Toric Projective Bundles

TORIC PROJECTIVE BUNDLES

by Jos´eL. Gonz´alez

A dissertation submitted in partial fulﬁllment of the requirements for the degree of Doctor of Philosophy (Mathematics) in The University of Michigan 2011

Doctoral Committee: Professor Mircea I. Mustat¸˘a,Chair Professor William E. Fulton Professor Karen E. Smith Professor Robert K. Lazarsfeld Associate Professor James P. Tappenden

ii ACKNOWLEDGEMENTS

I would like to thank the University of Michigan and its Department of Mathe- matics for providing me with a fantastic environment for my graduate studies.

I would also like to thank those who helped me and inspired me during my time at the University of Michigan.

I express my deepest gratitude to Mircea Mustat¸˘afor being a truly wonderful advisor. His great insight, his career enthusiasm and his constant guidance make me feel honored to have been his student.

I would like to thank William Fulton for his mentorship and his academic generos- ity. It has been a privilege to have attended his lectures and to have shared many inspiring conversations.

I am indebted to Karen Smith, Robert Lazarsfeld, Igor Dolgachev and Melvin

Hochster for their instruction, their generous advice, and for conveying with their example the love for mathematical research.

I would like to thank Tobias Staﬀord, Igor Kriz, Sergey Fomin and Alexander

Barvinok for their enjoyable classes. I would also like to thank James Tappenden for being a member of my dissertation committee.

I wish thank my collaborators Milena Hering, Sam Payne and Hendrik S¨ußfor a very enjoyable and productive joint research experience, and for allowing me to include our joint results as a conclusion to this dissertation.

I would like to thank my dear friends and student colleagues in the algebraic

iii geometry group of the University of Michigan, especially Dave Anderson, Eugene

Eisenstein, Mihai Fulger, Daniel Hern´andez,Kyle Hofmann, Shin-Yao Jow, Brian

Jurgelewicz, Kyungyong Lee, Victor Lozovanu, Yogesh More, Alan Stapledon, Kevin

Tucker, Giancarlo Urz´uaand Zhixian Zhu, as for me it was always a pleasure to participate in our active seminars and fruitful conversations.

To JoséManuel Gómez,Daniel Hernández,Gerardo Hernández,Fidel Jiménez,

Hyosang Kang, Marc Krawitz, Luis Núñez,Tomoki Ohsawa, Felipe Pérez,Felipe

Ram´ırez,Luis Serrano, Richard Vasques, Liz Vivas, Chelsea Walton and Emily Witt,

I extend my gratitude for being really good friends during all the years we shared as graduate students.

I am also grateful to Juan Diego Vélez,Federico Ardila and Jaime Muñozwho have been inspiring instructors at different stages of my mathematical career.

A few words cannot adequately express the thanks that I owe to my wife, Erin

Emerson, for her love and constant support.

Lastly, I want to express thanks to my parents Antonio Gonz´alezand Mar´ıa Nury

Zapata for their unconditional love and encouragement; to them I dedicate this work.

iv TABLE OF CONTENTS

DEDICATION ...... ii

ACKNOWLEDGEMENTS ...... iii

LIST OF FIGURES ...... vii

CHAPTER

I. Introduction ...... 1

II. Preliminaries ...... 8

2.1 Okounkov bodies ...... 9 2.2 Toric vector bundles and Klyachko filtrations ...... 11 2.3 The Klyachko filtrations for tensor products and Schur functors ...... 15 2.4 Cox rings ...... 19 2.4.1 An invitation to Mori dream spaces ...... 19 2.4.2 Definitions and basic results ...... 21

III. Okounkov bodies of projectivized rank two toric vector bundles ...... 29

3.1 Vanishing orders on P( )...... 29 3.1.1 A ﬂag of invariantE subvarieties in a projectivized rank two toric vector bundle ...... 30 3.1.2 Computing vanishing orders ...... 31

3.1.3 The image of νY• ...... 35 3.2 The global Okounkov body of P( )...... 39 3.2.1 Supporting hyperplanesE of the global Okounkov body of P( )... 40 3.2.2 The global Okounkov body of P( )...... E 42 3.3 Examples ...... E ...... 47

IV. The Cox ring of a projectivized rank two toric vector bundle ...... 52

4.1 Introduction ...... 52 4.2 The finite generation of the Cox ring of P( ) using its Okounkov body . . . 54 4.3 A combinatorial approach to the finite generationE of the Cox ring of P( ). 56 4.3.1 Preliminary lemmas ...... E . . . 57 4.3.2 The finite generation of the Cox ring of P( )...... 61 E V. Cox rings and pseudoeffective cones in higher ranks ...... 64

5.1 Preliminary Lemmas ...... 65 5.2 Cox rings of projectivized toric vector bundles ...... 74

v 5.3 Pseudoeﬀective cones of projectivized toric vector bundles ...... 76

BIBLIOGRAPHY ...... 81

vi LIST OF FIGURES

Figure

1.1 The global Okounkov body ...... 4

3.1 The Okounkov body of P(T )(1) ...... 48 O P2

vii CHAPTER I

Introduction

Toric projective bundles arise as projectivizations of toric vector bundles. These varieties are not toric in general, however, they are endowed with a torus action and they have a well-understood combinatorial description. In addition, these rational varieties enjoy some of the ﬁniteness properties of Mori dream spaces, such as the

finite generation of their nef and Mori cones (see [HMP10, Remark 2.5]). Toric vector bundles were classified by A. Klyachko in [Kly90], in terms of certain filtrations of a suitable vector space (see Theorem II.4), and they have been the focus of some recent activity, e.g. [Gon09], [Gon10], [GHPS10], [HMP10], [Pay09], [Pay08]. In [HMP10],

M. Hering, M. Mustat¸˘aand S. Payne raised one of the main questions regarding the geometric structure of their projectivizations. Namely, whether their Cox rings are indeed finitely generated. In this dissertation we study some invariants of these projectivized toric vector bundles such as their global Okounkov bodies, their Cox rings and their cones of pseudoeffective divisors. In the first of our main results we associate to each rank two toric vector bundle a flag of torus invariant subvarieties on its projectivization, and we describe the corresponding global Okounkov body in terms of the combinatorial data of the toric variety on the base and the data in the

Klyachko ﬁltrations of the toric vector bundle. Later on, we present two proofs of

1 2

the ﬁnite generation of the Cox rings of projectivizations of rank two toric vector

bundles, which use diﬀerent techniques from those used by J. Hausen and H. S¨ußin

[HS10] in their solutions to this ﬁnite generation question in the rank two case. In

one case our approach is direct and combinatorial; in the other, the argument relies

on our description of the global Okounkov bodies of these varieties. We conclude by

presenting my joint work with M. Hering, S. Payne and H. S¨uß,in which we give

negative answers to the ﬁnite generation of the Cox rings and pseudoeﬀective cones

of projectivizations of higher rank toric vector bundles. Our counterexamples have

two ﬂavors, some are very general in their moduli spaces, and some are determined

by the combinatorial data of the toric varieties such as their cotangent bundles.

Mori dream spaces were introduced by Hu and Keel in [HK00] as a class of varieties

with interesting features from the point of view of Mori theory. For instance, their

pseudoeﬀective and nef cones are both polyhedral, and the Mori program can be

carried out for any pseudoeﬀective divisor on these varieties. Additionally, their

pseudoeﬀective cones can be decomposed into ﬁnitely many closed convex chambers

that are in correspondence with the birational contractions of X having Q-factorial

image (see Proposition 1.11 [HK00]). A Mori dream space can be deﬁned as a normal

1 projective Q-factorial variety X, that satisfies Pic(X)Q = N (X)Q and has a finitely generated Cox ring (see Definition II.13). One basic example is that of toric varieties, in which the Cox ring is a polynomial ring in finitely many variables (see [Cox95]).

Thus, a projective simplicial toric variety is a Mori dream space. Another example is given by the (log) Fano varieties, which were recently proven to be Mori dream spaces

(see [BCHM08]). For one more example, if a vector bundle over a toric variety splits as a sum of line bundles, then its projectivization admits a toric variety structure

(see [Oda78, 7]), and therefore it is a Mori dream space. Some references for recent § 3

work on Cox rings of particular Mori dream spaces are [BP04], [Cas08], [Gon09],

[Gon10], [GHPS10], [HS10], [Ott10] and [STV07].

In his work on log-concavity of multiplicities, e.g. [Ok96], [Ok03], A. Okounkov

introduced a procedure to associate convex bodies to linear systems on projective

varieties. This construction was systematically studied by R. Lazarsfeld and M.

Mustat¸˘ain the case of big line bundles in [LM08]. The construction of these Ok-

ounkov bodies depends on a ﬁxed ﬂag of subvarieties and produces a convex compact

set for each Cartier divisor on a projective variety. The Okounkov body of a divisor

encodes asymptotic invariants of the divisor’s linear system, and it is determined

solely by the divisor’s numerical equivalence class. Moreover, these bodies vary as

ﬁbers of a linear map deﬁned on a closed convex cone as one moves in the space

of numerical equivalence classes of divisors on the variety. As a consequence, one

can expect to obtain results about line bundles by applying methods from convex

geometry to the study of these Okounkov bodies.

Let us consider an n-dimensional projective variety X over an algebraically closed

ﬁeld, endowed with a ﬂag X : X = Xn X0 = pt , where Xi is an i- • ⊇ · · · ⊇ { }

dimensional subvariety that is nonsingular at the point X0. In [LM08], Lazarsfeld

and Mustat¸˘aestablished the following:

(a) For each big rational numerical divisor class ξ on X, Okounkov’s construction

yields a convex compact set ∆(ξ) in Rn, now called the Okounkov body of ξ, whose

Euclidean volume satisﬁes

1 vol n ∆(ξ) = vol (ξ). R n! · X

The quantity volX (ξ) on the right is the volume of the rational class ξ, which is

deﬁned by extending the deﬁnition of the volume of an integral Cartier divisor D on 2 JOSELUISGONZ´ ALEZ´

Let us consider an n-dimensional projective variety X over an algebraically closed field, endowed with a flag X : X = Xn X0 = pt , where Xi is an i-dimensional subvariety • ⊇ ···⊇ { } that is nonsingular at the point X0. In [8], Lazarsfeld and Mustat¸˘aestablished the following: (a) For each big rational numerical divisor class ξ on X, Okounkov’s construction yields a convex compact set ∆(ξ)inRn, now called the Okounkov body of ξ, whose Euclidean volume satisfies 1 vol n ∆(ξ) = vol (ξ). R n! · X

The quantity volX (ξ)ontherightisthe! volume" of the rational class ξ, which is deﬁned by extending the deﬁnition of the volume of an integral Cartier divisor D on X, namely,

0 h (X, X (mD)) volX (D)=def lim O . m n →∞ m /n! We recall that the volume is an interesting invariant of big divisors which plays an important role in several recent developments in higher dimensional geometry. For basic properties of volumes we refer to [9]. n 1 (b) Moreover, there exists a closed convex cone ∆(X) R N (X)R characterized by the property that in the diagram ⊆ × ∆(X) / Rn N 1(X) × R

' t 1 N (X)R, the ﬁber ∆(X) Rn ξ = Rn of ∆(X) over any big class ξ N 1(X) is ∆(ξ). This is ξ 4 Q illustrated schematically⊆ × { in} Figure 1. ∆(X) is called the global Okounkov∈ body of X.

∆(ξ) ∆(X)

N 1(X) 0 ξ Figure 1. The global Okounkov body. Figure 1.1: The global Okounkov body.

Lazarsfeld and Mustat¸˘ahave used this theory to reprove and generalize results about volumesX, ofnamely, divisors, including Fujita’s Approximation Theorem. From (b), they can additionally give alternative proofs of properties ofh the0(X, volum(mDe function)) volX : Big(X) R, X → 1 deﬁned in the set of big classesvolX of(DR)-divisors. =def lim For example,O it follows. that vol is of class m mn/n! X and satisﬁes the log-concavity relation →∞ C

We recall that the volume is an1/n interesting invariant1/n of big1/n divisors which plays an vol (ξ + ξ#) vol (ξ) +vol (ξ#) , X ≥ X X important role in several recent developments in higher dimensional geometry. For

basic properties of volumes we refer to [Laz04].

(b) There exists a closed convex cone ∆(X) Rn N 1(X) characterized by the ⊆ × R property that in the diagram

∆(X) / Rn N 1(X) × R

( t 1 N (X)R,

the ﬁber ∆(X) Rn ξ = Rn of ∆(X) over any big class ξ N 1(X) is ∆(ξ). ξ ⊆ × { } ∈ Q This is illustrated schematically in Figure 1.1. ∆(X) is called the global Okounkov

body of X.

Lazarsfeld and Mustat¸˘ahave used this theory to reprove and generalize results

about volumes of divisors, including Fujita’s Approximation Theorem. Using (b),

they also give alternative proofs of properties of the volume function vol : Big(X) X →

R, deﬁned on the set of big classes of R-divisors. For example, it follows that volX 5

is of class 1 and satisﬁes the log-concavity relation C

1/n 1/n 1/n vol (ξ + ξ0) vol (ξ) + vol (ξ0) , X ≥ X X

1 for any two big classes ξ, ξ0 N (X) . It is worth mentioning that in [KK08], K. ∈ R Kaveh and A. Khovanskii use a similar procedure to associate convex bodies to ﬁnite

dimensional subspaces of the function ﬁeld K(X) of a variety X. In their work, they

use the bridge between convex and algebraic geometry provided by their construction

to obtain results in both areas.

The explicit description of Okounkov bodies in concrete examples can be rather

diﬃcult. One easy case is that of smooth projective toric varieties. If D is an

invariant divisor on such a variety, and if the ﬂag consists of invariant subvarieties,

then it is shown in [LM08] that the Okounkov body of D is the polytope PD that

one usually associates to D in toric geometry, up to a suitable translation.

As we mentioned, the goal of this dissertation is to study the Cox rings, the pseudoeﬀective cones and the global Okounkov bodies of projectivized toric vector bundles. We now summarize our main results and their organization in this dissertation.

In Chapter III, we present a description of the Okounkov bodies of all divisors on the projectivization P( ) of a rank two toric vector bundle on a smooth projective E E toric variety X. For the reader’s convenience, Klyachko’s classiﬁcation of toric vector bundles is reviewed in Chapter II, where we also include a description of the ﬁltrations corresponding to tensor products and Schur functors applied to toric vector bundles.

As we will see, these ﬁltrations can be used to compute the sections of all line bundles on P( ). For our main result concerning Okounkov bodies (see Theorem III.10 and E Remark III.11), we restrict to the case of rank two toric vector bundles, where the

Klyachko ﬁltrations are considerably simpler. Using the data from the ﬁltrations, we 6

construct a flag of torus invariant subvarieties on P( ) and produce finitely many E linear inequalities defining the global Okounkov body of P( ) with respect to this E flag. In particular, we see that this is a rational polyhedral cone. This description

of ∆(P( )) will be used in Chapter IV to give a proof of the finite generation of the E Cox ring of P( ). E In Chapter IV, we present two distinct proofs of the finite generation of the Cox rings of projectivized rank two toric vector bundles. While preparing the published versions of these two proofs, Milena Hering kindly brought to my attention the article [HS10] of J. Hausen and H. Süß,where they prove this finite generation using a different approach to ours, and also point out that it is possible to give yet another argument based on the main result in the paper [Kn93] of F. Knop. Our

ﬁrst proof (see Theorem IV.2) arises as an application of our description of the global

Okounkov body ∆(P( )) of a projectivized rank two toric vector bundle obtained in E Theorem III.10. Here, we prove that an appropriate Veronese subalgebra of the Cox ring of P( ) is isomorphic to the semigroup algebra obtained from the semigroup E of lattice points with suﬃciently divisible coordinates in the global Okounkov body

∆(P( )), which is finitely generated using Theorem III.10. For our second proof (see E Theorem IV.6), we consider a finer grading on Cox(P( )) induced by the torus action. E Next, we describe the graded pieces and the multiplication map in terms of the data appearing in the Klyachko filtrations of , and we obtain the finite generation by E exhibiting a finite generator set of a Veronese subalgebra of Cox(P( )). E In the final chapter, I present the results of my joint work [GHPS10], studying the Cox rings and pseudoeffective cones of projectivizations of toric vector bundles of higher rank, with M. Hering, S. Payne and H. Süß.Here we consider the projectivizations of a special class of toric vector bundles that includes cotangent bundles, 7 up to a twist. The definition of these special toric vector bundles is meant to guar- antee the existence of a T -invariant open subset of their projectivizations having a geometric quotient by the torus action that is isomorphic to a blow up of projective space at finitely many points of our choice. In this case, results of Hausen and Süß allow us to present the Cox rings of the projectivizations of this class of toric vector bundles as polynomial rings over the Cox rings of those blow ups. Using results of

Mukai, A.-M. Castravet and J. Tevelev, and B. Totaro concerning blow ups of projective spaces at ﬁnite collections of points, we obtain many new examples of Mori dream spaces as well as many examples of projectivized toric vector bundles whose

Cox rings and pseudoeﬀective cones are not ﬁnitely generated. In particular, we will see that for each d 3 there exist d-dimensional smooth projective toric varieties ≥ whose projectivized cotangent bundles are not Mori dream spaces. CHAPTER II

Preliminaries

In this chapter, we review the construction of Okounkov bodies and some basic facts about Cox rings and Mori dream spaces. We also review Klyachko’s classifica- tion of toric vector bundles, and we describe the Klyachko filtrations associated to tensor products and Schur functors applied to toric vector bundles. Unless explicitly stated otherwise, the notation introduced here will be used throughout this dissertation. All our varieties are assumed to be defined over a fixed algebraically closed

ﬁeld k. By a divisor on a variety Z we always mean a Cartier divisor on Z. We

denote the group of numerical equivalence classes of divisors on Z by N 1(Z), and we

denote the spaces N 1(Z) Q and N 1(Z) R by N 1(Z) and N 1(Z) , respectively. ⊗ ⊗ Q R By a line bundle on a variety Z, we mean a locally free sheaf of rank one on Z.

We follow the convention that the geometric vector bundle associated to the locally

m free sheaf is the variety V( ) = Spec m 0 Sym ∨, whose sheaf of sections F F ≥ F is . Also, by the ﬁber of over a pointL z Z, we mean the ﬁber over z of F F ∈ the projection f : V( ) Z. Lastly, by the projectivization P( ) of , we mean F → F F m the projective bundle Proj m 0 Sym over Z. This bundle is endowed with a ≥ F

projection π : P( ) Z andL an invertible sheaf P( )(1) (see II.7 in [Har77]). F → O F

8 9

2.1 Okounkov bodies

Let us consider a normal l-dimensional variety Z with a ﬁxed ﬂag Y : Z = Yl • ⊇ Y , where each Y is an i-dimensional normal subvariety that is nonsingular at · · · ⊇ 0 i

the point Y0. Given a big divisor D on Z, we will describe a procedure to assign a

l compact convex set with nonempty interior ∆Y (D) in R to D. First, given any divi- • sor F on Z and a nonzero section s = s H0(Y , (F )), we can associate an l-tuple l ∈ l OYl of nonnegative integers νY ,F (s) = (ν1(s), . . . , νl(s)) to s as follows. By restricting • to a neighborhood of Y0 we can assume that each Yi is smooth. We deﬁne ν1(s)

to be the vanishing order ordYl 1 (s) of s along Yl 1. Then, s determines a section − − 0 sl H (Yl, Y (F ) Y ( ν1(s)Yl 1)) that does not vanish along Yl 1. By restrict- ∈ O l ⊗ O l − − − 0 ing, we get a nonzero section sl 1 H (Yl 1, Yl (F ) Yl 1 Yl ( ν1(s)Yl 1) Yl 1 ), e − ∈ − O | − ⊗ O − − | − and we iterate this procedure. More precisely, assume that we have deﬁned nonneg-

0 ative integers ν1(s), . . . , νh(s), and nonzero sections sl H (Yl, Y (F )), . . . , sl h ∈ O l − ∈ 0 h H (Yl h, Yl (F ) Yl h i=1 Yl i+1 ( νi(s)Yl i) Yl h ), for some nonnegative integer − O | − ⊗ O − − − | − N h < l. We deﬁne νh+1(s) as the vanishing order ordYl h 1 (sl h) of sl h along Yl h 1; − − − − − −

then, sl h determines a section − h+1 0 sl h H (Yl h, Yl (F ) Yl h Yl i+1 ( νi(s)Yl i) Yl h ) − ∈ − O | − ⊗ O − − − | − i=1 O g that does not vanish along Yl h 1; and ﬁnally, by restricting, we get a nonzero section − − h+1 0 sl h 1 H (Yl h 1, Yl (F ) Yl h 1 Yl i+1 ( νi(s)Yl i) Yl h 1 ). − − ∈ − − O | − − ⊗ O − − − | − − i=1 O We repeat this procedure until we obtain nonnegative integers ν1(s), . . . , νl(s). This

construction gives us a function

0 l νY = νY ,F : H (Z, Z (F )) r 0 Z • • O { } −→ s (ν (s), . . . , ν (s)). 7−→ 1 l 10

We denote the image of νY ,F by either ν(F ) or ν( Z (F )). The function νY satisﬁes • O • the following valuation-like properties:

0 For any nonzero sections s1, s2 H (Z, Z (F )), we have that νY ,F (s1 +s2) lex • ∈ O • ≥ l min lex νY ,F (s1), νY ,F (s2) , where lex denotes the lexicographic order in Z . ≥ { • • } ≥

For any divisors F and F in Z, and nonzero sections s H0(Z, (F )) and • 1 2 1 ∈ OZ 1 0 s2 H (Z, Z (F2)), we have νY ,F1+F2 (s1 s2) = νY ,F1 (s1) + νY ,F2 (s2). ∈ O • ⊗ • •

Remark II.1. If W is a ﬁnite dimensional subspace of H0(Z, (F )), then the num- OZ

ber of vectors arising as images under νY of nonzero sections in W is equal to the • dimension of W . For example, when Z is complete, ν(F ) is a ﬁnite set with cardi-

nality dim H0(Z, (F )). A more general statement is proven by Lazarsfeld and k OZ Mustat¸˘aas Lemma 1.3 in [LM08].

Finally, ∆Y (F ) = ∆Y ( Z (F )) is deﬁned to be the following closed convex hull • • O in Rl: 1 ∆Y (F ) = Conv ν(mF ) . • m m Z+ [∈ We will denote the set ∆Y (F ) simply by either ∆(F ) or ∆( Z (F )) whenever the • O corresponding ﬂag is understood. In [LM08], Lazarsfeld and Mustat¸˘aproved that

when Z is a projective variety and D is a big divisor, ∆(D) is a compact convex

subset of Rl with nonempty interior, i.e. a convex body. In this case, ∆(D) is called

the Okounkov body of D.

+ Since ∆Y (mF ) = m∆Y (F ) for any divisor F and any m Z , this deﬁnition • • ∈ extends in a natural way to associate an Okounkov body to any big divisor with

rational coeﬃcients. As it turns out, the outcome depends only on the numerical

equivalence class of the divisor. We refer to (a) and (b) in the Introduction for

some of the main properties of this construction, including the existence of the global 11

Okounkov body of a projective variety Z. This global Okounkov body is a closed convex cone ∆(Z) Rl N 1(Z) characterized by the property that the ﬁber of ⊆ × R the second projection over any big class D N 1(Z) is the Okounkov body ∆(D). ∈ Q For proofs of these assertions, as well as of (a) and (b), we refer to [LM08].

l Example II.2. Let Z = P with homogeneous coordinates z0, . . . , zl. Let Y be the •

ﬂag of linear subspaces deﬁned by Yi = z1 = = zl i = 0 for each i. If D is the { ··· − } | |

linear system of hypersurfaces of degree m, then νY ,D is the lexicographic valuation • determined on monomials of degree m by

α0 αl νY (z0 zl ) = (α1, . . . , αl), • ···

and the Okounkov body ∆(D) is the simplex

l ∆(D) = (λ , . . . , λ ) Rl λ 0, . . . , λ 0, λ m . { 1 l ∈ | 1 ≥ l ≥ i ≤ } i=1 X 2.2 Toric vector bundles and Klyachko ﬁltrations

Let X be an n-dimensional toric variety corresponding to a fan ∆ in the lattice N.

We denote the algebraic torus acting on X by T , and the character lattice Hom(N, Z)

of T by M. Thus, T = Spec k[M] = Spec k[ χu u M ] and X has an open covering | ∈

given by the aﬃne toric varieties U = Spec k[ σ∨ M ] corresponding to the cones σ ∩

σ ∆, where for each such cone σ∨. We denote the rays in ∆ by ρ , . . . , ρ . For each ∈ 1 d

ray ρj, we denote its primitive lattice generator by vj and its associated codimension

one torus invariant subvariety by Dj. Let t0 denote the unit element of the torus.

For a detailed treatment of toric varieties we refer to [Ful93].

If T acts on a vector space V in such a way that each element of V belongs to

a finite dimensional T -invariant subspace, we get a decomposition V = u M Vu, ∈ where V = v V tv = χu(t)v for each t T . The spaces V Land their u def { ∈ | ∈ } u 12 elements are called isotypical summands and isotypical elements, respectively. This motivates the use of the following terminology. If T acts on the space of sections of a vector bundle on some variety, we say that a section s is T -isotypical if there exists u M such that ts = χu(t)s for each t T . Likewise, if T acts on a ∈ ∈ variety, we say that a rational function f on the variety is T -isotypical if there exists u M such that tf = χu(t)f for each t T , i.e. the domain dom(f) of f is ∈ ∈ 1 u T -invariant and (tf)(z) = f(t− z) = χ (t)f(z) for each z dom(f) and each def ∈ t T . When T acts algebraically on an affine variety Z, the induced action of T ∈ on H0(Z, ) satisfies the above finiteness condition, and we get a decomposition OZ 0 0 H (Z, Z ) = u M H (Z, Z )u as before (see I.6.3 in [LP97]). O ∈ O A toric vectorL bundle on the toric variety X is a locally free sheaf together with E an action of the torus T on the variety V( ), such that the projection f : V( ) X E E → is equivariant and T acts linearly on the fibers of f. In general, if is a toric vector E bundle, V( ) and P( ) need not be toric varieties. Given any T -invariant open E E subset U of X, there is an induced action of T on H0(U, ), defined by the equation E

1 (t s)(x) = t(s(t− x)), · def

for any s H0(U, ), t T and x X. This action induces a direct sum decompo- ∈ E ∈ ∈ sition

H0(U, ) = H0(U, ) , E E u u M M∈ where H0(U, ) = s H0(U, ) t s = χu(t)s for each t T , as before. E u { ∈ E | · ∈ }

Example II.3. For each torus invariant Cartier divisor D on a toric variety X, the

line bundle (D) has a natural toric vector bundle structure. For each w M, the OX ∈ isotypical summands in the decomposition of H0(U , (div χw)) over U are given σ OX σ 13 by

u kχ− if w u σ∨ M, 0 w − ∈ ∩ H (Uσ, X (div χ ))u =  O  0 otherwise,  for each cone σ ∆ and each u M.  ∈ ∈ 

A toric vector bundle over an aﬃne toric variety is equivariantly isomorphic to a direct sum of toric line bundles (see Proposition 2.2 in [Pay08]). Every line bundle L on X admits a T-equivariant structure, and choosing one such structure is equivalent to choosing a torus invariant divisor D such that = (D). The classiﬁcation of L ∼ OX toric vector bundles of higher rank is considerably more complicated.

Let E be the ﬁber over t of the toric vector bundle on X, and for any T -invariant 0 E open subset U, let ev : H0(U, ) E be the evaluation map at t . For each ray t0 E → 0 ρ ∆ and each u M, the evaluation map ev gives an inclusion H0(U , ) , j ∈ ∈ t0 ρj E u −→

E. If u, u0 M satisfy u, v u0, v , then ∈ h ji ≥ h ji

Im(H0(U , ) , E) Im(H0(U , ) , E). ρj E u −→ ⊆ ρj E u0 −→

Therefore the images of these maps depend only on u, v , or equivalently, only h ji

on the class of u in M/ρ⊥ M = Z. Hence, we may denote the image of the map j ∩ ∼ H0(U , ) , E simply by ρj ( u, v ). Note that for each u M the image of the ρj E u −→ E h ji ∈ evaluation map H0(X, ) , E is equal to ρ1 ( u, v ) ρd ( u, v ) E. The E u −→ E h 1i ∩· · ·∩E h di ⊆ ordered collection of finite dimensional vector subspaces ρj = ρj (i) i Z E def {E | ∈ } gives a decreasing filtration of E. The filtrations ρj j = 1, . . . , d are called the {E | } Klyachko filtrations associated to . For each σ ∆, by equivariantly trivializing E ∈ E over the affine open subset Uσ of X, one can show that there exists a decomposition

E E ρ i E ρ σ = u¯ M/σ M u¯, such that ( ) = u,v¯ ρ i u¯, for each ray and each ∈ ⊥∩ E h i≥ ⊆ i ZL. Klyachko proved in [Kly90] thatP the vector space E together with these ∈ 14

ﬁltrations, satisfying the above compatibility condition, completely describe . More E precisely,

Theorem II.4 (Klyachko). The category of toric vector bundles on the toric variety

X is equivalent to the category of ﬁnite dimensional k-vector spaces E with collections

of decreasing ﬁltrations ρ(i) i Z , indexed by the rays of ∆, satisfying the {E | ∈ } following compatibility condition: For each cone σ ∆, there is a decomposition ∈

E = u¯ M/σ M Eu¯ such that ∈ ⊥∩ L ρ(i) = E , E u¯ u,v¯ ρ i h Xi≥ for every ray ρ σ and every i Z. ⊆ ∈

Example II.5. Let m D + + m D be a torus invariant Cartier divisor on X, 1 1 ··· d d for some m , . . . , m Z. Denote by the line bundle (m D + + m D ) on 1 d ∈ L OX 1 1 ··· d d X. The Klyachko ﬁltrations for are given by L

k if i mj, ρj (i) =  ≤ L   0 if i > mj,

 for each ray ρj ∆.  ∈

Example II.6. The projective plane P2 can be represented as the toric variety

associated to the fan in N R = R2 with maximal cones σ = v , v , σ = v , v ⊗ 1 h 2 3i 2 h 3 1i and σ = v , v , where v = (1, 0), v = (0, 1) and v = ( 1, 1). The tangent 3 h 1 2i 1 2 3 − − 2 2 bundle TP2 of P is naturally a toric vector bundle on P . This bundle can be

equivariantly trivialized as

TP2 U = P2 (D2) U P2 (D3) U ,TP2 U = P2 (D3) U P2 (D1) U , | σ1 O | σ1 ⊕ O | σ1 | σ2 O | σ2 ⊕ O | σ2

TP2 U = P2 (D1) U P2 (D2) U . | σ3 O | σ3 ⊕ O | σ3 15

It follows that the Klyachko ﬁltrations associated to TP2 are

E if i 0, E if i 0, E if i 0,  ≤  ≤  ≤ ρ  ρ  ρ  T 1 (i) =  T 2 (i) =  T 3 (i) =  P2 V1 if i = 1, P2 V2 if i = 1, P2 V3 if i = 1,       0 if i 2, 0 if i 2, 0 if i 2,  ≥  ≥  ≥       where V1, V2and V3 are distinct one-dimensional subspaces of the ﬁber E of TP2 over

t0.

For any toric vector bundle over X of rank at least two, we have an isomorphism E N 1(X) Z = Pic X Z = Pic P( ) = N 1(P( )), which is induced by ( , m) ⊕ ⊕ ∼ E E L 7→

P( )(m) π∗ , where π is the projection map π : P( ) X. O E ⊗ L E → 2.3 The Klyachko ﬁltrations for tensor products and Schur functors

As we reviewed in II.4, Klyachko proved that the category of toric vector bundles § on a toric variety X is equivalent to the category of ﬁnite dimensional vector spaces

endowed with a collection of ﬁltrations that satisfy a certain compatibility condition.

Klyachko’s result allows us to carry out some explicit computations in this category,

including the description of the space of sections of a toric vector bundle over any

invariant open subset of X. Throughout this section X denotes an arbitrary toric

variety.

Using the notation introduced in Sections 2.1 and 2.2, each line bundle on P( ) is E

isomorphic to a line bundle of the form P( )(m) π∗( X (D)) for some T -invariant O E ⊗ O Cartier divisor D on X. For such an isomorphic representative we have a toric vector

m bundle structure on π P( )(m) π∗ X (D) = (Sym ) X (D), and we have ∗ O E ⊗ O E ⊗ O 0 0 m H P( ), P( )(m) π∗( X (D)) = H (X, (Sym ) X (D)) . E O E ⊗ O E ⊗ O From this, we see that the Klyachko filtrations associated to tensor products and 16 symmetric powers of toric vector bundles can be used to describe the spaces of global sections of line bundles on projectivized toric vector bundles. The goal of this section is to provide appropriate descriptions of these filtrations for toric vector bundles of arbitrary rank. We present the filtrations for tensor products in Lemma II.7 and its Corollary II.8. The filtrations for symmetric powers are given in Corollary II.11 to Lemma II.10. More generally, in that lemma we describe the filtrations for any

Schur functor, e.g. symmetric and wedge products.

We start by presenting the ﬁltrations for tensor products.

Lemma II.7. Let and be toric vector bundles on the toric variety X. Then the E F Klyachko ﬁltrations for their tensor product are given by E ⊗ F

ρ (i) = ρ(i ) ρ(i ), E ⊗ F E 1 ⊗ F 2 i1+i2=i X for each ray ρ ∆ and each i Z. ∈ ∈

Proof. Since the ﬁltration corresponding to the ray ρ only depends on Uρ, it suﬃces

to consider the case when X = U for some ray ρ ∆. Hence we can assume that ρ ∈ E and equivariantly trivialize as F

= (d D ) (d D ) (d D ) E OX 1 ρ ⊕ OX 2 ρ ⊕ · · · ⊕ OX r ρ = (e D ) (e D ) (e D ) F OX 1 ρ ⊕ OX 2 ρ ⊕ · · · ⊕ OX s ρ for some d , . . . , d , e , . . . , e Z. Now we note that 1 r 1 s ∈

ρ (d D ) (e D ) (i) = (d D )ρ(i ) (e D )ρ(i ) OX j1 ρ ⊗ OX j2 ρ OX j1 ρ 1 ⊗ OX j2 ρ 2 i1+i2=i X for each i Z and each j 1, . . . , r and j 1, . . . , s . Since the Klyachko ∈ 1 ∈ { } 2 ∈ { } ﬁltrations for a direct sum are the direct sums of the ﬁltrations for the summands,

the result follows. 17

Corollary II.8. Let ,..., be toric vector bundles on the toric variety X. Then E1 Es the Klyachko ﬁltrations for their tensor product are given by E1 ⊗ · · · ⊗ Es

ρ (i) = ρ(i ) ρ(i ), E1 ⊗ · · · ⊗ Es E1 1 ⊗ · · · ⊗ Es s i1+ +is=i X··· for each i Z and each ray ρ ∆. ∈ ∈

Proof. The conclusion follows from the previous lemma by induction on s.

Example II.9. Let be a toric vector bundle on the toric variety X, and let D = E m D + +m D be a torus invariant Cartier divisor on X, for some m , . . . , m Z. 1 1 ··· d d 1 d ∈ Let us denote the ﬁber over t of the line bundle (D) by G. From the previous 0 OX lemma and Example II.5, it follows that the Klyachko ﬁltrations of (D) are E ⊗ OX given by

( (D))ρj (i) = ρj (i m ) G, E ⊗ OX E − j ⊗ for each i Z and each ray ρ ∆. ∈ j ∈

In the following lemma we describe the Klyachko ﬁltrations for the toric vector bundle obtained by applying a Schur functor to another toric vector bundle. As a corollary, we state the case of symmetric products, which will be used in 3.2. For § the deﬁnition and basic properties of Schur functors we refer to 6 in [FH91]. §

Lemma II.10. Let be a toric vector bundle on the toric variety X, and let S be E λ the Schur functor associated to a Young tableau λ with m entries. Then the Klyachko

ﬁltrations for S are given by λE

ρ S (i) = Im ρ(i ) ρ(i ) S (E) , λE E 1 ⊗ · · · ⊗ E m −→ λ i1+ +im=i ···X for each ray ρ ∆ and each i Z. ∈ ∈ 18

m ρ Proof. Since S is a quotient of ⊗ , it follows that (S ) (i) is the image of λE E λE m ρ m ( ⊗ ) (i) under the natural map E⊗ S (E), for each ray ρ and each i Z. E → λ ∈ Now, the result follows at once from Corollary II.8.

Corollary II.11. Let be a toric vector bundle on the toric variety X. Then for E each m Z+, the Klyachko ﬁltrations for Symm are given by ∈ E

ρ Symm (i) = Im ρ(i ) ρ(i ) SymmE , E E 1 ⊗ · · · ⊗ E m −→ i1+ +im=i ···X for each ray ρ ∆ and each i Z. ∈ ∈

Proof. This is a particular case of the previous lemma.

Example II.12. Let D = m D + + m D be a torus invariant Cartier divisor 1 1 ··· d d on the toric variety X, and let us denote the ﬁber over t of the line bundle (D) 0 OX by G. The Klyachko ﬁltration associated to a rank two toric vector bundle on X E corresponding to a ray ρj has one of the following two forms:

E if i aj,  ≤ E if i aj,  ρj (i) =  ≤ ρj (i) =  E  E V if aj < i bj,   ≤ 0 if i > aj,  0 if i > bj,      where V is some one-dimensional subspace of the ﬁber E of over t , and a and b E 0 j j are some integers. For each positive integer m the corresponding Klyachko ﬁltration associated to (Symm ) (D) has respectively one of the forms: E ⊗ OX

m (Sym E) G if i ajm + mj, ((Symm ) (D))ρj (i) =  ⊗ ≤ E ⊗ OX   0 if i > ajm + mj,

  19

m (Sym E) G if i ajm + mj,  ⊗ ≤ i aj m mj m ρj  − − ((Sym ) (D)) (i) = m bj aj X Sym V − G if ajm+mj bjm + mj,    m c c (m c) where denotes the ceiling function, and Sym (V ) = Im(V ⊗ E⊗ − d e E def ⊗ → SymmE) for each integer 0 c m. This convenient notation will be generalized ≤ ≤ in the next chapter (see Notation III.2).

2.4 Cox rings

2.4.1 An invitation to Mori dream spaces

Mori dream spaces were deﬁned by Hu and Keel in the paper [HK00]. Their

essential feature is the ﬁnite generation of their total coordinate ring or Cox ring

(see Deﬁnition II.13), and some of their most remarkable properties come from an

interpretation of their Mori theory in terms of variation of geometric invariant theory

(GIT). In a Mori dream space the cones of pseudoeﬀective, movable and nef divisors

are rational polyhedral. Moreover, the pseudoeﬀective cone can be decomposed

into ﬁnitely many rational polyhedral chambers called Mori chambers, one of them

being the nef cone, and such that the movable cone is a union of some of these

Mori chambers. The diﬀerent equivalent interpretations of these Mori chambers

constitutes a particularly beautiful picture in algebraic geometry.

Given a line bundle on a variety X, for each n Z there is a rational map L ∈ φ : X P(H0(X, n)) given by the linear system n . For sufficiently divisible n 99K L |L | n, these maps stabilize to a rational map φ called Iitaka fibration of X (see 2.1 in L § [Laz04]). Two line bundles and are defined to be Mori equivalent if they give L1 L2 rise to equivalent Iitaka fibrations φ 1 and φ 2 . The relation of Mori equivalence can L L

be extended in a natural way to Pic(X)Q. On a Mori dream space the Mori chambers 20 are precisely the closures of the open equivalence classes under this relation.

Mori chambers can also be interpreted in terms of variation of GIT. Let G be a reductive algebraic group acting on a normal variety X. Two ample G-linearized line bundles are deﬁned to be GIT-equivalent if they have the same set of semistable points on X. Recall that in GIT one associates to an ample G-linearized line bundle on X, a good categorical quotient of the action of G on the G-invariant open subset of -semistable points of X, and moreover one gets a geometric quotient over the L G-invariant open subset of -stable points of X. The relation of GIT-equivalence L is the same as the relation of producing the same GIT quotient. The set of ample

G-linearized line bundles on X forms a cone in a ﬁnite dimensional vector space. If

X is a Mori dream space, after choosing line bundles ,..., that form a basis L1 Lm of the Neron-Severi space N1(X), one gets an action of the torus T with character lattice M = Z Z N1(X) on the Cox ring Cox(X) of X and on the ·L1 ⊕ · · · ⊕ ·Lm ⊆ variety Spec(Cox(X)). In the case of Mori dream spaces the T -ample cone for this action and the space containing it, can be identiﬁed with the cone of pseudoeﬀective divisors and the space N1(X). The Mori chambers are precisely the closures of the open GIT-equivalence classes for this action.

The Mori chambers on a Mori dream space X are in correspondence with the birational contractions of X having Q-factorial image, and those inside the movable cone correspond to those contractions that are isomorphic to X in codimension 1, i.e. small modiﬁcations of X. If X1,...,Xl are these distinct small modiﬁcations of

X, their Neron-Severi spaces can be identiﬁed in a natural way, and their nef cones are exactly the Mori chambers contained in the movable cone of X. Additionally, with the notation of the previous paragraph, the GIT quotient of Spec Cox(X) by the action of T , given by the trivial line bundle on Spec(Cox(X)) linearized by a 21

character of T in the interior of the cone Nef(Xi) recovers the variety Xi for each

1 i l. ≤ ≤ This deep understanding of the birational geometry of X permits to run the

minimal model program for any movable divisor on X. This is, one can ﬁnd a

birational model of X where the divisor becomes nef, and moreover semiample. In

this setting the ﬂips and divisorial contractions required, correspond to moving from

a cone to an adjacent cone in the fan decomposition of the movable cone induced by

the Mori chambers. Similarly, there is a version of the minimal model program that

can be run for eﬀective divisors with an analogous interpretation in terms of convex

geometry.

2.4.2 Deﬁnitions and basic results

Let us state the deﬁnition of the algebras that we will call Cox rings of varieties

in this dissertation, and let us brieﬂy compare it with the deﬁnition introduced by

Hu and Keel in [HK00].

Deﬁnition II.13. Let X be a variety such that Pic(X)Q is ﬁnite dimensional. Any k-algebra of the form

0 m1 m2 ms Cox X, ( 1, 2,..., s) =def H (X, 1⊗ 2⊗ s⊗ ) L L L s L ⊗ L ⊗ · · · ⊗ L (m1,m2,...,ms) Z M ∈ where , ,..., are line bundles on X whose classes span Pic(X) , will be called L1 L2 Ls Q a Cox ring of X.

Let , ,..., be line bundles on X whose classes span Pic(X) , and let be L1 L2 Ls Q L a a line bundle on X. If we ﬁx an isomorphism between ⊗ and a line bundle of the L

a1 a2 as + form ⊗ ⊗ ⊗ , for some a Z and some a , . . . , a Z, then we get L1 ⊗L2 ⊗· · ·⊗Ls ∈ 1 s ∈ 0 (m1 a1m) (m2 a2m) (ms asm) induced isomorphisms between H (X, ⊗ − ⊗ − s⊗ − L1 ⊗ L2 ⊗ · · · ⊗ L ⊗ 22

am 0 m1 m2 ms ⊗ ) and H (X, ⊗ ⊗ ⊗ ) for each tuple (m , m , . . . , m , m) L L1 ⊗ L2 ⊗ · · · ⊗ Ls 1 2 s ∈ Zs+1. These identiﬁcations induce an isomorphism of Zs+1-graded algebras between

a Cox X, ( ,..., , ⊗ ) and the localization Cox X, ( ,..., ) [x] , where x is L1 Ls L L1 Ls x

0 m1 m2 ms m an indeterminate, and where H (X, ⊗ ⊗ ⊗ ) x is the homoge- L1 ⊗ L2 ⊗ · · · ⊗ Ls neous component of degree (m a m, m a m, . . . , m a m, m) Zs+1, for each 1 − 1 2 − 2 s − s ∈ (m , m , . . . , m , m) Zs+1. Then the ﬁnite generation of Cox X, ( ,..., ) is 1 2 s ∈ L1 Ls a equivalent to the ﬁnite generation of Cox X, ( ,..., , ⊗ ) . Now, a basic result L1 Ls L on Veronese subalgebras of graded algebras (see [SYJ10, Lemma 1.4]) gives us that

a the finite generation of Cox X, ( ,..., , ⊗ ) is equivalent to the finite genera- L1 Ls L tion of Cox X, ( ,..., , ) . From this it follows that in the setting of Definition L1 Ls L II.13 the finite generation of any Cox ring of X is equivalent to the finite generation of every Cox ring of X. In [HK00], Hu and Keel introduced the Cox ring of any

1 projective variety X that satisfies Pic(X)Q = N (X)Q. Their definition is similar to Definition II.13, but they require the line bundles , ,..., to satisfy some L1 L2 Ls extra conditions. Namely, the classes of , ,..., are required to form a basis L1 L2 Ls for Pic(X)Q and their affine hull must contain the pseudoeffective cone of X. For

1 projective varieties that satisfy Pic(X)Q = N (X)Q, our previous discussion implies that a particular (equivalently every) Cox ring of X in the sense of Hu and Keel is finitely generated if and only if a particular (equivalently every) Cox ring of X in the sense of Definition II.13 is finitely generated. Since our interest lies in the

finite generation of these k-algebras, and this is independent of the one we choose, we follow the convention of calling any Cox ring of a given variety the Cox ring of the variety. And since this finite generation does not depend on which of these two definitions of Cox ring we use, we will use Definition II.13 since it applies in the case of the projectivization of a toric vector bundle over an arbitrary toric variety. 23

As we mentioned in the Introduction, toric varieties and log Fano varieties are examples of varieties with ﬁnitely generated Cox rings. In particular, for n 6, ≤

the moduli space M0,n of stable, rational curves with n marked points has a ﬁnitely

generated Cox ring, since in those cases it is log Fano (see [KM96]). In the case

of M0,6, a proof of the ﬁnite generation of the associated Cox ring by exhibiting

generators can be found in [Cas08]. Additionally, we remark that whether the Cox

ring of M is ﬁnitely generated for n 7 is an important open problem. On 0,n ≥ the other hand, we can get some examples of varieties with nonﬁnitely generated

Cox rings by using the results of Mukai, Castravet and Tevelev and Totaro that we describe in what follows. These results will be fundamental for our work in Chapter

r 1 A collection of d points q1, . . . , qd in the projective space P − is in Cremona

general position if they are in linear general position (i.e. for each i r, there are no ≤ i of these points contained in a linear subspace of dimension i 2), and in addition − this condition remains true for the images of these points when we apply a ﬁnite

sequence of standard Cremona transformations each one centered at a collection of r

of the partial images of the collection of d points (see 2 in [Tot08]). We recall that § r 1 the standard Cremona transformation of P − centered at an ordered collection of r

points in linear general position is the rational map given in the coordinates induced

1 1 by the r points by [x1 : : xr] [ : : ], where this expression is well deﬁned. ··· 7→ x1 ··· xr We also recall that this rational map can be extended to an automorphism of the

r 1 blow up of P − at the corresponding coordinate points. One can easily check that for collections of points in projective space, very general position implies Cremona general position.

One of the main results in the paper [Muk04] of Mukai can be rephrased by saying 24

r 1 that the Cox ring of the blow up of P − at d points in Cremona general position is

4 not ﬁnitely generated if d r + 2 + r 2 . Castravet and Tevelev prove a more general ≥ − version of this result and its converse in [CT06]. We now state the particular case of their result that we will use in Chapter V. Intuitively, it will say that we can think

4 r 1 of the quantity r + 2 + r 2 as a threshold for the behavior of the blow up of P − at − d points in very general position regarding the ﬁnite generation of its Cox ring and

semigroup of eﬀective divisors.

r 1 Theorem II.14 (Castravet-Tevelev). For d > r 2, let X be the blow up of P − ≥ at d points in very general position. The following statements are equivalent:

(a) The Cox ring of X is a ﬁnitely generated algebra.

(b) The semigroup of eﬀective divisors of X is ﬁnitely generated.

1 1 1 (c) r + d r > 2 . − Proof. This is a particular case of Theorem 1.3 in [CT06].

While it is a general fact that (a) implies (b), and the proof that (c) implies (a)

only uses that the given points are in linear general position, the proof that (b)

implies (c) in [CT06] uses that the points are in very general position. Indeed, this

is used to deduce that they are in Cremona general position and then proceed to

apply the results of Mukai in [Muk04]. In Chapter V we will construct examples

over more general ﬁelds, so we want to avoid restrictions on the cardinality and the

characteristic of the ground ﬁeld. In his paper on Hilbert’s 14th problem over ﬁnite

ﬁelds [Tot08], Totaro provided conﬁgurations of points in various projective spaces

that are in Cremona general position. In addition, he constructs conﬁgurations of

points that in spite of not satisfying that condition, still give rise to a variety with a

nonﬁnitely generated Cox ring when they are blown up. We refer to Totaro’s paper 25 for this interesting collection of examples, but we state here a particular one that we will use in Chapter V when constructing examples of smooth projective toric varieties whose projectivized cotangent bundles are not Mori dream spaces.

Theorem II.15. Let k be a field of characteristic different from 2 and 3. Let q1 . . . , q9 be the points in P2 corresponding to the normal hyperplanes to the columns of the following matrix 1 1 1 0 0 0 1 1 1  − − −  1 0 1 1 0 1 1 0 1 .  − − −       1 1 1 1 1 1 1 1 1     2  Then the Cox ring of the blow up of P along the points q1, . . . , q9 is not finitely generated.

Proof. This is part of the conclusion of Corollary 5.1 to Theorem 5.2 in [Tot08].

On the other hand, when the points q1, . . . , qd are not in general position, the

r 1 1 1 1 variety Bl q1,...,qd P − can be a Mori dream space even if r + d r 2 . For instance { } − ≤ r 1 if q1, . . . , qd lie in a rational normal curve, then Bl q1,...,q P − is a Mori dream space { d} r 1 [CT06, Theorem 1.2]. Also, if q1, . . . , qd are collinear then Bl q1,...,q P − is a rational { d} r 1 variety with a torus action with orbits of codimension one , and hence Bl q1,...,q P − { d} is a Mori dream space [EKW04], [HS10], [Ott10] (c.f. Lemma II.17).

Our main tool to study the Cox rings of projectivized toric vector bundles of higher ranks will be the following result of Hausen and S¨ußthat gives presentations of the

Cox rings for some varieties with torus actions over the Cox rings of appropriate quotients.

Proposition II.16. Let X be a smooth variety with an action of a torus T . Let us

assume that D1,...,Dh are the prime divisors of X that have positive dimensional 26

generic stabilizers. Also, suppose that the action of T on X r (D1 Dh) is ∪ · · · ∪ free with geometric quotient equal to a smooth variety Y , and that the class group of

Y is torsion free. Then the Cox ring of X is isomorphic to a polynomial ring in h

variables over the Cox ring of Y .

Proof. This is the special case of [HS10, Theorem 1.1] where X is smooth, the T -

action on the complement of D D is free, and the geometric quotient Y is 1 ∪ · · · ∪ h separated, with torsion free class group.

The following lemma relates the Cox ring of the blow up of projective space

at ﬁnitely many points lying on a hyperplane and the Cox ring of the blow up of

the hyperplane at these points. By applying the lemma repeatedly, we also get an

analogous result for blow ups at points lying on any linear subvariety of projective

space. This result appears as Lemma 3.5 in [GHPS10], although instances of it

can be found in [HT04, Example 1.8]. This lemma will be used in our proof of

Theorem V.16.

Lemma II.17. Let S be a ﬁnite set of points contained in a hyperplane H in Pn,

n and assume n > 2. Then the Cox ring of BlS P is isomorphic to a polynomial ring

in one variable over the Cox ring of BlS H.

Proof. Choose coordinates on Pn so that H is a coordinate hyperplane, and let

Gm = k∗ act by scaling on the coordinate that cuts out H. The action of Gm lifts to

n an action on BlS P , and we let Y be the locus of ﬁxed points of this action. Then

d Gm acts freely on BlS P r Y , with quotient BlS H. The strict transform of H is the

only divisor contained in Y , so the lemma follows by applying Proposition II.16.

We also mention the useful fact that if the blow up X0 of a variety X at a smooth

point x has a ﬁnitely generated Cox ring, then X itself has a ﬁnitely generated 27

Cox ring. This follows for instance from [CT06, Proposition 3.1], where the authors prove the following more general fact: If F denotes the exceptional divisor and

x H0(X,F ) Cox(X) denotes the corresponding section then Cox(X) is F ∈ ⊆ xF

isomorphic to Cox(X0)[x]x, where x is an indeterminate.

We conclude this chapter with the following lemma describing the cone of eﬀective

divisors of projectivized toric vector bundles which we will use in Chapter V when

constructing examples of such varieties with a nonﬁnitely generated pseudoeﬀective

cone. This result is a version in our setting of the well known fact that an eﬀective

divisor on a normal variety with a torus action is linearly equivalent to an eﬀective

torus invariant divisor.

Lemma II.18. Let be a toric vector bundle. The eﬀective cone Eﬀ(P( )) of its E E projectivization P( ) is given by E

1 Eﬀ(P( )) = Z 0 π− (Di) + Z 0 T C E ≥ · ≥ · · 1 i d C P(E) P( ) X≤ ≤ C⊆primeX⊆ divisorE Proof. Given a divisor D on P( ) such that H0(P( ), (D)) = 0, choose any hy- E E O 6 perplane H P( ). Clearly, ⊆ E|t0

1 1 P( )( T H ) = P( )(1) X (a1π− (D1) + + adπ− (Dd)), O E · ∼ O E ⊗ O ···

for some a , . . . , a Z. Then, 1 d ∈

1 1 P( )(D) = P( )(b) P( )(b1π− (D1) + + bdπ− (Dd)) O E ∼ O E ⊗ O E ··· 1 1 = P( )(b T H + c1π− (D1) + + cdπ− (Dd)) ∼ O E · ···

for some b Z 0 and some b1, . . . , bd, c1, . . . , cd Z. The space of global sections V ∈ ≥ ∈ 1 1 of the T -invariant divisor T H +c π− (D )+ +c π− (D ) decomposes as a direct · 1 1 ··· d d sum of T -eigenspaces. Let φ V K(P( )) be a nonzero section in one of those ∈ ⊆ E T -eigenspaces. Hence, D is linearly equivalent to the T -invariant eﬀective divisor 28

1 1 b T H + c1π− (D1) + + cdπ− (Dd) + div φ. Since P( ) r P( T ) = 1 i d Di, · ··· E E| ≤ ≤ the conclusion follows by noticing that any T -invariant prime divisor on PS( ) has E|T the form T C for some prime divisor C P(E) = P( ) P( ). · ⊆ E|t0 ⊆ E|T CHAPTER III

Okounkov bodies of projectivized rank two toric vector bundles

In this chapter we give an explicit description of the global Okounkov body of a projectivized rank two toric vector bundle over a smooth projective toric variety, with respect to a T -invariant ﬂag obtained by completing to a full ﬂag the pull back of a

T -invariant flag on the base. In particular we prove that this global Okounkov body is a rational polyhedral cone, and we provide inequalities that define it in terms of the data in the Klyachko filtrations of the toric vector bundle and the combinatorial data of the toric variety on the base (see Theorem III.10 and Remark III.11). We conclude the chapter with some examples that illustrate our main result. In Chapter

IV, this description of the global Okounkov body will be used to obtain a proof of the ﬁnite generation of the Cox rings of projectivized rank two toric vector bundles.

3.1 Vanishing orders on P( ) E

The description of the global Okounkov body of a projective variety Z, with

0 respect to a ﬂag Y , involves identifying the image of the map νY : H (Z, )r 0 • • L { } −→ Zdim Z for each line bundle on Z. In this section we study these images for a L suitable ﬂag Y , when Z is the projectivization P( ) of a rank two toric vector • E bundle on a smooth projective toric variety X. First, in 3.1.1 we introduce a E §

29 30

flag of torus invariant subvarieties Y in P( ), essentially by pulling back a flag of • E invariant subvarieties from X. Next, in Definition III.3 we present a collection of sections for each line bundle on P( ). We consider these sections since we WL L E can compute their images under νY using the formulas in Lemma III.4, and because • 0 they map onto νY (H (P( ), ) r 0 ), as we prove in Proposition III.8. In passing, • E L { } we prove that after choosing an isomorphic representative of so as to have an L induced torus action on H0(P( ), ), the isotypical sections with respect to this E L action also map onto the image of νY . Throughout we use the notation introduced • in Sections 2.1 and 2.2, and additionally assume that the toric variety X is smooth and projective.

3.1.1 A ﬂag of invariant subvarieties in a projectivized rank two toric vector bundle

Given a toric vector bundle of rank two, we construct a ﬂag of smooth T - E invariant subvarieties Y : P( ) = Yn+1 ... Y0 in P( ), as follows. Let X : X = • E ⊇ ⊇ E • X ... X be a ﬂag in X, where each X is an i-dimensional T -invariant n ⊇ ⊇ 0 i subvariety. By reordering the rays in ∆ if necessary, we can assume that Xn i = − i D for each i 1, . . . , n . Note that this implies that the rays ρ , . . . , ρ span j=1 j ∈ { } 1 n Ta maximal cone τ in ∆.

Let u and u in M be such that we can equivariantly trivialize over U as = 1 2 E τ E|Uτ ∼ (div χu1 ) (div χu2 ) . The lexicographic order in Zn induces an order OX |Uτ ⊕OX |Uτ ≥lex in M, via the isomorphism M = Zn induced by v , . . . , v . By reordering u ≥lex ∼ 1 n 1 and u if necessary, we can assume that u u . In other words, either u = u , 2 1 ≥lex 2 1 2 or the first nonzero number in the list u u , v ,..., u u , v is positive. h 1 − 2 1i h 1 − 2 ni 1 For each i 1, . . . , n + 1 , we define Yi = P( Xi 1 ) = π− (Xi 1) P( ). To ∈ { } E| − − ⊆ E define Y , note that the isomorphism = (div χu1 ) (div χu2 ) induces 0 E|Uτ ∼ OX |Uτ ⊕OX |Uτ an isomorphism Y = P( ). Hence, we get an isomorphism µ: Y 1 ∼ OX0 ⊕ OX0 1 → 31

P1 between Y and the projective space P1 = P( ) with homogeneous 1 ∼ OX0 ⊕ OX0 coordinates x, y. We take Y0 to be the point in Y1 corresponding under µ to the point

1 (0 : 1), deﬁned by the ideal (x) in P . Note that the ﬂag Y : P( ) = Yn+1 ... Y0 • E ⊇ ⊇ in P( ) consists of smooth T -invariant subvarieties. E 3.1.2 Computing vanishing orders

In this subsection we introduce the collection of T -isotypical sections and WL compute their vanishing vectors. We continue working in the setting of 3.1.1. §

Notation III.1. We denote by E and E the ﬁbers over t of (div χu1 ) and 1 2 0 OX (div χu2 ). We identify E and E with subspaces of the ﬁber E of over t in OX 1 2 E 0 the natural way. We denote by L1,...,Lp the distinct one-dimensional subspaces of

E that are different from E , but are equal to ρ(i) for some ray ρ ∆ and some 1 E ∈ i Z. We fix once and for all a one-dimensional subspace L of E, different from ∈ each of the subspaces E1,L1,...,Lp of E. This is done just as an alternative to ad hoc choices at different points in our discussion.

Notation III.2. Let V1,...,Vl be subspaces of a vector space V. For any nonnegative

m α1 α2 αl integers m, α1, . . . , αl, we deﬁne the notation SymV (V1 ,V2 ,...,Vl ) to represent either the subspace of SymmV equal to the image of the composition of the natural maps

Pl α1 α2 αl (m α ) m m V ⊗ V ⊗ V ⊗ V ⊗ − i=1 i V ⊗ Sym V, 1 ⊗ 2 ⊗ · · · ⊗ l ⊗ −→ −→ if m l α , or the subspace 0 of SymmV , otherwise. ≥ i=1 i P On the toric variety X, the map deﬁned by (m , . . . , m ) d m D n+1 d 7→ i=n+1 i i d n 1 induces an isomorphism between Z − and Pic X = N (X). Hence,P each line bundle on P( ) is isomorphic to a unique line bundle of the form P( )(m) L E O E ⊗ d π∗ ( m D ). OX i=n+1 i i P 32

d Definition III.3. Let be the line bundle P( )(m) π∗ X ( i=n+1 miDi) on P( ), L O E ⊗ O E where m, m , . . . , m Z. Let us consider the torus actionP on H0 P( ), = n+1 d ∈ E L H0 X, (Symm ) ( d m D ) , induced by the T -equivariant structure on E ⊗ OX i=n+1 i i (Sym m ) ( d Pm D ). Let Gbe the fiber over t of ( d m D ). We E ⊗ OX i=n+1 i i 0 OX i=n+1 i i define the followingP subsets of H0 P( ), : P E L = s H0 P( ), s H0 P( ), r 0 , for some u M VL ∈ E L | ∈ E L u { } ∈

m α0 α1 αp α = s s(t0) lies in the subspace SymE (E1 ,L1 ,...,Lp ,L ) G WL ∈ VL | ⊗ p m of Sym E, for some α0, . . . , αp, α Z 0, satisfying αj + α = m . ∈ ≥ j=0 X In the following lemma we give some formulas for the vanishing vector νY (s) of • a section s . ∈ WL

d Lemma III.4. Let be the line bundle P( )(m) π∗ X ( i=n+1 miDi) on P( ), L O E ⊗ O E for some m, m , . . . , m Z. Let G be the ﬁber over t ofP ( d m D ). Let n+1 d ∈ 0 OX i=n+1 i i s be a nonzero section in H0 P( ), = H0 X, (Symm ) P( d m D ) , E L u E ⊗ OX i=n+1 i i u n+1 for some u M, and let νY (s) = (ν1, . . . , νn+1) Z . Then: P ∈ • ∈ (a) For each j 1, . . . , n we have ν = ν u + (m ν )u u, v . ∈ { } j h n+1 1 − n+1 2 − ji

m α0 α1 αp α m (b) If s(t ) lies in the subspace Sym (E ,L ,...,Lp ,L ) G of (Sym E) G, 0 E 1 1 ⊗ ⊗ p for some α0, . . . , αp, α Z 0 such that αi + α = m, then νn+1 = α0. ∈ ≥ i=0 P Proof.

(a) The vector νY (s) can be computed in any neighborhood of Y0 in P( ). Hence • E we can assume that X = U , that = (div χu1 ) (div χu2 ), and that s is a τ E OX ⊕ OX

0 m m m (m i)u1+iu2 section in H X, Sym . Note that Sym = (div χ − ), and so E u E i=0 OX s corresponds to the section L

m u u u 0 (m i)u1+iu2 0 m c χ− , . . . , c χ− , . . . , c χ− H X, (div χ − ) = H X, Sym , 0 i m ∈ OX u E u i=0 M 33

for some c , . . . , c k. Let us denote by 0. By combining the nat- 0 m ∈ OX ⊕ OX E ural isomorphisms in each component, we get an isomorphism = (div χu1 ) E OX ⊕

u2 (div χ ) = = 0. This isomorphism induces a commutative diagram, OX ∼ OX ⊕ OX E

ϕ u1 u2 P( ) = P (div χ ) (div χ ) / P = P( 0) E OX ⊕ OX OX ⊕ OX E - X s

where the map ϕ is an isomorphism. Let Y 0 : Yn0+1 Yn0 Y00 be the T - • ⊇ ⊇ · · · ⊇ invariant flag in P 0 , as defined in 3.1.1. Note that the T -invariant flags in P( ) E § E and P( 0) correspond to each other under ϕ. On Yi0 = P( 0 Xi 1 ), let us denote E E | − (m) by (m) for each i 1, . . . , n + 1 . Under the isomorphism ϕ, s P( 0 Xi 1 ) Yi0 O E | − O ∈ { } corresponds to the section

m mu1 u (m i)u1+iu2 u mu2 u 0 s0 = s0 = c χ − , . . . , c χ − − , . . . , c χ − H X, n+1 0 i m ∈ OX i=0 M 0 m 0 0 = H X, Sym ( 0) = H P( 0), P( )(m) = H Yn0+1, Y 0 (m) . E E O E0 O n+1 Note that (ν1, . . . , νn+1) = νY (s) = νY 0 (s0). Let us set h = max i 0 i • • { | ≤ ≤

m and c = 0 , and let v∗, . . . , v∗ be the basis of M dual to the basis v , . . . , v of N. i 6 } 1 n 1 n

It is straightforward to see that when we follow the procedure to compute νY 0 (s0) • outlined in 2.1, for each 0 l n, the section obtained in the step when we restrict § ≤ ≤

to Yn0+1 l corresponds to the section −

Pl Pl mu1 u j=1 νj vj∗ (m i)u1+iu2 u j=1 νj vj∗ sn0 +1 l = c0χ − − Xn l , . . . , ciχ − − − Xn l ,..., − | − | − m Pl mu2 u j=1 νj vj∗ 0 cmχ − − Xn l H Xn l, Xn l | − ∈ − O − (3.1) i=0 M 0 m 0 = H Xn l, Sym ( 0 X ) = H P( 0 X ), (m) n l n l P( 0 Xn l ) − E | − E | − O E | − 0 = H Y 0 , (m) , n+1 l Yn0+1 l − O − under the natural identiﬁcation. 34

Assume now that for some 0 l < n, we have proven that ν = (m h)u + ≤ j h − 1 hu2 u, vj , for each j 1, . . . , l . Note that for each a Z 0 we have the following − i ∈ { } ∈ ≥ commutative diagram

0 ϕa 0 H Y 0 , (m) ( aY 0 ) H Y , (m) n+1 l Yn0+1 l Yn0+1 l n l n+1 l Yn0+1 l − O − ⊗ O − − − −−−→ − O − 0 m 0 m H Xn l, Sym ( 0 Xn l ) Xn l ( aXn l 1) H Xn l, Sym ( 0 Xn l ) , − E | − ⊗ O − − − − −−−→ − E | − and denote the map in its top row by ϕa. Next, we note that

(3.2)

νl+1 = max a Z 0 sn0 +1 l Im(ϕa) { ∈ ≥ | − ∈ } Pl (m i)u1+iu2 u j=1 νj vj∗ = max a Z 0 For each i = 0, . . . , mciχ − − − Xn l { ∈ ≥ | | − ∈ 0 0 Im(H (Xn l, Xn l ( aXn l 1)) , H (Xn l, Xn l )) − O − − − − → − O − }

= max a Z 0 a (m i)u1 + iu2 u, vl+1 for each i such that { ∈ ≥ | ≤ h − − i Pl (m i)u1+iu2 u j=1 νj vj∗ ciχ − − − Xn l = 0 . | − 6 }

Pl (m h)u1+hu2 u j=1 νj vj∗ We also note that chχ − − − Xn l = 0. Now, if u1 u2, vl+1 < 0, | − 6 h − i then there exists q 1, . . . , l , such that u u , v > 0. In this case, for each ∈ { } h 1 − 2 qi Pl (m i)u1+iu2 u j=1 νj vj∗ i 0, . . . , h 1 , it follows that ciχ − − − Xn q = 0. Hence, either ∈ { − } | − Pl (m i)u1+iu2 u j=1 νj vj∗ u1 u2, vl+1 0, or ciχ − − − Xn l = 0 for each i 0, . . . , h 1 . h − i ≥ | − ∈ { − } In either case, it follows from (3.2) that ν = (m h)u + hu u, v . Therefore l+1 h − 1 2 − l+1i we can iterate this procedure, and in this way we obtain that for each j 1, . . . , n , ∈ { }

ν = (m h)u + hu u, v . j h − 1 2 − ji

0 Now, νn+1 is equal to the vanishing order along Y 0 of the section s0 H (Y 0, Y (m)) 0 1 ∈ 1 O 10 1 described in (3.1) for l = n. We have a natural isomorphism µ: Y 0 P between 1 → 1 Y 0 = P( ) and the projective space P = P( ) with homogeneous 1 OX0 ⊕OX0 ∼ OX0 ⊕OX0 coordinates x, y. Recall that under µ, Y00 corresponds to the point (0 : 1), deﬁned by 35

1 the ideal (x) in P . Under µ, Y (m) corresponds to 1 (m). Depending on whether O 10 OP 0 1 or not u = u , there are two possibilities for the section in H (P , 1 (m)) that 1 2 OP m h h corresponds to s0 . Namely, s0 corresponds to c x − y if u = u , and it corresponds 1 1 h 1 6 2 m m i i to c x − y if u = u . In either case, we obtain that ν = m h, and then i=0 i 1 2 n+1 − partP (a) is proven.

(b) As in part (a), we can reduce to the case when X = U , = (div χu1 ) τ E OX ⊕ (div χu2 ), s H0 X, Symm , and s(t ) lies in the subspace Symm(Eα0 ,Lα1 , OX ∈ E u 0 E 1 1 αp α m ...,Lp ,L ) of Sym E. Let x, y E be such that E = kx and E = ky. Then ∈ 1 2 m i i m x and y form a basis for E, and x − y for i = 0, . . . , m form a basis for Sym E.

Let β , . . . , β , β k be such that L = k(βx + y) and L = k(β x + y) for each 1 p ∈ i i i 1, . . . , p . For c , . . . , c k, deﬁned as in part (a), we proved that max i 0 ∈ { } 0 m ∈ { | ≤ i m and c = 0 = m ν . On the one hand, we see that the image of s at t is ≤ i 6 } − n+1 0 m m i i m s(t ) = c x − y Sym E. On the other hand, 0 i=0 i ∈ m α0 1 P − − m α0 α1 αp α m i i α0 m α0 s(t ) Sym (E ,L ,...,L ,L ) = k( β0x − y + x y − ), 0 ∈ E 1 1 p i i=0 X

for some β0 , . . . , βm0 α0 1 k. From this it follows that νn+1 = α0 as desired. − − ∈

3.1.3 The image of νY•

In this subsection we prove that νY maps the collection of T -isotypical sections • 0 onto νY (H (P( ), ) r 0 ). We continue working in the setting of 3.1.1-3.1.2. WL • E L { } § We start by proving that if a line bundle and a flag Y on an affine variety Z are L • suitably compatible with the action of a torus T on Z, then the nonzero T -isotypical sections of map onto the image of νY . L • Lemma III.5. Let Z be an affine variety with an algebraic action of a torus T , and

a ﬂag Y : Z = Yl Yl 1 ... Y0, where each Yi is a normal i-dimensional T - • ⊇ − ⊇ ⊇ invariant subvariety. Assume that for each i 1, . . . , l , there is a T -isotypical ra- ∈ { } 36

tional function hi on Yi, such that Yi 1 = div hi. Let g be a T -isotypical rational func- − tion on Z and let s , . . . , s H0(Z, (div g)) be nonzero T -isotypical sections cor- 1 q ∈ OZ

responding to distinct characters of T . Then νY (s1+ +sq) νY (s1), . . . , νY (sq) . • ··· ∈ { • • }

Proof. We proceed by induction on the dimension of Z. Let Z : Yl 1 = Zl 1 • − − ⊇

Zl 2 ... Z0 be the flag of normal T -invariant subvarieties in Yl 1 defined by − ⊇ ⊇ − Z = Y , for each i. Using the natural isomorphism H0(Z,O (div g)) = H0(Z, ), i i Z ∼ OZ we can reduce to the case when g = 1 and the sections are identified with regular

functions. Let s = s1 + + sq. For each a Z 0 the natural inclusion map ··· ∈ ≥ 0 0 ϕa : H (Yl, Y ( aYl 1)) , H (Yl, Z ) is compatible with the decomposition of O l − − −→ O these spaces into T -isotypical summands. It follows that

ν1(s) = ordYl 1 (s) = min ordYl 1 (s1),..., ordYl 1 (sq) = min ν1(s1), . . . , ν1(sq) . − { − − } { }

If we reorder the sections so that ν (s) = ν (s ) for 1 i e, and ν (s) < ν (s ) for 1 1 i ≤ ≤ 1 1 i ν1(s) e ν1(s) − − e + 1 i q, for some e 1, . . . , q , then (hl s) Yl 1 = i=1(hl si) Yl 1 and ≤ ≤ ∈ { } | − | − using the induction hypothesis we get P

ν1(s) ν1(s) − − νY (s) = ν1(s), νZ (hl s) Yl 1 ν1(s), νZ (hl si) Yl 1 i = 1, . . . , e • • | − ∈ • | − | = νY (s1), . . . , νY (se) νY (s1), . . . , νY (sq) . { • • } ⊆ { • • }

Recall that any line bundle on P( ) is isomorphic to a unique line bundle of the E d form P( )(m) π∗ X ( i=n+1 miDi). In the following proposition, we prove that O E ⊗ O for a line bundle on P( )P of that form, the T -isotypical sections map onto the image E of νY . •

d Proposition III.6. Let be the line bundle P( )(m) π∗ X ( i=n+1 miDi) on L O E ⊗ O P( ) for some m, m , . . . , m Z, and let s be a nonzero globalP section of . E 1 d ∈ L 37

Let s , . . . , s H0(P( ), ) be the unique nonzero T -isotypical sections corre- 1 q ∈ E L

sponding to distinct characters of T such that s = s1 + + sq. Then νY (s) ··· • ∈

νY (s1), . . . , νY (sq) . { • • }

Proof. It is enough to consider the case X = Uτ . There is a natural choice of coor-

dinates X = Spec k[x , . . . , x ] = An and P( ) = Spec k[x , . . . , x ] Proj k[x, y] = 1 n E 1 n × An P1, which is induced by the ordering of the rays of τ and the trivialization of × E over U . In these coordinates we have that X = (x , . . . , x ) An x = 0 for 1 τ i { 1 n ∈ | j ≤ j n i , and Y = X P1 for 0 i n and Y = X (0 : 1) . We also ≤ − } i+1 i × ≤ ≤ 0 0 × { } n n have that T = Spec k[x1, . . . , xn]x1 xn = (k∗) acts on A by componentwise mul- ··· tiplication, and that an element t = (t , . . . , t ) T acts on P = (x , . . . , x ), (x : 1 n ∈ 1 n n 1 u1,v1 u1,vn u2,v1 u2,vn y) A P by tP = (t x , . . . , t x ), (th i tnh ix : th i tnh iy) . If ∈ × 1 1 n n 1 ··· 1 ··· U P1 is the complement of (1 : 0), it is enough to prove that in the T -invariant ⊆ aﬃne open set Z = An U the restriction of the ﬂag Y and the line bundle × • L satisfy the hypotheses of Lemma III.5. We show that P( )(1) Z = Z (div g) for O E | O some T -isotypical rational function g on Z, since from this all the assertions follow at once. The surjective map (div χu2 ) corresponds to a geometric section θ E → OX of the projection π : P( ) X, i.e. a morphism θ : X P( ) such that π θ = id . E → → E ◦ X

u2 If we set X0 = θ(X), then π ( P( )(1) X ) = X (div χ ) and we have the exact ∗ O E ⊗ O 0 O sequence

(3.3) 0 P( )(1) P( )( X0) P( )(1) P( )(1) X 0. −→ O E ⊗ O E − −→ O E −→ O E ⊗ O 0 −→

1 By Grauert’s theorem (see III.12.9 in [Har77]) R π ( P( )(1) P( )( X0)) = 0. ∗ O E ⊗ O E −

u1 Then applying π to (3.3) gives π ( P( )(1) P( )( X0)) = X (div χ ). Since ∗ ∗ O E ⊗ O E − O

P( )(1) P( )( X0) has degree zero along the ﬁbers of π, O E ⊗ O E −

u1 P( )(1) P( )( X0) = π∗π ( P( )(1) P( )( X0)) = π∗ X (div χ ). O E ⊗ O E − ∗ O E ⊗ O E − O 38

n u1 And since in local coordinates X0 = A (0 : 1), we can take g = (x/y)π∗(χ ). ×

The following lemma will be used in the proofs of Proposition III.8 and Theorem

III.10.

Lemma III.7. Let V1,...,Vl be distinct one-dimensional subspaces of a two-dimen-

sional vector space V . Let m, α1, . . . , αl be nonnegative integers. Then

Symm(V α1 ) Symm(V α2 ) Symm(V αl ) = Symm(V α1 ,V α2 ,...,V αl ). V 1 ∩ V 2 ∩ · · · ∩ V l V 1 2 l

Furthermore, this subspace of SymmV is nonzero precisely when m l α , and ≥ i=1 i in that case its dimension is m + 1 l α . P − i=1 i P h Proof. We ﬁx an isomorphism of k-algebras between h 0Sym V and the polynomial ⊕ ≥

ring in two variables k[x, y]. The subspaces V1,...,Vl of V correspond to the linear

spans of some distinct linear forms f , . . . , f . The subspaces l Symm(V αi ) and 1 l ∩i=1 V i m α1 αl m SymV (V1 ,...,Vl ) of Sym V both correspond to the homogeneous polynomials

of degree m divisible by f α1 f αl . From this observation the conclusion follows. 1 ··· l

In the next proposition we prove that for every line bundle on P( ), in order L E

to ﬁnd the image of νY , we can restrict our attention to the sections in . • WL

d Proposition III.8. Let be the line bundle = P( )(m) π∗ X ( i=n+1 miDi) L L O E ⊗ O on P( ), for some m, m , . . . , m Z. Then we have the followingP equality of E n+1 d ∈ subsets of Zn+1:

0 νY (H P( ), r 0 ) = νY ( ) = νY ( ). • E L { } • VL • WL 0 Proof. From their deﬁnitions, we have that νY (H P( ), r 0 ) νY ( ) • E L { } ⊇ • VL ⊇ 0 νY ( ). The sets νY (H P( ), r 0 ) and νY ( ) are equal by Proposition III.6. • WL • E L { } • VL 0 Let us consider νY (s) νY (H P( ), r 0 ) = νY ( ). We can assume that • ∈ • E L { } • VL 39

0 0 s H P( ), , for some u M. By Remark II.1, the set νY (H (P( ), )u r 0 ) ∈ E L u ∈ • E L { } is ﬁnite with cardinality dim H0 P( ), . Let us denote the ﬁber over t of k E L u 0 ( d m D ) by G. From Example II.12 and Lemma III.7 we see that there OX i=n+1 i i existPα0, . . . , αp Z 0 such that ∈ ≥ Im H0 P( ), = H0 X, π , (SymmE) G E L u ∗L u −→ ⊗ ρ1 ρ2 ρd = (π ) ( u, v1 ) (π ) ( u, v2 ) (π ) ( u, vd ) ∗L h i ∩ ∗L h i ∩ · · · ∩ ∗L h i = (Symm(Eα0 ) G) (Symm(Lα1 ) G) (Symm(Lαp ) G) E 1 ⊗ ∩ E 1 ⊗ ∩ · · · ∩ E p ⊗ = Symm(Eα0 ,Lα1 ,Lα2 ,...,Lαp ) G. E 1 1 2 p ⊗ Using again Lemma III.7 we see that α = m p α is a nonnegative integer and def − i=0 i 0 0 α + 1 = dimk H P( ), . For each j 0, . . .P , α , let sj H X, π r 0 = E L u ∈ { } ∈ ∗L u { } H0 P( ), r 0 be such that E L u { } m α0+j α1 α2 αp α j s (t ) Sym (E ,L ,L ,...,L ,L − ) G. j 0 ∈ E 1 1 2 p ⊗

0 From Lemma III.4 it follows that νY (s0), . . . , νY (sα) νY (H P( ), r 0 ) are • • ∈ • E L u { } pairwise distinct, so

0 νY (s) νY (H P( ), r 0 ) = νY (s0), . . . , νY (sα) νY ( ), • ∈ • E L u { } { • • } ⊆ • WL and this completes the proof of the proposition.

3.2 The global Okounkov body of P( ) E

In this section we describe the global Okounkov body of the projectivization of a rank two toric vector bundle over a smooth projective toric variety, with respect to the ﬂag of invariant subvarieties constructed in 3.1.1. We introduce the relevant § terminology in 3.2.1, and we prove our result describing this global Okounkov body § in terms of linear inequalities in 3.2.2. Throughout this section we use the notation § and constructions introduced in 2.1, 2.2 and 3.1.1-3.1.2. § § § 40

3.2.1 Supporting hyperplanes of the global Okounkov body of P( ) E

Let be a toric vector bundle of rank two over the smooth projective toric variety E X. Let u , u M, with u u , be as defined in 3.1.1, and let the subspaces 1 2 ∈ 1 ≥lex 2 § E ,L ,...,L of the fiber E of over t be as defined in 3.1.2. Let us classify the 1 1 p E 0 § filtrations ρj j = 1, . . . , d associated to by defining {E | } E

A = j 1, . . . , d dim ρj (i) = 1 for all i Z def { ∈ { } | k E 6 ∈ } B = j 1, . . . , d ρj (i) = E for some i Z def { ∈ { } | E 1 ∈ } C = j 1, . . . , d ρj (i) = L for some i Z h def { ∈ { } | E h ∈ }

for each h 1, . . . , p . And let us deﬁne a nonempty set J 1, . . . , d to be ∈ { } ⊆ { } admissible if it has one of the following three forms:

J = j for some j A. • { } ∈

J = j for some j B. • { } ∈

J = j , . . . , j for some j , . . . , j 1, . . . , d such that there exist distinct • { 1 l} 1 l ∈ { } indices i , . . . , i 1, . . . , p with j C , for each h 1, . . . , l . 1 l ∈ { } h ∈ ih ∈ { }

Note that each admissible subset of 1, . . . , d is contained in exactly one of the sets { } A, B and C = p C . For each ray ρ ∆ we deﬁne integers a and b as follows. def ∪i=1 i j ∈ j j Let a = max i Z ρj (i) = E , and let j { ∈ | E }

a + 1 if j A, j ∈ bj =   max i Z dim ρj (i) = 1 if j B C. { ∈ | k E } ∈ ∪   Example III.9. In the case of TP2 , the tangent bundle of the projective plane (see

Example II.6), if we take the T -invariant ﬂag P2 D D D in P2, we get ⊇ 1 ⊇ 1 ∩ 2

τ = σ3, u1 = (1, 0) and u2 = (0, 1). We get a1 = a2 = a3 = 0 and b1 = b2 = b3 = 1. 41

We also get E = V , L = V and L = V , and then A = , B = 1 and 1 1 1 2 2 3 {∅} { } C = 2, 3 . Hence, in this case the admissible subsets of 1, 2, 3 are 1 , 2 , 3 { } { } { } { } { } and 2, 3 . { }

1 1 1 d n The isomorphisms N (P( )) = N (X) Z and N (X) = Z − , described in 2.2 E ∼ ⊕ ∼ § 1 d n+1 and 3.1.2, induce an isomorphism between N (P( )) and R − , which we use § E R to identify these spaces hereafter. Likewise, we identify Rn+1 N 1(P( )) with × E R d+2 R , with coordinates (x1, . . . , xn+1, wn+1, . . . , wd, w). Let v1∗, . . . , vn∗ be the basis of

M = M R dual to the basis v , . . . , v of N = N R. Note that we have R def ⊗ 1 n R def ⊗ an isomorphism

d+2 d n+2 ψ : R M R − −→ R × n (x , . . . , x , x , w , . . . , w , w) x v∗ + x u + (w x )u , 1 n n+1 n+1 d 7−→ − i=1 i i n+1 1 − n+1 2 P xn+1, wn+1, . . . , wd, w For each j 1, . . . , d we deﬁne the linear function: ∈ { } d n+2 γ ,j : MR R − R E × −→ u,v a w w h j i− j − j (u, xn+1, wn+1, . . . , wd, w) b a 7−→ j − j for any u M , and any x , w , . . . , w , w R, and where w = 0 for each ∈ R n+1 n+1 d ∈ j j n. We will denote this function simply by γ , when no confusion is likely to ≤ j arise. Finally, for each admissible set J 1, . . . , d , we deﬁne the linear function ⊆ { } I : Rd+2 R by declaring its value at P = (x , . . . , x , w , . . . , w , w) Rd+2 J → 1 n+1 n+1 d ∈ to be:

γj ψ(P ), if J = j A,  ◦ { } ⊆  I (P ) =  J  γj ψ(P ) xn+1, if J = j B,  ◦ − { } ⊆ 

j J γj ψ(P ) w + xn+1, if J C.  ∈ ◦ − ⊆  For notational convenience, P we deﬁne for each admissible set J 1, . . . , d the  ⊆ { } d n+2 1 linear function I0 : M R − R to be I0 = I ψ− . J R × → J J ◦ 42

3.2.2 The global Okounkov body of P( ) E

Theorem III.10. Let be a toric vector bundle of rank two on the smooth projective E toric variety X. The global Okounkov body ∆(P( )) of P( ) is the rational polyhedral E E cone in Rn+1 N 1(P( )) Rd+2 given by × E R '

∆ = (x , . . . , x , w , . . . , w , w) Rd+2 w x 0 and 1 n+1 n+1 d ∈ | ≥ n+1 ≥ n I (x , . . . , x , w , . . . , w , w) 0 for all admissible J 1, . . . , d . J 1 n+1 n+1 d ≤ ⊆ { } o Proof. From the characterization of the global Okounkov body in terms of its ﬁbers

over big classes in N 1(P( )) , it suﬃces to show the following stronger assertion: E Q 1 For every class N (P( ))Q, the ﬁber ∆ of ∆ over is equal to ∆( ) . L ∈ E L L L × {L} 1 To prove the assertion, we consider a class N (P( ))Q. Note that ∆ m = L ∈ E L m∆ and ∆( m) m = m(∆( ) ), for each m Z+. Hence, we can assume L L ×{L } L ×{L} ∈ that N 1(P( )). Let w , . . . , w , w Z be the unique integers such that L ∈ E n+1 d ∈ d

= P( )(w) π∗ X wiDi . L O E ⊗ O i=n+1 X For notational convenience, we set w = 0 for each i 1, . . . , n . We ﬁrst show that i ∈ { } ∆( ) ∆ . For this, it is enough to see that the set L × {L} ⊆ L 1 ∆( ) = Conv ν m (w , . . . , w , w) L × {L} m L × n+1 d m Z+ [∈ is contained in ∆. Since ∆ is closed and convex, it suﬃces show that

1 ν( m) (w , . . . , w , w) ∆, m L × n+1 d ⊆ for each m Z+. Furthermore, since ∆ is a cone, it is enough to prove that this ∈ inclusion holds when m = 1. With this in mind, we consider

P = (x , . . . , x , w , . . . , w , w) ν( ) (w , . . . , w , w). 1 n+1 n+1 d ∈ L × n+1 d 43

Note that the existence of P implies that w 0. Let Q = ψ(P ), i.e. ≥ n d n+2 Q = ( x v∗ + x u + (w x )u , x , w , . . . , w , w) M R − . − i i n+1 1 − n+1 2 n+1 n+1 d ∈ R × i=1 X By replacing with a suitable tensor power, we can assume that γ (Q) Z for all L j ∈ w d j 1, . . . , d . By the projection formula we have π = (Sym ) X ( wiDi). ∈ { } ∗L E ⊗O i=1 If we denote the ﬁber of ( d w D ) over the unit of the torus byPG, then by OX i=1 i i Example II.9 we get P

ρj w ρj (π ) (i) = (Sym ) (i wj) G, ∗L E − ⊗ for all j 1, . . . , d and all i Z. Since (x , . . . , x ) ν( ), Proposition III.8 im- ∈ { } ∈ 1 n+1 ∈ L 0 0 plies that there exist u M and a nonzero section s H (P( ), )u = H (X, π )u ∈ ∈ E L ∗L such that νY (s) = (x1, . . . , xn+1), and such that •

s(t ) V = Symw (Eα0 ,Lα1 ,...,Lαp ,Lα) G, 0 ∈ def E 1 1 p ⊗ for some one-dimensional subspace L of E diﬀerent from E1,L1,...,Lp and for some

p α0, . . . , αp, α Z 0 with αi + α = w. Note now that ∈ ≥ i=0 P d 0 w ρi ρj 0=V Im H X, π , (Sym E) G = (π ) ( u, vi ) (π ) ( u, vj ), 6 ⊆ ∗L u −→ ⊗ ∗L h i ⊆ ∗L h i i=1 \ for each j 1, . . . , d . By Lemma III.4, we have that α = x and x = u + ∈ { } 0 n+1 i h− α u + (w α )u , v for each i 1, . . . , n . In particular, we see that Q = 0 1 − 0 2 ji ∈ { }

(u, xn+1, wn+1, . . . , wd, w). From α0 = xn+1 we get

(3.4) w x 0. ≥ n+1 ≥

Hence, we are reduced to proving that I (P ) 0 for each admissible set J J ≤ ⊆

1, . . . , d , or equivalently, to proving that I0 (Q) 0 for every such J. { } J ≤ Let us consider an admissible set J 1, . . . , d . Then either J = j for some j A, ⊆ { } { } ∈ 44

J = j for some j B, or J = j , . . . , j for some j , . . . , j C such that there { } ∈ { 1 l} 1 l ∈ exist distinct indices i , . . . , i 1, . . . , p , with j C for each h 1, . . . , l . 1 l ∈ { } h ∈ ih ∈ { }

ρj In the ﬁrst case 0 = V (π ) ( u, vj ) gives u, vj ajw + wj, and then 6 ⊆ ∗L h i h i ≤

(3.5) I0 (Q) 0. J ≤

ρj w max 0,γj (Q) In the second case 0 = V (π ) ( u, vj ) = SymE(E1 { }) G, and this 6 ⊆ ∗L h i ⊗ implies

w max 0,γj (Q) w max 0,γj (Q) 0 = V = V (Sym (E { }) G) (Sym (E { }) G) 6 ∩ E 1 ⊗ ⊆ E 1 ⊗ ∩ (Symw (Lα1 ) G) (Symw (Lαp ) G) (Symw (Lα) G), E 1 ⊗ ∩ · · · ∩ E p ⊗ ∩ E ⊗

and then from Lemma III.7 we get γ (Q) max 0, γ (Q) w p α α = x , j ≤ { j } ≤ − i=1 i− n+1 so P

(3.6) I0 (Q) 0. J ≤

In the third case, we have γ (Q) α for each h 1, . . . , l . Otherwise we would jh ≤ jh ∈ { } γj (Q) ρjh w h have 0 = V (π ) ( u, vjh ) = SymE(Li ) G, and from Lemma III.7 we 6 ⊆ ∗L h i h ⊗ would get

γ (Q) V = V (Symw (L jh ) G) = (Symw (Eα0 ) G) (Symw (Lα1 ) G) ∩ E ih ⊗ E 1 ⊗ ∩ E 1 ⊗ ∩ · · · ∩

αi 1 γj (Q) αi +1 w h− w h w h (SymE(Li 1 ) G) (SymE(Li ) G) (SymE(Li +1 ) G) h− ⊗ ∩ h ⊗ ∩ h ⊗ ∩ · · · ∩ (Symw (Lαp ) G) (Symw (Lα) G) = 0, E p ⊗ ∩ E ⊗

which is a contradiction. By adding these inequalities over j J, we get h ∈ l p γ (Q) α α = w α α w x , j ≤ jh ≤ i − 0 − ≤ − n+1 j J h=1 i=1 X∈ X X and therefore

(3.7) I0 (Q) 0. J ≤ 45

From (3.4), (3.5), (3.6) and (3.7), it follows that P ∆. ∈ As we have completed the proof of ∆( ) ∆ , we now prove that ∆ L × {L} ⊆ L L ⊆ ∆( ) . For this, we note that L × {L}

n+1 1 ∆ = ∆ (Q N (P( ))Q), L L ∩ × E

since ∆ is deﬁned by rational linear inequalities. Thus, it suﬃces to show that L

n+1 1 ∆ Q N (P( ))Q ∆( ) . L ∩ × E ⊆ L × {L} To prove this, let us consider

n+1 1 P = (x1, . . . , xn+1, wn+1, . . . , wd, w) ∆ Q N (P( ))Q , ∈ L ∩ × E and deﬁne Q to be ψ(P ), i.e.

n 1 Q = ( x v∗+x u +(w x )u , x , w , . . . , w , w) M Q N (P( )) . − i i n+1 1 − n+1 2 n+1 n+1 d ∈ Q× × E Q i=1 X By replacing with a suitable tensor power, we can assume that Q M Z L ∈ × × N 1(P( )) and γ (Q) Z for each j 1, . . . , d . We have w x 0, and E j ∈ ∈ { } ≥ n+1 ≥

I0 (Q) = I (P ) 0 for each admissible set J 1, . . . , d . Let us deﬁne J J ≤ ⊆ { } n u = x v∗ + x u + (w x )u M, − i i n+1 1 − n+1 2 ∈ i=1 X and note that Q = (u, xn+1, wn+1, . . . , wd, w). We will show the existence of a nonzero

0 0 section s H (P( ), )u = H (X, π )u, satisfying ν (s) = (x1, . . . , xn+1), which ∈ E L ∗L • will give us

P ν( ) (w , . . . , w , w) ∆( ) . ∈ L × n+1 d ⊆ L × {L}

w d By the projection formula we have π = (Sym ) X ( wiDi). If we denote ∗L E ⊗ O i=1 the ﬁber of ( d w D ) over the unit of the torus by PG, then by Example II.9 OX i=1 i i we get P

ρj w ρj (π ) (i) = (Sym ) (i wj) G, ∗L E − ⊗ 46

for all j 1, . . . , d and all i Z. For each i 1, . . . , p , deﬁne αi Z 0 by ∈ { } ∈ ∈ { } ∈ ≥

α = max ( 0 γ (Q) j C ) . i def { } ∪ { j | ∈ i}

We claim that α = w x p α is a nonnegative integer. Indeed, this is clear def − n+1− i=1 i if α = 0 for each i 1, . . . , p .P On the other hand, if α = 0 for some i 1, . . . , p , i ∈ { } i 6 ∈ { } let i , . . . , i 1, . . . , p be the distinct indices such that for i 1, . . . , p , α = 0 1 l ∈ { } ∈ { } i 6 if and only if i i , . . . , i . For each h 1, . . . , l , let us choose j 1, . . . , d ∈ { 1 l} ∈ { } h ∈ { } such that j C and α = γ (Q). Then the set J = j . . . , j is admissible, and h ∈ ih ih jh { 1 l}

we have I0 (Q) 0. Hence J ≤ p l α = α = γ (Q) w x . i ih j ≤ − n+1 i=1 h=1 j J X X X∈ In either case, it follows that α is a nonnegative integer. Let L be a one-dimensional

subspace of E diﬀerent from E1,L1,...,Lp. From Lemma III.7, we see that

V = Symw (Exn+1 ,Lα1 ,...,Lαp ,Lα) G def E 1 1 p ⊗

w ρj is a one-dimensional subspace of (Sym E) G. We now prove that V (π ) ( u, vj ) ⊗ ⊆ ∗L h i for each j 1, . . . , d , considering separately the cases j A, j B and j C. If ∈ { } ∈ ∈ ∈

j A, then J = j is admissible, and I0 (Q) 0. This gives u, v a w + w , ∈ { } J ≤ h ji ≤ j j and therefore

w ρj V (Sym E) G = (π ) ( u, vj ). ⊆ ⊗ ∗L h i

If j B, then J = j is admissible, and I0 (Q) 0. This gives ∈ { } J ≤

u, v a w + w + (b a )x , h ji ≤ j j j − j n+1

and therefore

w xn+1 ρj ρj V SymE(E1 ) G = (π ) (ajw + wj + (bj aj)xn+1) (π ) ( u, vj ). ⊆ ⊗ ∗L − ⊆ ∗L h i 47

If j C, then there exists i 1, . . . , p such that j C , and ∈ ∈ { } ∈ i

w αi w max 0,γj (Q) ρj V SymE(Li ) G SymE(Li { }) G = (π ) ( u, vj ). ⊆ ⊗ ⊆ ⊗ ∗L h i Therefore d ρj 0 w V (π ) ( u, vj ) = Im H X, π , (Sym E) G . ⊆ ∗L h i ∗L u −→ ⊗ j=1 \ 0 0 We can now choose a nonzero section s H (P( ), )u = H (X, π )u such that ∈ E L ∗L s(t ) V . By Lemma III.4, the section s satisﬁes 0 ∈

νY (s) = u + xn+1u1+(w xn+1)u2, v1 ,..., • h− − i u + x u + (w x )u , v , x = (x , . . . , x ). h− n+1 1 − n+1 2 ni n+1 1 n+1 Thus P = νY (s) (wn+1, . . . , wd, w) ∆( ) . It follows that ∆ ∆( ) , • × ∈ L ×{L} L ⊆ L ×{L} and this completes the proof.

Remark III.11. Explicitly, the inequalities that deﬁne the global Okounkov body

∆(P( )) of P( ) given in Theorem III.10 are w x 0 together with: E E ≥ n+1 ≥ For each j A, ∈ n v∗, v x + u u , v x + (a u , v )w + w 0, h i ji i h 2 − 1 ji n+1 j − h 2 ji j ≥ i=1 X for each j B, ∈ n v∗, v x + ( u u , v + b a )x + (a u , v )w + w 0, h i ji i h 2 − 1 ji j − j n+1 j − h 2 ji j ≥ i=1 X and for each admissible set J C, ⊆ 1 n vi∗, vj xi + u2 u1, vj xn+1 + (aj u2, vj )w + wj + w xn+1 0. bj aj h i h − i −h i − ≥ j J " i=1 # X∈ − X 3.3 Examples

We recall that the explicit description of Okounkov bodies in concrete examples

can be rather diﬃcult. The main result of this chapter, Theorem III.10, allows us 22 JOSELUISGONZ´ ALEZ´

Example 6.1. We consider TP2 , the tangent bundle of the projective plane (see Example 2.5). From Remark 5.3 (see Example 5.1), we get inequalities for the Okounkov body of each line bundle on P(TP2 ). For instance, by setting w =1andwj = 0 for each j, we deduce 3 that the Okounkov body ∆( P(T )(1)) is deﬁned inside R by the inequalities: O P2 1 x x 0 x 0 ≥ 3 3 ≥ 1 ≥ x2 02x1 + x2 + x3 1 x1 ≥ ≥ 48 ≥ In particular, we see that the volume of P(T )(1) is volR3 ∆( P(T )(1)) 3! = 6. O P2 O P2 ·

x3 ! "

(0,0,1)

(1,0,1) (0,1,1)

(1,0,0)

(1,1,0) (0,2,0) x2

Figure 2. The Okounkov body of P(T 2 )(1). Figure 3.1: The Okounkov body of P(T P)(1). OO P2

In the next example we see that our description gives the expected answer for line bundlesto explicitly that are pulledcompute back the from Okounkov the base. bodies of all line bundles on projectivizations of Examplerank two 6.2. toricThe vector inequalities bundles for overthe Okounkov smooth projective body of a toric line bundle varieties, on P with( )oftheform respect to d E π∗ ( m D )arex =0and OX i=n+1 i i n+1 the flag from 3.1.1, by substitutingn combinatorial data into the inequalities given in # § v∗,v x + m 0, " i j# i j ≥ Remark III.11. In this sectioni=1 we present some examples to illustrate this theorem. $ for each j 1,...,d . Furthermore, from the description of the Okounkov body of a toric ∈ { } line bundleExample on X III.12.given inWe [8], consider we seeT thatP2 , the tangent bundle of the projective plane (see d d Example II.6). From Remark III.11 (see Example III.9), we get inequalities for the ∆Y π∗ X ( miDi) = P d m D 0 = ∆X X ( miDi) 0 . • O i=n+1 i i × { } • O × { } i=n+1 i=n+1 # Okounkov! body$ of each line" bundle on P(TP2 ). For instance,! $ by setting" w = 1 and In the next example, we see that our description gives the expected answer when the wj = 0 for each j, we deduce that the Okounkov body ∆( P(T )(1)) is defined inside toric vector bundle equivariantly splits. O P2 3 ExampleR by 6.3. the inequalities:When equivariantly splits as the sum of two toric line bundles 1 and 2, the variety P( ) is a toricE variety. The subvarieties in our flag in P( ) are also invariantL withL E E respect to the torus T " of1 P(x)., Hence we x have0 two, descriptions x of0, the Okounkov bodies ≥ E3 3 ≥ 1 ≥ x 0, 2 x + x + x , 1 x 2 ≥ ≥ 1 2 3 ≥ 1

(see Figure 3.1). In particular, we see that the volume of P(T )(1) is O P2

volP(T ) P(T )(1) = volR3 ∆( P(T )(1)) 3! = 6. P2 O P2 O P2 · In the next example we see that our description gives the expected answer for line

bundles that are pulled back from the base. 49

Example III.13. The inequalities for the Okounkov body of a line bundle on P( ) E d of the form π∗ ( m D ) are x = 0 and OX i=n+1 i i n+1 P n v∗, v x + m 0, h i ji i j ≥ i=1 X for each j 1, . . . , d . Furthermore, from the description of the Okounkov body of ∈ { } a toric line bundle on X given in [LM08, Proposition 6.1], we see that

d d

∆Y π∗ X ( miDi) = PPd m D 0 = ∆X X ( miDi) 0 . • O i=n+1 i i × { } • O × { } i=n+1 i=n+1 X X In the next example, we see that our description gives the expected answer when the toric vector bundle equivariantly splits.

Example III.14. When equivariantly splits as the sum of two toric line bundles E and , the variety P( ) is a toric variety. The subvarieties in our ﬂag in P( ) L1 L2 E E

from 3.1.1 are also invariant with respect to the torus T 0 of P( ). Hence, in this § E case we have two descriptions of the Okounkov bodies of line bundles on P( ) with E respect to this ﬂag of invariant subvarieties, namely, the one given by our theorem,

and the one given by Lazarsfeld and Mustat¸˘ain [LM08] in the case of toric varieties.

It is good to see that these two descriptions agree, as expected.

Let h and h be the piecewise linear functions associated to and (see 1 2 L1 L2 3.4 in [Ful93]). In particular, and correspond to the T -invariant divisors L1 L2 d h (v )D and d h (v )D . Let Φ: N N R be the piecewise − j=1 1 j j − j=1 2 j j R → R × linearP map deﬁned by Φ(Pv) = (v, h (v) h (v)). For each cone σ ∆, let σ+ and 1 − 2 ∈

σ− be the cones in N R spanned by Φ(σ) and (0, 1), and by Φ(σ) and (0, 1), R × − + respectively. Let ∆ be the fan in N R consisting of the faces of σ and σ− for R × all σ ∆. The toric variety associated to the fan ∆ is isomorphic to P( ) (see 7 ∈ e E § + in [Oda78]). The rays of ∆ are ρ = R 0 (0, 1), ρ−e= R 0 (0, 1) and ρj = Φ(ρj) ≥ · ≥ · − e e 50

for each j 1, . . . , d . Let us denote the corresponding T 0-invariant divisors by ∈ { } + D , D− and D for each j 1, . . . , d , respectively. The ﬂag in P( ) given by our j ∈ { } E construction from 3.1.1 is f §

Y : P( ) D1 D1 D2 D1 D2 Dn D1 D2 Dn D−. • E ⊇ ⊇ ∩ ⊇ · · · ⊇ ∩ ∩ · · · ∩ ⊇ ∩ ∩ · · · ∩ ∩ f f f f f f f f f Note that Φ(v ),..., Φ(v ) and (0, 1) span the maximal cone τ − ∆. Let us 1 n − ∈ change the reference ordered basis in N R to Φ(v ),..., Φ(v ), (0, 1) . In R × { 1 n e− } + these new coordinates the rays are given by ρ = R 0 (0, 1), ρ− = R 0 (0, 1) ≥ · − ≥ · and ρj = R 0 (vj, h2(vj) h1(vj) + u2, vj u1, vj ) for each j 1, . . . , d . Using ≥ · − h i − h i ∈ { } + the argument in the proof of Proposition III.6, we see that P( )(1) = P( )(D ) e O E O E ⊗ + d π∗ 2 = P( )(D−) π∗ 1. We set D = D + j=1( h2(vj) u2, vj )Dj, and note L O E ⊗ L − − h i that the T -invariant divisor D satisﬁes (1)P = (D) and D = 0. 0 P( ) P( ) Uτ f O E O E | − d Let us consider a line bundle = P( )(m) π∗ X ( i=n+1 miDi) on P( ). L O E ⊗ O E Let us identify the dual of N R with Rn+1 by identifyingP the ordered basis R × Φ(v ),..., Φ(v ), (0, 1) of N R with the coordinates x , . . . , x on Rn+1. We { 1 n − } R × 1 n+1 set m = 0 for each i 1, . . . , n . On the one hand, the description in [LM08, Propo- i ∈ { }

sition 6.1] says that with this identiﬁcation ∆Y ( ) is the polytope PmD+Pd m D . • L i=n+1 i fi This polytope is deﬁned as a subset of Rn+1 by the inequalities

xn+1 0, m xn+1,  ≥ ≥  n  v∗, vj xi + (h2(vj) h1(vj) + u2, vj u1, vj )xn+1 mh2(vj)  i=1h i i − h i − h i −  P m u2, vj + mj 0, for each j 1, . . . , d .  − h i ≥ ∈ { }   On the other hand, the admissible subsets of 1, . . . , d associated to a toric vector { } bundle that equivariantly splits are exactly the singletons. From Remark III.11, our 51

inequalities for ∆Y ( ) are • L

m xn+1 0,  ≥ ≥  n  v∗, vj xi + u2 u1, vj xn+1 + (h2(vj) h1(vj))xn+1 mh2(vj)  i=1h i i h − i − −  P  m u2, vj + mj 0, for each j 1, . . . , d .  − h i ≥ ∈ { }   Therefore, the two descriptions of the Okounkov body ∆Y ( ) coincide. • L CHAPTER IV

The Cox ring of a projectivized rank two toric vector bundle

4.1 Introduction

In this chapter we present two distinct proofs of the ﬁnite generation of the Cox rings of projectivized rank two toric vector bundles. These two arguments were obtained in our study of this problem for arbitrary ranks (c.f. Chapter V). There are yet two more proofs of this ﬁnite generation that can be found in the literature

(see [HS10]). In their work [HS10], J. Hausen and H. Süßprovide a different argument for this finite generation and point out that one can obtain yet another argument based on the main theorem in the paper [Kn93] of F. Knop. The proof in [HS10] is based on some interesting results where the authors find presentations of Cox rings of varieties with torus actions over Cox rings of some associated quotient prevarieties, and in turn they find presentations of those Cox rings over the Cox rings of some other varieties associated to the aforementioned quotient prevarieties which they call separations. In the additional argument suggested by Hausen and Süß,starting from a variety X in a certain class of varieties that includes projectivizations of rank two toric vector bundles, one constructs a unirational T -variety of complexity one that

has the Cox ring of X as its function ﬁeld. We recall that a T -variety is a normal

variety with an action of a torus, and that its complexity is deﬁned as the smallest

52 53 codimension of a T -orbit. From this the conclusion follows from the ﬁnite generation as algebras of the function ﬁelds of T -varieties of complexity one, which was proved by Knop in [Kn93].

Our ﬁrst proof arises as an application of our description of the global Okounkov body ∆(P( )) of a projectivized rank two toric vector bundle , with respect to E E our ﬂag of T -invariant subvarieties from 3.1.1 as a rational polyhedral cone in a

finite dimensional real vector space. The finite generation will follow after we prove that there is a Veronese subalgebra of a Cox ring of P( ) that is isomorphic to E the semigroup algebra associated to the semigroup of lattice points with sufficiently

divisible coordinates in ∆(P( )), which is clearly ﬁnitely generated. E For our second proof (see Theorem IV.6), we consider a particular set of generators

of the group Pic(P( )), and describe a ﬁnite set of generators for the associated Cox E ring. For this, we will reﬁne the grading of Cox(P( )) using the induced torus action E on the global sections of invariant T -divisors. Next, we describe the graded pieces

and the multiplication map in terms of the data appearing in the Klyachko ﬁltrations

of . In this case, we obtain the ﬁnite generation by exhibiting a ﬁnite generator set E of a Veronese subalgebra of the Cox ring of P( ). E In this chapter we use the notation for toric varieties and toric vector bundles

introduced in Section 2.2 unless we state otherwise. We conclude the introduction

to this chapter with the following remark that shows that the question of ﬁnite

generation of Cox rings of projectivized toric vector bundles over arbitrary toric

varieties can be reduced to the case when the base is smooth.

Remark IV.1. Given a toric variety X there exists a toric resolution of singularities,

i.e. there exists a smooth toric variety X0 and a proper birational toric morphism

f : X0 X. Given a toric vector bundle on X, the induced map f 0 : P(f ∗ ) → E E → 54

P( ) is also proper and birational. In this case f ∗ is a toric vector bundle on X0 E E and the finite generation of a Cox ring of P(f ∗ ) implies the finite generation of any E Cox ring of P( ). To see this we consider line bundles , ,..., on P( ) whose E L1 L2 Ls E classes span Pic(P( ))Q and line bundles 0 , 0 ,..., 0 on P(f ∗ ) whose classes E L1 L2 Ls0 E span Pic(P(f ∗ )) . The finite generation of the algebra E Q

Cox P(f ∗ ), ( 0 , 0 ,..., 0 , f 0∗ 1, f 0∗ 2, . . . , f 0∗ s) E L1 L2 Ls0 L L L implies the ﬁnite generation of the subalgebra

0 m1 ms H (P(f ∗ ), f 0∗ 1⊗ f 0∗ s⊗ ) = Cox P( ), ( 1,..., s) , s E L ⊗ · · · ⊗ L E L L (m1,...,ms) Z M ∈ since 0 Zs is a ﬁnitely generated saturated subsemigroup of Zs0 Zs (see Lemma × × 4.8 in [ELMNP05]). Therefore, to prove the ﬁnite generation of the Cox rings of projectivizations of rank two toric vector bundles over arbitrary toric varieties it is enough to consider the case when the base is a smooth toric variety.

4.2 The ﬁnite generation of the Cox ring of P( ) using its Okounkov body E

In the next theorem we show that the projectivization of a rank two toric vector bundle over a simplicial projective toric variety is a Mori dream space (c.f. [HS10] and Theorem IV.6 for alternative arguments).

Theorem IV.2. Any Cox ring of the projectivization P( ) of a rank two toric vector E bundle over the projective simplicial toric variety X is ﬁnitely generated and P( ) E E is a Mori dream space.

Proof. Any simplicial toric variety is Q-factorial, and a projective bundle over a Q- factorial variety is again Q-factorial, hence the we are reduced to prove the ﬁnite 55 generation of any Cox ring of P( ) in the sense of Hu and Keel. By Remark IV.1 E we can assume that X is smooth. Let us prove that the semigroup deﬁned by

S = (x , . . . , x , w , . . . , w , w) Rd+2 = Rn+1 N 1(P( )) 1 n+1 n+1 d ∈ × E R | d 0 There exists s H P( ), P( )(w) π∗ X ( wiDi) ∈ E O E ⊗ O i=n+1 X

such that νY (s) = (x1, . . . , xn+1) •

is finitely generated. Since S Zd+2, it is enough to prove that the semigroup ⊆ S (c Zd+2) is finitely generated, where c = lcm b a j = 1, 2, . . . , d . And for ∩ · { j − j | } this it suffices to prove that S (c Zd+2) = ∆(P( )) (c Zd+2), since ∆(P( )) is a ∩ · E ∩ · E rational polyhedral cone. From the definition of ∆(P( )), we have that S (c Zd+2) E ∩ · ⊆ ∆(P( )) (c Zd+2). Let (x , . . . , x , w , . . . , w , w) ∆(P( )) (c Zd+2). E ∩ · 1 n+1 n+1 d ∈ E ∩ · Proceeding exactly as in the second part of the proof of Theorem III.10, it follows

0 d that there exists a nonzero section s H P( ), P( )(w) π∗ X ( i=n+1 wiDi) ∈ E O E ⊗ O that satisﬁes νY (s) = (x1, . . . , xn+1). Therefore the semigroup S is ﬁnitelyP generated • as we claimed. Now we prove that the Cox ring of P( ) associated to the line bundles E

π∗ X (Dn+1), . . . , π∗ X (Dd) and P( )(1)) on P( ) is ﬁnitely generated. This Cox O O O E E ring is equal to

R = R(mn+1,...,md,m), d n+1 (mn+1,...,m ,m) Z Md ∈ − d n+1 where for each (m , . . . , m , m) Z − , n+1 d ∈

R = H0(X, (Symm ) (m D ) (m D )). (mn+1,...,md,m) def E ⊗ OX n+1 n+1 ⊗ · · · ⊗ OX d d

Let g , g , . . . , g be generators of S. For each j 1, 2, . . . , l , there exist m(j) , { 1 2 l} ∈ { } n+1 . . . , m(j), m(j) Z and a nonzero section d ∈ d 0 (j) (j) sj H P( ), P( )(m ) π∗ X ( mi Di) ∈ E O E ⊗ O i=n+1 X 56

(j) (j) (j) d n+1 such that gl = (νY (sj), mn+1, . . . , md , m ). For each (mn+1, . . . , md, m) Z − , • ∈ since g1, g2, . . . , gl generate S, it follows that

νY (k[s1, s2, . . . , sl] R(mn+1,...,md,m)) r 0 = νY R(mn+1,...,md,m) r 0 . • ∩ { } • { } By Remark II.1 the ﬁnite dimensional vector spaces k[s , s , . . . , s ] R 1 2 l ∩ (mn+1,...,md,m)

and R(mn+1,...,md,m) have the same dimension, and thus they are equal. Therefore

R = k[s1, s2, . . . , sl] and this completes the proof.

4.3 A combinatorial approach to the ﬁnite generation of the Cox ring of P( ) E

As before, we use Remark IV.1 to reduce the question of ﬁnite generation of the Cox rings of projectivized toric vector bundles of a given rank over arbitrary toric varieties, to the case when the base is smooth. Let X be a smooth toric variety and let π : P( ) X be the projectivization of the rank two toric vector E → bundle over X. The classes of the line bundles (D ), (D ),..., (D ) E OX 1 OX 2 OX d 1 span Pic(X)Q = N (X)Q. Since Pic(P( )) = Pic(X) Z [ P( )(1)], the classes of E ⊕ · O E

the line bundles P( )(1), π∗ X (D1), π∗ X (D2), . . . , π∗ X (Dd) span Pic(P( ))Q = O E O O O E N 1(P( )) . Therefore the following algebra is a Cox ring of P( ) in the sense of E Q E Deﬁnition II.13:

0 C = H (P( ), P( )(m) π∗ X (m1D1) π∗ X (mdDd)). E d+1 E O ⊗ O ⊗ · · · ⊗ O (m,m1,...,m ) Z Md ∈ By the projection formula, C is isomorphic to the algebra

0 m R = H (X, Sym X (m1D1) X (mdDd)), d+1 E ⊗ O ⊗ · · · ⊗ O (m,m1,...,m ) Z Md ∈ where Symm = 0 for m < 0. For each m = (m , . . . , m ) Zd, let us denote the E 1 d ∈ toric line bundle (m D ) (m D ) on X by m, and the ﬁber of m OX 1 1 ⊗ · · · ⊗ OX d d L L 57

d over t by L . Note that for each m, m0 Z, each m, m0 Z and each u, u0 M, 0 m ∈ ∈ ∈ the product of H0(X, Symm m) and H0(X, Symm0 m0 ) in the algebra R E ⊗ L u E ⊗ L u0 is contained in H0(X, Symm+m0 m+m0 ) . Therefore we get a ﬁner grading for E ⊗ L u+u0 R given by

R = R = H0(X, Symm m) . (u,m,m) E ⊗ L u (u,m,m) M Z Zd (u,m,m) M Z Zd M∈ × × M∈ × × Note that each of the homogeneous components R = H0(X, Symm m) (u,m,m) E ⊗ L u is a ﬁnite dimensional k-vector space, as each of them is isomorphic to a subspace of

E via ev . For each l Z+, let R(l) be the Veronese subalgebra of R given by t0 ∈

(4.1) R(l) = R = H0(X, Symlm lm) . (lu,lm,lm) E ⊗L lu (u,m,m) M Z Zd (u,m,m) M Z Zd M∈ × × M∈ × × Since R is a domain and H0(X, ) is a finitely generated k-algebra, it follows OX from general considerations that the finite generation of R is equivalent to the finite

generation of R(l) for any l Z+ (see [SYJ10, Lemma 1.4]). ∈ 4.3.1 Preliminary lemmas

Let W1,W2,...,Wl be subspaces of a vector space W. For every collection of

nonnegative integers m, c , c , . . . , c , we denote by Symm (W c1 ,W c2 , ,W cl ) the 1 2 l W 1 2 ··· l subspace of SymmW equal to the image of the composition of the natural maps

Pl c1 c2 cl (m c ) m m W ⊗ W ⊗ W ⊗ W ⊗ − i=1 i W ⊗ Sym W, 1 ⊗ 2 ⊗ · · · ⊗ l ⊗ −→ −→

if m l c , or the subspace 0 of SymmW , otherwise. ≥ i=1 i P Lemma IV.3. Let W1,W2,...,Wq be distinct subspaces of a vector space W . Let

q q m, m0, c , c , . . . , c , c0 , c0 , . . . , c0 be nonnegative integers. If c m and c0 1 2 q 1 2 q i=1 i ≤ i=1 i ≤

m0, then P P

c c +c m c1 cq m0 c10 q0 m+m0 c1+c10 q q0 µ Sym W ,...,W Sym W ,...,Wq = Sym W ,...,Wq , W 1 q ⊗ W 1 W 1 58 where µ: SymmW Symm0 W Symm+m0 W is the multiplication map. ⊗ →

Proof. The conclusion follows at once from the commutativity of the diagram

c P c0 (m P c ) N ⊗ i (m ci) N ⊗ i 0 i0 / m m0 / m m0 ( Wi ) W ⊗ − ( Wi ) W ⊗ − W ⊗ W ⊗ Sym W Sym W i ⊗ ⊗ i ⊗ ⊗ ⊗ µ o (c +c ) P N ⊗ i i0 ((m+m0) (ci+ci0 )) / (m+m0) / m+m0 ( Wi ) W ⊗ − W ⊗ Sym W. i ⊗

Let be a toric vector bundle of rank two on X. As before, let E be the fiber E of over the unit element t of the torus, and let ρ1 , ρ2 ,..., ρd be the Kly- E 0 E E E achko filtrations associated to . Let V ,V ,...,V be the distinct one-dimensional E 1 2 p subspaces of E that appear in the Klyachko filtrations of , i.e. each V is equal E l to ρj (i) for some j 1, 2, . . . , d and some i Z. We now define the subsets E ∈ { } ∈ A ,A ,...,A of 1, 2, . . . , d , which intuitively classify the filtrations according to 0 1 p { } their one-dimensional subspace, as follows. For each l 1, 2, . . . , p , we define ∈ { }

A = j 1, 2, . . . , d ρj (i) = V for some i Z , l { ∈ { } | E l ∈ }

and we also deﬁne

A0 = 1, 2, . . . , d r Al. { } 1 l p ≤[≤ For each j 1, 2, . . . , d let l(j) be the unique element of 0, 1, 2, . . . , p such that ∈ { } { } j A . We next introduce the following collection of possibly empty subsets of ∈ l(j) 1, 2, . . . , d : { }

= J J A , and J A has at most one element, for each l = 1, 2, . . . , p . J { | ⊆ l ∩ l } 1 l p ≤[≤ Note that is a ﬁnite set. For each j 1, 2, . . . , d , we deﬁne the integers a and J ∈ { } j 59 b by a = max i Z ρj (i) = E and j j { ∈ | E }

ρj max i Z dimk (i) = 1 if j 1 l p Al { ∈ | E } ∈ ≤ ≤ bj =   S a + 1 if j A . j ∈ 0  To each ﬁltration ρj we associate the linear functional λ on M Rd+1 deﬁned by E j R ×

λ : M Rd+1 R j R × −→ u,v a m m h j i− j − j (u, m, m1, . . . , md) b a 7−→ j − j for any u M and any m, m , m , . . . , m R. ∈ R 1 2 d ∈ Remark IV.4. To motivate the preceding deﬁnitions, we note that for each j 1, ∈ { d 2, . . . , d , and for each m Z 0, m Z and u M, we have } ∈ ≥ ∈ ∈

m m ρj m ρ max 0, λj (u,m,m) ((Sym ) ) ( u, v ) = Sym ( j (b )) { d e} L , E ⊗ L h ji E E j ⊗ m where x = min l Z l x for any x R. This equality is a reformulation of d e { ∈ | ≥ } ∈ Example II.12, and it can also be veriﬁed through direct computation.

We now deﬁne a rational polyhedral cone Q in M Rd+1 for each J , as J R × ∈ J follows. Given such a set J = j , j , . . . , j , the cone Q is deﬁned as the set of { 1 2 q} J

elements (x, w, w1, . . . , wd) satisfying the linear inequalities

(4.2) w 0, ≥ q (4.3) λ (x, w, w , . . . , w ) w, jh 1 d ≤ Xh=1

(4.4) λj(x, w, w1, . . . , wd) 0 for each j 1, 2, . . . , d r Al(jh), ≤ ∈ { } 1 h q ≤[≤ (4.5) λ (x, w, w , . . . , w ) 0 for each h 1, 2, . . . , q , jh 1 d ≥ ∈ { } (4.6) λ (x, w, w , . . . , w ) λ (x, w, w , . . . , w ) for each h 1, 2, . . . , q jh 1 d ≥ j 1 d ∈ { } and each j A , ∈ l(jh) 60 where (x, w, w , . . . , w ) are the coordinates in M Rd+1. Intuitively, the cones Q 1 d R × J were deﬁned precisely to satisfy the conclusions of the following lemma.

Lemma IV.5. For each g = (u, m, m) M Z Zd we have: ∈ × × (a) If (u, m, m) belongs to the cone Q , for some J = j , j , . . . , j , then J { 1 2 q} ∈ J

0 m m m λj1 (g) λj2 (g) λjq (g) evt H (X, Sym )u = Sym V d e,V d e,...,V d e Lm. 0 E ⊗ L E l(j1) l(j2) l(jq) ⊗ (b) If H0(X, Symm m) = 0, then (u, m, m) belongs to the cone Q for some E ⊗ L u 6 J J . ∈ J (c) Assume that (u, m, m) c (M Z Zd) belongs to the cone Q , for some J = j , ∈ × × J { 1 j , . . . , j , where c = lcm b a j = 1, 2, . . . , d . Then H0(X, Symm 2 q} ∈ J { j − j | } E ⊗ m) = 0. L u 6

Proof.

(a) From the deﬁnition of the Klyachko ﬁltrations of Symm m, we have that E ⊗ L

ev H0(X, Symm m) = (Symm m)ρj ( u, v ). t0 E ⊗ L u E ⊗ L h ji 1 j d ≤\≤

For each j 1, 2, . . . , d r 1 h q Al(jh), the inequality λj(g) 0 implies that ∈ { } ≤ ≤ ≤ (Symm m)ρj ( u, v ) = SymS mE L . For each h 1, 2, . . . , q , the inequality E ⊗ L h ji ⊗ m ∈ { } λj (g) m m ρj m h λj (g) 0 implies that (Sym ) h ( u, vj ) = Sym (V d e) Lm. Sim- h ≥ E ⊗ L h h i E l(jh) ⊗ ilarly, for each h 1, 2, . . . , q and each j A the inequality λ (g) λ (g) ∈ { } ∈ l(jh) jh ≥ j implies that

m m ρ m m ρ (Sym ) j ( u, v ) (Sym ) jh ( u, v ). E ⊗ L h ji ⊇ E ⊗ L h jh i 61

Therefore,

ev H0(X, Symm m) t0 E ⊗ L u = (Symm m)ρj ( u, v ) = (Symm m)ρj ( u, v ) E ⊗ L h ji E ⊗ L h ji 0 l p j A 1 h q j A ≤\≤ \∈ l ≤\≤ ∈\l(jh) λj (g) m m ρj m h = (Sym ) h ( u, vj ) = Sym (V d e) Lm E ⊗ L h h i E l(jh) ⊗ 1 h q 1 h q ≤\≤ ≤\≤ m λj1 (g) λj2 (g) λjq (g) = Sym V d e,V d e,...,V d e Lm. E l(j1) l(j2) l(jq) ⊗ (b) We deﬁne

I = l 1, 2, . . . , p max λ (u, m, m) j A 0 . ∈ { } | { j | ∈ l} ≥ We can assume that I = φ, since otherwise (u, m, m) belongs to Q for J = φ . 6 J ∈ J Let l , l , . . . , l be the distinct elements of I, and for each h 1, 2, . . . , q let us 1 2 q ∈ { } choose j A such that λ (u, m, m) = max λ (u, m, m) j A . Clearly, the h ∈ lh jh { j | ∈ lh } set J = j h = 1, 2, . . . , q belongs to . Then (u, m, m) satisﬁes (4.2) and def { h | } J (4.3) since

m λjh (u,m,m) 0 m m Sym V d e Lm = evt H (X, Sym )u = 0, E l(jh) ⊗ 0 E ⊗ L 6 1 h q ≤\≤ and it satisfies (4.4)-(4.6) by the definitions of I and J. Thus, (u, m, m) Q . ∈ J λ (u,m,m) λ (u,m,m) λ (u,m,m) (c) By (a), it suffices to show that Symm V j1 ,V j2 ,...,V jq is E l(j1) l(j2) l(jq) nonzero, which is true by (4.3) and Lemma III.7.

4.3.2 The ﬁnite generation of the Cox ring of P( ) E

In the next theorem we prove that any Cox ring of P( ), in the sense of Definition E II.13, is finitely generated. As a corollary we obtain that P( ) is a Mori dream space E as defined by Hu and Keel in [HK00] (c.f. [Kn93], [HS10] and Theorem IV.2), if

the toric variety X is projective and ∆ is simplicial (i.e. each cone in ∆ is spanned 62 by as many vectors as its dimension). Throughout this section we use the notation from 2.2 and the notation previously introduced in this chapter, in particular the § deﬁnitions from 4.3.1 will play an important role. §

Theorem IV.6. Any Cox ring of the projectivization P( ) of a rank two toric vector E bundle over an arbitrary toric variety X is ﬁnitely generated as a k-algebra. E

Proof. By Remark IV.1 we can assume that X is smooth. It suffices to find a finite set

of generators for the k-algebra R(c) from (4.1), where c = lcm b a j = 1, 2, . . . , d . { j − j | } For each set J , let G c (M Zd+1) be a ﬁnite set of generators for the ∈ J J ⊆ × semigroup Q c (M Zd+1). For each g M Zd+1, let β be a k-basis of R . J ∩ × ∈ × g g We claim that the ﬁnite set

β =def βg J g G [∈J [∈ J generates R(c) as a k-algebra. In order to prove the claim, consider (u, m, m) ∈ c (M Z Zd) such that H0(X, Symm m) = 0. By Lemma IV.5 (a) and (b), × × E ⊗ L u 6 there exist J = j , j , . . . , j , such that (u, m, m) Q and { 1 2 q} ∈ J ∈ J

0 m m m λj1 (u,m,m) λj2 (u,m,m) λjq (u,m,m) evt H (X, Sym )u = Sym V ,V ,...,V Lm. 0 E ⊗ L E l(j1) l(j2) l(jq) ⊗ 

Now, ﬁx an expression (u, m, m) = g G cgg, where cg Z 0 for each g GJ . Let ∈ J ∈ ≥ ∈ g = (u , m , m ) for each g G , beP the corresponding coordinates in M Z Zd. g g g ∈ J × × From Lemma IV.5 (a) applied in the cone Q , we get that for each g G , J ∈ J

λ (g) 0 mg m mg λj1 (g) λj2 (g) jq evt H (X, Sym g )u = Sym V ,V ,...,V Lm . 0 E ⊗ L g E l(j1) l(j2) l(jq) ⊗ g 

Since g G cgλjh (g) = λjh (u, m, m) for each h 1, 2, . . . , q , it follows by Lemma ∈ J ∈ { } IV.5 (c)P and Lemma IV.3 that in the commutative diagram 63

(R ) cg / R g ⊗ (u,m, m) g G _ ∈ J N evt0 mg cg / m (Sym E Lmg )⊗ Sym E Lm g G ⊗ ⊗ ∈ J N cg m the images of g G (Rg)⊗ and R(u,m,m) in Sym E Lm coincide. The injectivity ∈ J ⊗ N cg of evt0 implies that g G (Rg)⊗ surjects onto R(u,m,m), and this completes the ∈ J proof. N

Corollary IV.7. The projectivization P( ) of a rank two toric vector bundle over E E the projective simplicial toric variety X is a Mori dream space.

Proof. The ﬁnite generation of any Cox ring of P( ) in the sense of Hu and Keel is E a consequence of Theorem IV.6. Any simplicial toric variety is Q-factorial, and a projective bundle over a Q-factorial variety is again Q-factorial, hence the additional conditions follow at once from the hypotheses.

In spite of having these several distinct approaches giving a positive answer to the ﬁnite generation of the Cox rings of projectivized rank two toric vector bundles

(namely, the argument in [HS10], the proof based on the main result of [Kn93] as pointed out in [HS10], and our proofs in Theorem IV.2 and Theorem IV.6), they all failed to extend to the higher rank case. Now we understand that the situation for higher ranks is quite diﬀerent. In the next chapter I present my joint work with

Hering, Payne and Süß,that gives negative answers to the finite generation question for Cox rings and pseudoeffective cones of for projectivized toric vector bundles of higher rank. CHAPTER V

Cox rings and pseudoeﬀective cones in higher ranks

In this chapter we give negative answers to the questions of ﬁnite generation of the Cox rings and pseudoeﬀective cones of projectivized toric vector bundles (see

Question 7.2 in [HMP10]). All results included in this chapter were obtained in collaboration with M. Hering, S. Payne and H. S¨uß,see [GHPS10]. Our examples and counterexamples will arise by studying a particular family of toric vector bundles that includes cotangent bundles, up to a twist. This family will consist of those toric vector bundles whose Klyachko ﬁltrations satisfy a condition that we will call (?)

(see Definition V.1). Our motivation for this definition comes from two independent collections of results. One the one hand we are motivated by the results of Hausen and Süßin [HS10] that under suitable conditions provide a presentation of the Cox ring of a variety with a torus action as an algebra over the Cox ring of another variety closely related to a certain Chow quotient prevariety associated to the given action. For our work in this chapter the special case of their result presented in

Proposition II.16 will suﬃce. On the other hand, we are motivated by results of

Mukai in [Muk04], Castravet and Tevelev in [CT06], and Totaro in [Tot08], regarding the finite and nonfinite generation of the Cox rings and pseudoeffective cones of blow ups of projective space at finite collections of points. Our goal will be to prove that

64 65 the projectivizations of toric vector bundles satisfying (?) contain a T -invariant open

subset where the restricted action is free, and such that this restricted action has

as geometric quotient equal to a projective space blown up at a ﬁnite collection of

points. Moreover, we will be able to vary the collection of points being blown up on

projective space by varying the initial data in the Klyachko ﬁltrations of the toric

vector bundle while still preserving the condition (?). Ultimately, using the results

of Hausen and S¨uß,we will obtain a presentation of the Cox ring of our projectivized

toric vector bundle as a polynomial ring over a blow up of projective space at ﬁnitely

many points. Therefore we get a complete understanding of the Cox rings associated

to projectivizations of toric vector bundles whose ﬁltrations satisfy the condition

(?), and in this way we obtain many new examples of both finite generation and nonfinite generation of these algebras. In addition, in some examples we will use the quotient map that we mentioned before to identify the Neron-Severi vector spaces of the projectivized toric vector bundle and the blow up of projective space, and we show that in those cases their pseudoeffective cones also get identified. Again, using the results of Mukai, Castravet-Tevelev, and Totaro that we mentioned before, we will also get examples of projectivized toric vector bundles where the pseudoeffective cone is not finitely generated. This second approach also gives a negative answer to the question about the finite generation of Cox rings, nonetheless the first approach additionally provides presentations of the Cox rings of many new varieties which is a fact of independent interest.

5.1 Preliminary Lemmas

In this Chapter we use the notation for toric varieties and toric vector bundles introduced in Section 2.2. Then, X = X(∆) will denote an n-dimensional toric 66 variety with d torus invariant divisors and will denote a toric vector bundle of rank E r 3 over X. ≥

Deﬁnition V.1. Let ρj j = 1, . . . , d be a collection of decreasing ﬁltrations of {E | } a vector space E, each of them indexed by the integer numbers. We say that the

ﬁltrations satisfy the condition (?) if they have the form

E for i 0,  ≤ ρj (5.1) (i) =  Ej for i = 1, E   0 for i > 1,    where each Ej is either 0 or a codimension one subspace of E, and all of the nonzero

Ej are distinct.

Now we prove that ﬁltrations of this form deﬁne a toric vector bundle, i.e. satisfy

Klyachko’s condition, exactly when for each maximal cone of the fan the hyperplanes

that occur in the ﬁltrations associated to its rays intersect transversely.

Lemma V.2. Let ρj j = 1, . . . , d be a collection of ﬁltrations as in Deﬁnition {E | } V.1 that satisfy the condition (?). Then, over a smooth complete nondegenerate toric

variety X these ﬁltrations satisfy Klyachko’s compatibility condition if and only if,

for each maximal cone σ ∆, the nonzero hyperplanes E such that ρ σ meet ∈ j j ⊆ transversely.

Proof. Let σ be a maximal cone in ∆. If the ﬁltrations satisfy Klyachko’s compat-

ibility condition, then there is a splitting E = G G such that if ρ σ 1 ⊕ · · · ⊕ r j ⊆ and E is nonzero, then E is the sum of r 1 of the G . Since the nonzero E are j j − i j distinct by hypothesis, after renumbering all the rays we can assume that

E = G G G . j 1 ⊕ · · · ⊕ j ⊕ · · · ⊕ r b 67

For each nonzero E such that ρ σ, and therefore they intersect transversely. j j ⊆ Conversely, let σ ∆ be any maximal cone. Since the nonzero E such that ρ σ ∈ j j ⊆ intersect transversely, after renumbering all rays in ∆ if necessary, we can choose

an ordered basis of E so that the one-dimensional coordinate subspaces L1,...,Lr

satisfy E = L L L , for each nonzero E such that ρ σ. Since j 1 ⊕ · · · ⊕ j ⊕ · · · ⊕ r j j ⊆ X(∆) is smooth, we can choose u , . . . , u M, so that for each ρ σ, the product b 1 r ∈ j ⊆ u , v is equal to one if E is nonzero and i = j, or equal zero otherwise. Then the h i ji j 6 characters u , . . . , u and the decomposition E = L L satisfy Klyachko’s 1 r 1 ⊕ · · · ⊕ r compatibility condition.

Remark V.3. Since at most r hyperplanes can meet transversely in a vector space of dimension r, for a collection of ﬁltrations satisfying (?) to also satisfy Klyachko compatibility condition over a smooth nondegenerate toric variety X(∆), it is necessary that for each cone in the fan ∆ at most r of the Ej associated to the rays of the cone

can be diﬀerent from zero. In particular, if all the Ej are nonzero, in order for the

filtrations to define a toric vector bundle over X(∆), it is necessary that r n. In ≥ this same setting, if the hyperplanes Ej are chosen in general position the condition r n is also sufficient. ≥

Lemma V.4. Let X be an arbitrary toric variety and let be a toric vector bundle E over X satisfying (?), or more generally satisfying that ev : H0(X, ) E is sur- t0 E 0 → 1 jective. Then the isomorphism ev− canonically induces a rational map ψ : P( ) t0 E 99K P(E). Moreover ψ is constant on T -orbits and it is a retraction of the inclusion

P(E) , P( ). → E|Uρ

Proof. For each T -invariant aﬃne open subset U of X, the evaluation map evt0 gives an isomorphism φ = ev : H0(U, ) E. For each such U, the composition of U t0 E 0 → 68

1 0 0 φ− with the inclusion H (X, ) , H (X, ), induces a rational map U E 0 → E

ψ : P( ) = Proj Sym Proj Sym E = P(E). U E|U U E|U 99K

The rational maps ψ := ψ can be glued to a rational map ψ : P( ) P(E), ρ Uρ E 99K

since they agree with ψT over some nonempty open subset. Fix a ray ρ and ﬁx

an equivariant trivialization = (div χu1 ) (div χur ) for some E|Uρ ∼ OX |Uρ ⊕ · · · ⊕ OX |Uρ u , . . . , u M. We have 1 r ∈

0 u1 ur H (U , ) = χ− k[ρ∨ M] χ− k[ρ∨ M] ρ E ∩ ⊕ · · · ⊕ ∩

= x k[ρ∨ M] x k[ρ∨ M] 1 ∩ ⊕ · · · ⊕ r ∩

u 0 where x = (0, . . . , χ− i ,..., 0) H (U , ) for each i = 1, . . . , r. The local trivi- i ∈ ρ E

alization induces isomorphisms P( ) = Proj k[ ρ∨ M ][x , . . . , x ] and P(E) = E|Uρ ∩ 1 r

Proj k[y1, . . . , yr], and in this coordinates ψρ corresponds to the algebra homomor-

phism

ψρ∗ : k[y1, . . . , yr] k[ ρ∨ M ][x1, . . . , xr] (5.2) −→ ∩ y χui x . i 7−→ i

ui This description implies that ψρ is constant on T -orbits since each χ xi is T - invariant, and that ψ is a retraction of the inclusion P(E) , P( ), as this ρ → E|Uρ last map corresponds to the algebra homomorphism

ψ∗ : k[ ρ∨ M ][x , . . . , x ] k[y , . . . , y ] ρ ∩ 1 r −→ 1 r w χ 1 for all w ρ∨ M, 7−→ ∈ ∩ x y . i 7−→ i

Notation V.5. When we have a toric vector bundle associated to some ﬁltrations E

satisfying (?) over the toric variety X = X(∆), we will renumber the rays ρ1, . . . , ρd of 69

∆ so that the E are distinct hyperplanes for 1 j s and are zero for s+1 j d. j ≤ ≤ ≤ ≤ For each 1 j s the hyperplane E of E corresponds to a point in the projective ≤ ≤ j space P(E) that we will denote by pj. We will denote by S the set whose elements

are the distinct points p , . . . , p . In addition, for each 1 j s we will denote by 1 s ≤ ≤

Fj each of the exceptional divisors corresponding to pj in the blow ups Blpj P(E) and BlS P(E).

To simplify the next part of our discussion, we assume that the fan ∆ of the toric variety X has rays as maximal cones. This does not change the Cox ring of

X and under the assumption of smoothness it will preserve the Picard group and the semigroup of effective divisors. Now let us consider a toric vector bundle over E X satisfying (?). For each ray ρ ∆ such that the ρ(1) = E is nonzero, fix ∈ E ρ an equivariant trivialization = (div χu1 ) (div χur ) for some E|Uρ ∼ OX |Uρ ⊕ · · · ⊕ OX |Uρ u , . . . , u M. Since satisfies (?), we can reorder u , . . . , u and assume that 1 r ∈ E 1 r u , v = 0 and h r ρi

u1, vρ = = ur 1, vρ = 1. h i ··· h − i Let us deﬁne the subvariety

u1 ur 1 Zρ = P ( X (div χ ) X (div χ − )) Uρ Dρ P( Uρ Dρ ) P( Uρ ). O ⊕ · · · ⊕ O | ∩ ⊆ E| ∩ ⊆ E| Form our work it will follow that Zρ is independent of the choice of equivariant

trivialization. Also, if p P(E) is the point corresponding to the hyperplane ρ ∈ ρ(1) = E of E, we define W = T p P( ), that is, W is the closure of E ρ ρ · ρ ⊆ E ρ the T -orbit of p P(E) P( ). Finally, we define Z = Z P( ) and ρ ∈ ⊆ E ρ ρ ⊆ E W = W P( ). Note that the closed subset Z W has codimensionS at least ρ ρ ⊆ E ∪ two inSP( ). The motivation behind these definitions is that Z is the indeterminacy E locus of the map ψ in the Lemma V.4 and after we remove W the inverse image of 70 the ideal sheaf of p ρ(1) = 0 is locally principal. { ρ | E 6 } Lemma V.6. Let X be a toric variety corresponding to a fan whose maximal cones

are rays and let be a toric vector bundle over X satisfying (?). Then the rational E map ψ from Lemma V.4 is a morphism on the open subset P( ) r Z and there E

exists a unique surjective morphism θ : P( ) r (Z W ) BlS P(E) that factors the E ∪ →

restriction of ψ to P( ) r (Z W ) through the blow up BlS P(E). E ∪ Proof. Let ρ be a ray in ∆ and let us consider the equivariant trivialization of E|Uρ

that we fixed earlier, and local coordinates P( ) = Proj k[ ρ∨ M ][x , . . . , x ] E|Uρ ∩ 1 r and P(E) = Proj k[y , . . . , y ] as in Lemma V.4. Since satisfies (?) the subspace 1 r E ρ(1) = E in filtration ρ is either zero or a hyperplane of E, and we will now E ρ E consider these two cases separately. In the first case, we have that χu1 , . . . , χur are units in k[ ρ∨ M ] and that P( ) is T -equivariantly isomorphic to U P(E) = ∩ E|Uρ ρ ×

Proj k[ ρ∨ M ][y , . . . , y ] with ψ corresponding to the projection. In particular, ∩ 1 r ρ

ψ is a morphism over P( Uρ ) r Zρ = P( Uρ ) r Z. We deﬁne Bρ = P(E) r S E| E| ⊆ 1 BlS P(E). We have that ψρ− (Bρ) = P( Uρ ) r (Z W ). In this case, it is clear E| ∪ that ψ : P( Uρ ) r (Z W ) Bρ factors through the blow up BlS P(E) via a | E| ∪ → unique morphism that we call θρ : P( Uρ ) r (Z W ) Bρ. We note that θρ is E| ∪ → 1 surjective and that θρ− (Bρ) = P( Uρ ) r (Z W ). In the second case, i.e. when E| ∪ the ﬁltration ρ contains a hyperplane, let us again choose coordinates P( ) = E E|Uρ

Proj k[ ρ∨ M ][x , . . . , x ] and P(E) = Proj k[y , . . . , y ] as in Lemma V.4. We ∩ 1 r 1 r

u1 ur will denote W P( ) by W 0. In these coordinates Z = V (χ x , . . . , χ x ) ρ ∩ E|Uρ ρ ρ 1 r and Wρ0 = V (x1, . . . , xr 1). Since satisﬁes the (?) condition, it follows that Zρ = − E

u1 V (χ , xr). The rational map ψρ is surjective, and from (5.2) its domain of deﬁnition is P( Uρ) r Zρ = P( Uρ) r Z. Then ψ is a morphism over P( ) r Z. Again E| E| E from (5.2), the inverse image under ψ of the ideal sheaf of p P(E) is locally ρ ρ ∈ 71

principal on P( Uρ) r (Zρ Wρ0), therefore by the universal property of blow ups E| ∪

there exists a unique morphism θρ : P( Uρ) r (Zρ Wρ0) Blpρ P(E) such that E| ∪ →

πρ θρ = ψρ P( Uρ)r(Zρ W ), where πρ : Blpρ P(E) P(E) is the blow up morphism. ◦ | E| ∪ ρ0 → 1 Let Bρ = Blpρ P(E) r pρ ρ0 ray of ∆, ρ0 = ρ and Eρ = 0 . Note that θρ− (pρ) = { 0 | 6 0 6 } 1 π− (Dρ) (P( Uρ) r (Zρ Wρ)), and also that for each ray ρ0 ∆, diﬀerent from ∩ E| ∪ ∈

ρ, and such that Eρ0 is nonzero we have

1 θρ− (pρ ) = Wρ (P( Uρ) r (Zρ Wρ)) = T pρ (P( Uρ) r (Zρ Wρ)). 0 0 ∩ E| ∪ · 0 ∩ E| ∪

1 Therefore θρ− (Bρ) = P( Uρ) r (Z W ). Now we consider the maps θρ 1 as E| ∪ { |θρ− (Bρ)} rational maps from P( ) to Bl P(E). These rational maps agree on the common E S 1 open subset π− (T ), since they clearly agree after composing them with the blow up map Bl P(E) P(E). Therefore they can be glued to a surjective morphism S →

θ : P( Uρ)r(Z W ) BlS P(E) that factors the restriction of ψ through BlS P(E) E| ∪ → as required.

In the next lemma we collect some properties of the morphism θ : P( ) r (Z E ∪ W ) Bl P(E) from Lemma V.6 that follow directly from its construction. This → S lemma will be used in Section 5.3 to relate the pseudoeﬀective cones of P( ) and E

BlS P(E).

Lemma V.7. The morphism θ : P( ) r (Z W ) BlS P(E) from Lemma V.6 E ∪ → satisﬁes the following properties:

1 (a) θ∗(Fi) = π− (Di) (P( ) r (Z W )), for each 1 i s. ∩ E ∪ ≤ ≤

(b) θ∗p∗(H) = (1), where H is the hyperplane class in P(E). O

(c) θ∗(C) = T C, for any prime divisor C on P(E) = P( ) P( ) with strict · E|t0 ⊆ E transforme C on BlS P(E).

e 72

Proof. Property (a) follows from the construction of θ, as we have seen that the

inverse image under ψ of the ideal sheaf of the point p P(E) is the ideal sheaf | i ∈ 1 of π− (Di) (P( ) r (Z W )) in P( ) r (Z W ). Property (b) follows from the ∩ E ∪ E ∪ functoriality of the Proj functor. Let C and C be as in (c). For each 1 i s, since ≤ ≤ 1 1 the divisor π− (Di) (P( )r(Z W )) maps onto Fi, then π− (Di) (P( )r(Z W )) ∩ E ∪ e ∩ E ∪

has coeﬃcient zero in θ∗(C). Since θ∗(C 1 ) = ψ∗(C 1 ), we just need to show |π− T |π− T

that ψ∗ (C) = T C. As in the construction of the map ψ in Lemma V.4, choose T · e e an equivariant trivialization = (div χu1 ) (div χur ) for some E|T ∼ OX |T ⊕ · · · ⊕ OX |T u , . . . , u M. We have that χu1 , . . . , χur are units in k[ M ] and it follows that 1 r ∈ P( ) is T -equivariantly isomorphic to T P(E) with ψ corresponding to the E|T × T projection. From this the conclusion of (c) follows.

Let X be a toric variety corresponding to a fan whose maximal cones are rays and

let be a toric vector bundle over X satisfying (?). Let U be the T -invariant subset E of P( ) obtained by removing the proper closed subsets Z, W and the preimages E

under π of the divisors Ds+1,...,Dd of X, i.e.

1 U = P( ) r (Z W π− (Di)) E ∪ ∪ s+1 i d [≤ ≤ Now we achieve our ﬁrst goal by proving that the action of T on U is free and that a projective space blown up at a ﬁnite collection of points is a geometric quotient for this restricted action.

Lemma V.8 (Geometric Quotient). The action of T on U is free and the morphism

θ : U Bl P(E) is a geometric quotient for this action. |U → S Proof. Let ρ be one of the rays ρ ∆, for some 1 i s. We continue using the i ∈ ≤ ≤ notion introduced in the proofs of the Lemmas V.4 and V.6. We claim that the map

θρ is the geometric quotient of the action of T on P( Uρ) r (Zρ Wρ). To see this E| ∪ 73

we ﬁrst show that θρ is a toric map. The variety P( Uρ) r (Zρ Wρ) admits a toric E| ∪ r 1 variety structure given as follows. Let e1, . . . , er 1 be the canonical basis of R − , and − r 1 let er = e1 er 1 and vρ = (vρ, e1 + + er 1) N R − . Then P( Uρ) is − − · · · − − ··· − ∈ × E| r 1 isomorphic to the toric variety associated to the fan in N R − whose maximal cones e × are the cones spanned by v together with any r 1 of the vectors (0, e ),..., (0, e ) ρ − 1 r (see 7 in [Oda78]). Note that both Z and W are invariant subvarieties for the toric § e ρ ρ variety structure of P( Uρ). The fan Σρ for the toric variety P( Uρ)r(Zρ Wρ) is the E| E| ∪ r 1 fan in N R − obtained from the fan of P( ) by removing the cones containing × E|Uρ either both of the vectors vρ and (0, er) or all of the vectors (0, e1),..., (0, er 1). − Let Λ be the complete fan in Rr 1 whose maximal cones are those cones spanned ρ e − by any r 1 of the vectors e , . . . , e . The toric variety associated to the fan Λ − 1 r ρ r 1 is isomorphic to P − . We get a toric structure on P(E) = Proj k[y1, . . . , yr] by identifying it with this toric variety via the unique isomorphism between them that

y y1 r 1 e1∗ er∗ 1 extends the isomorphism Spec k[ ,..., − ] Spec k[χ , . . . , χ − ], induced by yr yr → e yi the unique algebra map satisfying χ i∗ for each i = 1, . . . , r 1. In this way, 7→ yr − since the ideal of pρ in this coordinates is (y1, . . . , yr 1), the variety Blpρ P(E) is − r 1 identiﬁed with the toric variety associated to the fan Λρ0 in R − obtained from Λρ by the star subdivision corresponding to the vector (1,..., 1). It is now easy to see that the action of T on P( Uρ) r (Zρ Wρ) is free and that the map θρ is the toric E| ∪ map P( Uρ ) r (Zρ Wρ) = X(Σρ) X(Λρ0 ) = Blpρ P(E) induced by the projection E| ∪ → r 1 r 1 N R − R − . By Proposition 3.2 in [AHN99] this map is the geometric quotient × → of the action of the torus T on P( Uρ)r(Zρ Wρ). Consider now the open set Bρ of E| ∪ 1 BlS P(E) deﬁned in Lemma V.6 and satisfying that θρ− (Bρ) = P( Uρ) r (Z W ). E| ∪ Therefore we get a geometric quotient

θ = θρ : P( Uρ) r (Z W ) Bρ. | | E| ∪ −→ 74

We note that for ρ ρ , . . . , ρ , the open subsets B cover the variety Bl P(E) ∈ { 1 s} ρ S when put together, and the open subsets P( ) cover U when put together. Since E|Uρ the freeness of an action is a local property and the property of a map being a geometric quotient is local on the base, the desired conclusions follow.

5.2 Cox rings of projectivized toric vector bundles

In this section we use our lemmas from Section 5.1 to construct examples of toric

vector bundles of higher rank whose projectivizations have ﬁnitely and nonﬁnitely

generated Cox rings. We start with a result that gives presentations for the Cox

rings of projectivized toric vector bundles that satisfy the condition (?).

Theorem V.9. Let be a toric vector bundle over the smooth toric variety X E satisfying the condition (?). Then the Cox ring of P( ) is isomorphic to a polynomial E ring in d s variables over the Cox ring of Bl P(E). − S

Proof. Codimension two modiﬁcations to a smooth variety do not change its Cox ring.

Thus, we can assume that the fan of X has rays as its maximal cones. Likewise, we

can canonically identify the Cox rings of P( ) and P( ) r (Z W ), for the closed E E ∪ 1 sets Z and W that we introduced before. For each s + 1 i d the divisor π− (D ) ≤ ≤ i of P( ) is stabilized pointwise by the one parameter subgroup of T -corresponding to E 1 vj N. By Lemma V.8 the action of T on U = P( ) r (Z W π− (Ds+1) ∈ E ∪ ∪ ∪ · · · ∪ 1 π− (D )) is free and there exists a geometric quotient morphism θ : U Bl P(E). d | → S The desired relation between the Cox rings now follows from Proposition II.16.

The following theorem gives a negative answer to the question of ﬁnite generation

for the Cox rings of projectivized toric vector bundles (see Question 7.2 in [HMP10]).

1 1 1 Theorem V.10. Suppose that k is uncountable, d > r n, and r + d r 2 . Then ≥ − ≤ 75 there is a nonsplit toric vector bundle of rank r on X(∆) such that the Cox ring E of the projectivization P( ) is not ﬁnitely generated. E

Proof. We can take as in Theorem V.9 such that the Cox ring of Bl P(E) is not E S ﬁnitely generated [Muk04], and then the conclusion follows from that theorem.

Remark V.11. As pointed out by Hering, Mustat¸˘aand Payne in [HMP10], the ﬁnite

generation of the Cox rings of all projectivized toric vector bundles is equivalent to

the ﬁnite generation of the section rings of the canonical line bundles (1) for all O projectivized toric vector bundles (see Question 7.1 and Remark 7.3 in [HMP10]).

Therefore, Theorem V.10 also implies the existence of a toric vector bundle so E that the section ring H0(P( ), (m)) H0(X, Symm ) of the m Z 0 = m Z 0 ∈ ≥ E O ∼ ∈ ≥ E line bundle (1) on PL( ) is not finitely generated.L If is a toric vector bundle O E F and ,..., are line bundles on X that generate Pic(X) as a group, then the L1 Lh finite generation of the Cox ring of P( ) is equivalent to the finite generation of the F section ring of the line bundle (1) on the projectivization of the toric vector bundle O = . By choosing such that the Cox ring of its projectivization E F ⊕ L1 · · · ⊕ Lh F is not finitely generated we get the desired example.

Remark V.12. We notice that the Theorem V.10 holds when the Klyachko ﬁltrations

of the toric vector bundle contain suﬃciently many nontrivial subspaces in very gen-

eral position. In the following corollary to Theorem V.9, we show that Theorem V.10

1 1 1 is sharp in the sense that if r + d r > 2 and the nontrivial subspaces are in general − position, then the projectivization of any bundle of this form is a Mori dream space.

Corollary V.13. Suppose is given by ﬁltrations satisfying (?) with the hyperplanes E 1 1 1 Ei in general position. If r + d r > 2 then P( ) is a Mori dream space. − E

1 1 1 r 1 Proof. Suppose r + d r > 2 . Then the blow up of P − at d points in general position − 76

r 1 is a Mori dream space [CT06, Theorem 1.3], and then so is the blow up Bls P − of

r 1 P − at s points in general position, where s is the number of rays ρj such that Ej is nonzero. The corollary then follows immediately from Theorem V.9, which says

r 1 that the Cox ring of P( ) is ﬁnitely generated over the Cox ring of Bl P − . E s 5.3 Pseudoeﬀective cones of projectivized toric vector bundles

In this section we use our lemmas from Section 5.1 to construct examples of toric

vector bundles whose projectivizations have ﬁnitely and nonﬁnitely generated Cox

rings and pseudoeﬀective cones. We start with a result that relates the eﬀective cone

of a projectivized toric vector bundles that satisﬁes the condition (?) to the eﬀective

cone of the associated blow up of projective space that we have been considering.

Let X = X(∆) be a smooth toric variety and let be a toric vector bundle E over X that satisﬁes the condition (?). In Section 5.1 we constructed a canonical

rational map ψ : P( ) P(E), and proved the existence of a unique rational map E 99K θ : P( ) Bl P(E) that factors ψ through Bl P(E). The map θ is a morphism E 99K S S on the complement of a closed set of codimension at least two, and it is surjective

as a map from its domain to BlS P(E). Therefore, via pullback, θ induces a group

homomorphism

θ∗ : Pic(Bl P(E)) Pic(Domain(θ)) = Pic(P( )). S → E

Theorem V.14. Let X = X(∆) be a smooth toric variety and let be a toric vector E bundle over X that satisﬁes the condition (?). Then the eﬀective cone of P( ) is E

generated by the image under θ∗ of the eﬀective cone of BlS P(E) together with the

1 classes of the divisors π− (Di) such that Ei is zero, i.e.

1 Eﬀ(P( )) = θ∗(Eﬀ BlS P(E)) + Z 0 π− (Di) E ≥ · s+1 i d X≤ ≤ 77

Proof. We can assume that the maximal cones of the fan ∆ are rays. Since the map

θ is surjective it pulls back eﬀective divisors to eﬀective divisors, thus

1 θ∗(Eff Bl P(E)) + Z π− (D ) Eff(P( )). S · i ⊆ E s+1 i d X≤ ≤ 1 1 By Lemma II.18, the cone Eff(P( )) is generated by π− (D ), . . . , π− (D ) and by E 1 d the closures of the T -orbits of prime divisors of P(E) considered as subvarieties of of

P( ) via the identiﬁcation P(E) = P( ). Therefore, parts (a) and (b) of Lemma E E|t0 V.7 imply that

1 Eﬀ(P( )) θ∗(Eﬀ Bl P(E)) + Z π− (D ), E ⊆ S · i s+1 i d X≤ ≤ and the result follows.

In the next theorem we construct examples of projectivized toric vector bundles

whose pseudoeﬀective cone and Cox ring are not ﬁnitely generated. We show that

such toric vector bundles exist over any smooth toric variety whose fan has suﬃciently

many rays and has a cone containing each ray or its negative. For instance, one can

construct smooth projective toric varieties whose fan has arbitrarily many rays and

satisfy this condition on cones through sequences of iterated blowups of (P1)n.

1 1 1 Theorem V.15. Suppose that k is uncountable, d n > r n, and r + d n r 2 , − ≥ − − ≤ and assume that there is some cone σ ∆ such that every ray of ∆ is contained in ∈ either σ or σ. Then there is a nonsplit toric vector bundle of rank r on X(∆) − E such that the pseudoeﬀective cone and the Cox ring of P( ) are not ﬁnitely generated. E

Proof. We can assume that the cone σ is maximal and that it is spanned by the rays

ρd n+1, . . . , ρd. Choose a collection of ﬁltrations satisfying (?), such that Ed n+1,..., − −

Ed are zero and such that E1,...,Ed n are hyperplanes on E. Moreover, we choose − the hyperplanes E1,...,Ed n in very general position, so that we can assume that − 78 these filtrations satisfy the Klyachko compatibility condition (see Lemma V.2), and that the blow up BlS P(E) of P(E) at the collection of points S = p1, . . . , pd n has − a nonfinitely generated pseudoeffective cone (see Theorem II.14). The Neron-Severi

1 space N (BlS P(E))R = Pic(BlS P(E))R has a basis given by the pullback of the hyperplane class in P(E) and the classes F1,...,Fd n of the exceptional divisors, − and the Neron-Severi space N 1(P( )) = Pic(Bl P( )) has a basis given by the E R S E R 1 1 class of P( )(1) and the classes of the divisors π− (D1), . . . , π− (Dd n). By Lemma O E − 1 1 V.7, it follows that the map θ∗ : N (Bl P(E)) N (P( )) induced by the S R → E R

map θ∗ from Theorem V.14 is an isomorphism. Moreover, the form of the fan ∆

ensures that each divisor Dd n+1,...,Dd is an eﬀective combination of the divisors − 1 1 D1,...,Dd n, and hence that each divisor π− (Dd n+1), . . . , π− (Dd) is an eﬀective − − 1 1 combination of the divisors π− (D1), . . . , π− (Dd n). Theorem V.14 together with − property (a) from Lemma V.7, imply that

1 Eff(P( )) = θ∗(Eff Bl P(E)) + Z π− (D ) E S · i d n+1 i d − X≤ ≤ 1 θ∗(Eff Bl P(E)) + Z π− (D ) ⊆ S · i 1 i d n ≤X≤ −

θ∗(Eﬀ Bl P(E)) Eﬀ(P( )). ⊆ S ⊆ E

Hence, the isomorphism θ∗ identiﬁes both the semigroup of eﬀective divisors and the

pseudoeffective cones of Bl P(E) and P( ). It follows that the pseudoeffective cone S E and Cox ring of P(E) are not finitely generated as desired.

The constructions in our proofs of Theorem V.10 and Theorem V.15 involve the

choice of toric vector bundles that are very general in their moduli spaces, which in

principle leaves the ﬁnite generation questions open over more general ﬁelds and for

toric vector bundles that are determined by the combinatorial data of the fan. In

this direction, we know that the projectivization of the tangent bundle has a finitely 79 generated Cox ring as it was proved in [HS10, Theorem 5.8]. Nonetheless, cotangent bundles behave quite differently. In the final result of this dissertation, we show that even over more general fields there are smooth projective toric varieties whose projectivized cotangent bundle is not a Mori dream space.

Theorem V.16. Suppose that d 3 and the characteristic of k is not two or three. ≥ Then there exists a smooth projective toric variety X(∆) of dimension d over k

such that the Cox ring of the projectivized cotangent bundle on X(∆) is not ﬁnitely

generated.

Proof of Theorem V.16. We construct the required examples inductively, starting in dimension d = 3. The vectors v1 = (0, 0, 1), v2 = (0, 1, 0), v3 = (1, 1, 1) and v = ( 1, 2, 2) span the rays of a unique smooth projective fan in R3. We 4 − − − sequentially perform the star subdivisions of fans, associated to the rays spanned by v = v + v = (1, 1, 2), v = v + v + v = (0, 1, 1), v = v + v + v = (1, 0, 1), 5 1 3 6 1 4 5 − 7 3 4 5 v = v + v + v = (1, 1, 1), v = v + v = ( 1, 2, 1), v = v + v + v = 8 4 5 7 − 9 1 4 − − − 10 1 2 9 ( 1, 1, 0), v = v + v = ( 1, 1, 1), v = v + v = ( 1, 0, 1), v = v + v = − − 11 1 10 − − 12 2 11 − 13 2 12 v + v = ( 1, 1, 1) and v = v + v = (0, 1, 1), which correspond to successive 2 12 − 14 1 2 blow ups along smooth invariant subvarieties. Then, the vectors v1, . . . , v14 span the rays of a smooth projective fan ∆ in R3, which clearly does not contain any pair of opposite rays. Let X(∆) be the corresponding toric variety over the given ﬁeld k.

Let be the tensor product of the cotangent bundle Ω and the anticanonical E X(∆) class K of X(∆). The ﬁltrations corresponding to are given by − X(∆) E

M Z k for j 0,  ⊗ ≤ ρ (j) =  ρ⊥ for j = 1, E   0 for j > 1,    80

for each ray ρ ∆. Since ∆ contains no pair of opposite rays, the hyperplanes ρ⊥ ∈ are distinct, and this collection of ﬁltrations satisﬁes (?). By Theorem V.9, the Cox ring of P(Ω ) = P( ) is isomorphic to the Cox ring of P2 = P(M k) blown X(∆) ∼ E ⊗Z up at the collection of points corresponding to the collection ρ⊥ ρ ray of ∆ of { | } hyperplanes of M k. By Theorem II.15, the blow up of P2 at the nine points ⊗Z corresponding to v11⊥ , v12⊥ , v13⊥ , v6⊥, v1⊥, v14⊥ , v8⊥, v7⊥ and v3⊥, is not a Mori dream space,

and therefore the Cox ring of P(ΩX(∆)) is not ﬁnitely generated.

d Now, given a smooth projective fan ∆0 in R such that the Cox ring of P(ΩX(∆0)) is not ﬁnitely generated, there is a smooth projective fan ∆ in Rd+1 = Rd R with × rays spanned by (1,..., 1, 1), (1,..., 1, 1), and (v, 0) for each primitive generator − v of a ray in ∆0. The points (v, 0)⊥, for each primitive generator v of a ray in ∆0,

d lie on a hyperplane H in P . We let S0 be the set of points in H corresponding

d to ρ0⊥ ρ0 ray of ∆0 , and we let S be the set of points in P corresponding to { | }

ρ⊥ ρ ray of ∆ . As before, by Theorem V.9 the Cox ring of Bl H is isomorphic { | } S0 to the Cox ring of P(ΩX(∆0)), so it is not ﬁnitely generated. Then, by Lemma II.17

d we conclude that the Cox ring of BlS P is not ﬁnitely generated. By applying

Theorem V.9 again, we have that the Cox ring of P(ΩX(∆)) is isomorphic to the Cox

d ring of BlS P , and therefore it is not ﬁnitely generated. BIBLIOGRAPHY

81 82

BIBLIOGRAPHY

[AHN99] A. A’Campo-Neuen and J. Hausen, Quotients of toric varieties by the action of a subtorus, Tohoku Math. J. (2) 51 (1999), no. 1, 112. [BP04] V. Batyrev and O. Popov, The Cox ring of a del Pezzo surface, Arithmetic of higher- dimensional algebraic varieties, Progr. Math. 226 (2004), 85-103. [BCHM08] C. Birkar, P. Cascini, C. Hacon and J. McKernan, Existence of minimal models for varieties of log general type, preprint, arXiv:math/0610203.

[Cas08] A.-M. Castravet, The Cox ring of M¯ 0,6, preprint, to appear in Trans. Amer. Math. Soc., arXiv:0705.0070. [CT06] A.-M. Castravet and J. Tevelev, Hilberts 14th problem and Cox rings, Compos. Math. 142 (2006), no. 6, 14791498. [Cox95] D. Cox, The homogeneous coordinate ring of a toric variety, J. Algebraic Geom. 4 (1995), 17-50. [ELMNP05] L. Ein, R. Lazarsfeld, M. Mustat¸˘a,M. Nakamaye and M. Popa, Asymptotic invariants of base loci, Annales de l’institut Fourier, 56 no. 6 (2006), 1701-1734. [EKW04] J. Elizondo, K. Kurano and K. Watanabe, The total coordinate ring of a normal projective variety, J. Algebra 276 (2004), no. 2, 625637. [Ful93] W. Fulton, Introduction to toric varieties, Annals of Math. Studies, Vol. 131, Princeton Univ. Press, Princeton, 1993. [FH91] W. Fulton, J. Harris, Representation theory: a first course, Springer-Verlag, New York, 1991, Graduate Texts in Mathematics, No. 129. [Gon09] J. Gonzalez, Okounkov bodies on projectivizations of rank two toric vector bundles, preprint, to appear in J. Algebra, arXiv:0911.2287, 2009. [Gon10] J. González,Projectivized rank two toric vector bundles are Mori dream spaces, to appear in Comm. Algebra, preprint, arXiv:1001.0838v1, 2010. [GHPS10] J. Gonzalez, M. Hering, S. Payne and H. Süß,Cox rings and pseudoeffective cones of projectivized toric vector bundles, preprint, arXiv:1009.5238, 2010. [Har77] R. Hartshorne, Algebraic Geometry, Springer-Verlag, New York, 1977, Graduate Texts in Mathematics, No. 52. [HT04] B. Hassett and Y. Tschinkel, Universal torsors and Cox rings, Arithmetic of higher- dimensional algebraic varieties, Progr. Math., vol. 226, BirkhäuserBoston, Boston, MA, 2004, pp. 149173. [HS10] J. Hausen and H. Süß,The Cox ring of an algebraic variety with torus action, preprint, to appear in Adv. Math., arXiv:0903.4789v4, 2010. 83

[HMP10] M. Hering, M. Mustat¸˘aand S. Payne, Positivity for toric vector bundles,Ann. Inst. Fourier (Grenoble) 60 (2010), no. 2, 607640. [HK00] Y. Hu and S. Keel, Mori dream spaces and GIT, Michigan Math. J. 48 (2000) 331-348, Dedicated to William Fulton on the occasion of his 60th birthday.

[SYJ10] Shin-Yao Jow, A Lefschetz hyperplane theorem for Mori dream spaces, preprint, to appear in Math. Zeit., arXiv:arXiv:0809.4036v2. [KK08] K. Kaveh, A. Khovanskii, Convex bodies and algebraic equations on aﬃne varieties, preprint, arXiv:0804.4095.

[KM96] S. Keel and J. McKernan, Contractible extremal rays of M0,n, preprint (1996), arXiv:alggeom/9707016. [Kly90] A. Klyachko, Equivariant vector bundles on toral varieties, Math. USSR-Izv. 35 (1990), 337-375. [Kn93] F. Knop, Uber¨ Hilberts vierzehntes Problem f¨urVariet¨atenmit Kompliziertheit eins, Math. Z. 213 (1993) 33-35. [LP97] J. Le Potier, Lectures on vector bundles, Cambridge studies on advanced mathematics, No. 54, Cambridge university press, Cambridge, 1997. [LM08] R. Lazarsfeld and Mircea Mustat¸˘a,Convex bodies associated to linear series, preprint, arXiv:0805.4559. [Laz04] R. Lazarsfeld, Positivity in Algebraic Geometry, I & II, Ergebnisse der Mathematik und ihrer Grenzgebiete, Vols. 48 & 49, Springer Verlag, Berlin, 2004. [Muk04] S. Mukai, Geometric realization of T-shaped root systems and counterexamples to Hilberts fourteenth problem, Algebraic transformation groups and algebraic varieties, Encyclopaedia Math. Sci., vol. 132, Springer, Berlin, 2004, pp. 123129. [Oda78] Tadao Oda, Lectures on torus embeddings and applications, Tata Inst. Fund. Research, Bombay, No. 58, Springer-Verlag, Berlin-Heidelberg-New York, 1978. [Ok96] A. Okounkov, Brunn-Minkowski inequality for multiplicities, Invent. Math. 125 (1996) 405-411. [Ok03] A. Okounkov, Why would multiplicities be log-concave?, The orbit method in geometry and physics, Progr. Math. 213 (2003) 329-347. [Ott10] J. Ottem, On the Cox ring of P2 blown up in points on a line, To appear in Math. Scand. arXiv:0901.4277v5, 2010.

[Pay09] Sam Payne, Toric vector bundles, branched covers of fans, and the resolution property, J. Algebraic Geom. 18 (2009) 1-36. [Pay08] Sam Payne, Moduli of toric vector bundles, Compos. Math. 144 (2008) 1199-1213. [STV07] M. Stillman, D. Testa and M. Velasco, Gröbnerbases, monomial group actions, and the Cox rings of del Pezzo surfaces, J. Algebra 316 (2007), No. 2, 777-801. [Tot08] B. Totaro, Hilberts 14th problem over finite fields and a conjecture on the cone of curves, Compos. Math. 144 (2008), no. 5, 11761198.