved in the following cases: projective space

Pn [Sch79, Pey95], toric varieties [BT96a, BT98a, Sal98], equivariant compactifica-

tions of algebraic groups [CLT02, STBT07, ST16], flag varieties [FMT89], low degree

complete intersections [Bir62, BHB17], some classes of geometrically rational surfaces

[dlBBD07, Bro09, Bro10, dlBBP12], Le Rudulier’s example [LR14].

Manin’s conjecture was first formulated in [FMT89, BM90] by analyzing examples

of Pn, flag varieties, complete intersections. The set Z in Manin’s conjecture is

called the exceptional set Initially, in [FMT89, BM90], the conjecture was stated

with Z being contained a proper closed subset of X. But counterexamples due to

[BT96b, LR14] show that the closed set version is false. In [Pey03], Peyre was the

33

first to suggest that the exceptional set must be a thin set. There are currently no

known counterexamples to the thin set in Conjecture 2.5.1.

The counterexamples to the closed set version of Manin’s conjecture arise due the

presence of subvarieties and covers with higher values of the geometric invariants (a

and b-constants). In the following two examples we recall the counterexamples due

to [BT96b, LR14] that exhibits this phenomenon.

Example 2.5.2 [BT96b] Let F = Q(√−3). Let X ⊂ P3

x × P3y be the hypersurface

defined over F by a form of bidegree (1, 3):

3∑j=0

xjy3j = 0.

Let p : X → P3x and q : X → P3

y be the projection morphisms. By the Lefschetz

hyperplane section theorem, we know that Pic(X) = Z2, with the basis given by the

pullback of hyperplane classes of P3x and P3

y, i.e. the Picard rank ρ(X) = 2. The

anticanonical class is given by

−KX = (3, 1).

Let L = −KX , which is ample. We see that b(F,X,L) = 2 as a(X,L) = 1. The

projection q exhibits X as a P2-fibration over P3y and the projection q gives a Mori

fiber space structure on X where the fibers Xx of q are diagonal cubic surfaces. Now

ρ(Xx) = 7 whenever the coordinates xj of x are cubes in F . Since KX |Xx = KXx we

see that b(F,Xx, L) = 7 as a(Xx, L) = 1. Therefore, we see that

(a(X,L), b(F,X,L)) < (a(Xx, L), b(F,Xx, L)).

By [Tsc09, Section 4.4], we know that

N(X◦x, L,B) ∼ B(logB)6

34

for all such fibers and all dense Zariski open subsets X◦x of the fibers. But Conjecture

2.5.1 for closed sets would imply that

N(X◦, L,B) ∼ B(logB).

for some Zariski open subset X◦ ⊂ X. Since every open subset X◦ intersects infinitely

many of the fibers with b(F,Xx, L) = 7, we see that Conjecture 2.5.1 can not hold

for Z being contained a proper closed subset.

Example 2.5.3 [LR14] In this example we discuss Le Rudulier’s example. We refer

to [LR14] for a detailed study. Let F = Q and S denote the surface P1 × P1 defined

over F . Consider the Hilbert scheme X = Hilb2(S). The variety X is a weak Fano

variety of dimension 4, i.e. −Kx is big and nef. Let L = −KX . Then, we have

a(X,L) = 1 and b(F,X,L) = 3 since the Picard rank of X is 3. Let W be the

blow-up of S × S along the diagonal. We have a commutative diagram

W S × S

X Chow2(S)

f g

Here g is the natural quotient by the Z/2Z-action and f is a cover of degree 2. We

have a(W, f ∗L) = 1 and b(F,W, f ∗L) = 4. In [LR14], Conjecture 2.5.1 was proved

for W and L. Since we have the following inequality

(a(X,L), b(F,X,L)) < (a(W, f ∗L), b(F,W, f ∗L))

we see that the closed set version of Conjecture 2.5.1 can not be true for X. However,

[LR14] shows that Conjecture 2.5.1 holds for X after removing the thin exceptional

set Z = W (F ).

35

Therefore, if we have a breaking thin map f : Y → X and if Conjecture 2.5.1 holds

for both (X,L) and (Y, f ∗L), then the rational points in f(Y (F )) must be contained

in the thin exceptional set Z ⊂ X(F ), as shown in the examples above. Motivated

by this the following statement was conjectured in [LT17].

Conjecture 2.5.4 (Geometric consistency.) [LT17, Conjecture 1.1] Let X be a

smooth geometrically uniruled geometrically integral projective variety defined over

a number field F . Let L be a big and nef Q-divisor on X. As we vary over all thin

maps f : Y −→ X with Y geometrically integral, such that

(a(X,L), b(F,X,L)) < (a(Y, f ∗L), b(F, Y, f ∗L))

the set of rational points ⋃f

f(Y (F ))

is contained in a thin subset of X(F ).

In [LST18] we prove the conjecture about geometric consistency above, which is

the content of Theorem 1.3.1.

Remark 2.5.5 [LST18] also deals with the case of thin maps when there is an

equality of the geometric invariants and proposes a conjectural description of the

thin exceptional set Z ([LST18, Conjecture 4.1]). The conjectural description agrees

with many known examples for which Manin’s conjecture is known to be true (see

[LST18, Section 4] for details).

2.6 Twists

In this section we prove Theorem 1.3.3. Let us assume that the ground field is a

number field F . Let X be a smooth projective variety over F and L a big and nef

36

Q-divisor on X.

Let f : Y −→ X be a generically finite cover of quasi-projective varieties defined

over F . A twist of f : Y −→ X is a generically finite cover f ′ : Y ′ −→ X such that,

after base change to the algebraic closure F , we have an isomorphism g : Y∼−→ Y ′

with f = f ′ ◦ g.

Y Y ′

X

∼g

f f ′

All the twists of f : Y −→ X is parametrized by the Galois cohomology of Aut(Y /X).

Precisely, there is a bijection between the set of isomorphism classes of twists of f

and the Galois cohomology group H1(Gal(F/F ),Aut(Y /X)) (see [Ser02, Chapter

III, Proposition 5]). The bijective correspondence works as follows:

Let f ′ : Y ′ → X be a twist of f : Y → X such that we have an X-isomorphism

φ : YF ′∼−→ Y ′F ′ over a finite extension F ′/F . We obtain the corresponding 1-cocyle

in Hom(Gal(F ′/F ),Aut(YF ′/XF ′)) as the map which sends s ∈ Gal(F ′/F ) to the

automorphism φ−1 ◦ φs ∈ Aut(YF ′/XF ′) where φs denotes conjugate of φ by id ⊗ s.

Conversely, let σ ∈ Hom(Gal(F ′/F ),Aut(YF ′/XF ′)) be a 1-cocyle. Then we obtain

an action of Gal(F ′/F ) on YF ′ where s ∈ Gal(F ′/F ) acts by σ(s)◦ (id⊗s). By taking

the quotient by this action we obtain the corresponding twist Y σ = YF ′/Gal(F ′/F ).

Note that the quotient exists because of quasi-projectivity.

Galois morphisms appear prominently in the literature. We use the following

definition.

37

Definition 2.6.1 A finite surjective morphism f : Y → X is called Galois cover if

there is a finite group G ⊂ Aut(Y ) such that f is isomorphic to the quotient map.

Remark 2.6.2 [GKP16, Theorem 3.7] Let f : Y → X be a finite surjective morphism

of quasi-projective varieties. Then there exists a normal quasi-projective variety W

with a finite surjective morphism g : W → Y such that the following holds:

(1) There exist groups H ≤ G such that the morphisms f ◦ g and g are Galois with

group G and H respectively.

(2) The branch locus of the two morphisms agree, i.e. Branch(f ◦ g) = Branch(f).

The morphism W → X is called the Galois closure of f .

Before proving Theorem 1.3.3, we prove a few auxiliary results.

Lemma 2.6.3 [Che04, LST18] Let f : Y 99K X be a dominant generically finite

rational map of normal projective varieties over a number field F . Let L be a big and

nef Q-divisor on X. Then there is a birational modification f ′ : Y ′ → X of f , such

that

(1) Y ′ is smooth,

(2) we have Bir(Y ′/X) = Aut(Y ′/X),

(3) the rational map π : Y 99K Z to the canonical model for KY ′ + a(Y ′, f ′∗L)f ′∗L

is a morphism.

Proof. We may assume that f : Y → X is a morphism by passing to a resolution. We

may replace f : Y → X by the Stein factorization to assume that f is finite. Then

we have Bir(Y /X) = Aut(Y /X). Indeed, let φ ∈ Bir(Y /X) and W be a resolution

of Y with morphisms p, q : W → Y which commute with φ over X. Since f is the

38

Stein factorization of the composition f ◦p = f ◦ q, by the universal property of Stein

factorization, we obtain that φ is an automorphism.

Note that Y is only normal so far. We would like to replace Y by a resolution while

preserving the condition (2). Let G = Aut(Y /X). Let F ′/F be a finite extension

such that Aut(Y /X) = Aut(YF ′/XF ′). Then, there is a natural action of Gal(F ′/F )

on G and the semi-direct product G o Gal(F ′/F ) is a finite group acting on YF ′ .

Note that the canonical model map π : YF ′ 99K ZF ′ is equivariant for the G o

Gal(F ′/F ) action. By [AW97, Theorem 0.1], we have a resolution of singularities

YF ′ → YF ′ equivariant with respect to the G o Gal(F ′/F ) action such that we have

a morphism to the canonical model. Any birational automorphism φ ∈ Bir(YF ′/X)

induces an automorphism in G, by the universal property of Stein factorization as

above. Since action of G lifts to YF ′ , we see that φ is an automorphism. Hence we

have Bir(YF ′/X) = Aut(YF ′/X). We let Y ′ be the quotient of YF ′ by Gal(F ′/F ).

Then Y ′ defined over F is smooth and satisfies (2) and (3).

Remark 2.6.4 Let f : Y → X be a dominant generically finite morphism of normal

projective varieties over a number field F . Let f ′ : Y ′ → X be the smooth birational

modification of f obtained in Lemma 2.6.3. Then any twist Y σ → X of f is birational

to a twist Y ′σ → X of f ′ : Y ′ → X. Indeed, σ induces a 1-cocyle defined by the

morphism Gal(F/F )σ−→ Aut(Y /X) ⊂ Bir(Y ′/X) = Aut(Y ′/X). Thus we obtain a

corresponding twist f ′σ : Y ′σ → Y σ → X. Note that if B ⊂ Y is the locus such that

Y ′ → Y is an isomorphism over U = Y \ B, then the morphism Y ′σ → Y σ is an

isomorphism over Uσ.

The following result is an adaptation of Hilbert’s Irreducibility Theorem [Ser92,

Proposition 3.3.1].

Lemma 2.6.5 [LST18] Let f : Y → X be a generically finite surjective morphism

of normal geometrically integral projective varieties over a number field F . Suppose

39

the extension of geometric function fields F (Y )/F (X) is Galois with Galois group G.

Then there is a thin set Z ⊂ X(F ) such that for x ∈ X(F ) \Z, the scheme-theoretic

fiber f−1(x) is irreducible and the corresponding extension of residue fields is Galois

with Galois group G.

Proof. By Lemma 2.6.3 we may replace Y by a birational model such that

Bir(Y /X) = Aut(Y /X). Therefore we have Aut(Y /X) = G, as Bir(Y /X) =

Gal(F (Y )/F (X)). Let F ′/F be a finite extension such that Aut(Y /X) =

Aut(YF ′/XF ′). Let V ⊂ XF ′ be the complement of the branch locus of f and

U = f−1(V ). Then f : U → V is etale. We show that f : U → V is a Galois

cover with Galois group G as follows. Note that any automoprhism g ∈ Aut(U/V )

base changes to an element of Bir(Y /X). Therefore, g extends to an element of

Aut(YF ′/XF ′). Hence we have Aut(U/V ) = Aut(YF ′/XF ′) = G. Since the geometric

function field extension F (Y )/F (X) is Galois with Galois group G, we see that U/V

is an etale Galois cover with Galois group G. Therefore, the group G = Aut(U/V )

acts transitively on the geometric fibers of f : U → V . Hence f : U → V is Galois

with Galois group G, i.e. V = U/G.

By applying Hilbert Irreducibility theorem [Ser92, Proposition 3.3.1] to f : U →

V , we obtain a thin set ZV ⊂ V (F ′) satisfying the conclusion. Let Z ′ = ZV ∪ (XF ′ \

V )(F ′). Then Z ′ ⊂ X(F ′) is a thin set and Z = Z ′ ∪ X(F ) is a thin set satisfying

the desired properties.

The following result will be used in Chapter 5 for proving Theorem 1.3.1. It

shows that if f : Y → X is a breaking thin map and Y ′ → Y is a base change of the

canonical model map for (KY + a(Y, f ∗L)f ∗L), then the twists of f ′ : Y ′ → X are

also breaking thin maps.

40

Lemma 2.6.6 Let Y be a geometrically uniruled smooth projective variety over a

number field F and let L be a big and nef Q-divisor on Y . Suppose κ(KY +a(Y, L)L) >

0 and let π : Y 99K Z denote the canonical model. Suppose that h : T → Z is any

dominant generically finite map from a projective variety T . Let U be the locus where

π is defined and set Y ′ to be a projective closure of the main component of U ×Z T .

Let g : Y ′ → Y denote the corresponding dominant map.

Then for every twist gσ : Y ′σ → Y of g with Y ′σ irreducible, the induced map

g∗ : FY ′σ → FY is surjective. In particular we have b(F, Y ′σ, gσ∗L) ≥ b(F, Y, L).

Proof. Since any twist gσ : Y ′σ → Y is dominant, we have a(Y ′σ, gσ∗L) = a(Y, L). By

Lemma 2.6.3, we have a birational model Y → Y ′ such that π ◦ g is a morphism and

Bir(Y /Y ) = Aut(Y /Y ). By Lemma 2.4.12, FY ' FY ′ , hence we may replace Y ′ by

Y . Let T σ → Z be the Stein factorization of Y ′σ → Y → Z. Let t ∈ T σ be a general

closed point and z = gσ(t). We have a commutative diagram of push-forward map of

1-cycles

FY ′σt FY ′σ

FYz FY

The horizontal arrows are surjective by Lemma 2.4.15 and the left vertical ar-

row is surjective Lemma 2.4.12. Therefore the right vertical arrow is surjective and

b(F, Y ′σ, gσ∗L) ≥ b(F, Y, L).

Proof of Theorem 1.3.1.

Let X be a geometrically uniruled smooth projective variety over a number field

F . Suppose f : Y −→ X is a dominant generically finite morphism from a normal

41

projective variety. We want to construct a thin set Z ⊂ X(F ) such that, for any

twist fσ : Y σ → X of f , with Y σ irreducible and

(a(Y σ, fσ∗L), b(F, Y σ, fσ∗L)) > (a(X,L), b(F,X,L))

we have f(Y σ(F )) ⊂ Z.

We may assume that X(F ) is non-empty. Since X has a smooth rational point,

X is geometrically integral. If there is a twist Y σ with a smooth rational point,

then Y σ (consequently, Y ) is geometrically integral. Otherwise, we may take Z =

f(Sing(Y ))(F ). Therefore we may assume that Y is geometrically integral. If the

extension of geometric function fields F (Y )/F (X) is not Galois, the statement follows

from [LT17, Proposition 8.2]. Hence we assume that F (Y )/F (X) is Galois with Galois

groupG. By Lemma 2.6.3 and Remark 2.6.4, we may replace Y by a smooth birational

model such that Aut(Y /X) = Bir(Y /X) = G. Indeed, if Y ′ is the birational model

produced by Lemma 2.6.3 and Z ′ is the corresponding thin set, then we may take

Z = Z ′ ∪ f(B)(F ), where B ⊂ Y such that Y ′ → Y is an isomorphism over Y \B.

By Lemma 2.4.4, we may assume a(Y, f ∗L) = a(X,L). Let F1/F be a finite

extension such that N1(Y ) = N1(YF1) and G = Aut(YF1/XF1). By Lemma 2.6.5, we

obtain a thin set Z1 ⊂ X(F1) such that for any point x ∈ X(F1) \Z the fiber f−1(x)

is irreducible over F1 and the corresponding extension of residue fields is Galois with

Galois group G. Let Z = Z1 ∩X(F ) which is a thin set. It is enough to show that if

a twist σ satisfies fσ(Y σ(F )) 6⊂ Z then b(F, Y σ, fσ∗L) ≤ b(F,X,L).

Let KY = KX + E, where E is an effective divisor with irreducible components

E1, · · · , En. Note that the space of invariant divisors N1(Y )G is spanned by f∗N1(X)

and (Span({Ei}))G. Let FX be the minimal face of Eff1(X) containing a(X,L)L+KX

and FY be the minimal face of Eff1(Y ) containing a(X,L)f ∗L + KY . Note that

42

Ei ∈ FY and the action of G preserves the subspace Span(FY ). Therefore we obtain

a surjective morphism,

N1(X)/Span(FX)→ N1(Y )G/Span(FY )G.

Let y ∈ Y σ(F ) with fσ(y) = x ∈ X(F ) \ Z. By construction, the fiber

f−1(x) is irreducible over F1 and the corresponding extension of residue fields

k(f−1(x))/F1 is Galois with Galois group G. We have a natural map Aut(YF1/XF1)∼−→

Aut(k(f−1(x))/F1) by restriction to the fiber over x. Since Aut(YF1/XF1) = G =

Aut(k(f−1(x))/F1), the map is an isomorphism. Now the composition with the map

given by σ

Gal(F/F1)σ−→ Aut(YF1/XF1)

∼−→ Gal(k(f−1(x))/F1) = G

is the natural quotient map of Galois groups, hence surjective. Therefore σ is surjec-

tive. Note that for any s ∈ Gal(F/F1) the action of σ(s) on N1(YF1) is the same as

the Galois action of s. Therefore we have

(N1(Yσ)G/Span(FY

σ

)G)Gal(F/F ) = (N1(Yσ)/Span(FY

σ

))Gal(F/F )

= N1(Y σ)/Span(FY σ).

Since we have a surjection

N1(X)/Span(FX)→ N1(Yσ)G/Span(FY

σ

)G,

by taking Galois invariants of Gal(F/F ), we obtain a surjection

N1(X)/Span(FX)→ N1(Y σ)/Span(FY σ).

43

Therefore b(F, Y σ, fσ∗L) ≤ b(F,X,L).

44

Chapter 3

The b-constant in families

In this chapter we prove Theorem 1.1.1 and Theorem 1.1.2 about the behaviour of

the b-constant in families.

3.1 Global invariant cycles.

Let π : X −→ Z be a morphism of complex algebraic varieties. Then, by Verdier’s

generalization of Ehresmann’s theorem [Ver76, Corolaire 5.1], there exists a Zariski

open U ⊂ Z such that π−1(U) −→ U is a topologically locally trivial fibration (in

the analytic topology), i.e. every point z ∈ U has a neighbourhood N ⊂ U in the

analytic topology, such that there is a fiber preserving homeomorphism

π−1(N) N × F

N

∼

where F = π−1(z). Consequently we have a monodromy action of π1(U, z) on the

cohomology of the fiber H i(Xz,R).

Let π : X −→ Z be a morphism of normal projective varieties. Note that by

generic smoothness and the discussion above, given any resolution of singularities

45

µ : X −→ X, we may choose a Zariski open U ⊂ Z such that π ◦ µ is smooth over U

and (π ◦µ)−1(U) −→ U and π−1(U) −→ U are topologically locally trivial fibrations.

The following result is an adaptation of Deligne’s global invariant cycles theo-

rem [Del71] to the case of singular varieties, which helps us to compute the b-constant.

Lemma 3.1.1 Let π : X −→ Z be a morphism of normal projective varieties over C

where X is Q-factorial. Let µ : X −→ X be a resolution of singularities. Let U ⊂ Z

be a Zariski open subset such that π ◦µ is smooth over U and (π ◦µ)−1(U) −→ U and

π−1(U) −→ U are topologically locally trivial fibrations (in the analytic topology).

Suppose for general z ∈ U , the fiberXz := π−1(z) is rationally connected with rational

singularities. Then

im(N1(X)R −→ N1(Xz)R) ' H2(Xz,R)π1(U,z)

for general z ∈ U , where H2(Xz,R)π1(U,z) is the monodromy invariant subspace.

Proof. Let Xz be the fiber of π ◦ µ over z. For z ∈ U general, µz : Xz −→ Xz is

a resolution of singularities. Since Xz is rationally connected, Q-linear equivalence

and numerical equivalence of Q-Cartier divisors coincide, i.e. Pic(Xz)Q ' N1(Xz)Q.

We know h1(Xz,OXz) = h2(Xz,OXz) = 0 since Xz is smooth rationally connected.

We also have h1(Xz,OXz) = h2(Xz,OXz) = 0, because Xz has rational singularities.

Therefore H2(Xz,Q) ' N1(Xz)Q and H2(Xz,Q) ' N1(Xz)Q.

Consider the natural restriction map on cohomology groups H2(X,Q) −→

H2(Xz,Q). By Deligne’s global invariant cycles theorem ([Del71] or [Voi07, 4.3.3])

46

we know that for z ∈ U ,

im(H2(X,Q) −→ (H2(Xz,Q)) = H2(Xz,Q)π1(U,z).

and if α ∈ H2(Xz,Q)π1(U,z) is a Hodge class then there is a Hodge class α ∈ H2(X,Q)

such that α restricts to α. Since H2(Xz,Q) ' N1(Xz)Q, we see that

im(H2(X,Q) −→ H2(Xz,Q)) ' im(N1(X)Q −→ N1(Xz)Q)

for z ∈ U . In particular

im(N1(X)R −→ N1(Xz)R) ' H2(Xz,R)π1(U,z)

for z ∈ U .

Now the following diagram of pull-back morphisms commutes

N1(X)R N1(Xz)R

N1(X)R N1(Xz)R

i∗

µ∗ µ∗z

i∗

Since µ : X −→ X and µz : Xz −→ Xz are resolutions of singularities for general

z ∈ U , the vertical morphisms are injective. Therefore

im(i∗) ' im(µ∗z ◦ i∗) = im(i∗ ◦ µ∗)

Since X is Q-factorial, we have N1(X)R ' µ∗N1(X)R⊕⊕

j REj where Ej are the

µ-exceptional divisors. For z ∈ U general, the restriction of a µ-exceptional divisor

Ej to Xz is µz-exceptional. In N1(Xz)R, we have im(µ∗z) ∩ ⊕jREzj = 0 where Ez

j are

47

µz-exceptional. Therefore

im(i∗ ◦ µ∗) = im(i∗) ∩ im(µ∗z).

Recall that we have the isomorphisms given by first Chern classN1(Xz)R ' H2(Xz,R)

and N1(Xz)R ' H2(Xz,R). We know that im(i∗) ' H2(Xz,R)π1(U,z) and the mon-

odromy actions on H2(Xz,R) and H2(Xz,R) commute with the pullback map µ∗z.

Hence

im(i∗) ∩ im(µ∗z) ' H2(Xz,R)π1(U,z).

Therefore

im(N1(X)R −→ N1(Xz)R) = im(i∗) ∩ im(µ∗z) ' H2(Xz,R)π1(U,z)

for general z ∈ U .

3.2 Constancy on very general fibers

Let f : X −→ T be a projective morphism and L is an f -big Q-Cartier divisor. We

denote Lt := L|Xt , the restriction to the geometric fiber of t.

Lemma 3.2.1 Let X −→ T be a smooth projective family of varieties and s, t ∈ T

such that s is a specialization of t.

(a) Eff1(Xt) maps into Eff

1(Xs) under the specialization morphism spt,s :

NSR(Xt) −→ NSR(Xs).

(b) Suppose a(Xt, Lt) = a(Xs, Ls) and ρ(Xt) = ρ(Xs). Then b(Xt, Lt) ≥ b(Xs, Ls).

Proof. (a) Let D be an effective divisor in NS(Xt)R. We may pick a discrete valuation

ring R with a morphism φ : SpecR = {s′, t′} −→ T where s′ and t′ map to s and

48

t respectively and t′ is the generic point. By Remark 2.1.1 we have isomorphisms

NS(Xt)∼−→ NS(Xt′) and NS(Xs)

∼−→ NS(Xs′). Therefore we may assume T is the

spectrum of a discrete valuation ring R and t is the generic point t′. Now D is defined

over a finite extension L of k(t′). We can replace R by a discrete valuation ring RL

with quotient field L. Then the image of D under Pic(Xt′)∼−→ Pic(φ∗X) −→ Pic(Xs′)

is effective by semi-continuity. After passing to the algebraic closure and taking

quotient by algebraic equivalence we conclude that, spt,s maps D to an effective

divisor class.

(b) Since ρ(Xt) = ρ(Xs), we have an isomorphism NS(Xt)R −→ NS(Xs)R. Let

a := a(Xs, Ls) = a(Xt, Lt). Note that spt,s maps KXt+ aLt to KXs + aLs. Let F be

a supporting hyperplane of Eff1(Xs) corresponding to the minimal supporting face

containing KXs + aLs. Since Eff1(Xt) ⊂ Eff

1(Xs), we see that F is a supporting

hyperplane of Eff1(Xt) containing KXt

+ aLt. Therefore,

b(Xs, Ls) = codim(F ∩ Eff1(Xs)) ≤ codim(F ∩ Eff

1(Xt)) ≤ b(Xt, Lt).

Lemma 3.2.2 Let X −→ T a smooth projective family. Let η ∈ T be the generic

point. We denote a = a(Xη, Lη), n = ρ(Xη) and b = b(Xη, Lη). For m ∈ N, define

Tm := {t ∈ T |a(Xt, Lt) ≤ a− 1

m}

T0 := {t ∈ T |ρ(Xt) > n}

and

T∞ := {t ∈ T |a(Xt, Lt) = a, ρ(Xt) = n, b(Xt, Lt) < b}.

We let ZT := ∪mTm ∪ T∞ ∪ T0. Then

49

(a) ZT is closed under specialization.

(b) If we base change by a morphism of schemes g : T ′ −→ T , then ZT ′ = g−1(ZT ).

Proof. (a) Let t ∈ ZT and s a specialization of t in T . If t ∈ Tm for some

m ∈ N, then Lemma 3.2.1(a) implies that KXs + a(Xt, Lt)Ls ∈ Eff1(Xs). Therefore,

a(Xs, Ls) ≤ a(Xt, Lt) and hence s ∈ Tm. If t ∈ T0, then by Proposition 2.1.2(a),

ρ(Xs) ≥ ρ(Xt) and s ∈ T0. If t /∈ T0 ∪ ∪mTm, then ρ(Xt) = n and a(Xt, Lt) = a.

Then Lemma 3.2.1(b) implies b(Xs, Ls) ≤ b(Xt, Lt) < b. Therefore s ∈ T∞ and ZT is

closed under specialization.

(b) This follows from the fact that the Picard number and a,b-constants are

invariant under algebraically closed base extension.

Proof of Theorem 1.1.1: By passing to a resolution of singularities and using

generic smoothness, we may exclude a closed subset of T to assume the family f :

X −→ T is smooth and T is affine. Since our base field k is algebraically closed, we

may find a subfield k′ ⊂ k which is the algebraic closure of a field finitely generated

over Q, and there exists a finitely generated k′-algebra A such that our familyX −→ T

and L are a base change of a family XA −→ SpecA and a line bundle LA on XA.

Now B = SpecA is countable and hence ZB = ∪b∈B{b} is a countable union of closed

subsets by Lemma 3.2.2(a). Now Lemma 3.2.2.(b) implies that ZT is a countable

union of closed subsets.

3.3 Family of uniruled varieties

In this section we prove Theorem 1.1.2. Let f : X −→ T be a projective family of

uniruled varieties over an algebraically closed field k of characteristic 0 and L an

50

f -nef and f -big Q-Cartier Q-divisor.

We recall that when the fibers are adjoint-rigid, constancy of the b-constant was

proved in [LT17].

Proposition 3.3.1 (see [LT17, Prop. 4.4]) Let f : X −→ T be a smooth family of

projective varieties. Suppose L is an f -big and f -nef Cartier divisor on X. Assume

that for a general member Xt, we have κ(KXt + a(Xt, Lt)Lt) = 0. Then b(Xt, Lt) is

constant for general t ∈ T .

By a standard argument using the Lefschetz principle, it is enough to prove the

statement for k = C. We will henceforth assume that k = C.

We can reduce to the statement for closed points only, as follows. Let us assume

that there is an open U ⊂ T such that b(Xt, Lt) = b is constant for all closed points

t ∈ U . Let s ∈ U and Z = {s}∩U . By applying Theorem 1.1 to the family over Z, we

may find F = ∪iFi ⊂ Z a countable union of closed subvarieties such that b(Xt, Lt)

is constant on Z \ F . Since C is uncountable, there exists a closed point t ∈ Z \ F .

Now s ∈ Z \F , since s is the generic point of Z. Therefore, b(Xs, Ls) = b(Xt, Lt) = b.

Since s ∈ U was arbitrary, we conclude that b(Xt, Lt) = b for all t ∈ U . Therefore it

is enough to prove the statement for closed points.

Proof of Theorem 1.1.2 for closed points when k = C: We may replace

X by a resolution, and by generic smoothness, we may exclude a closed subset of

the base to assume that f : X −→ T is a smooth family. By Theorem 2.4.6, we can

shrink T such that a(Xt, Lt) = a for all t ∈ T and κ(KXt + aLt) is independent of t.

We may assume that T is affine. Since L is f -big and f -nef, we can replace L by a

51

Q-linearly equivalent divisor to assume that (X, aL) is klt.

Since the local system GN 1(X/T ) has finite monodromy, we can base change to

a finite etale cover of T to assume that GN 1(X/T ) has trivial monodromy.

If κ(KXt + aLt) = 0 then we can conclude by Proposition 3.3.1. Let us assume

that κ(KXt + aLt) = k > 0 for all t ∈ T .

Since KX + aL is f -pseudo-effective and aL is f -big, we may run a (KX + aL)-

MMP over T to obtain a relative minimal model φ : X 99K X ′. Let π : X ′ −→ Z

be the morphism to the relative canonical model over T . By Proposition 2.4.13, we

may replace T by an open subset to assume that the base change φt : Xt 99K X ′t is

a Q-factorial minimal model and πt : X ′t −→ Zt is the canonical model for (Xt, aLt)

for all t ∈ T .

For z ∈ Z, we denote the image of z in T by t and let X ′z denote the fiber of

π : X ′ −→ Z over z.

X ′z X ′t X ′

Speck(z) Zt Z

Speck(t) T

πt π

gt g

Let µ : X −→ X ′ be a resolution of singularities. We may replace T by an open

subset to assume that X −→ T is smooth. Let Xz be the fiber of π : X −→ Z over

z ∈ Z. By [Ver76, Corolaire 5.1] we can find a Zariski open UZ ⊂ Z such that π is

smooth over UZ and π−1(UZ) −→ UZ and π−1(UZ) −→ UZ both are topologically

locally trivial fibrations (in the analytic topology). Again we may replace T by

52

a Zariski open V ⊂ T to assume that UZ −→ T is a topologically locally trivial

fibration (in the analytic topology). Let Ut ⊂ Zt denote the fiber of UZ over t ∈ T .

For all z ∈ UZ , there is a monodromy action of π1(Ut, z) on H2(X ′z,Z) acting

by an integral matrix Mz on the free part. Now for any two points z and z′ in UZ ,

the fundamental groups π1(Ut, z) and π1(Ut′ , z′) are isomorphic, since UZ −→ T is a

locally trivial fibration. Also, the cohomology groups H2(X ′z,Z) and H2(X ′z′ ,Z) are

isomorphic, because π−1(UZ) −→ UZ is a locally trivial fibration. Since the mon-

dromy actions depend continuously on z ∈ UZ , we see that Mz is constant. Therefore

the monodromy invariant subspaces have constant rank, i.e. rkH2(X ′z,R)π1(Ut,z) is

constant for all z ∈ UZ .

By [HM07] we know that a general fiber X ′z is rationally connected and has ter-

minal singularities. Since X ′t is Q-factorial, Lemma 3.1.1 implies that

rk(im(N1(X ′t)R −→ N1(X ′z)R) = rkH2(X ′z,R)π1(Ut,z).

for general z ∈ Ut. Now using Proposition 2.9.(3) we have

b(Xt, Lt) = rkH2(X ′z,R)π1(Ut,z)

for general z ∈ UZ . Since rkH2(X ′z,R)π1(Ut,z) is constant for z ∈ UZ , we may conclude

that b(Xt, Lt) is constant for general t ∈ T .

53

Chapter 4

Fujita invariant of finite covers

In this chapter we prove Theorem 1.2.1. First we prove that the degree of the

covers under consideration are bounded. Then we study positivity properties of the

ramification divisors and we show that the a-constant of a branch divisor is higher

than that that of the ambient variety. This enables us to prove the main result using

Theorem 2.4.8.

We form the following definition for convenience.

Definition 4.0.1 (cf. [LT17, Section 4.1]) Let X be a smooth uniruled variety and

L a big and nef Q-Cartier Q-divisor. We say that a morphism f : Y −→ X is an

adjoint-rigid cover preserving the a-constant if,

(1) f : Y −→ X is a generically finite surjective morphism from a normal variety

Y ,

(2) a(Y, f ∗L) = a(X,L),

(3) κ(KY + a(Y, f ∗L)f ∗L) = 0.

54

Note that the conditions (1)-(3) are preserved under taking a resolution of

singularities.

4.1 Boundedness of degree of covers

As a consequence of the Weak BAB-conjecture we obtain the following result. It shows

that the degrees of all adjoint-rigid covers preserving the a-constant are bounded by

a constant.

Proposition 4.1.1 LetX be a smooth uniruled variety and L a big and nef Q-divisor.

Then there exists a constant M > 0 such that, if f : Y −→ X is an adjoint-rigid

cover preserving the a-constant, then deg(f) < M .

Proof. By Lemma 2.2.5, we may replace Y by a resolution to assume that there exists

∆ ∼ af ∗L with (Y,∆) terminal and we have a morphism ψ : Y −→ Y ′ to a minimal

model (Y ′,∆′) with Q-factorial terminal singularities. Now κ(KY + ∆) = 0 implies

that κ(KY ′ + ∆′) = 0. As KY ′ + ∆′ is semi-ample ([BCHM10, Corollary 3.9.2]), we

have KY ′ + ∆′ ≡ 0. Since ∆′ is big, we can write ∆′ ≡ A+ E where A is ample and

E is effective. Now for 0 < t� 1, (Y ′, (1− t)∆′ + tE) is terminal and

KY ′ + (1− t)∆′ + tE ≡ −tA

is anti-ample. Therefore (Y ′, (1− t)∆′ + tE) is terminal log Fano. In particular, it is

ε-klt for ε = 12. Therefore by the Weak BAB conjecture (Corollary 2.3.4), there exists

M > 0 such that

vol(∆′) = vol(−KY ′) < M.

55

Since f : Y −→ X is generically finite, we have vol(af ∗L) = andeg(f)vol(L). Now

we have the following inequality

andeg(f)vol(L) = vol(∆) ≤ vol(∆′) < M.

Therefore the degree of f is bounded.

4.2 Finiteness of adjoint-rigid covers

If X is a smooth surface and E is a curve contracted by the KX-MMP, then KE is

not pseudo-effective. The following proposition is a generalization of this fact and it

is a key step for proving Theorem 1.2.1. For a smooth projective uniruled variety

X and a big and nef divisor L, this proposition enables us to compare a(X,L) with

the a-constants of L under restriction to the exceptional divisors contracted by a

KX + a(X,L)L-MMP.

Proposition 4.2.1 Let X be a normal variety with canonical singularities and ∆

an effective R-Cartier R-divisor which is nef. Suppose ψ : X 99K X ′ is a KX + ∆-

minimal model obtained by a running a KX + ∆-MMP. Let E be an exceptional

divisor contracted by ψ. Let π : E −→ E be a resolution of singularities. Then

KE + π∗(∆|E) is not pseudo-effective.

Proof. Note that it enough to prove the statement for one resolution of singularities

of E. In particular, let π : X −→ X be a log resolution of (X,E + ∆) such

that we have a morphism φ = ψ ◦ π : X −→ X ′. Let E = π−1∗ E be the strict

transform. We reduce to the case when X and E are smooth and ∆ is ample

56

as follows. Let H be a general ample divisor on X. Then ψ : X 99K X ′ is also

a KX + ∆ + εH-MMP for ε > 0 sufficiently small. Hence, by replacing ∆ with

∆ + εH, we may assume that ∆ is ample. Note that, since X has canonical sin-

gularities, φ : X −→ X ′ is a KX + π∗∆-minimal model. As π∗∆ is big and nef,

we may choose ∆ ∼R π∗∆ such that (X, ∆) is klt ([Xu15, Proposition 2.3]). Now

φ : X −→ X ′ is a minimal model for (X, ∆) and since minimal models of klt pairs

are isomorphic in codimension one ([BCHM10, Corollary 1.1.3]), we know that E

will be contracted by any KX + ∆-MMP. Therefore we may assume that X and

E are smooth and ∆ is ample. We need to show that KE+∆|E is not pseudo-effective.

Since ∆ is ample, we may choose ∆0 ∼R ∆ such that (X,E + ∆0) is simple

normal crossing and divisorially log terminal, (X,∆0) is kawamata log terminal and

(E,∆0|E) is canonical, by using the Bertini theorem ([Xu15, Lemma 2.2]).

As (X,E+∆0) is dlt and ∆0 is ample, by [BCHM10], we may run a KX +E+∆0-

MMP. Since we know that the ACC holds for log canonical thresholds [HMX14] and

special termination holds for dlt flips ([BCHM10, Lemma 5.1]), the KX + E + ∆0-

MMP terminates with a minimal model θ : X 99K Xm by [Bir07, Theorem 1.2]. Since

E is contained in the negative part of the Zariski decomposition of KX + E + ∆0,

the MMP given by θ contracts E. Let θk : Xk −→ Xk+1 be the divisorial contraction

step of the KX + E + ∆0-MMP that contracts the push-forward of E on Xk. Let

Θk = X 99K Xk be the composition of the steps of the KX +E+∆0-MMP. We denote

∆k = Θk∗∆0 and Ek = Θk∗E. Note that Ek is normal ([KM98, Proposition 5.51]).

By [AK17, Theorem 7], the restriction map Θk|E : (E,DiffE∆0) 99K (Ek,DiffEk∆k)

is a composition of steps of a KE + DiffE∆0-MMP. As DiffE∆0 = ∆0|E, it is enough

to show that KEk + DiffEk∆k is not pseudo-effective. This follows from the fact that

Ek is covered by curves C such that (KEk + DiffEk∆k) ·C = (KXk +Ek + ∆k) ·C < 0.

57

Note that the assumption about ∆ being nef is necessary. For example, let Y be

a minimal surface and X = Bl4(BlyY ) be the blow-up of BlyY at four distinct points

yi ∈ E, 1 ≤ i ≤ 4, where E ⊂ BlyY is the exceptional curve corresponding to y ∈ Y .

Let Ei ⊂ X be the exceptional curve corresponding to yi for 1 ≤ i ≤ 4 and E0 be the

strict transform of E on X. Let ∆ = 12E1 + 1

2E2 + 1

2E3 + 1

2E4. Note that ∆ is not nef

as ∆ ·E4 = −12. Now E0 is contracted by the KX +∆-MMP but deg(KE0 +∆|E0) = 0.

Corollary 4.2.2 Let X be a smooth projective uniruled variety and L a big and

nef Q-divisor. Let f : Y −→ X be a generically finite cover with Y smooth and

a(Y, f ∗L) = a(X,L). Let R ⊂ Y be a component of the ramification divisor of f

(i.e. the strict transform of a component of the ramification divisor for the Stein

factorization of f) and B be the component of the branch divisor on X which is the

image of R. If R is contracted by a KY + a(X,L)f ∗L-MMP, then

a(B,L|B) > a(X,L)

Proof. We may assume that L|B is big. We have a generically finite surjective map

f |R : R −→ B. Therefore, by Lemma 2.4.4, we have a(R, f ∗L|R) ≤ a(B,L|B).

Now Proposition 4.2.1 implies that a(R, a(X,L)f ∗L) > 1 and hence a(R, f ∗L|R) >

a(X,L).

Proof of Theorem 1.2.1: Let X be a smooth uniruled variety of dimension

n and L a big and nef Q-divisor on X. Suppose a(X,L) = a. We need to show

that, upto birational equivalence, there exist finitely many varieties Y that admit

a morphism f : Y −→ X which is an adjoint-rigid cover preserving the a-constant.

The statement is obvious if X is a curve. We assume that n ≥ 2. By passing to a

resolution it is enough to show that, upto birational equivalence, there exist finitely

58

many smooth varieties Y with a morphism f : Y −→ X which is an adjoint-rigid

cover preserving the a-constant. By Proposition 4.1.1, we know that there exists a

constant M > 0 such that deg(f) < M for any adjoint-rigid cover preserving the

a-constant f : Y −→ X. Now, for an open U ⊂ X, there are finitely many etale

covers (upto isomorphism) of U of a given degree d. Hence it is enough to show that

there is a proper closed subset V ( X, such that if f : Y −→ X is an adjoint-rigid

cover preserving the a-constant and Y is smooth, the branch locus of f is contained

in V .

Suppose f : Y −→ X is an adjoint-rigid cover preserving the a-constant and

Y is smooth. Let Yπ−→ Y

f−→ X be the Stein factorization of f . Let B ⊂ X be

a component of the branch divisor of f . Note that, by the Zariski-Nagata purity

theorem, the branch locus is a divisor. Let∑

j rjRj ⊂ Y be the ramification divisor,

i.e. KY = f∗KX +

∑j rjRj. Let R ⊂ Y be a component of the ramification divisor

mapping to B and R ⊂ Y be the strict transform π−1∗ (R).

We have the following equation

KY + af ∗L ≡ f ∗(KX + aL) + π−1∗ (

∑i riRi) +

∑i aiEi

where ai > 0. Note that, as KX + aL is pseudo-effective and L is big and nef, by

non-vanishing ([BCHM10, Theorem D]) we have

KX + aL ∼R D ≥ 0.

Therefore we have

KY + af ∗L ≡∑

j cjFj ≥ 0

59

Now, we may find a ∆ ≡ af ∗L such that, (Y,∆) is terminal. As KY + ∆ is

pseudo-effective, we can run a KY + ∆-MMP

ψ : (Y,∆) 99K (Y1,∆1) 99K · · · 99K (Ym,∆m) = (Y ′,∆′)

to obtain a KY +∆-minimal model (Y ′,∆′). Since κ(KY +∆) = 0, we have KY ′+∆′ ≡

0. Hence the KY + ∆-MMP contracts all components of the divisor∑

j cjFj. As

R = Fj for some j, Corollary 4.2.2 implies that a(R, af ∗L|R) > 1 and hence

a(B,L|B) > a = a(X,L).

Therefore, by Theorem 2.4.8, there exists a proper closed subset V ( X such that

B ⊂ V ′ = V ∪ B+(L). Then, for any adjoint-rigid cover preserving the a-constant

f : Y −→ X with Y smooth, the branch locus of f is contained in V ′. Therefore we

have the desired conclusion.

60

Chapter 5

Geometric consistency of Manin’s

conjecture

In this chapter we prove Theorem 1.3.1. As discussed in the Introduction, first we

prove a finiteness statement (Theorem 1.3.2) for breaking thin maps and then use

Theorem 1.3.3 to obtain the desired conclusion. The results in this chapter are from

the joint work [LST18] with B. Lehmann and S. Tanimoto.

5.1 Boundedness of breaking thin maps

In this section we prove Theorem 1.3.2. First we show that subvarieties with

a-constant at least a(X,L) are parametrized by finitely many families. Then we

construct finitely many thin maps from these finitely many parametrizing families

and show that they satisfy the desired factoring property.

61

5.1.1 Boundedness of subvarieties

Let X be a smooth projective geometrically uniruled variety defined over a field F

of characteristic 0. Let L be a big and nef Q-divisor on X. Using the Weak BAB-

conjecture (Corollary 2.3.4), we prove the following result that, subvarieties Y ⊂ X

with a(Y, L) ≥ a(X,L) form a bounded family over F . The idea of the proof is

essentially from [HJ17] and [LT17].

Theorem 5.1.1 [LST18] LetX be a smooth projective geometrically uniruled variety

defined over F and let L be a big and nef Q-divisor on X. There exists a constructible

bounded subset T ⊂ Hilb(X), a decomposition T = ∪Ti of T into locally closed

subsets, and smooth projective morphisms pi : Ui → Ti equipped with morphisms

si : Ui → X such that

(1) over F , each fiber of pi : U i → T i is an integral uniruled variety which is mapped

birationally by si onto the subvariety of X parametrized by the corresponding

point of Hilb(X);

(2) every fiber Y of pi is a smooth variety satisfying a(Y, s∗iL|Y ) ≥ a(X,L) and is

adjoint rigid with respect to s∗iL|Y , and

(3) for every subvariety Y ⊂ X not contained in B+(L) which satisfies a(Y, L|Y ) ≥

a(X,L) and which is adjoint rigid with respect to L, there is some index i such

that Y is birational to a fiber of pi under the map si.

Furthermore, if si : Ui → X is dominant then si must be generically finite.

Proof. Consider the set T of subvarieties Y ′ ⊂ X such that (Y ′, L|Y ′) is adjoint rigid,

Y ′ 6⊂ B+(LX), and a(Y, L|Y ′) ≥ a(X,L). We show that T is bounded as follows.

By applying [LTT18, Theorem 4.7] to each irreducible component of X, it is enough

to show that there is a constant C such that vol(L|Y ′) < C for all Y ′ ∈ T . Note

that is enough to prove the volume bound for a resolution Y of Y ′ (by Lemma 2.3.1).

62

By [LT17, Theorem 3.5], there is a birational contraction φ : Y 99K Y1 where Y1 is

a Q-factorial terminal weak Fano variety such that KY1 + a(Y ′, L)φ∗(L|Y ) ≡ 0. By

[Bir16b, Theorem 2.11], there is constant v such that we have

a(Y ′, L)dim(Y ′)vol(L|Y ) = vol(a(Y ′, L)L|Y ) ≤ vol(a(Y ′, L)φ∗(L|Y ) = vol(−KY1) < v.

Therefore, it is enough to take C = max{v/a(X,L)d|0 ≤ d ≤ dim(X)} as a(Y ′, L) ≥

a(X,L).

Let M ⊂ Hilb(X) be the finite union of irreducible components which contain the

points parametrized by T . Let T ⊂M be the locus parametrizing subvarieties in T .

We show that T is constructible and it descends to F as follows. Let p : U → M

be the universal family. Let ∪iN i = N ⊂ M denote the constructible subset over

which the universal family has irreducible and reduced fibers. Let π : Ui → U|N ia

resolution of singularities. By applying generic smoothness and Theorem 2.4.7 to the

morphism p ◦ π, we obtain an open subsets N◦i ⊂ N i such that the universal family

over N◦i parametrizes subvarieties with fixed a-constant and fixed Iitaka dimension.

By Noetherian induction, we obtain a new stratification of ∪jN′j = N such that each

N′j parametrizes subvarieties with fixed a-constant and fixed Iitaka dimension. Then

T = ∪iT i where T i = N′j for some j such that the subvarieties parametrized by

N′j have a-constant at least a(X,L). Thus T is constructible. Now, if a subvariety

Y ′ ⊂ X parametrized by a point of T , then σ(Y ′) is also parametrized in T for any

σ ∈ Gal(F/F ) (by Proposition 2.4.5). Thus T descends to a subset T ⊂ Hilb(X)

over F .

Note that B+(LX) descends to B+(L). Now T ⊂ Hilb(X) is bounded (i.e. finitely

many irreducible components) since T is bounded. Now over the base field F we

pick a resolution of the total space of the universal family U → U and carry out the

construction in the previous paragraph to obtain a stratification ∪Ti = T into finitely

63

many locally closed subsets. We let Ui → Ti be the family U |Ti → Ti. Then Ui → Ti

satisfies the conditions (1), (2) and (3). Furthermore, by [LT17, Proposition 4.14] we

see that the evaluation morphism si : Ui → X is generically finite.

We obtain the following result as a byproduct of the construction above. Here we

take projective closures of the universal families over the locus ∪iTi.

Theorem 5.1.2 [LST18] LetX be a smooth projective geometrically uniruled variety

and let L be a big and nef Q-divisor on X. Then there exist a proper closed subset

V and finitely many families pi : Ui → Wi of closed subschemes of X where Wi is a

projective subscheme of Hilb(X) such that

(1) over F , pi : U i → W i generically parametrizes integral uniruled subvarieties of

X;

(2) for each i, the evaluation map si : Ui → X is generically finite and dominant;

(3) for each i, a general member Y of pi is a subvariety of X such that (Y, L|Y ) is

adjoint rigid and a(X,L) = a(Y, L|Y ), and;

(4) for any subvariety Y such that (Y, L|Y ) is adjoint rigid and a(Y, L|Y ) ≥ a(X,L),

either Y is contained in V or Y is a member of a family pi : Ui → Wi for some

i.

Proof. Let Wi be the projective closure of Ti ⊂ Hilb(X) given by Theorem 5.1.1. Let

pi : Ui → Wi be the universal family with evaluation maps si : Ui → X. Then by

construction, each family pi : Ui → Wi satisfies the condition (1). Let V ′ ⊂ X be the

proper closed subset given by Theorem 2.4.8. Let V = B+(L)∪⋃j sj(Uj)∪ V ′ where

j is such that sj : Uj → X is not dominant. If si : Ui → X is dominant then the

a-constant of the subvarieties parametrized by Wi is equal to a(X,L) by Theorem

2.4.8. Therefore the families pi : Ui → Wi with si dominant satisfy conditions (2), (3)

and (4).

64

5.1.2 Finiteness of covers

In this section we work over C.

Definition 5.1.3 A good family of adjoint rigid varieties is a morphism p : U → W

of smooth quasi-projective varieties and a relatively big and nef Q-divisor L on U

satisfying the following properties:

1. The map p is projective, surjective, and smooth with irreducible fibers.

2. The a-value a(Uw, L) is constant for the fibers Uw over closed points and KUw +

a(Uw, L)L is rigid for each fiber.

3. Let Q denote the union of all divisors D in fibers Uw such that a(D,L) >

a(Uw, L). Then Q is closed and flat over W and p : U\Q → W is a locally

trivial fibration in the Euclidean topology.

Note that the invariance of the a-value implies that the restriction of L to every fiber

of p is big. A base change of a good family is defined to be the good family induced

via base change by a map g : T → W . We say that p has a good section if there is a

section W → U\Q, i.e. there is a section avoiding Q.

A good morphism of good families is a diagram

Y U

T W

f

q p

g

and a relatively big and nef Q-divisor L on U such that p and q are good families of

adjoint rigid varieties (with respect to L and f ∗L respectively), the relative dimensions

of p and q are the same, and a(Yt, f ∗L) = a(Ug(t), L) for any point t ∈ T .

65

Lemma 5.1.4 [LST18] Let p : X → W be a surjective morphism of projective

varieties with connected fibers and let L be a big and nef Q-Cartier divisor on X.

Suppose that a general fiber Xw of p satisfies a(Xw, L) > 0. Fix an ample divisor H

on W . Then there is an open subset W ◦ ⊂ W and a positive integer m such that

a(X,L+mp∗H) = a(Xw, L)

for any w ∈ W ◦.

Proof. We may assume that X is smooth by passing to a resolution. By Lemma 2.4.7

there is an open subset W ◦ ⊂ W such that p is a smooth over W ◦ and the a(Xw, L)

is constant for w ∈ W ◦. For any positive integer m, by Lemma 2.4.9

a(X,L+mp∗H) ≥ a(Xw, L) = a.

for a general w ∈ W . We next show that KX + aL + `p∗H is pseudo-effective for

some sufficiently large `. As L is big and nef, we may write a(X,L)L ∼ A+ ∆ where

X,A + ∆) is terminal. Then, by [Leh12, Theorem 1.3], there are only finitely many

extremal rays {αi}ri=1 of Nef1(X) with negative intersection against KX + aL

The following argument of [Pet12] shows that none of these extremal rays αi

satisfies p∗αi = 0. Fix an ample divisor A on X and fix ε > 0. Since KX + aL is

pseudo-effective on a general fiber, KX + aL + εA is p-big, there is some positive cε

such that KX + aL+ εA+ cεp∗H is big. In particular, for every i

(KX + aL+ εA+ cεp∗H) · αi > 0.

If p∗αi = 0 then we obtain 0 < (KX+a(Y, L)+εA)·αi for every ε > 0, a contradiction.

66

Thus, we may choose ` sufficiently large so that

`p∗H · αi > −(KX + a(Y, L)L) · αi

for every i. This verifies that KX + aL + `p∗H is pseudo-effective. Then for any

m ≥ `/a, we have

a(X,L+mp∗H) ≤ a = a(Xw, L)

proving the reverse inequality.

Remark 5.1.5 Let p : X → W be a surjective morphism of projective varieties and

let L be a big and nef Q-Cartier divisor on X. Suppose that each component of a

general fiber Xw of p satisfies a(Xw, L) > 0. Note that the conclusion of Lemma 5.1.4,

still holds with Xw replaced by each component of Xw as follows. We may replace X

by a resolution. Then let Xp′−→ W ′ g−→ W be the Stein factorization. Then by Lemma

5.1.4, we obtain an open subset W ′◦ ⊂ W ′ such that a(X,L+mp′H) = a(Xw′ , L) for

all w′ ∈ W ′◦. Now let U ⊂ W be the open subset such that g is etale over U . Then

we let W ◦ = g(W ′◦ ∩ g−1(U)). Then, if Y ⊂ Xw is an irreducible component of a

fiber, we have a(X,L+mp∗H) = a(Y, L) for w ∈ W ◦.

The following lemma shows that we can shrink the base of a morphism to obtain

a good family.

Lemma 5.1.6 [LST18] Suppose p : X → W is a projective morphism of varieties

such that X is smooth and p has connected fibers. Let L be a p-relatively big Q-

Cartier divisor which is the restriction to X of a nef Q-Cartier divisor on a projective

compactification X ′. Suppose furthermore that the general fiber Xw of p is adjoint

rigid with respect to L. Then there is a non-empty open subset W ◦ ⊂ W with

preimage X◦ = p−1(W ◦) such that p : X◦ → W ◦ is a good family of adjoint rigid

varieties.

67

Proof. Let W ◦ ⊂ W be the open subset such that p is smooth over W ◦ and the

a-invariant of the fibers is constant and Xw is adjoint-rigid for all w ∈ W ◦. Let Q+

denote the union of the subvarieties Y ⊂ Xw over W ◦ such that a(Y, L) > a(Xw, L).

We claim that after shrinking W ◦ the set Q+ is closed. Note that we may replace X

and W projective closures to assume that X and W are projective. We may replace

L by L + p∗A, where A is an ample line bundle on W , to assume that L is big and

nef. Let H be an ample divisor on W , by Lemma 5.1.4, we may assume that

a(Xw, L) = a(X ′, L′ +mp∗H ′)

for every fiber Xw over W ◦. By Theorem 2.4.8 the union of all subvarieties Y of

X ′ satisfying a(Y, L′ + mp′∗H ′) > a(X ′, L′ + p′∗H ′) is closed. We let Q∗ denote this

closed set. Note that Q+ ⊂ Q∗. We claim that if we increase m and shrink W ◦

further then Q∗ ∩X◦ coincides with Q+. First note that by Noetherian induction Q∗

must eventually stabilize as m increases. After shrinking W ◦, we may suppose that

each component of Q∗ that intersects X◦ dominates W ′. By applying Lemma 5.1.5

to the finitely many components of Q∗ that surject onto W ′ and are not contained

in B+(L′), we see that if we increase m and shrink W ◦ further then Q∗ ∩X◦ = Q+

as claimed. In particular, this implies that Q+ is a closed set. Set Q to be the

codimension 1 components of Q+. By shrinking W ◦ we may ensure that p : Q→ W ◦

is flat. By [Ver76, Corollaire 5.1] after further shrinking W ◦ we may guarantee that

p : X◦\Q → W ◦ is topologically trivial. Note that this set now coincides with Q as

defined in the definition of a good family and satisfies all the necessary properties.

Remark 5.1.7 Suppose that p : U → W is a good family of adjoint rigid varieties.

Let V ⊂ U be the complement of the set Q as in Definition 5.1.3. Suppose that p ad-

mits a good section ζ. Then using the fibration exact sequence as in [Shi, Proposition

68

5.5.1] we obtain

π1(V) = π1(Vw) o π1(W )

for a fiber Vw = V ∩ Uw. Indeed, we have an exact sequence

π2(V , ζ(w))→ π2(W,w)→ π1(Vw, ζ(w))→ π1(V , ζ(w))→ π1(W,w)

and the good section ζ induces sections of the first map and the last map. Thus we

have the exact sequence

0→ π1(Vw, ζ(w))→ π1(V , ζ(w))→ π1(W,w)→ 0

which splits and we conclude that the middle group is a semi-direct product of other

two groups.

Lemma 5.1.8 [LST18] Let p : U → W be a good family of adjoint rigid varieties

with a good section. Let V ⊂ U be the complement of the set Q.

Fix a subgroup Ξ ⊂ π1(Vw). Consider the set of subgroups

{G ⊂ π1(V)|G = Ξ oH for some H ⊂ π1(W )}.

This set contains a unique maximal subgroup Υ = ΞoN where N is the normalizer of

Ξ in π1(W ). Furthermore Υ is stable under base change: for any morphism g : T → W

from a smooth variety T , the corresponding subgroup for the family U ×W T is the

preimage of Υ under the natural map π1(V ×W T )→ π1(V).

Proof. It is clear that Υ is actually a subgroup and that if G = Ξ oH is any other

subgroup of the desired form, then H must be contained in N .

We need to show that Υ is stable under base change. Let g : T → W be a

morphism from a smooth variety T inducing g∗ : π1(T )→ π1(W ). It suffices to show

69

that the normalizer of Ξ in π1(T ) is exactly g−1∗ N . To see this, recall from [Shi,

Section 5] that the action of a loop γ ∈ π1(W ) on π1(Vw) can be computed using the

restriction of the family to γ. In particular, any loop in π1(T ) will act in the same

way as its image in π1(W ), proving the theorem.

Remark 5.1.9 Note that since the fundamental group of an algebraic variety is

finitely generated, there will only be finitely many subgroups of a given finite index.

Thus, if a subgroup Ξ as above has finite index, its normalizer has finite index in

π1(W ) and the maximal subgroup Ξ oN also has finite index.

Given a good family U → W of adjoint rigid varieties with a good section, the

next construction produces a finite set of associated good families which will be used

to prove the finiteness statement in Theorem 1.3.2.

Construction 5.1.10 [LST18] Let p : U → W be a good family of adjoint rigid

varieties with a good section and L be the divisor associated to the family, as in

Definition 5.1.3. Suppose that the divisor L on U is the restriction of a nef Q-Cartier

divisor on a projective compactification of U . We construct a finite set of groups and

good families and a closed subset as follows.

Associated groups: Let V = U \Q as in Definition 5.1.3. Note that by Remark

5.1.7, we have π1(V) = π1(Vw)oπ1(W ). Since V → W is locally trivial in the analytic

topology, we know that π1(Vw) is isomorphic to a fixed group G for all w ∈ W .

Note that any adjoint rigid cover of Uw which preserves the a-constant is etale

over Vw, by Corollary 4.2.2. Also, the degree of any adjoint rigid cover preserving

the a-constant is bounded by Proposition 4.1.1. Therefore, to each adjoint rigid

cover of Uw that preserves the a-constant there is an associated finite index subgroup

Ξj ⊂ π1(Vw) = G. For each such Ξj we use Nj to denote its normalizer as in Lemma

5.1.8; we also denote Υj = ΞjoNj. As remarked earlier Υj will always be a subgroup

70

of π1(V) of finite index, by Remark 5.1.9.

Associated good families Yj → Tj and the closed set D: Let Ej denote the

etale cover of V corresponding to Υj. By the lifting property, Ej admits a morphism

to the etale cover Rj → W defined by Nj, since Nj is the image under the map

π1(Ej) → π1(V) → π1(W ). By construction we know Ej → Rj has at least one

irreducible fiber, and by local topological triviality we deduce that every fiber is

irreducible. Let rj : Ej → Rj be a resolution of a completion of Ej to a projective

family over Rj. Let J be the set of indices j such that a general fiber of rj : Ej → Rj

is adjoint rigid with repect to the pull-back of L. Then by Theorem 2.4.7 and Lemma

5.1.6, we obtain an open subset Tj ⊂ Rj such that r−1j (Tj) → Tj is a good family of

adjoint rigid varieties for all j ∈ J . We denote these good families of adjoint varities

by qj : Yj → Tj. Note that by construction we have good morphisms of good families

{fi : Yi → U}. Let D ⊂ W be the closed subset which is the union of the images of

Rj \ Tj.

The next lemma proves that we have a factoring proerty for the associated good

familes constructed above.

Lemma 5.1.11 [LST18] Let p : U → W be a good family of adjoint rigid varieties

with a good section. Suppose that the divisor L on U is the restriction of a nef

Q-Cartier divisor on a projective compactification of U . Then there is a finite set

of dominant generically quasi-finite good morphisms of good families {fi : Yi → U}

with family maps qi : Yi → Ti and a closed proper subset D ( W such that the

following holds.

Suppose that q : Y → T is a good family of adjoint rigid varieties admitting a

good morphism f : Y → U . Then either,

(a) f(Y) is contained in p−1D,

71

or

(b) there is a base change q : Y → T of q by a generically finite surjective morphism

T → T such that the induced f : Y → U factors rationally through the map fi for

some i.

Furthermore, in case (b), we may ensure there is an open subset T ◦ such that

Y◦ = q−1(T ◦) admits a good morphism of good families f ◦ : Y◦ → Yi where a general

fiber of q over T ◦ is birational to a fiber of qi over Ti.

Proof. Let Yj → Tj and D ⊂ W be the finite set of associated good families and the

closed set constructed in Construction 5.1.10. Set W ◦ = W \D and let (Tj)◦ and V◦

denote the preimages of W ◦ in Tj and V . Note that by the construction, (Tj)◦ → W ◦

is proper etale so that Ξj o π1((Tj)◦) is a finite index subgroup of π1(V◦).

Now we show that the set {fi : Yi → U} satisfies the condition in the statement

of the lemma as follows. Suppose we have a morphism f : Y → U as in the statement

of the lemma. The good family q : Y → T might not admit a good section. Therefore

we take an appropriate base change to obtain a good section as follows. Let T ′ be a

general intersection of hyperplanes in Y that maps generically finitely onto T and is

not contained in the f -preimage of Q. We let q′ : Y ′ → T ′ denote the base change

over T ′ → T , and let f ′ : Y ′ → U denote the induced map. After shrinking T ′ we

may ensure that q′ admits a good section ζ ′.

Let VY ′ = Y ′ \ QY ′ , where QY ′ is the closed subset as in Definition 5.1.3. Since

q′ : Y ′ → T ′ admits a good section, by Remark 5.1.7, we have

π1(VY ′) = π1(VY ′,t′) o π1(T ′)

for t′ ∈ T ′. Note however that this semidirect product structure need not be com-

patible with the semidirect product structure of π1(V) since there is no relationship

72

between the two sections used in the constructions.

Let t′ be a general point in T ′ and w = p ◦ f ′(ζ ′(t′)). Since f ′ : Y ′ → U is a

morphism of good families, by definition we know that Y ′t′ → Uw is an adjoint-rigid

cover preserving the a-constant. Hence, by Corollary 4.2.2, f ′ : f ′−1(V)t′ → Vw

is proper and etale. Therefore the image of π1(f ′−1(V)t′) in π1(Vw) is one of the

associated subgroups Ξj ⊂ π1(Vw) constructed in Construction 5.1.10. We will show

that after a further base change by T → T ′, the morphism of good families Y ′T→ U

rationally factors through the morphism Yj → U , for the associated good family

Yj → Tj corresponding to Ξj.

Let (T ′)◦ denote the preimage of W ◦ in T ′. If (T ′)◦ is empty then the image of

Y ′ is contained in p−1(D), so we may assume otherwise. We have the composition

morphism (T ′)◦ ↪→ T ′ζ′−→ VY ′

f ′−→ V . The group π1((T ′)◦) maps into π1(V◦) under

this composition moprhism. Consider the finite index subgroup M ⊂ π1((T ′)◦) which

is the pullback of Ξj o π1((Tj)◦) by this map. Let T ◦ → (T ′)◦ be the etale cover

corresponding to M . We extend it to a generically finite surjective morphism T → T ′.

Now the base change q : Y → T satisfies the desired factoring property. Since

f−1(V◦)→ T is a locally topologically trivial fibration, by using the section ζ induced

from ζ ′ by base change we have an identification

π1(f−1(V◦)) = π1(f−1(V)t) o π1(T ◦).

Note that every element in f∗π1(f−1(V◦)) will be a product of an element in

Ξjo{1} ⊂ π1(V◦) with an element in f∗π1(T ◦), so by construction this set is contained

in Ξjoπ1((Tj)◦). By the lifting property for fundamental groups, the map f−1(V)→ V

factors through the cover defined by Ξj o π1((Tj)◦). Hence Y → U rationally factors

through Yj. we use the lifting property again to get a natural map from T ◦ to (Tj)◦.

73

For any fiber of q, if the corresponding fiber of rj is irreducible then it must be adjoint

rigid with respect to the pullback of L and have the same a-value as a fiber of p. Such

a fiber of rj will either be a fiber of the good family Yj or will map into p−1(D). We

conclude that the map f will either factor rationally through one of the fj or will

map into p−1(D).

The following result shows that base changes of the associated good families from

Construction 5.1.10 also satisfy the conclusion of Lemma 5.1.11.

Corollary 5.1.12 [LST18] Let p : U → W be a good family of adjoint rigid varieties

with a good section. Suppose that the divisor L on U is the restriction of a nef Q-

Cartier divisor on a projective compactification of U . Consider the good morphisms

of good families {fi : Yi → U} with family maps qi : Yi → Ti and the closed proper

subset D ( W constructed in Construction 5.1.10.

Suppose that for each i we fix a proper generically finite dominant map T ′i → Ti

from a smooth variety T ′i and replace qi : Yi → Ti by the base change Y ′i := T ′i ×Ti Yi

(with the natural induced maps f ′i and q′i). Construct a closed subset D′ ( W by

taking the union of D with the branch locus of each map T ′i → W .

Then the good families Y ′i (equipped with the morphisms f ′i and q′i) and the closed

subset D′ ( W again satisfy the conclusion of Lemma 5.1.11.

Proof. Let q : Y → T be a good family of adjoint rigid varieties admitting a good

morphism f : Y → U . If f(Y) ⊂ D, then f(Y) ⊂ D′ as D ⊂ D′ . Suppose f : Y → U

rationally factors through after a base change. Let W ◦ = W \D′ and T ◦i , T ′◦i and T ◦

be the corresponding inverse images of W ◦. Then T ′◦i ×T ◦i T◦ is an etale cover of T ◦.

Let R be any irreducible component of this product which dominates T ◦. Then the

basechange of q : Y → T over the map R→ T gives the desired rational factoring.

By Noetherian induction and applying Lemma 5.1.11 on the family over the closed

set D repeatedly, we obtain the following result.

74

Theorem 5.1.13 [LST18] Let p : U → W be a good family of adjoint rigid varieties

with a good section. Suppose furthermore that the divisor L on U is the restriction

of a nef Q-Cartier divisor on a projective compactification of U . There is a finite set

of generically quasi-finite good morphisms of good families {fi : Yi → U} with family

maps qi : Yi → Ti such that the following holds.

Suppose that q : Y → T is a good family of adjoint rigid varieties admitting a

good morphism f : Y → U . Then there is a base change q : Y → T of q such that the

induced f : Y → U factors rationally through the map fi for some i and a general

fiber of q is birational to a fiber of qi.

Proof of Theorem 1.3.2 over C.

Let X be a smooth projective uniruled variety over C. Let L be a big and nef Q-

divisor on X. We want to construct a finite set of breaking thin maps f` : Y` −→ X

such that for any breaking thin map f : Y −→ X there is an Iitaka base change Y

of Y with a morphism f : Y −→ X such that f factors rationally through one of the

f`’s.

Construction 5.1.14 (Associated thin maps.) Let pi : Ui → Wi be the families

constructed in Theorem 5.1.1. If si(Ui) ⊂ B+(L) then we ignore the corresponding

family pi from now on. Since L is nef, by applying Lemma 5.1.6 and shrinking

the base we obtain a good family with a good section. By repeating the process

using Noetherian induction on the base Wi and throwing away the families with

images contained in B+(L), we obtain a finite collection of good families with good

sections. By applying Theorem 5.1.13 to each of the families we obtain finitely many

{qi,j : Yi,j → Ti,j} with maps gi,j : Yi,j → X such that the image of gi,j is not

contained in B+(L).

We replace the families qi,j : Yi,j → Ti,j by their base changes over Ti,j as follows.

By taking a base change we may assume that gi,j is not birational. By a further base

75

change we may assume that the monodromy action of π1(Ti,j) on the Neron-Severi

group of a general fiber of qi,j is trivial. Finally, we take a base change over a cyclic

cover of Ti,j whose branch divisor is very ample.

We construct the thin maps {f` : Y` → X} as follows. Let Di,j ⊂ X be the closure

of gi,j(Yi,j). Note that Di,j 6⊂ B+(L). We consider the following two cases:

(1) Allowable family. Suppose a(Di,j, L|Di,j) is equal to the a-value of the fibers of

qi,j, then we say that Yi,j is an allowable family. Let Y` be a resolution of a projective

closure of Yi,j such that we have a morphism f` : Y` → X extending gi,j and a

morphism r` : Y` → R` extending the family map qi,j. Note that in this case, by

[LT17, Proposition 4.14], gi,j is generically quasi-finite. Since gi,j is not birational, f`

is a thin map. Note that by Remark 5.1.15 below, we know that r` is birationally

equivalent to the canonical model map h` : Y` 99K S` for KY` + a(Y`, f∗` L)f ∗i L.

(2) If a(Di,j, L|Di,j) is larger than the a-values of the fibers of Yi,j, then Di,j must

be a proper closed subvariety of X by Theorem 2.4.8. We let the inclusion Di,j ↪→ X

to be one of the f`.

Remark 5.1.15 Let h` : Y` 99K S` denote the canonical map for KY`+a(Y`, f∗` L)f ∗i L.

Recall that in the construction of Y` we took a cyclic cover of the corresponding Ti,j

branched over a very ample divisor. Hence there is an ample divisor H on R` such that

KY` + a(Y`, f∗` L)f ∗` L− r∗`H is Q-linearly equivalent to an effective divisor. Therefore

we have an inclusion R(Y`, r∗`H) ⊂ R(Y`, KY` + a(Y`, f

∗` L)f ∗` L) of the corresponding

section rings. Therefore by taking Proj of the section rings, we see that r` factors

rationally through h`. However, since the general fiber of r` is adjoint rigid with

respect to the restriction of a(Y`, f∗` L)f ∗` L, we deduce that r` is birationally equivalent

to h`.

Proposition 5.1.16 [LST18] Let f : Y → X is any thin map satisfying a(Y, f ∗L) ≥

a(X,L) such that f(Y ) 6⊂ B+(L). There is an Iitaka base change Y of Y such that

the induced map f : Y → X rationally factors through some f` constructed above.

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Proof. We may assume that Y is smooth and admits a morphism q : Y → T which is

the canonical model for KY + a(Y, f ∗L)f ∗L, by passing to a resolution. By Lemma

5.1.6, we may assume that q : Y → T is a good family, after possibly shrinking T .

Now, for a general t ∈ T , we know that (Yt, f∗L) is adjoint-rigid and a(Yt, f

∗L) =

a(Y, L) ≥ a(X,L). Then f(Yt) ⊂ X is a subvariety which is adjoint rigid with

respect to L and has a-constant at least as large as a(X,L). Therefore, by applying

Theorem 5.1.1 and possibly shrinking T , we obtain a good morphism of good families

Y → Ui for some i. By the construction of the families qi,j : Yi,j → Ti,j above, we

see that after possibly shrinking T and making an additional base change we obtain

Y , such that the map Y → Ui rationally factors through Yi,j for some j. If Yi,j is an

allowable family, then f will factor rationally through the corresponding Y`. If Yi,j is

not allowable, then f factors through the inclusion Di,j ↪→ X

Proposition 5.1.17 [LST18] Let f : Y → X is any thin map satisfying a(Y, f ∗L) ≥

a(X,L) such that f(Y ) 6⊂ B+(L). Let Y → Y denote the Iitaka base change and

f` : Y` → X the corresponding thin map constructed in Proposition 5.1.16. Then we

have,

(1) a(Y, f ∗L) = a(Y , f ∗L) ≤ a(Y`, f∗` L),

(2) if equality of a-invariants is achieved in (1), then b(Y, f ∗L) ≤ b(Y , f ∗L) ≤

b(Y`, f∗` L).

Proof. (1) After passing to a resolution we may assume that we have a morphism to

the canonical model Y → T for KY + a(Y, f ∗L)f ∗L. Let Yt and Yt be the general

fibers of the canonical model maps of (Y , a(Y , f ∗L)f ∗L) and Y, a(Y, f ∗L)f ∗L. Note

that, since Y is obtained by an Iitaka base change, we have an isomorphism Yt∼−→ Yt.

Therefore,

a(Y, f ∗L) = a(Yt, f∗L) = a(Yt, , f

∗L) = a(Y , f ∗L).

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Note that maps Yt maps birationally onto a fiber Yi,j,ti,j of qi,j : Yi,j → Ti,j. Therefore,

a(Y , f ∗L) = a(Yi,j,ti,j , g∗i,jL|Yi,j,ti,j ).

If Yi,j is an allowable, then a-invariant is constant for the fibers of qi,j and Y` is

dominated by subvarieties with the same a-value as Y . By Lemma 2.4.9, a(Y , f ∗L) ≤

a(Y`, f∗` L). If Yi,j is not allowable, f factors through Di,j ↪→ X where Di,j has higher

a-value than the members of the family Yi,j, and thus also higher a-value than Y .

Thus in either case

a(Y , f ∗L) ≤ a(Y`, f∗` L).

(2) If we have an equality of a-constants then Y` must be an allowable family.

Since we have an isomoprhism Yt∼−→ Yt of the general fibers of the canonical model

maps, by Lemma 2.4.15, we have

b(Y, f ∗L) ≤ b(Y , f ∗L) ≤ b(Yt, f∗L|Yt)

where the first inequality is by 2.4.15(1) and the second one is by part (3). Since Yt

maps birationally onto a general fiber Y`,s` of the canonical model map h` : Y` → S`

for (Y`, a(Y`, f∗` L)f ∗` L), we have

b(Yt, f∗L|Yt) = b(Y`,s` , f

∗` L|Y`,s` ).

Therefore it is enough to show that b(Y`,s` , f∗` L|Y`,s` ) = b(Y`, f

∗` L). By Remark

5.1.15, the canonical model map for (Y`, a(Y`, f∗` L)f ∗` L) is birationally equivalent to r`.

Therefore by Lemma 2.4.12, it is enough to show that b(Y`,t` , f∗` L|Y`,t` ) = b(Y`, f

∗` L)

for a general fiber Y`.t` of r`.

Recall that, by construction, the monodromy action is trivial on the N1(Y`,t`).

Let {Ek} denote the finite set of irreducible divisors such that KY`,t`+ f ∗` L|Y`,t` − cEk

78

is pseudo-effective for some c > 0. By [Nak04, III.1.10 Proposition] the classes of the

Ek are linearly independent in the Neron-Severi space. Since the monodromy action

is trivial, {Ek} lift to linearly independent divisors {Ek} on Y`. Now Ek dominate

the base of the canonical map and KY` +f ∗` L−cEk is pseudo-effective for some c > 0.

By Lemma 2.4.15, we conclude that b(Y`,t` , f∗` L|Y`,t` ) = b(Y`, f

∗` L) for a general fiber

Y`.t` of f`, since the monodromy action is trivial.

Proof of Theorem 1.3.2. Let X be a smooth projective uniruled variety over C.

Let L be a big and nef Q-divisor on X. Let f` : Y` −→ X be the finite set of associated

thin maps constructed in Construction 5.1.14. Let f : Y → X be a breaking thin

map. Then, by Proposition 5.1.16, we know that after an Iitaka base change Y of

Y such that the induced map f : Y → X rationally factors through some f`. Since

(a(Y, f ∗L), b(Y, f ∗L)) > (a(X,L), b(X,L)), we see that the corresponding thin map

f` : Y` → X is also a breaking thin map, by Proposition 5.1.17. Therefore the thin

maps f` : Y` → X with (a(Y`, f∗` L), b(Y`, f

∗` L)) > (a(X,L), b(X,L)) satisfy the desired

property.

Over subfields of C.

Let k ⊂ C be a subfield. We extend the notion of good families (see Definition 5.1.3)

to the field k as follows:

Definition 5.1.18 A good family of adjoint rigid varieties over k is an k-morphism

p : U → W of smooth quasi-projective varieties and a relatively big and nef Q-divisor

L on U such that the base-change to C is a good family of adjoint rigid varieties over

C.

Note that the set QC ⊂ UC as in Definition 5.1.3 descends to a closed set Q over

k. We define V = U\Q.

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Remark 5.1.19 Note that the conclusion of Lemma 5.1.6 still holds over the field

F . Indeed, by applying Lemma 5.1.6 after base changing to C and descending to F ,

we may prove the result over the algebraic closure. Then by taking Galois conjugates

of the complement of the open subset and descending to F , we can conclude.

Remark 5.1.20 By the Lefschetz principle, Construction 5.1.10 and Construction

5.1.14 work over the algebraic closure k. Indeed, we may define the associated groups

in Construction 5.1.10, by replacing the topological fundamental group by the etale

fundamental group, since subgroups of πet1 (V) of finite index are in bijection with

subgroups of π1(VC) of finite index. Since fundamental groups over C are finitely

generated, semidirect products commute with profinite completions so we can use the

comparison theorem of etale and topological fundamental groups over C. Therefore,

if p : U → W is a good family of adjoint rigid varieties with a good section and Uw is

a fiber of p, we have

πet1 (V) = πet

1 (V ∩ Uw) o πet1 (W ).

Note that the homotopy lifting property is true for etale fundamental groups by

[Gro71, Expose V Theoreme 4.1, Corollaire 5.3]. Therefore the construction of asso-

ciated good families and thin maps as well as the factoring properties work over k.

Therefore the statements in the previous subsection still hold over any algebraically

closed subfield k.

5.2 Thinness of rational points from breaking thin

maps

In this section we prove Theorem 1.3.1. Throughout we assume that the ground

field is a number field F . First, we prove analogues of the finiteness results from

80

the previous section over F . Then we construct a thin set and show that it contains

rational points contributed by all breaking thin maps.

Let f : Y → X be a breaking thin map. Note that we have breaking thin maps

f` : Y` → X from Construction 5.1.14 which satisfy the factoring property of Theorem

1.3.2 over F . Now these families might not descend to the ground field F . Therefore,

we construct finitely many covers defined over the ground field, by working with etale

fundamental groups and keeping careful track of the rational points. Then we show

that the twists of such covers satisfy a desired factoring property over F . Finally

using Theorem 1.3.3, we can conclude that the rational points contributed by all such

breaking thin f : Y → X.

In the following construction we carry out the analogues of Construction 5.1.10

and Construction 5.1.14 over F .

Construction 5.2.1 [LST18] Let X be a geometrically uniruled smooth projective

variety defined over F and let L be a big and nef Q-divisor on X. Let p : U → W

be a morphism between projective varieties where U is equipped with a morphism

s : U → X which is birational.. Suppose that there exists an open subset W ◦ ⊂ W

such that p : U◦ → W ◦ is a good family of adjoint rigid varieties over F (where U◦

denotes the preimage of W ◦) and that any fiber over W ◦ has the same a-invariant as

X does. Let Q denote the closed subset of U◦ as in Definition 5.1.3 and let V denote

its complement. Suppose that V admits a rational point.

In this construction we will produce a proper closed subset C ( X and a finite

set of dominant generically finite morphisms {fj : Yj → U} defined over F that fit

into commutative diagrams

Yj U

Tj W

fj

qj p

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such that the following properties hold.

(1) Yj and Tj are smooth and projective and qj : Yj → Tj is generically a good

family of adjoint rigid varieties,

(2) the canonical model for a(X,L)f ∗j L+KYj is birational to qj;

(3) the morphism Tj → W is Galois,

(4) we have Bir(Yj/X) = Aut(Yj/X),

(5) after shrinking W ◦, for any twist Yσj over X and for any closed point t ∈ T σ◦j

we have b(F,Yσj , sσ∗j L) = b(F,Yσj,t, sσ∗j L|Yσj,t) where sj : Yj → X denotes the

composition of fj : Yj → U and s : U → X, T ◦j denotes the preimage of W ◦,

and Yσj,t denotes the fiber over t;

Notation. During the construction we will shrink W ◦ several times and U◦ will

continue to denote its preimage and V will continue to denote U◦\Q. This abuse

of notation is justified since we will shrink W ◦ is such a way that the modified V

still contains the rational point. If W µ → W is a morphism, we let W µ◦ denote the

preimage of W ◦. Let pµ : Uµ → W µ denote the base change and Vµ denote V×W ◦W µ◦.

Step 1: (Base changes of W .) We take a finite Galois base change W µ → W

and shrink W ◦, such that pµ : Vµ → W µ◦ admits a good section ζ. We replace W µ

by cyclic cover ramified over the pull-back of an ample divisor on W . Then by the

argument in Remark 5.1.15, we may assume that pµ : Uµ → W µ is birational to the

canonical model for (Uµ, s∗L). We shrink W ◦ so that W µ◦ → W ◦ is proper and etale.

Step 2: (A closed subset C.) After base changing to F and applying

Lemma 5.1.11 to the good family pµ : Uµ◦ → W µ◦, we obtain a proper closed

subset D ⊂ Wµ◦

. By taking Galois conjugates of D we descend to D defined over

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F . Let C ⊂ X be the closure of sµ((pµ)−1(D ∪ R) ∪ Q) where R is the ramification

locus of W µ → W . We will replace C later by a larger closed subset, which we will

continue to denote by C.

Step 3: (Associated etale covers.) In this step, we construct analogues of the

associated groups and the associated families in Construction 5.1.10. By assumption,

V contains a rational point. Since W µ → W is Galois we may ensure that Vµ admits

a rational point after replacing W µ by its twist. (Note that after this change the

section ζ may not be defined over the ground field but is still defined after base

change to F . The other properties of W µ are preserved by replacing by a twist.)

Let wµ denote a rational point on W µ◦ which is the image of a rational point in

Vµ and define the geometric point vµ = ζ(wµ). Consider the associated finite index

subgroups Ξj ⊂ πet1 (Vµ ∩ Uwµ , vµ) corresponding to adjoint-rigid covers preserving

the a-constant (see Construction 5.1.10). As in Lemma 5.1.8, we have a normalizer

Nj ⊂ πet1 (W

µ◦, wµ) for each Ξj. We take the maximum subgroup Nj ⊂ Nj such that

the monodromy of Nj on

πet1 (Vµ ∩ Uwµ , vµ)

/ ⋂g∈πet

1 (V∩Uwµ ,vµ)

gΞjg−1

is trivial. We define Υj = Ξj o Nj. Each Υj defines an etale cover E j → Vµ, and by

composing with the etale map Vµ → V we obtain an etale cover sj : E j → V .

Note that the covers E j → V might not descend to F . Even if a cover descends

it is not clear that it contains a rational point. If E j → V descends to a morphism

Ej → V over F such that Ej contains a rational point, we use it to construct the

covers fj : Yj → U with the desired properties (see Step 4). Otherwise, we will use

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the covers to just update the closed set C (see Step 5).

Step 4: (Associated thin maps fj : Yj → U .) In this step we construct the

finite set of covers fj : Yj → U which are analogues of the finite set of thin maps in

Construction 5.1.14. Consider the covers E j → V that descend to a morphism Ej → V

over F such that Ej contains a rational point. For each such cover, let us fix a choice

Ej of such a model over F with a rational point.

Let Yj be a compactification of Ej over F such that we have a morphism fj : Yj →

U extending the covering map Ej → V . Let qj : Yj → Tj be the Stein factorization of

Yj → W . By applying Lemma 5.1.6 to the base change of qj to the algebraic closure

and excluding Galois conjugates of the complement of the open subset of the base, we

know that qj is generically a good family of adjoint rigid varieties over F . From now

on we let T ◦j , Y◦j denote the preimages of W ◦. Now we modify the families Yj → Tj

so that they satisfy conditions (1)-(4) as follows:

By Lemma 2.6.3 we may replace Yj by a birational model to ensure that

Bir(Yj/X) = Aut(Yj/X). After shrinking W ◦, qj is a good family of adjoint rigid

varieties over T ◦j . Note this cause Ej to shrink. Now we have two cases:

Case (a): Suppose that no twist of the Ej contains a rational point, then we add

sj(Yj \ Ej) ∪ s(p−1(Bj)) ∪ Ej to C, where Ej is the branch locus of sj : Yj → X and

Bj is the branch locus of Tj → W . To distinguish such types of families, we will

henceforth denote it as qk : Pk → Sk with the evaluation map sk : Pk → X.

Case (b): Suppose that some twist of Ej contains a rational point. Then we

replace the Ej by this twist. Let bj ∈ T ◦j be the image of the rational point on Ej.

We take a base change defined over F which kills the geometric monodromy action

on the geometric Neron-Severi space of the fiber over bj as follows. Let G is the kernel

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of the geometric monodromy action on the geometric Neron-Severi space of the fiber

over bj. We replace T ◦j by the etale cover of T ◦j defined by GoGal(F/F ) ⊂ πet1 (T ◦j , bj)

and replace Tj by a projective closure of the etale cover. By replacing Tj by a Galois

closure we may assume that the geometric monodromy of πet1 (T ◦j , bj) on the Neron-

Severi space of a general fiber of qj is trivial and Tj/W is Galois.

By Lemma 2.6.3, we replace Yj by a birational model to such that Bir(Yj/X) =

Aut(Yj/X). After shrinking W ◦ we have a good family T ◦j . By shrinking W ◦ further

we may ensure that there is an effective divisor numerically equivalent to KYj +

a(X,L)s∗jL which does not contain any fiber over T ◦j . After possibly shrinking W ◦

further, we may guarantee that T ◦j is proper and etale over an open set W ◦ and that

fj : Ej = f−1j (V)→ V is etale.

If no twist of Ej contains a rational point again, then we add sj(Yj \ Ej) ∪

s(p−1(Bj)) ∪ Ej to C and then we relabel this family as one of the qk : Pk → Sk

with the evaluation map sk : Pk → X.

If some twist of Ej does contain a rational point, then we replace Ej with this

twist and we also replace Yj, Tj by the corresponding twists. We enlarge C by adding

sj(Yj \ Ej)∪ s(p−1(Bj))∪Ej where Y◦j is the preimage of W ◦ in Yj, Ej is the branch

locus of sj : Yj → X and Bj is the branch locus of Tj → W . Note that the covers

fj : Yj → U satisfy the desired properties (1)-(4).

Now we prove that the families also satisfy the condition (5). Let b ∈ T σ◦j be a

closed point. Let b ∈ Tσ◦j denote a geometric point above b. By construction Yσj

has a birational model with a trivial geometric monodromy action. We let Γ be the

smooth geometric fiber over b of this birational model so that Γ is birational to the

smooth fiber Yσj,b. Let {Ei}ri=1 be the divisors on Yσj,b which lie in the support of the

rigid divisor a(X,L)s∗jL|Yσj,b +KYσj,b and let {E ′j}sj=1 denote the analogous divisors on

85

Γ. By Lemma 2.4.12 we have an isomorphism

N1(Yσj,b)/Span({Ei}ri=1) ∼= N1(Γ)/Span({E ′j}sj=1).

Thus we see that the geometric monodromy acts trivially on N1(Yσj,b)/Span({Ei}).

By Lemma 2.4.15 (1) we obtain the desired equality.

Step 5: Suppose E j → V is a cover constructed in Step 3 that fails to descend to F

in such a way that it admits a rational point. Over F we can repeat the construction

of Step 4 to obtain a morphism of varieties Pk → U over F with structure maps

Pk → Sk. Let C ′ = sk(Pk \P◦k)∪sk(p−1(Bk))∪Ek where as usual P◦k is the preimage

of W◦. By taking Galois conjugates we may assume that C ′ descends to F and replace

the closed set C by C ∪ C ′.

Now we show that the twists of the covers fj : Yj → U from Construction 5.2.1

above satisfy a factoring property.

Lemma 5.2.2 [LST18] Let X be a geometrically uniruled smooth projective variety

defined over F and let L be a big and nef Q-divisor on X. Let p : U → W be a

morphism between projective varieties where U is equipped with a morphism s : U →

X which is birational. Suppose that there exists an open subset W ◦ ⊂ W such that

p : U◦ → W ◦ is a good family of adjoint rigid varieties over F (where U◦ denotes the

preimage of W ◦) and that any fiber over W ◦ has the same a-invariant as X does. Let

qj : Yj → Tj be the families and C ⊂ X be the closed set from Construction 5.2.1.

Then the following holds.

86

Suppose that q : Y → T is a projective surjective morphism of varieties over F

and that we have a diagram diagrams

Y U

T W

f

q p

satisfying the following properties:

(a) There is some open subset T ◦ ⊂ T such that Y is a good family of adjoint rigid

varieties over T ◦ and the map f : q−1(T ◦)→ U is a good morphism.

(b) There is a rational point y ∈ Y(F ) contained in the smooth locus of Y such

that f(y) 6∈ C.

Then for some index j there will be a twist fσj : Yσj → U such that f(y) ∈ fσj (Yσj (F )).

Furthermore, after a base change over T the induced map f : Y → U will factor

rationally through fσj and a general geometric fiber of the structure map for Y will

map birationally to a geometric fiber of the structure map for Yσj .

Proof. We set t = q(y), where y ∈ Y(F ) is the rational point above. We will construct

a twist Yσj such that f(y) ∈ fσj (Yσj (F )). We follow the idea of the proof of the

factoring property in Lemma 5.1.11.

Step 1: We base change the family Y → T as follows. Let W µ → W be the Galois

cover in Step 1 of Construction 5.2.1. We may replace W µ by a twist to assume

that it carries a rational point whose image in W is the same as the image of t ∈ T .

Since f(y) 6∈ C, we know that T ×W W µ is etale over an open neighborhood of t

(as C contains the branch divisor by construction and the property of being etale

is unchanged after passing to a twist). Thus there is some component of T ×W W µ

which maps dominantly to T and which admits a rational point in its smooth locus

87

mapping to t. Let T µ be the normalization of this dominant component. Let Yµ be a

smooth resolution of Y ×T T µ. Since Yµ is etale over an open neighborhood of y, Yµ

admits a rational point mapping to y which we denote by yµ. We denote the induced

morphism by qµ : Yµ → T µ and let tµ = qµ(yµ).

Step 2: Now we make a further base change to assume that we have a rational

section compatible with the rational points as follows. Let T ′ be a general intersection

of hyperplanes through yµ such that T ′ → T µ is generically finite. We take the base

change of qµ : Yµ → T µ by the morphism T ′ → T µ. Now there is a rational point

t′ ∈ T ′(F ) mapping to tµ such that T ′ is smooth at t′ and the main component YT ′

admits a rational section τ such that (yµ, t′) = τ(t′). By the generality of T ′ we

may furthermore ensure that the image of τ intersects the smooth locus of YT ′ . Let

T → T ′ be the blow up at t′ and consider the main component YT of the base change

by T . Let YT be a resolution of YT chosen in such a way that YT still admits a

rational section τ . Note that τ is well-defined on the generic point of the exceptional

divisor lying over t′. Thus by taking the image under τ of a suitable rational point

t in the exceptional divisor we obtain a rational point y′ ∈ YT mapping to y and t.

Let v be the image of y′ in Vµ and set wµ = pµ(v) and vµ = ζ(wµ).

Step 3: In this step we show that f : Y → U factors through one of the fj : Yj → U

after passing to the algebraic closure of F . Note that the geometric families E j → T◦j

(of Step 3 of Construction 5.2.1) are base changes of the families constructed in

Cosntruction 5.1.10. Therefore, by Lemma 5.1.11 and Corollary 5.1.12, we know that

after some Iitaka base change of Y T , the induced map to Uµ factors rationally through

Yj for some j or Pk for some k.

We will show that the latter case can not occur. Assume for a contradiction that

the map factors rationally through Pk. First we prove the the following.

88

Claim: If we take the Stein factorization Y ′ of the map of fibers Yt → Uwµ and

then base change to F the result is birational to the adjoint rigid variety Pk,s where

s is some suitably chosen preimage of wµ.

Proof. Let T◦⊂ T an open subset such that the image of this set in W is contained

in W◦

and the τ -image of this set lies in the preimage of V (Here τ is the section

constructed in Step 2 above). Let Tν

denote the etale cover of T◦

defined by the finite

index subgroup of πet1 (T

◦, t) constructed by pulling back under τ the subgroup of π1(V)

corresponding to the etale cover defined by Ek. For the open subset of T◦

over which

we have a good family, just as in Lemma 5.1.11 we know that the main component of

the base change Yν

over Tν

admits a rational map to Pk. Since the map Tν

to T◦

is

etale, Yν

is smooth in a neighborhood of the fiber Yν

tν where t

νis a geometric point

mapping to t. Let Y∗ denote a smooth resolution of the rational map to Pk. The fiber

Y∗tν maps to some fiber Pk,s. Since the induced map Y∗ → Pk ×Sk Tν

is birational

and the target is smooth, it has connected fibers. In particular the map Y∗tν → Pk,s

has connected fibers and we deduce that the Stein factorization of Y∗tν → Uwµ is

birational to Pk,s. Note that this map also factors through our original fiber Yt

and

that the first step of this factoring has connected fibers. Thus the Stein factorization

of our original fiber over F is birational to Pk,s. Since Stein factorization commutes

with base change to the algebraic closure our assertion follows.

Now we obtain a contradiction from the assumption that the rational factoring of

the morphism from the Iitaka base of Y T is through Pk. Note that the claim above

This implies that the subgroup Ξk which corresponds to the cover Pk,s → Uwµ admits

an extension Ξk ⊂ πet1 (Vw, vµ) corresponding to the cover Y ′ → Uwµ defined over the

ground field. We next show that this means that f−1k (V) must descend to the ground

field. Indeed, using the fact that we constructed πet1 (T

◦k, tk) ⊂ πet

1 (Vw, vµ) to have a

trivial monodromy action on the cosets of the intersection of conjugates of Ξk, one

89

can show that πet1 (T

◦k, tk) · Ξk = Ξk · πet

1 (T◦k, tk) so that one may define an etale cover

f−1k (V) → Vµ using πet

1 (T◦k, tk) · Ξk which is an extension of Υk. Moreover we claim

that f−1k (V) admits a fiber birational to Y ′ and this birational map is an isomorphism

on an open neighborhood of the image of y. Indeed, let T ◦k be the Stein factorization

of f−1k (V) → Vµ → W µ◦. Then the cover T ◦k → W µ◦ corresponds to an extension of

πet1 (T

◦k, tk) by Gal(F/F ). Note the fundamental group of T ◦k admits a splitting which

is compatible with the splitting of πet1 (W µ◦, wµ) coming from wµ. On the other hand

since Tk is Galois over W µ, T ◦k admits a twist T ◦σk with a rational point mapping to

wµ. Moreover the fundamental group of T ◦σk also has a compatible splitting so T ◦k and

T ◦σk must be isomorphic to each other. Altogether we conclude that T ◦k comes with a

rational point tk mapping to wµ. By comparing fundamental groups, we see that the

fiber over tk is birational to the variety defined by Ξk as claimed. Furthermore, this

birational map is an isomorphism on a neighborhood of y because y maps to V . We

conclude that f−1k (V) admits a rational point coming from y. However, the fact that

the geometric model descends to the ground field with a rational point contradicts

our definition of the Pk. We deduce that this case cannot happen; in other words,

some base change of Y T admits a rational map to Yj for some j.

Step 4: Now we show that some twist of Yj contains a rational point yj mapping

to v. As we discussed before, the Stein factorization Y ′ of Yt → Uwµ is birational to

an etale cover of Vµ ∩ Uwµ defined by Ξj ⊂ πet1 (Vµ ∩ Uwµ , vµ). Just as in the previous

paragraph, we construct a cover Yσj → U with the family structure Yσj → T σj . This

comes with a rational point tj on T σj mapping to wµ and the two varieties Y ′ and

Yσj,tj are birational over the ground field. Thus Yσj,tj comes with a rational point yj

mapping to v.

Step 5: In this step we prove the factoring property over F . Recall that we have

constructed a rational point t ∈ T and a rational point yj ∈ Yσj such that the image

of yj in U is the same as the image of t under the rational map T 99K U given by

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the composition of the rational section to YT and the map to U . Let T † be the open

subset where the rational map to U is defined. Consider the base change

T † ×U Yσj Yσj

T † U

Since Yσj → U is etale on a neighborhood of the image of t and admits a rational

point mapping to the image of t, there is a component T ν of T ×U Yσj which maps

dominantly to T † and admits a rational point tν mapping to t. Furthermore, the base

change YT ν is smooth at any point of the fiber over tν . By Lemma 5.1.11 and Corollary

5.1.12, after base changing to F the map YT ν → X factors rationally through the

twist Yσj .

We claim that the map YT ν → X factors rationally through Yσj over the ground

field. Indeed, by the lifting property over F one may find a rational map h : YT ν 99K

Yσj mapping (y′, tν) to the point yj constructed above. Let s be an element of the

Galois group. Then both h and hs

are lifts of the same map to U and they both map

(y′, tν) to yj. Thus h = hs

by the uniqueness of the lift. Thus our assertion follows.

Now we construct a thin set that will contain the rational points contributed by

all breaking thin maps.

Construction 5.2.3 (A thin set Z.) Let X be a geometrically uniruled geometrically

integral smooth projective variety over a number field F . Let L be a big and nef Q-

divisor on X. We start with Z = ∅ ⊂ X(F ). In the following steps we will increase

Z by adding rational points and at the end of this construction we will end up with

a thin set Z ⊂ X(F ).

91

Step 1: We apply Theorem 5.1.2 to (X,L) to obtain a proper closed subset V ⊂ X

and finitely many projective families pi : Ui → Wi with generically finite surjective

evaluation maps si : Ui → X. Set Z = V (F ).

Step 2: If Ui is not geometrically integral, then Ui(F ) is contained in the singular

locus of Ui. Therefore the closure si(Ui(F )) is contained in the proper closed subset

si(Sing(Ui)) = Vi ( X. We update Z by including the rational points in Vi(F ).

Step 3: If Ui is geometrically integral and deg(si) > 1, then Ui(F ) is a thin subset

of X(F ). We update Z by including such Ui(F ).

Step 4: Suppose Ui is geometrically integral and si : Ui → X is birational. We

replace Ui by a resolution of singularities and include the rational points Vi(F ) in Z

where Vi = si(Sing(Ui)) ( X.

Step 5: By applying Lemma 5.1.6 and Remark 5.1.19, to pi : Ui → Wi, we obtain

an open subset W ◦i ⊂ Wi such that pi : p−1

i (W ◦i ) → W ◦

i is a good family of adjoint

rigid varieties. Let Qi be the closed subset associated to this good family and define

Vi = p−1i (W ◦

i ) \ Qi (see Definition 5.1.3). Let Ei denote the ramification divisor of

si and Si denote the proper closed subset si(Ui \ Vi) ∪ si(Ei) ( X. We update Z by

including the rational points in the Si(F ) .

Step 6: By applying Construction 5.2.1 to the families pi : Ui → Wi from Step 5

above, we obtain finitely many families qi,j : Yi,j → Ti,j with thin maps si,j : Yi,j → X

and proper closed subsets Ci ( X. We update Z by including the rational points in

Ci(F ).

Step 7: For each of the thin maps si,j : Yi,j → X from Step 6, we consider all

twists σ ∈ H1(F,Aut(Y i,j/X)) such that

(a(X,L), b(X,L)) < (a(Yσi,j, (sσi,j)∗L), b(F,Yσi,j, (sσi,j)∗L)).

92

Note that, by construction, Yi,j is geometrically irreducible since Ui is. As Yi,j is

geometrically integral, all the twists Yσi,j are irreducible. By Theorem 1.3.3, we obtain

a thin sets

Zi,j =⋃σ

sσi,j(Yσi,j(F )) ⊂ X(F ).

Finally we update Z by include all the rational points in ∪i,jZi,j.

Proof of Theorem 1.3.1.

Let X be a smooth geometrically uniruled geometrically integral projective variety

defined over a number field F . Let L be a big and nef Q-divisor onX. Let f : Y −→ X

be a breaking thin map with Y geometrically integral, i.e.

(a(X,L), b(F,X,L)) < (a(Y, f ∗L), b(F, Y, f ∗L)).

We will show that f(Y (F )) ⊂ Z where Z ⊂ X(F ) is the thin set constructed in

Construction 5.2.3.

Let us define d(X,L) as

d(X,L) = dim(X)− κ(X,KX + a(X,L)L).

Note that d(X,L) is the dimension of a general fiber of the map to the canonical

model for (KX + a(X,L)L). Similarly we define d(Y, f ∗L) = dim(Y ) − κ(Y,KY +

a(Y, f ∗L)f ∗L). We deal with the following two cases depending on the fiber dimen-

sion.

Case 1: Suppose d(Y, f ∗L) > d(X,L). We will show that f(Y ) ⊂ V , where

V ⊂ X is the proper closed subset given by Theorem 5.1.2. Note that by Step 1 of

Construction 5.2.3, we know that V (F ) ⊂ Z.

93

Suppose that f(Y ) 6⊂ V . We reach a contradiction as follows.

We may replace Y by a resolution and assume that Y admits a morphism to the

canonical model for (Y, a(Y, L|Y )L|Y ). Let Γ ⊂ Y be a general fiber of the canonical

model for (Y, a(Y, L|Y )L|Y ). Then a(Γ, f ∗L|Γ) = a(Y, L) and Γ is adjoint rigid with

respect to f ∗L|Γ. Note that f |Γ is generically finite. Hence

a(f(Γ), L|f(Γ)) ≥ a(Γ, f ∗L|Γ) ≥ a(X,L).

By assumption, we have f(Γ) 6⊂ V . Then we have a(f(Γ), L|f(Γ)) = a(X,L), since by

Theorem 5.1.2, V contains all subvarieties with higher a-constant. Since Γ is adjoint

rigid with respect to f ∗L|Γ and the a-invariant is the same as for f(Γ), we deduce

that f(Γ) is also adjoint rigid with respect to L|f(Γ). Since f(Γ) ⊂ V , by Theorem

5.1.2 the images f(Γ) are parametrized by a family pi : Ui → Wi and the evaluation

map si for this family must be dominant. Note that dim f(Γ) = dim Γ = d(Y, f ∗L).

Therefore, we will obtain the desired contradiction if we prove that there is no

dominant family of subvarieties U ⊂ X such that (U,L|U) is adjoint rigid, a(U,L|U) =

a(X,L) and dim(U) > d(X,L).

By replacing X by a resolution we may assume that canonical model map for

KX + a(X,L)L is a morphism π : X → T . Thus there is an ample Q-divisor H on

T and an effective Q-divisor E such that KX + a(X,L)L is numerically equivalent to

π∗H + E. Suppose we have a diagram of smooth varieties

U X

W

g

p

94

such that g is generically finite and dominant and the fibers of p are smooth varieties

Uw satisfying a(Uw, g∗L|Uw) = a(X,L) and dim(Uw) > d(X,L). We can write

KU + a(X,L)g∗L = g∗(KX + a(X,L)L) +R = g∗π∗H + (g∗E +R)

for some effective divisor R. Now for a general fiber,

(KU + a(X,L)g∗L)|Uw = KUw + a(Uw, g∗L|Uw)g∗L|Uw .

If Uw is adjoint rigid with respect to g∗L then,

0 ≤ κ(Uw, g∗π∗H) ≤ κ(KUw + a(Uw, g

∗L|Uw)g∗L|Uw) = 0.

This would imply that κ(Uw, g∗π∗H) = 0 and Uw would be contracted by the

morphism π : X → T . But this is a contradiction since dim(Uw) > d(X,L). This

concludes the proof in Case 1.

Case 2: Suppose d(Y, f ∗L) ≤ d(X,L). Let y ∈ Y (F ). We may assume that

y 6∈ V . If a(Y, f ∗L) > a(X,L), then a(f(Y ), L|f(Y )) > a(X,L) and consequently

f(Y ) ⊂ V . Therefore we may assume that a(Y, f ∗L) = a(X,L).

Now we modify Y so that we can apply Lemma 5.2.2. We may replace Y by a

resolution and replace y by a preimage in the resolution to assume that we have a

morphism φ : Y → T to the canonical model for KY +a(Y, f ∗L)f ∗L. If the a-values of

the images of the fibers of φ were larger than a(X,L) then we would have f(Y ) ⊂ V ,

so by assumption we must have an equality instead. Similarly, since the fibers of the

canonical map for Y are adjoint rigid with respect to f ∗L, their images also must be

adjoint rigid with respect to L. Thus we have a rational map g : T 99K Wi for some i,

where pi : U〉 → Wi are the families in Theorem 5.1.2. Passing to a birational model

95

and replacing y by pre-image, we may assume that we have a morphism T → Wi.

Then f : Y → X rationally factors through f ′ : Y 99K T ×WiUi → Ui. After again

replacing Y by a birational model and replacing y by any preimage, we may suppose

that f ′ is a morphism.

Y T ×WiUi Ui

T Wi

φ

By Lemma 5.1.6 and Remark 5.1.19, we may assume that we have a morphism

of good families of adjoint rigid varieties over an open subset of T . We may assume

that Ui is geometrically irreducible and si : Ui → X is birational as otherwise, by

Construcion 5.2.3, we have y ∈ Z.

By Lemma 5.2.2, there exists a twist σ such that

f(y) ∈ sσi,j(Yσi,j(F )),

and f : Y → X factors through sσi,j : Yσi,j → X after an Iitaka base change. Therefore

by Construction 5.2.3, it is enough to show that

(a(X,L), b(X,L)) < (a(Yσi,j, (sσi,j)∗L), b(F,Yσi,j, (sσi,j)∗L))

We may assume that a(X,L) = a(Yσi,j, (sσi,j)∗L), otherwise we would have sσi,j(Yσi,j) ⊂

V and consequently f(y) ∈ Z. Therefore, to show the above inequality, it is enough

to show that b(F, Y, f ∗L) ≤ b(F,Yσi,j, (sσi,j)∗L)). Let s ∈ T be a general closed point.

By Lemma 2.4.15 (2) and the birational invariance of the b-constant (Lemma 2.4.12),

we have

b(F, Y, f ∗L) ≤ b(F, Ys, f∗L).

96

Since by assumption f(y) 6∈ V and Ys is general, Ys will map to a fiber of Yσi,j lying

over T σ◦i,j , Where T σ◦i,j is as in property (4) of Construction 5.2.1. Let t ∈ T σ◦i,j denote

the image of s and let Yσi,j,t denote the corresponding fiber of qi,j. By construction

every geometric component of Ys is birational to a geometric component of Yσi,j,t.

Hence, by Lemma 2.4.12, we have

b(F, Ys, f∗L) = b(F,Yσi,j,t, sσ∗i,jL|Yσi,j,t)

Finally, by property (4) of Construction 5.2.1 we have

b(F,Yσi,j, sσ∗i,jL) = b(F,Yσi,j,t, sσ∗i,jL|Yσi,j,t).

Therefore we conclude that

b(F,X,L) < b(F, Y, f ∗L) ≤ b(F,Yσi,j, (sσi,j)∗L))

and the thin map sσi,j : Yσi,j → X is a breaking thin map. Hence by Construction 5.2.3,

we have f(y) ∈ sσi,j(Yσi,j(F )) ⊂ Z. This concludes the proof of Theorem 1.3.1.

97

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