Introduction to Toric Varieties. (AM-131), Volume 131 9781400882526

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Annals o f Mathematics Studies

N u m b er 131

T H E WILLIAM H. ROEVER LE C TU R E S IN GEO M ETRY

The William H. Roever Lectures in Geometry were established in 1982 by his sons William A. and Frederick H. Roever, and members of their families, as a lasting memorial to their father, and as a continuing source of strength for the department of mathematics at Washington University, which owes so much to his long career. After receiving a B.S. in Mechanical Engineering from Washington University in 1897, William H. Roever studied mathematics at Harvard University, where he received his Ph.D. in 1906. After two years of teaching at the Massachusetts Institute of Technology, he returned to Washington University in 1908. There he spent his entire career, serving as chairman of the Department of Mathematics and Astronomy from 1932 until his retirement in 1945. Professor Roever published over 40 articles and several books, nearly all in his specialty, descriptive geometry. He served on the council of the American Mathematical Society and on the editorial board of the Math­ ematical Association of America and was a member of the mathematical societies of Italy and Germany. His rich and fruitful professional life remains an important example to his Department.

This monograph is an elaboration of a series of lectures delivered by William Fulton at the 1989 William H. Roever Lectures in Geometry, held on June 5-10 at Washington University, St. Louis, Missouri.

Introduction to Toric Varieties

by

William Fulton

THE WILLIAM H. ROEVER LECTURES IN GEOMETRY W a s h in g t o n U n iv e r sit y

S t. L o u is

PR IN C ETO N UNIVERSITY PRESS PRINCETO N , N EW JER SEY 1993

Copyright © 1993 by Princeton University Press A L L RIGH TS R E SE R V E D Library o f Congress Cataloging-in-Publication Data Fulton, William. Introduction to toric varieties / by William Fulton, p. cm.— (Annals o f mathematics studies ; no. 131) Includes bibliographical references and index. ISBN 0-691-03332-3— ISBN 0-691-00049-2 (pbk.) 1. Toric varieties. I. Title. II. Series QA571.F85 1993 516.3*53— dc20 93-11045

The publisher would like to acknowledge the authors o f this volume for providing the camera-ready copy from which this book was printed Princeton University Press books are printed on acid-free paper and meet the guidelines for permanence and durability o f the Committee on Production Guidelines for Book Longevity o f the Council on Library Resources Second printing, with errata sheet, 1997 http://pup.princeton.edu Printed in the United States o f America 3 5 7 9

10 8 6 4 2

W illiam H. Roever 1874-1951

Dedicated to the memory of Jean-Louis and Yvonne Verdier

CONTENTS

C h a p te r 1

Definitions and exam p les

1.1 1.2

Introduction C onvex polyhedral cones

1.3

A ffin e toric va rieties

1.4

Fans and toric va rieties

20

1.5

Toric varieties from polytopes

23

C h a p te r 2

3 8 15

Singularities and com pactness

2.1

Local properties of toric varieties

28

2.2

Surfaces; quotient singularities

31

2.3

O n e - p a r a m e t e r subgroups; lim it points

36

2.4

Compactness and properness

39

2.5

Nonsingular surfaces

42

2.6

Resolution of singularities

45

C h a p te r 3

Orbits, topology, and line bundles

3.1

O rbits

51

3.2

F u n d a m en tal groups and Euler characteristics

56

3.3

Divisors

60

3.4 3.5

Line bundles Cohomology of line bundles

63 73

C h a p te r 4

M o m e n t m a p s and the tangent bun d le

4.1

The m an ifold w ith singular corners

78

4.2

M om ent m ap

81

4.3

Differentials and the tangent bundle

85

4.4

Serre d u a lity

87

4.5

Betti n u m bers

91

C h a p te r 5

In tersection th e o ry

5.1

Chow groups

96

5.2

Cohomology of nonsingular toric varieties

101

5.3

R ie m a n n -R o c h theorem

108 114

5.4

Mixed vo lu m es

5.5

Bezout th eore m

121

5.6

Stanley's th eore m

124

N o te s

131

R eferen ces

149

Index

151

ofN otation

Index

155 vii

PREFACE

A lge b ra ic g e o m e t r y has developed a great deal of m a c h in e r y for stu dyin g higher dimensional nonsingular and singular varieties; for ex a m p le , all sorts of cohom ology theories, resolution of singularities, Hodge t h e o r y , intersection th e o r y , R ie m a n n -R o c h theorem s, and va n is h in g theorem s.

There has been real progress r e c e n tly t o w a r d at

least a rough classification of higher dimensional varieties, p a rtic u la rly b y M o ri and his school. For all this — and for a n y on e learning algebraic g e o m e t r y — it is im p o r ta n t to h a v e a good source of exam ples. In in t r o d u c t o r y courses this can be done in several w ays.

One

can stu dy algebraic cu rves, w h e r e m u ch of the story of their linear sy s te m s (line bundles, p r o je c t iv e embeddings, etc.) can be w orked out ex plicitly for low genus/D

For surfaces one can w o r k out some of

the classification, and w o r k out some of the corresponding facts for the special surfaces one finds/2) A n o th e r approach is to study va rieties th a t arise in "classical" p r o je c t iv e g e o m e tr y : Grassmannians, flag va rieties, Veron ese embeddings, scrolls, quadrics, cubic surfaces, e tc/ 3) Toric v a rieties provide a quite different y e t e le m e n t a r y w a y to see m a n y ex am ples and ph enom ena in algebraic g e o m e tr y .

In the

general classification scheme, these va rieties are v e r y special. For ex a m p le , t h e y are all rational, and, although t h e y m a y be singular, the singularities a re also rational.

N evertheless, toric va rieties h a v e

provided a r e m a r k a b ly fertile testing ground for general theories. Toric va r ie tie s correspond to objects m u ch like the simplicial complexes studied in e le m e n t a r y topology, and all the basic conceptss on toric v a r ie tie s — m aps be tw ee n th em , line bundles, cycles, etc. (at least those com p atible w ith the torus action) — correspond to simple "simplicial" notions.

This m akes e v e r y t h in g m u ch m o re com putable

and co n crete than usual. For this reason, w e believe it provides a good com p an ion for an introduction to algebraic g e o m e t r y (but c e r ta in ly not 1

The n um bers re fer to the notes at the back of the book.

X

PREFACE

a substitute for the study of curves, surfaces, and p r o je c t iv e g eo m e try !). In addition, th ere are applications the other w a y , and interesting relations w ith c o m m u t a t iv e algebra and lattice points in polyhedra. The g e o m e t r y of toric varieties also provides a good m odel for h o w some of the compactifications of s y m m e t r ic va rie tie s look; indeed, this w as the origin of their study. Although w e w on 't stu dy c o m p a c ­ tifications in this book, knowing about toric v a rieties m akes t h e m easier to understand. The goal of this m ini-course is to develop the foundational m ateria l, w ith m a n y examples, and then to c o n ce n tra te on the topology, intersection th eory , and R ie m a n n -R o c h problem on toric varieties. These are applied to count lattice points in polytopes, and study vo lu m es of co n ve x bodies. The notes conclude w ith Stanley's application of toric va rieties to the g e o m e tr y of simplicial polytopes. Relations betw ee n algebraic g e o m e tr y and oth er subjects are em phasized, ev e n w h e n proofs w ith o u t algebraic g e o m e t r y are possible. W hen this course w as first planned th ere w as no accessible text containing foundational results about toric varieties, although th ere w as the excellent in trod u ctory s u r v e y by Danilov, as w ell as articles b y Brylinski, Ju rk iew itz, and Teissier, and m o r e technical m onographs by D em azu re, K em pf - Knudsen - M u m fo r d - Saint-Donat, A s h M u m fo r d - Rapoport -T a i, and Oda, w h e r e most of the results about toric va rieties appeared for the first t i m e . ^

Since then the excellent

book of Oda [Oda] has appeared. This allows us to choose topics based on their suitability for an in tro d u cto ry course, and to present t h e m in less than their m a x im u m generality, since one can find com p lete a rg u m en ts in [Oda]. Oda's book also contains a w e a lth of references and attributions, w h ich frees us fro m a ttem p tin g to g iv e com p lete references or to assign credits. In no sense are w e t r y in g to s u r v e y the subject.

Alm ost all of the m ateria l, including solutions to m a n y of the

exercises, can be found in the references.

W e m a k e no claim s for

originality, beyond hoping that an occasional proof m a y be sim p ler than the original; and some of the intersection t h e o r y on singular toric va rieties has not appeared before. These notes w e r e prepared in connection w it h the 1989 W illia m H.

PREFACE

xi

R o e v e r Lectures in G eo m etry at Washington U n iv e rs ity in St. Louis.

I

thank D. W rig h t for organizing those lectures. T h ey are based on courses taught at Brow n and the U n iv e rs ity of Chicago.

I a m grateful

to C. H. Clemens, D. Cox, D. Eisenbud, N. Fakhruddin, A. Grassi, M. Goresky, P. Hall, S. K im u ra, A. L andm an, R. Lazarsfeld, M. McConnell, K. Matsuki, R. Morelli, D. Morrison, J. P o m m e rs h e im , R. Stan ley, and B. Totaro for useful suggestions in response to these lectures, courses, and p r e lim in a r y versions of these notes. Readers can also thank H. Kley and oth er avid proofreaders at Chicago for finding m a n y errors. P a r tic u la r thanks are due to R. MacPherson, J. Harris, and B. Stu rm fels w ith w h o m I h a v e learned about toric va rieties, and V. Danilov, whose s u r v e y provided the model for these courses.

The

a uthor has been supported by the NSF. In r e w r itin g these notes several yea rs later, w e h a v e not a tte m p te d to include or s u r v e y recent w o r k in the s u b je c t .^ )

In the

notes in the back, h o w e v e r , w e Have pointed out a fe w results w h ich h a v e appeared since 1989 and are closely related to the text.

These

notes also contain references for m o r e com plete proofs, or for needed facts fr o m algebraic g eo m e try .

W e hope this will m a k e the text m o r e

accessible both to those using toric va rieties to learn algebraic g e o m e t r y and to those interested in the g e o m e t r y and applications of toric varieties.

In addition, these notes include hints, solutions, or

references for some of the exercises.

M arch, 1993

W illiam Fulton The U n iv e rs ity of Chicago

E R R A T A I thank J. Cheah, B. Harbourne, T. Kajiwara, and I. Robertson for these corrections.

Page

For

Read

12, line 12

Gordon

Gordan

15, line 16

convex subset

convex cone

30, line 7

integrally closed

integrally closed in C[M]

37, line 9

0*

38, line 13

|A|

63, line 23

abelian

a cone in A

abelian

if A contains a cone

of maximal dimension

Replace lines 13-15 with;

64

The group DivT (X) is torsion free since it is a subgroup of ©M/M(a).

Since some M(a) is 0, the embedding M-*.DivT (X)

must split,

so the cokernel Pic(X) is torsion free.

67,

line 26

-&2X

a 2x

70,

line 28

ma^

ma2

lines

mD

D

79,

19, 23, 24

September,

1996

William Fulton

Introduction to Toric Varieties

CHAPTER DEFINITIONS A N D

1.1

1 EXAMPLES

In trod uctio n

Toric va r ie tie s as a subject c a m e m o r e or less independently fr o m the w o r k of seve ra l people, p r im a r ily in connection w ith the study of co m p ac tifica tion prob lem s/1^ This com pactification description gives a simple w a y to say w h a t a toric v a r i e t y is: it is a n orm a l v a r i e t y th a t contains a torus action

T x X —» X

T as a dense open subset, together w ith an

of T on X

on itself. The torus

X

that extends the n a tu ral action of T

T is the torus

C*x . . . x C* of algebraic groups,

not the torus of topology, although the la tter will play a role here as well.

The simplest com p act exam ple is p r o je c tiv e space

P n, regarded

as the com p actifica tion of Cn as usual: (C * )n C

Cn C

lPn .

S im ila rly , a n y product of affin e and p r o je c t iv e spaces can be realized as a toric v a r ie t y . Besides its b r e v it y , this definition has the v ir t u e that it explains the original n a m e of toric va rieties as "torus embeddings."

U n fo r­

tu n a te ly , this n a m e and description m a y lead one to think that one would be interested in such va rieties only if one has a torus one w an ts to c o m p a c tify ; indeed, one m a y w o n d e r if there w ou ld n ’t h a v e been m o r e general in terest in this subject, at least in the West, if this na m e had been avoided.

The action of the torus on a toric v a r i e t y w ill be

im p o r t a n t, as w ell as the fact that it contains the torus as a dense open orbit, but the problem w ith this description is that it co m p letely ignores the relation w it h the simplicial g e o m e tr y that m akes their stu dy so interesting.

A t a n y rate, w e fa r p refer the n a m e "toric

varieties," w h ic h is becom ing m o r e com m on. In this in tr o d u c to r y section w e g ive a brief definition of toric va r ie tie s as w e w ill study them ; in the following sections these notions

3

4

SECTION

1.1

will be m a d e m o r e co m plete and precise, and the basic facts assumed h ere w ill be proved.

A toric v a r i e t y will be constructed fro m a la ttic e

N (w h ic h is isomorphic to

Z n for some

n),

and a fan

A

in

N,

w h ich is a collection of “strongly co n vex rational polyhedral cones" in the real v e c t o r space

a

Njr = NzlR, satisfying the conditions

analogous to those for a simplicial complex: e v e r y face of a cone in is also a cone in of each.

A,

and the intersection of tw o cones in

A s tr o n g ly c o n v e x r a tio n a l polyh e d ra l cone

a

A

A

is a face

in Njr is a

cone w it h apex at the origin, generated by a finite n u m b e r of vectors; "ra tio n a l” m ea n s th at it is generated by vectors in the lattice, and "strong" c o n v e x it y that it contains no line through the origin.

W e often

abuse notation by calling such a cone sim ply a "cone in N". Let denoted

M = Hom (N ,Z ) ( , >. If

v e cto rs in

denote the dual lattice, w ith dual pairing

cr is a cone in N,

the dual cone

Mg* th at are non n ega tive on

is the set of

a. This determ in es a

c o m m u t a t i v e sem igroup S q- =

av nM

=

{ u € M : ( u , v ) > 0 for all

v € a } .

This sem igrou p is fin ite ly generated, so its corresponding "group algebra"

C[SCT] is a fin itely generated c o m m u t a t iv e (C-algebra. Such

an algebra corresponds to an affine v a r ie t y : set Ua = If

t

is a face of

subalgebra of

a,

then

Spec(C[S(J]) .

SCT is contained in ST, so CfS^] is a

C[ST], w h ich gives a m a p

UT

Uff. In fact,

principal open subset of UCT: if w e choose u € Sa so that then

t

UT is a = a f l u x,

UT = {x € UCT : u (x ) * 0}. W ith these identifications, these affine

v a rieties fit tog eth er to fo rm an algebraic v a r ie t y , w h ich w e denote by X (A ).

(The "embedding" notation for this is T Ne m b ( A ) ,

follow this co nvention.)

but w e w o n ’t

Note that sm aller cones correspond to sm aller

open sets, w h ich explains w h y the g e o m e tr y in N is p re ferred to the eq u iva len t g e o m e t r y in the dual space W e tu rn to some simple examples.

M. For these, the la ttice

N is

taken w ith a fixed basis e^, . . . , en, w ith

Xj., . . . , X n the elem ents in

C[M]

n < 3, w e usually w r it e

Y,

corresponding to the dual basis. For

and

Z for the first three of these.

W e first consider some affine

X,

I NTRODUCTION

exam ples, w h e r e and

X (A )

A

consists of a cone

is the affine v a r i e t y

The origin

{0}

sem igrou p is all of

a

5

together w ith all of its faces,

UCT.

is a cone, and a face of e v e r y other cone. The dual M,

w ith generators

±e£, . . . , ± e * ,

so the

corresponding group algebra is

CtM]

=

C [X i, x r 1, X 2, X 2_1

X n> X n-1] ,

w h ic h is the affine ring of the torus: U { q} = T = ( C * )n. So e v e r y toric v a r i e t y contains the torus as an open subset. If

a

is the cone generated by

e j, . . . , e n, then

Sa is generated

b y the dual basis, so c [ s CTi =

m

l f X 2, . . . , X n] ,

w h ic h is the affine ring of affin e space: UCT = Cn. For a n o th er ex am ple take and

n = 2, and take

a

generated by

2ei - e2.

Sem igrou p generators for C[Sa ] = Hence,

Sa are

^

^

^

e^, e^ + e 2 and

C[X, X Y , X Y 2] =

X

X

e^ + 2e2, so

C [U ,V ,W ]/ (V 2 - U W ) .

is a q u a d r i c cone, i.e., a cone o v e r a conic:

e2

6

SECTION

1.1

N ext w e look at a fe w basic exam ples w h ich are not affine. For n = 1,

the on ly n on -affin e ex am ple has

IR>0, iR and C, and

{0},

A

consisting of the cones

w h ich correspond to the affine toric va rieties

€,

C*. These t h ree cones fo r m a fan, and the corresponding toric

v a r i e t y is constructed fr o m the gluing: — ----- •------------ ►

^

C [x - 1 ] C* C tX .X - 1 ]

x of €

Ua3 and

w ith

C. The

Ua4) patch together to the

C * IP*, so all together w e h a v e a

P * - b u n d le o v e r

rational ruled surfaces are som etim es denoted

IP*. These

F a, and called

H irze b ru c h surfaces. Exercise.

Id e n tify the bundle

Fa

P*

w ith the bundle

P (0 (a )® ll)

of lines in the v e c t o r bundle that is the sum of a t r iv ia l line bundle and the bundle

0 (a )

on

P *.^ )

Each of the four ra y s

determ in es a c u r v e

t

DT in the surface.

Such a c u r v e w ill be contained in the union of the t w o open sets for the t w o cones

a

of w h ich

t

is a face, m eetin g each of th em in a

c u r v e isomorphic to C, glued together as usual to fo rm equation for

Dx n U a in

g en e ra to r of

SCT that does n o t vanish on

the r a y through on

U„

= Spec(C [X,Y]),

Exercise. V e r i f y that in

s C2 is % u = 0, w h e r e

e 2 , the c u r v e and

t

P*.

The

u is the

. For exam ple, if

t

DT is defined by the equation

is Y = 0

X aY = 0 on UCT4 = Spec(£[XaY , X " 1]).

DT s IP1. Show that the n orm a l bundle to

F a is the line bundle

is -a.

Ua

0 (- a ),

so the self-intersection n u m b e r

DT

(D*D)

Find the corresponding nu m bers for the other three r a y s . ^

Beginners are encouraged to ex perim en t before going on. See if you can find fans to construct the following varieties as toric varieties: P n, the b lo w -u p of Cn at a point, € * P *, pa x pb

P* x p *,

, and

vVhat ar? all the one-dim ensional toric varieties?

Construct

some oth er tw o-d im en s ion a l toric varieties.

1.2

C on vex p o ly h e d ra l cones

W e include h ere the basic facts about co n vex polyhedral cones that will be needed. These results can be found in their n a tu ral g en e ra lity in a n y book on c o n v e x i t y / 4^ but the proofs in the polyhedral case are so simple that it is n e a r ly as easy to p ro v e th e m as to quote texts.

We

include proofs also because t h e y show how to find generators of the semigroups, w h ic h is w h a t w e need for actual computations.

CONVEX

Let

V

P OLYHEDRAL

CONES

9

be the v e c t o r space N(r, w ith dual space V * = M|r. A

c o n v e x p o ly h e d ra l cone is a set a

=

( r i v * + . . . + r s v s C V : r* > 0 }

generated by a n y finite set of vectors

vj_,... , v s in V.

Such vectors,

or som etim es the corresponding r a y s consisting of positive multiples of some

vj,

are called ge n e ra to rs for the cone

a.

W e will soon see a dual description of cones as intersections of h a lf­ spaces. The dim ension d i m ( a ) space

lR*a = a + ( - a )

of

spanned by

a

is the dimension of the linear

a. The dual

a v of a n y set

a

is

the set of equations of supporting hyperplanes, i.e., ctv

=

(u € V * : > 0 for all

v € a} .

E v e r y th in g is based on the following fu n d am en ta l fact fro m the th eory of co n v e x sets/5^ (*) som e

If o

is a c o n v e x p olyh edra l cone and vq t cr,

uq €

w ith

then th e re is

< 0.

W e list some consequences of ( * ) .

A direct translation of ( * ) is the

d u a lity t h e o r e m : (1 )

(crv )v

A face hyperplane:

=

cr. t t

of

a

is the intersection of

a

w ith a n y supporting

= a f l u 1 = { v € a : < u ,v> = 0 }

for some

u in

a v. A

cone is regarded as a face of itself, w hile others are called p r o p e r faces. Note that a n y lin e a r subspace of a cone is contained in e v e r y face of the cone. (2)

A n y face is also a c o n v e x po ly h e d ra l cone.

10

SECTION

The face for

a

1.2

a f l u 1 is generated by those vectors

such that

< u , V } > = 0.

vj

in a generating set

In particular, w e see that a cone has

o nly fin it e ly m a n y faces. (3)

A n y in te rs e c tio n of faces is also a face.

This is seen from the equation (4)

PKanu^) = a fld u jh

for

uj € 0 and

take

on the a,

and so

= 0, so < 0,

a contradiction. The proof gives a practical procedure for finding generators for the dual cone

a v. For each set of n-1

independent vectors am ong

the generators of

cr, solve for a v e c to r

neith er

is non nega tive on all generators of

u or

-u

carded; otherw ise either the

n-1

uT above. (9)

u or

ve ctors are in a facet

-u t

,

u annihilating the set; if a

it is dis­

is taken as a generator for

a v; if

this v e c to r will be the one denoted

From (8) w e deduce the fact known as Farkas' Theorem :

The dual of a c o n v e x po ly h e d ra l cone is a c o n v e x po lyh e d ra l

cone. If

cr spans V,

the vectors

uT generate

w e r e not in the cone generated by the th ere is a v e c t o r

v

in V

w ith

crv'; indeed, if u in

uT, applying ( * ) to this cone,

> 0 for all facets

t

and

12

SECTION

1.2

< u ,v> < 0, and this contradicts (8). If W = IR’ cr, then

a ''

a

spans a sm aller linear space

is generated b y lifts of generators of the dual cone

in W * = V*/W-L, together w ith ve ctors a basis for W x.

u and

-u

as u ranges ov e r

This shows that polyhedral cones can also be given a dual definition as the i n t e r s e c t i o n of h a l f - s p a c e s : f o r g e n e r a t o r s u l t of

. ,u t

o'*, cr *

( v € V : > 0 , . . . , > 0 } .

If w e now suppose can be taken fr o m

N,

cr is rational, m eaning that its generators then

crv is also rational; indeed, the above

p rocedure shows h ow to construct generators P r o p o s it io n 1.

(Gordon’s L e m m a )

p o ly h e d ra l cone, then P r o o f.

Take

K fiM

Indeed, if u is in w it h

mj

is a fin ite ly genera ted semigroup.

cr^OM

K = { S t j U j : 0 < tj < 1). Since intersection

that generate

K is com pact and

is finite. Then a v OM, w r it e

KnM

u* and

u' = S tjU j

as a cone. Let M

is discrete, the

generates the semigroup.

u = Z r^ i,

a n on n eg a tiv e in teger and

w it h each

cr^nM.

I f cr is a ra tio n a l c o n v e x

Sa = a v H M

u*, . . . , u s in

uj in

rj > 0, so ri = m j + tj

0 < tj < 1. Then

u = Z m jU j + u1,

in K n M .

It is often necessary to find a point in the r e la t iv e i n t e r io r of a cone

a,

i.e., in the topological interior of

spanned b y dim(cr)

a.

If

in the space

t

a

is a face of cr,

then

faces of ctv . The smallest face of o

This sets up a o n e - t o - o n e

is

To see this, note first that the faces of v € a = ( a v ) v. If

r e la tiv e interior, then o'*

In

cr^OT-1- is a face of o'*, with

o r d e r - r e v e r s i n g c orresp ond ence betw een the faces of a

a v f l v x for

a.

is rational, w e can find such points in the lattice.

d i m ( x ) + d i m ( a v' n T x ) = n = d im (V ).

face of

{R»cr

lin e a r ly independent ve ctors am ong the generators of

p a rtic u la r, if (10)

a

This is a chieved by taking a n y positive com bination of

t

and the

a O (-a ). o”

are ex a c tly the cones

is the cone containing

v

in its

a v f l v x = a v f l ( T v/O v x) = a v f l x x , so e v e r y

has the asserted form .

The m a p

t

»-»

t

*

=

cle a rly o r d e r -re v e rs in g , and fr o m the obvious inclusion

a v f l x x is t

C

(t *)*

it

CONVEX

POLYHEDRAL

CONES

13

follows f o r m a l ly that

t * = ((t*)*)*,

o n e -to -o n e and onto.

It follows from this that the smallest face of

o

and hence that the m a p is

is ( a ' T f l t a T = ( a v')J- = a n ( - a ) .

In particular, w e see that

d i m ( a n ( - c r ) ) + d i m ( a v ) = n. The corresponding equation for a general face a,

t

can be deduced by putting

t

in a m a x im a l chain of faces of

and co m p arin g w ith the dual chain of faces in

(11)

I f u € crN/, and x = a f l u x, then

xv =

o'*.

+ IR>q « ( - u ).

Since both sides of this equation are co n ve x polyhedral cones, it is enough to show that their duals are equal. The dual of the left side is t

, and the dual of the right is c r n ( - u )v = a D u A, as required.

P r o p o s it io n 2. L e t u be in

o

Sa = a v f)M .

p o ly h e d r a l cone.

be a ra tio n a l c o n v e x p o ly h e d ra l cone, and let Then

x = crflu x is a ra tio n a l c o n v e x

All faces of a ST =

P r o o f.

If

in terior of rational.

is a face, then

t

a v n T A, and

Given

and taking

h a v e this f o r m , and SCT +

.

= a f l u x for a n y

t

u can be taken in

w € ST,

then

w + p-u

M

is in

p to be an integer shows that

w

u in the r e la tiv e since

a v D T A is

for large

positive p,

is in Sa + Z> q* (- u ).

Finally, w e need the following strengthening of ( * ) , known as a S eparation L e m m a , that separates co n vex sets by a hyperplane: (12) t

If o

and

cr* a re c o n v e x polyh e d ra l cones whose in te rs e c tio n

is a face o f each, then th e re is a u in t

=

a flu x =

This is p rov ed by looking at the cone that for a n y face of

a N/n ( - a ,)v' w ith

cr'flu1 .

y = a - a ‘ = a + (-cr*). W e know

u in the r e la tiv e interior of

yv , y flu x is the smallest

y: y nuA =

The claim is that this

u

y n( - y )

= ( a - a ,) n ( a ' - a ) .

works.

Since o

a v , and since

x

C onversely, if

v € a (1u A, then

v = w' - w,

w ' € a',

is contained in

y,

uis in

yn(-y), x is contained in a(1uA.

is contained in v

w € (j. Then

is in

o ' - a,

v + w

so there is an equation

is in the intersection

x

of

14

SECTION

cr and

1.2

cr', and the sum of t w o elem ents of a cone can be in a face

only if the su m m an d s a re in the face, so v

is in

oT lu x =

shows that

t

, and the sam e a rg u m e n t for

P r o p o s it io n 3.

If a

whose in te rs e c tio n

and

ST 3 SCT +

(13)

.

is obvious. For the other inclu­ u in

a v O (-cj,) v n M

= cjn u x = a ' f l u x. By Proposition 2 and the fact that

we have

t

SCT + Sa» .

sion, b y the proof of (12) w e can take t

a 'flu -1* =

is a face of each, then

t

One inclusion

. This shows that

a 1 a re r a tio n a l c o n v e x p o ly h e d ra l cones

ST = P r o o f.

-u

t

so that

-u

is in Sa',

ST C Sa + Z>q*(-u) C Sa + Sa«, as required.

F o r a c o n v e x p o ly h e d ra l cone a ,

the following conditions are

e q u iv a le n t: (i)

a n ( - c r ) = (0 );

(ii)

a

con tains no n o n ze ro lin e a r subspace;

(iii)

th e re is a

u

(iv)

crv spans

V*.

in

a'"' w ith

The first t w o are eq u iva len t since a;

a f l u x = (0);

CTfl(-a)

the second t w o are equ ivalent since

is the largest subspace in

a fl(-a )

is the smallest face of

cr. The first and last are equ iva lent since dim( 0

AFFI NE

for all (v )

u in

for all

w ith

av '

a x; (iii)

x € a

TORIC

V ARI ET IES

a v f l v x = a x; (i v )

15

a + IR>q* (- v ) = IR-cr;

th ere is a positive nu m b er p and a y

in cr

p-v = x + y. (6)

Exercise.

If

v e cto rs in

is a face of a cone

t

cr can

be in

cr, show that the sum of tw o

only if both of the su m m and s are in

t

Show c o n v e rs e ly that a n y co n ve x subset of a cone

t

.

cr satisfying this

condition is a face. Since w e a re m a in ly concerned w ith these cones, w e w ill often say

"a

is a cone in N"

polyh edral cone in to m e a n th at

t

to m ea n that

cr is a strongly co n ve x rational

Njr. W e will som etim es w r it e

is a face of

" t •< cr" or

"cr >

t

"

cr. A cone is called simplicial, or a

sim plex, if it is generated by linearly independent generators. Exercise.

If

cr spans

N jr, m ust

cr and

cr''

h a v e the sam e m in im a l

n u m b e r of generators? ^

1.3

A ffin e to ric v a r ie tie s

When th at

a

is a strongly co n v e x rational polyhedral cone, w e h a v e seen

SCT = a v flM

is a fin itely generated semigroup. A n y additive

sem igroup

S de te rm in es a "group ring"

C-algebra.

As a com plex v e c to r space it has a basis

over

C[S], w hich is a c o m m u t a t iv e %u, as u varies

S, w ith m ultiplication determ in ed by the addition in

S:

__ ^ u + u' The unit

1 is

generators

X°*

Generators

( X Ui) f ° r the

{uj}

C-algebra

for the semigroup

S d e te rm in e

C[S].

A n y fin ite ly generated c o m m u t a t iv e C-algebra co m p lex affine v a r ie t y , w h ich w e denote by

Spec(A).

A

determ in es a

We r e v i e w this

construction and its related notation /8^ If generators of A chosen, this presents then space

Spec(A )

A

as C[X*, . . . ,Xm]/I, w h e r e

can be identified w ith the s u b v a r ie ty

Cm of c o m m o n zeros of the polynom ials in

are

I is an ideal; V (I )

of affin e

I, but as usual for

m od e rn m a th e m a tic ia n s , it is convenient to use descriptions that are

16

SECTION

independent of coordinates. so S p ec(A )

1.3

In our applications,

w ill be an irreducible v a r ie t y .

includes all p r im e ideals of

A

A

Although

will be a domain, Spec(A )

officially

(corresponding to subvarieties of V (I )),

w h e n w e speak of a p o in t of Spec(A)

w e will m ea n an o r d in a r y closed

point, corresponding to a m a x im a l ideal, unless w e specify otherwise. These closed points a re denoted A

B of

varieties.

A n y h om om orp h ism

C-algebras determ in es a m orphism

f € A

of

A

to €. If

X = Spec(A),

for each nonzero

the principal open subset Xf

=

S pec(A f)

c

X

=

Spec(A)

corresponds to the localization hom om orp h ism For

Spec(B) -> Spec(A )

In particular, closed points correspond to C-algebra

h o m o m o rp h ism s fr o m ele m en t

S pecm (A ).

A = C[S]

A »-» Af.

constructed fr o m a semigroup, the points are easy

to describe: t h e y correspond to hom om orp h ism s of semigroups fr o m to

C, w h e r e

C = C* U {0}

S

is regarded as an abelian semigroup via

m ultiplication: S p e c m (C [S ]) For a sem igroup h o m o m o rp h ism

=

x fr o m

v a lu e of th e corresponding function Specm (C[S]) When cone, w e set

H o m sg( S , C ) . u in S, the

%u at the corresponding point of

is the im ag e of u by the m a p

x:

%u(x ) = x(u).

S = SCT arises fr o m a strongly co n ve x rational polyhedral A c = C[Sa], and U CT -

SpectClS*])

=

the corresponding affine to ric v a rie ty . sub-sem igroups of the group and

S to C and

e£, . . . , e*

As a sem igroup,

M

C[M]

=

has generators = =

All of these semigroups will be

M = S{q}. If e*, . . . , e n is a basis for

is the dual basis of Xj

S pec(A a) ,

M, w r it e € C[M] . ±e£, . . . , ± e * ,

so

C [X 1, X i - 1.X 2 .X 2- 1, . . . . X n . X n " 1] C [ X i , . . . , X n] X r

w h ic h is the ring of L a u r e n t p o ly n o m ia ls in

.Xn ,

n variables. So

N,

AFFINE

U {0}

=

TORI C

Spec(C[M])

s

V ARI ET IES

C* x . . . x C*

is an a ffin e algebraic torus. All of our semigroups sem igroups of a lattice

M,

17

=

(C * )n

S will be sub­

so C[S] will be a subalgebra of C[M];

p a rtic u la r, C[S] will be a domain.

W hen a basis for

M

in

is chosen as

a bove, w e usually w r it e elem ents of C[S] as Lauren t polynom ials in the corresponding variables

Xj. Note that all of these algebras are

g en erated b y m o n o m ia ls in the variables The torus

T = Tjsj

X\.

corresponding to M

or

N can be w r it t e n

intrinsically: Tn

=

Spec(C[M ])

For a basic exam ple, let e l> •• • » e k

f ° r some k,

S0'e2 + • • • + 2 >o*e^ +

2 -e £ +1+ . . . + 2 -e* .

A „ = S' determ in es a h o m o m o r p h is m Spec((C[S'])

C[S] —> €[S'] of algebras, hence a m orphism

Spec(C[S])

->

then

of affine varieties.

In particular, if

contained in

a,

a m o r p h is m

Ux -* UCT. For exam ple, the torus Tjsj = U{o)

of the a ffin e toric va rieties

UCT that com e fr o m cones

Lem m a.

a,

If

t

is a

t

is

SCT is a sub-sem igroup of ST, corresponding to

face of

then the m a p

a

m aps to all in N.

UT —> Ua embeds

UT as a p rin c ip a l open subset of U^. P r o o f.

By Proposition 2 in §1.2, th ere is a u € SG w ith

t

= o flu 1

and ST =

Sq- + Z>o*(~u) .

This implies im m e d ia t e ly that each basis elem ent for C[ST] w r it t e n in the fo rm

x w _ p u = X W / ( X U) P for A-j- -

can be

w € SCT. Hence

( A a) ^ u f

w h ic h is the algebraic version of the required assertion. Exercise.

Show th at if

em bedding, then

t

c a

and the m apping

m ust be a face of

M o re generally, if that

t

UT -» UCT is an open

a. ^

tp: N' -> N is a h om om orp h ism of lattices such

cpm m aps a (rationa l strongly co n vex polyh edral) cone

into a cone

a in N, then the dual

d e te rm in in g a h o m o m o rp h ism Uff' -

A CT

q)v : M -> M' maps A a n d

hence

a'

in N'

SCT to SCT»,

a m orphism

Uff.

Exercise.

Show th at if

corresponding m a p S and

S'

i.e., if

are sub-semigroups, the

Spec(C[S']) -» Spec(€[S])

gen e rate the sam e subgroup of

The semigroups respects.

S C S' C M

is birational if and only if

M.

Sa arising fr o m cones are special in several

First, it follows fr o m the definition that

p*u

is in Sa for some positive integer

p,

SCT is saturated, then u is in Sa.

In

AFFINE

TORIC

V ARI ET IES

19

addition, the fact that

cr is strongly co n vex implies that

M|r,

M

Sct generates

so

M Exercise. of

M

= Sa + (- S a) .

Show c o n ve rs ely that a n y fin itely generated sub-sem igroup

th a t generates

crv n M

a v spans

as a group, i.e.,

M

as a group and is saturated has the fo rm

for a unique strongly co n ve x rational polyhedral cone

a

in N.

The following exercise shows that affine toric va rieties are defined b y m o n o m i a l equations. Exercise.

If

Sa is generated b y A ct =

show th at

u j, . . . , u*,

C[%ul, . . . , %ut] = C

Y

Y

t] / 1 ,

I is generated by polynomials of the fo rm Y i al . Y 2a2- . . . - Y tat -

w here

[

so

Y ! b l -Y 2b2- . . . - Y tbt ,

a^, . . . , a*, b j, . . . , bt are non negative integers satisfying the

equation a i u i + . . . + a t ut = b j u i + . . . + btut . If

a

is a cone in N, the torus

Tj^ acts on

TN x U ct “♦ as follows.

A point

groups, and a point product

t-x

x € UCT w ith a m ap

is the m a p of semigroups

The dual m a p on algebras, X UXU for

C[SCT]

Exercise.

a

If

of semigroups; the

SCT -» C given by

C[Sff]® C [M ], a = (0),

is given by m apping

this is the usual product

T jsj. These maps are compatible w ith inclusions

of open subsets corresponding to faces of Tn

SCT -» €

M —» (Cw of

t(u )x (u ) .

u € Sa. W hen

in the algebraic group

the action of

,

t € Tfyj can be identified w ith a m a p

u

X u to

Uff,

a.

In particular, t h e y extend

on itself. is a cone in

N and

a'

is a cone in

N',

show that

a x a 1 is a cone in N 0 N 1, and construct a canonical isomorphism

20

SECTION

1.4

^ f f x a 1 ~’ ^ r for all

v

in K; K is

usually included as an im p ro p er face. W e assume for sim p licity that K is n -dim ensional, and that fa c e t of

K contains the origin in its interior.

K is a face of dimension

A

n-1. The results of §1.2 can be

used to deduce the corresponding basic facts about faces of co n v e x polytopes. E x IR.

For this, let

The faces of

faces of

cr be the cone o v e r

K x 1 in the v e c t o r space

cr are easily seen to be e x a c tly the cones o v e r the

K (w it h the cone

{0}

corresponding to the e m p t y face of K);

fr o m this it follows that the faces satisfy the analogues of properties (2 ) - ( 7 ) of §1.2. As fo r the d u a lity t h e o r y of polytopes, the p o la r set (or p o la r) of K is defined to be the set K° (Often

=

(u € E* : < u , v > > -1

{u € E* : < 1

V v € K} = -K°

set, but this does not change the results.) octahed ron in (0,0,±1)

P r o p o s it io n .

v € K) .

is taken to be the polar

For exam ple, the polar of the

IR3 w ith v e rtic es at the points (±1,0,0), (0,±1,0),

is the cube w ith ve rtic es

p o la r of K°.

for all

The p o la r set K°

is a c o n v e x polytope, and K is the

I f F is a face of K, F*

is a face o f K°,

=

and

(±1 ,±1 ,±1 ).

then

{ u € K° : < u , v > = -1

and the corresp ond ence F

V v € F} F*

is a o n e -t o -o n e ,

o r d e r - r e v e r s i n g corre s po n d e n ce betw een the faces of K and the faces of K°,

w ith

d im (F ) + d im (F * ) = dim (E) - 1. I f K is rational,

i.e., its v e rtic e s lie in a la ttic e in

E,

then

K° is also ratio n a l, w ith

its v e rtic e s in the dual lattice. P r o o f.

W ith

a

the cone o v e r

K x 1, the dual cone

a '/ consists of

TORI C

those

u xr

in

V A R I ET I E S

F RO M

P OL Y T O F E S

25

E* x (R such that + r > 0 for all v

follows th at

is

the cone o v e r

K° x 1

in

in K. It

E* x (R. The assertions of

the proposition are now easy consequences of the results in §1.2 for cones.

For exam ple, the du ality

(crN') N' = a. dual

For a face

F of K, if

a v f l T x is the cone o v e r

faces of Exercise.

K and Let

K°

t

is the cone o v e r

F x l f then the

F* x 1, from w hich the du ality betw een

follows.

K be a c o n v e x po ly h e d ro n in

K = for som e

(K °)° = K follows fro m the duality

E, i.e.,

( v € E : < u i , v > > -a *, . . . , > - a r )

u i, . . . , u r

in E* and real n u m bers

a*, . . . , a r . Show that

K is bounded if and only if K is the co n ve x hull of a finite set/16^ A rational co n ve x polytope

K in N|r determ in es a fan

A

whose

cones a re the cones o v e r the proper faces of K. Since w e assume that K contains the origin in its interior, the union of the cones in be all of

A

will

N jr. All of the fans w e h a v e seen so fa r whose cones c o v e r

Nr

h a v e this form . M o re generally, if K1 is a subdivision of the bo u nd a ry of K, i.e., K1 is a collection of co n ve x polytopes whose union is the bo u n d a ry of K, and the intersection of a n y t w o polytopes in K1 is a polytope in K1, then the cones o v e r the polytopes in K* fo rm a fan. Here are some exam ples of such

K1:

Note th a t the second can be "pushed out", so that the cone o v e r it is the cone o v e r a co n v e x polytope, but the third cannot. T h ere are m a n y fans, h o w e v e r , that do not com e fr o m a n y c o n v e x polytope, h o w e v e r subdivided. To see one, start w ith the fan o v e r the faces of the cube w ith ve rtic es at

( ± 1 ,± 1 ,± 1 )

in

Z3. Let

A

be the fan w ith cones spanned by the sam e sets of generators except

26

SECTION

that the v e r t e x

(1,1,1)

1.5

is replaced b y

(1,2,3).

It is impossible to find

eight points, one on each of the eight positive r a y s through the vertices, such that for each of the six cones generated by four of these vertices, the corresponding four points lie on the sam e affine plane: Exercise. num ber ua in

Suppose for each of the eight v e rtic es r v , and for each of the six large cones

M|r = IR3, such that

four v e rtic es in w ith

v

t h ere is a real

cr t h ere is a v e c t o r

= r v w h e n e v e r

v

is one of the

a. Show that there is then one v e c t o r

= r v

for all

v. In particular, the points

u in

Mr

pv = ( l / r v )«v

cannot h a v e each quadruple corresponding to a cone lying in a plane unless all eight points are coplanar/17^ A p a rtic u la rly im p orta n t construction of toric v a rieties starts w ith a rational polytope

P in the dual space

M[r. W e assume th at

P

is n-dim ensional, but it is not necessary that it contain the origin. From

P

of

ctq

a fan denoted

Ap

Ap

for each face

crq =

is constructed as follows.

Q of

P, defined by

{ v € N|r : < -* (p* ,

is an isomorphism. (A m apping betw een tori induces a corresponding m apping on o n e -p a r a m e t e r subgroups.) (b)

Composition gives a pairing Hom(TN,Gm ) x Hom(Gm ,TN)

-*

Hom(Gm ,Gm ) .

Show that, by the above identifications, this is the du a lity pairing < , > : M x N

-*

Z .

Note in particular that the above prescription shows h o w to r e c o v e r the lattice

N from the torus

w a n t to see how to r e c o v e r

a

The key is to look at lim its

T n - Given a cone

lim Av (z)

for variou s

z ~*° the com plex va riab le z approaches the origin. suppose

cr

x ( C * ) n~k. por

Then

Av (z )

v = ( m i , ... , m n) € Z n,

has a lim it in

w e next

Tn

c

v € N,

LfCT. as

For exam ple,

is generated by part of a basis e^, . . . ,e^

is

a,

fro m the torus em bedding

for

N,

so Ua

Av (z ) = ( z m i , . . . , z mn).

if and only if all

mj

are n o n n eg a tiv e

and

m, = 0 for

is in

a.

i > k. In other words, the limit exists e x a c tly w h e n

and

8j = 0 if rrtj > 0. Each of these lim it points is one of the

In this case, the limit is (8j_, ... ,8n), w h e r e

distinguished points

xT

for some face

t

of

cr.

v

8j = 1 if lrq = 0

SECTION

38

In general, for each cone distinguished point contained in

2.3

in a fan

t

x T in UT. If

A,

w e h a v e defined the

is a face of

t

cr, then

UCT, so w e must be able to realize

Ux is

x T as a h o m o m o r ­

phism of semigroups from

Sa to C. This h o m o m o rp h ism is

w h ich is w ell defined since

x An a v is a face of

that the resulting point t

xT of X ( A )

cjv. F rom this it follows

is independent of

a; th a t is, if

< a < y, then the inclusion of Ua in Uy takes the point defined in

UCT to the point defined in Uy. W e note also that these points are all distinct; this follows fro m the fact that proper face of

t

.

in each orbit of Tn Claim 1. v

If v

x T is not in

UCT if

cr is a

As w e will see later, there is e x a c tly one such point on

is in

X (A ).

|A|, and

is the cone of A

t

in its r e la t iv e i n t e r i o r , then

lim Av (z )

=

th a t con ta in s

xT .

z -»0

For the proof, look in identify

Av (z )

for a n y

ct containing

w ith the h om om orp h ism from

u to z^UiV^. For ex a c tly w h e n

u in Sa, w e h a v e

u belongs to

h om om orp h ism from

t

Sa C M

x.

as a face, and

t

M

to C*

that takes

> 0, w ith eq u a lity

It follows that the lim iting

to £ is precisely that w h ich defines

xT. (One should check that this is the topological lim it, say b y choosing m

generators

%u for

C laim 2. I f v

SCT to em bed

UCT in Cm .)

is n o t in a n y cone of A ,

then

lim Av (z ) z

exist in

X ( A ).

In fact, if v

is not in

as z approaches (possible since

a,

v

Av (z )

the points Av (z )

0. To see this, take

a = (crv ) v'). Then

W ith these claims, w h ich

a flN

A,

u in

h a v e no lim it points in UCT ctv

w ith

< 0

%u(Av (z )) = z^u , v ^ -> »

as z

is ch ara cterized as the set of

v

0. for

has a limit in UCT as z -* 0, and the lim it is x a if

is in the r e la tiv e interior of

|A| of

does n o t

0

a.

For those

i.e., the union of the cones in

co n vergin g subsequence).

A,

v

not in the s u p p o rt

th ere is no lim it (or

C O MP A C T N E S S

Exercise.

For

v € N,

m orphism fro m

show that

C to X ( A )

AN D

PROPERNESS

Xv : C* —» Tjsj

if and only if v

and

Xv

extends to a m orphism from

if v

and

2.4

Compactness and properness

-v

39

extends to a

belongs to |A|,

IP1 to X ( A )

if and only

belong to |A|.

Recall that a com plex v a r i e t y is com pact in its classical topology e x a c tly w h e n it is com plete (prop er) as an algebraic v a r ie t y .

For a

toric v a r ie t y , w e can see this in term s of the fan: A to r ic v a r i e t y X ( A )

is c o m p a c t if and o nly if its s u p p o rt

|A|

is the whole space Njr. Because of this w e say that a fan

A

is c o m p le t e if

|A| = Njr.

One im plication is easy: if the support w e r e not all of N r , since finite, there would be a lattice point Xv (z )

v

A

is

not in a n y cone; the fact that

has no limit point as z -* 0 contradicts compactness. Before proving the converse, w e state the appropriate

generalization. m aps a fan

Let

A'

(jr. N ‘ -» N be a h om om orp h ism of lattices that

into a fan

A

as in §1.4, so defining a m orp h is m

cp*: X ( A ') -> X ( A ). Proposition.

The m a p

K*

Spec(K)

f = tp*; assume

to U is given by a

of groups. W e w a n t to find

cr* m apping

cr so w e can fill in the diagram

K «—

€[M‘] D C[Sffi]

R «---- C[SCT] The fact that

Spec(R )

maps to Ua says that, if ord

function of the discrete valuation, then cr^nM;

ord°oc°(p*

is the order

is n o n n eg a tiv e on

e q u iva len tly , ip(ord°oc)

= ord°ot°cp*

By the assumption, th ere is a cone ord°cx € cr'. This says that

ord°oc

€ ( a v )v =

a1

so that

a .

cp(cr‘) c cr

is non n ega tive

on

and

a ,v, w h ic h

is precisely the condition needed to fill in the diagram . A fu n d a m en ta l exam ple of a proper m a p is b low ing up.

We

saw the first ex am p le of this in the construction of the b lo w -u p of Uq- = C2 at the origin

x a = (0,0)

suppose a cone

A

Set

a

in

in Chapter 1. M o re g en erally ,

is generated by a basis v^,

vq = v i + . . . + v n, and replace

a

ated b y those subsets of (vo, v * , . . . , v n ) This gives a fan

. . . , v n for N.

by the cones that not containing

A 1, and the resulting proper m a p

the b lo w -u p of X ( A )

at the point

changed except o v e r

UCT, w e m a y assume

a re g e n e r ­ ( v j , . . . , v n ).

X ( A ') - > X ( A )

is

x a. To see this, since nothing is A

consists of

a

and all

COMPACTNESS

AN D

of its faces, so X ( A ) = UCT = € n, w ith 1 < i < n. Then w here

X ( A ')

41

x CT the origin, and

v* = e* for

is covered by the open affin e va rieties

a 4 is the cone generated by

and

P ROPERNESS

is generated by

U ^ .,

eg, e i, . . . , e if . . . , e n, 1 < i < n,

e i * , e * * - e * , . . . , e n* - e^ .

The c o rre s­

ponding coordinate ring is A CT. =

CtXi( X i X f 1,

X n X f 1] .

On the other hand, the blow -u p of the origin in Cn is the s u b v a r ie ty of £ n x (Pn-1

defined b y the equations

are homogeneous coordinates on Xj = Xi«(Tj/Tj), for

w here

Uj w h e r e

so Uj is Cn, w ith coordinates

j * i. So Uj = UCT. w ith

Exercise.

XjTj = XjTj,

P n“ *. The set

X* and

T i, . . . , Tn Tj =* 0 has

Tj/Tj = Xj/Xj

ctj as above, and w ith the sam e gluing.

For a nonsingular affine toric v a r i e t y of the fo rm

x ( C * ) n“ k, show how to construct the blow -u p along

{0} x ( £ * ) n~k

as a toric v a r ie t y . Exercise. even if

If

A

is an infinite fan, show that

|A| = Njr and all

Av (z )

criterion for properness not apply? A

in

X (A )

h a v e limits in

Z 2 whose support is all of

is n e v e r com p act,

X (A ).

W h y does the

Give an ex a m p le of an infinite fan

IR2. Show that, for infinite fans, the

proposition rem ains true w ith the added condition that t h e r e a re only fin itely m a n y cones in Exercise.

A'

(F ib e r bundles)

m apping to a given cone in Suppose

0 -» N 1 -» N -» N"

exact sequence of lattices, and suppose N,

and

A.

A ',

A,

and A "

0 is an are fans in N',

N “ that are compatible w ith these mappings as in §1.4, giving

rise to maps X ( A ' ) -> X ( A ) -> X ( A " ) . Suppose th ere is a fan

A M in N that lifts

is the isomorphic im age of a unique cone in a

in

A

a'

A ",

i.e., each cone in

a cone in

= A'

+ a" and

=

{v* + v " : v' € a', v " € a " }

cr" a cone in

sequence is a locally triv ia l fibration.

A ".

A"

such that the cones

are of the fo rm a

for

A ",

Show that the a bove

SECTION

42

Exercise.

2.5

Construct the p r o je c t iv e bundle P ( 0 ( a 1) © e ( a 2) e . . • ® 0 ( a r ))

->

IPn

as a toric v a r i e t y / 12^ Exercise.

Let

cp*: X ( A ') -» X ( A )

h om om orp h ism of lattices Show that for a the point contains Exercise. and (i)

cone a'

x CT of X ( A ) ,

be the m a p arising fr o m a

cp: N* —* N m apping in

w here

A 1 to

A 1, cp* m aps the point cr is

A

as in §1.4.

x CT« of X ( A ' ) to

the smallest cone of

A

that

ct‘.

Find a subdivision of the cone generated by

(0,0,1)

in

the resulting

proper m a p

Z3, including the r a y through toric v a r i e t y

X'

X'

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

(1,1,1),

isnonsingular;

C3 is an isomorphism o v e r

(ii)

such that: the resulting

X ( A )

A

A'

is a union of cones in

N ow w e fix the

is a r e f in e m e n t of

A,

A'. The m orphism

induced by the id e n tity m ap of N is b ira tio n a l and

p r o p e r ; indeed, it is an isomorphism on the open torus

Tjsj contained

in each, and it is proper b y the proposition in §2.4. This construction can be used on singular toric varieties to resolve singularities.

Consider the exam ple w h e r e

cx is the cone in

Z 2 gen erated by

3 e i - 2 e 2 and e 2 , and insert the edges through the points e i and 2 e ± - e 2 - The indicated subdivision

gives a nonsingular

X ( A ')

m apping birationaily and prop erly to UCT.

This can be generalized to a n y tw o-d im ensional toric singularity. Given a cone

ct that is not generated by a basis for

that w e can choose generators gen erated by

v = e 2 and

r e l a t iv e ly prim e.

and

e 2 for

v' = m e i - k e 2 , 0 < k < m ,

Insert the line through

N, w e h a v e seen

N so that w ith

cr is k and

m

e*:

(m ,- k ) The cone generated by

e^

and

e 2 corresponds to a nonsingular open

set, w h ile the other cone generated by

e^ and

m e ^ - k e 2 corresponds

46

SECTION

2.6

to a v a r i e t y whose singular point is less singular than the original one. To see this, ro ta te the picture by 90°, m o v in g

e*

to e 2 , and then

tra n s late the other basic v e c t o r v e r tic a lly (b y a m a t r ix §2.2) to put it in the position and

k i = aj_k - m

(m ^ -k i),

for some integer

i)

as

m i = k, 0 < k^ < m i ,

a^ > 2:

This corresponds to a smooth cone w h en m /k = a^ - k^/m i = a^ - l / ( m j / k i ) ,

w ith

k^ = 0. Otherwise

and the process can be repeated.

The process continues as in the Euclidean a lgorithm , or in the construction of the continued fraction, but w ith altern ating signs:

a2

"

JL_ ar

w ith integers f ra c tio n of E x ercise,

a* > 2. This is called the H i r z e b r u c h - J u n g c o n tin u e d

m/k. (a ) Show that the edges d r a w n in the a bove process are

e x a c tly those through the ve rtic es on the edge of the polygon th at is the c o n v e x hull of the nonzero points in

ctDN:

R E S OL UT I ON

(b)

S h o w that there are

given vertices (c)

v =

r

and

vq

OF

added vertices v' = v r + i ,

47

v^, . . . , v r

and

b e t w e e n the

a j Vj = v ^ i + v i + i.

S h o w t h a t these a d d e d r a y s c o r r e s p o n d to e xc e pt i on a l di vi sor s

Ej = IP*,

fo rm in g a chain

w ith self-intersection nu m bers (d)

SINGULARITIES

Show th at

sem igroup Exercise.

{v

,

q

. . • , v r +i )

(E j-Ej) = -aj. is a m in im a l set of generators of the

a ON. Show that the algebra

g en erators

A a = C[Sa] has a m in im a l set of

1 < i < e), w h e r e the embedding dimension e and

{U^V**,

the exponents are dete rm in ed as follows. Let integers (each at least fra ction of

m / (m - k ).

b 2 , . . . , be_ i

be the

2) arising in the Hirzebruch-Jung continued Then

si

= m , S2 = m - k ,

sj+i = bpSj - Si_i

for 2 < i < e -1 ;

ti

= 0,

tj+i = bj*11 - t j_ i

for 2 < i < e -1 S 21^

Let a

Exercise. Show that

t2 = 1 ,

be the cone generated by

Sa is generated by

u 3 = e^ + e^, w ith

e2

and (k + l ) e i - k e 2 -

u* = e£, U2 = ke£ + (k + D e^,

and

(k + l ) u 3 = u* + U2 - Deduce that

A a = C [ Y i , Y 2, Y 3] / ( Y 3k + 1 - Y i Y 2) . w h ic h is the ra tio n a l double po in t of type A^. Show that the resolution of singularities given by the above toric construction has k exceptional divisors in a chain, each isomorphic to intersection Exercise. let

a'

- 2 . ^22^

Let

a

be generated by

be generated by

r e la t i v e ly prim e. m' = m,

and

IP* and w ith self­

k' = k or

Given a fan one can subdivide

e 2 and

Show that

A A

e 2 and

m e j - k e 2 as above, and

m 'e^ - k 'e 2 , w ith

0 < k' < m'

UCT» is isomorphic to UCT if and only if

k'*k = 1 (mod m). ^23^

in a n y lattice to a fan

A'

N,

and a n y lattice point

v

in N,

as follows: each cone that contains

48

v

SECTION

2.6

is replaced by the joins (sum s) of its faces w ith the r a y through

each cone not containing

v

is left unchanged.

Since

A 1 has the sam e support as A ,

X (A ')

to

X (A )

v;

the induced m apping fr o m

is proper and birational. The goal is to choose a

succession of such subdivisions to get to a nonsingular toric v a r ie t y . E xercise.

Show that one can subdivide a n y fan, by successively

adding ve c to r s in larger and larger cones, until it becomes simplicial. N o w if

cr is a k-dim ensional simplicial cone, and

the first lattice points along the edges of

defined to be the index of the lattice generated by the generated b y

Vj

are

ct is

in the lattice

a: m u lt(a )

Note th at

vj_, . . . , v^

cr, the m u l t i p li c it y of

= [N a : Zv^ + . . . + Z v ^ ] .

UCT is nonsingular precisely w h en the m u ltiplicity of

cr is

one. E xercise. fo rm

Show that if

v = E tjV i,

0
1 th ere is a lattice point of the < 1. For such

v,

taken m in im a l along its ra y ,

show that the m ultiplicities of the subdivided k-dim ensional cones are tp m u lt(a ),

w ith one such cone for each nonzero

tj.

From the preceding tw o exercises, one has a procedure for resolving the singularities of a n y toric v a r i e t y — n e v e r leavin g the w orld of toric varieties: P r o p o s itio n . of

A

so t h a t

F o r a n y to ric v a r i e t y X ( A ) -» X ( A )

X (A ),

th e re is a r e f in e m e n t A

is a resolution of singularities.

In particular, the resulting resolution is e q u iv a ria n t, i.e., the m ap c o m m u te s w ith the action of the torus.

R E SOL UT I ON

Exercise.

OF

SI NGULARITI ES

49

For surfaces, show that this procedure is the sam e as that

described at the beginning of this section. Show that the integer found t h ere is the m u ltip lic ity of the cone generated by

vj_i

a*

and

v i+i. Let v e cto rs

N

be a lattice of rank

vj_, V 2 , V 3 , and

3, w ith

a

V 4 that generate

the cone generated by N as a lattice and satisfy

v i + v 3 = v 2 + V 4 , as w e considered in §1.3:

There a re th ree obvious w a y s to resolve the sin gularity b y subdividing: Ai

D raw the plane through

v^ and

V 3 (take

v = v^ or V 3 );

A2

D raw the plane through

v 2 and

V 4 (take

v = v 2 or V 4 );

A3

Add a line through

v = v^ + V 3 =

The first t w o of these replace

a

v 2 + V4 .

by tw o simplices, the third by four.

Since the third refines each of the first two, the corresponding resolution m aps to each of them : X ( A 3)

^

\

X (A i)

X ( A 2)

\

^ X

50

SECTION

2.6

This description of t w o different m in im a l resolutions is well known for this cone o v e r a quadric surface: each of X ( A i ) has fiber

X and

X (A 2)

X

IP1 o v e r the singular point, corresponding to the t w o rulings

of the quadric, w hile

X ( A $ ) —> X

The tra n s fo r m a tio n fr o m

X (A i)

has fiber

IP^xlP1, the quadric itself.

to X ( A 2 ) is an ex am p le of a "flop",

w h ic h is a basic tra n s fo rm a tio n in higher-dim ensional birational g e o m e tr y .

In fact, toric varieties h a v e provided useful models for

M o r i’s p r o g r a m / 25^ Let us consider a global example. faces of the cube w it h ve rtic es at sublattice of

1?

Let

A

(± 1 ,± 1 ,± 1 );

IR3 o v e r the

be fan in take

N to be the

generated by the ve rtices of the cube, as in Example

(3) at the end of Chapter 1. The six singularities of X ( A )

can be

resolved b y doing a n y of the above subdivisions to each of the face of the cube.

The following is a p a rtic u la rly pleasant w a y to do it:

These added lines d e te rm in e a tetra hedron; the fan

A

faces of this tetra h e d ro n determ in es the toric v a r i e t y Since

A

is also a re fin e m e n t of

X (A ) Exercise.

«-

A,

X ( 2 ) -*

Show that the m orph ism

of cones o v e r X( A ) = (P3.

w e h a v e m orphism s X( A ) = IP3 . X( A ) -* X( A )

IP3 along the four fixed points of the torus. In

X( A )

is the b lo w -u p of the proper

tra n s fo r m s of the six lines joining these points are disjoint. the m orp h is m points of

X (A ).

X( A ) -* X ( A )

Show that

contracts these lines to the six singular

CHAPTER

3

ORBITS, TOPOLOGY, AND LINE BUNDLES

3.1

Orbits

A s w i t h a n y set on w h i c h a g r o u p acts, a toric v a r i e t y

X = X (A)

d i s j oi nt u n i o n of its or bi t s b y the acti on of t he t or u s T = T n - W e see t h a t t h e r e is one s u c h orbi t

0 T for e a c h cone

or bi t c o n t a i n i n g t he di s t i ngu i s he d point

in

t

A;

it

is a will

is t he

x T t h a t w a s d e s c r i b ed in §2.1.

Moreover, 0T If

t

is

=

« C * ) n ~k

n-dime nsi ona l, then

0 T = T n - W e wi l l see t h a t w h i c h is de no t e d

V (t).

0 T is t he point

x T. If

t

= (0),

t he n

0 T is a n open s u b v a r i e t y of its closure,

The v a r i e t y

t h a t is a g a i n a toric v a r i e t y . those o r b i t s

dim (T) = k .

if

for w h i c h

is a closed s u b v a r i e t y of

V (t )

In fact,

V (t )

y c o n t a i ns

t

X

will be a di sj oi nt u n i o n of as a face.

B e f o r e w o r k i n g this out, let us look at t he si mpl est e x a m p l e : T = (C*)n

a c t i n g as u s u a l on

X = C n . The

{ ( z i , . . . , z n ) € C n : zj = 0 as

I

r a n g e s o v e r all s ub s e t s of

x T, w h e r e

t

for

or bi t s a r e t he sets

i € I,

{1, . . . , n) .

zj # 0 for i £ 1}

This is t h e orbit c o n t a i ni ng

is g e n e r a t e d b y t he basic v e c t o r s

a b o v e a s s e r t i o n s a r e e v i d e n t in this e x a m p l e . Ox

=

Horn ( t

xD

,

e*

for

i € I.

All of the

No t e also t h a t

M , (C*) ,

w h i c h is a f o r m u l a t h a t wi ll be t r u e g e n e r a l l y as wel l.

The g e n e r a l

c ase of a n o n s i n g u l a r a f f i ne v a r i e t y is o b t a i n e d b y cr ossi ng this e x a m p le w it h a torus

(€*)*.

For a c o m p a c t e x a m p l e , cons i de r the p r o j e c t i v e s pa ce c o r r e s p o n d i n g to the {vq,

If

t

. . • , v n ),

where

lPn

f a n of cones g e n e r a t e d b y p r o p e r s ubs e t s of t he v e c t o r s g e n e r a t e t he l attice a n d a d d tozero.

is g e n e r a t e d b y t h e sub s et

{ v j : i € I),

51

then

V (t )

is t he

52

SECTION

i nt e r s e c t i on of t he h y p e r p l a n e s t h e poi nts of

V (t )

zj = 0

3.1

f or

i € I,

and

0T

w h o s e o t h e r c o o r d i na t e s a r e no n z e r o.

In the general case w e w ill first describe the orbits closures

V (t )

For each

x

group) b y

0 T and their

ab strac tly , and then show how to em bed th e m in X (A ).

w e defined t

consists of

flN,

Nx to be the sublattice of N generated (as a

and N (x)

=

N/NT ,

M (t )

=

t

-'-d

m

the q uotient lattice and its dual. Define 0 T to be the torus corresponding to these lattices: °x

=

Tn (t)

-

H o m (M (x ), C * )

This is a torus of dimension

=

n -k ,

w here

acts t r a n s it iv e ly via the projection The s ta r of a cone cones

a

in

A

t

S p e c (€ [M (T )])

=

k = d im (x ),

N(x)z C* . on w h ich

Tn

T n -* T n (t )*

can be defined a b strac tly as the set of

that contain

as a face. Such cones

t

cr are

d e te r m in e d b y their im ages in N ( t ), i.e. by o7 =

a + ( N t )|r / ( N t ) r

These cones

{ct : x < a )

by

(W e think of S t a r (x )

S t a r (x ).

C

N|r / (N x )|r

fo rm a fan in N (x ),

=

N ( t )|r .

and w e denote this fan

as the cones containing

realized as a fan in the quotient la ttice

t

, but

N ( t ).)

Set V (t ) =

X (S ta r(x )) ,

the corresponding (n -k )-d im en s io n a l toric v a r ie t y . em b edd ing

Ox = T n ( t ) c V ( x )

Note that the torus

corresponds to the cone

(0) = T

in

ORBITS

53

N(t ).

This toric v a r i e t y

V ( t ) has an affin e open co verin g

cr va ries o v e r all cones in

A that contain

U(j(t ) = Spec(C[av nM(T)l)

Note that duality).

a v n T x is a face of For

a =

To em bed

t

as

as a face:

= Spec(C[avriT xn M]) .

crv (th e face corresponding to

t

by

, U t ( t ) = 0 T.

V (x )

closed em bedding

t

(U ct( t )},

as a closed s u b v a r ie ty of X ( A ) ,

of Ua(T )

inUa

for each

cr >

w e construct a

. Regarding the

t

points as semigroup hom om orphism s, the em bedding U ct( t ) = H om Sg ( a v flT'Ln M , C )

H o m sg( a v n M , C ) = UCT

is given by extension b y zero; again, the fact that

a v O T 1 is a face of

ctv implies that the extension by zero of a sem igroup h o m o m o rp h ism is a semigroup hom om orp hism .

The corresponding su rjection of rings

dcr^H M]---- > C [a v n T x n M] , is the obvious p ro je c t io n : it takes X u to and it takes

%u to

These maps are compatible: if of

%u if u is in

a v flT xn M ,

0 otherwise. t

is a face of

a,

and

cr a face

cr', the diagram U ct( t ) ^ £ UCT ^

U ct»( t ) £ Uff.

co m m u te s, since it comes fro m the c o m m u t a t iv e d ia g ram Hom Sg(avriT 1 n M ,C )

Homsg(a lv n T xnM,(C)

1

1

H o m sg( Tjsj, so an

Show that the set of fixed points of this action for w hich

v

is in Ny.

F u n d a m e n t a l g ro u p s a n d E u ler c h a r a c t e r i s t i c s

First w e look at the fu n d a m en ta l group

tti ( X ( A ) )

of a toric v a r ie t y .

Base points will be om itted in the notation for fu n d a m e n ta l groups; th e y m a y be taken to be the origins of the em bedded tori. The m ain fact is that com plete toric varieties are sim p ly connected. Pr o p o s itio n . Then

X (A )

Pro o f.

Let

A

In fact,

be a fan th a t con tains an n - d im e n s io n a l cone.

is s im p ly connected.

The first observation is that the inclusion

Tn

X (A )

gives a

sur jection tti (T n )

----~

T r i(X (A )) .

This is a general fact for the inclusion of a n y open s u b v a r ie ty of a n o r m a l v a r ie t y ; the point is that a n orm a l v a r i e t y is locally irreducible as an a n a ly tic space, so that its u niversal co v e r in g space cannot be disconnected by th row in g a w a y the inv erse im ag e of a closed s u b v a r ie ty / 3^ N ow for a n y torus

T n , th ere is a canonical isomorphism N

n i ( T N) >

FUNDAMENTAL

defined by taking is the m a p

Av

GROUP S

AND

EULER

S 1 c (C* —» Tjsj, w h e r e

v € N to the loop

defined in §2.3. If

v

is in

the loop can be contracted in UCT, since in fact, w e h a v e seen that

Av

CHARACTERISTICS

ctHN

57

C* —> T^j

for some cone

cr,

lim Av (z ) = x a exists in

z —» 0

extends to a m a p fr o m

C to

UCT;

Ua. The

contraction is given by

X y ’t ( z )

If

a

=

f Xv (t z )

z € S1, 0 < t < 1

1 [ xa

z € cSl ,

is n-dimensional, then

a flN

t =0

generates

N as a group, so the

fact that such loops are triv ia l in U CT implies that all loops a re trivial. C o r o lla r y .

If o

is ak -d im e n s io n a l cone, then

Pro o f. This follows

fr o m

the fact that

TTi(Ua)

= Z n” k.

= U^' x (C^)n_k, and

u 1(C*‘) - uiCS1) = 2. M ore intrinsically, if by

a,

the fibration

a'

is the cone in the la ttice

Ua' -»

N CT generated

-> T n (ct) induces a canonical

isomorphism TTi(Ua) E xercise.

Let

A

T tid N ^ )) IR2

be the fan in

origin, and the t w o ray s through tti(X (A ))

=

N (a ) .

consisting of th r e e cones: the

2 e^+e2 and e i + 2 e 2 - Show that

Z/3Z.

s

In com plete generality, if N ‘ is the subgroup of all

crON,

as

cr va ries o v e r

A,

then

tt 1 ( ( X ( A ) ) To see this, note that for each

a,

N generated by

=

N/N' .

Tr^Utf) = N/NCT. By the general va n

Kam pen th eorem , t t i ( ( X ( A ) ) ) = t i 1( U U (T) = l i m u i t U j = lim N/Nq. = N/XNa = N/N' . For affin e toric varieties, a similar a rg u m e n t shows m ore: P ro p o sitio n . con tractible.

If a

is an n -d im e n s io n a l cone, then

Ua is

58

SECTION

P r o o f.

3.2

W e w a n t to define a hom otop y H : UCT x [0,1] -4 UCT

b etw een the retrac tion lattice point

v

r: UCT —» x CT and the iden tity map.

in the interior of

cj .

semigroup hom om orp hism s from H(x x t )(u ) and

Regarding the points of

SCT to C, define

t •x(u )

H(x x 0) = xa. It is easy to see that

hom om orp h ism w h e n e v e r all

=

t. For

u €

follows that

S CT

x [o],

H(cp x t)

x is. For

Choose a

for

H(x x t)

Ua as

H by

t >0 , is a sem igroup

u = 0, H(x * t )(u ) = x(0) = 1 for

> 0, so H(x x t ) ( u ) -> 0 as t -» 0. It

approaches

x CT as t —> 0, and, since generators

of Sa d e te rm in e an embedding of

UCT in some

Cm , the resulting

m apping is continuous. C o r o lla ry . P ro o f.

If dim(cr) = k,

then

Oa c UCT is a d e fo r m a tio n r e t r a c t .

One can use the sam e proof as in the proposition, or an

isomorphism C o r o lla ry .

Ua = UCT' x 0 CT.

There is a canon ical is o m o rp h is m

w h e re

M ( a ) = a ^ flM .

P ro o f.

Since

0 CT is the torus

algebra on the dual

M (a )

Ti\|(ff),

HHUq-jZ) = A ^ M t a ) ) ,

its cohom ology is the e x te rio r

of its first hom ology group

N (a ).

Knowing the cohom ology of the basic open sets Ua can g ive some in form a tion about the cohom ology of X (A ). covering

W hen one has an open

X = U iU . . . UU$ of a space, if all intersections of the open

sets are sim ply connected, the cohom ology of the Cech cohom ology of the covering.

X can be co m p u ted as

In general one has a spectral

sequence

=

® io < ■--< ip

Hq (Ui n . - . n u , ) =* Hp+q( x ) . (4) p

A pply this to the co verin g by open sets Uj = UCT. , w h e r e the the m a x im a l cones of

A:

aj

are

FUNDAMENTAL

E1p-q =

GROUPS

©

AND

EULER

CHARACTERISTICS

A q M(CTjn . . .fieri )

=* Hp+q( X ( A ) ) .

i0 < - - < ip

59

p

In particular, this gives a calculation of the tcpological E uler c h a r a c t e r is t ic than

n,

%( X(A)).

For e v e r y cone

the a ltern ating sum

the dimension is n, X (X (A ))

A

= X ( - l ) p+q rank

is complete.

q

Since each

=# n -d im e n s io n a l cones in

0

A.

A are n-dimensional, as is the case

U^.

E1 ° 'q =

iscontractible, for

q > 1 .

E^’0 is

In addition, the com plex

0 —> © ZCT —> © ZCT.flCT: “ * i

vanishes, w hile if

this altern ating sum is one. Therefore

Assume all m a x im a l cones in if

that has dimension less

t

X ( - l ) q r a n k ( A qM ( x ) )

i giving a h om om orp h ism from

to the group

M

D iv j X

of

T-C artier divisors. The following proposition shows that

Pic(X )

can be com puted by using only T-C artier divisors and functions,

and s im ila rly for

A n_^(X)

w ith T-W eil divisors.

In addition, it gives a

recipe for calculating these groups for a com plete toric v a r i e t y Pr o p o s itio n .

Let X = X ( A ) ,

a n y p r o p e r subspace of N|r.

w h e re

A

X.

is a fan n o t c on ta in e d in

Then th e re is a c o m m u t a t i v e d iagram

with e x a c t rows: 0

->

M ->

DivTX

II

£

->

Pic (X )

-> 0

s

d 0

M —>

©

Z-Dj

A n_ i ( X )

-* 0

i= l In p a r t ic u la r ,

r a n k (P ic ( X )) < r a n k ( A n_ i ( X ) ) = d - n,

the n u m b e r of edges in the fan. In addition, P roo f.

First note that

X ' U Dj = T n

the unique factorization ring

w h e re

d is

is free abelian.

is affine, w ith coordinate ring

C [ X i , X i _1, . . . , X n, X n_1], so all Cartier

divisors and Weil divisors on Tn sequ en ce'(9)'

P ic (X )

are principal.

This gives an exact

64

SECTION

A n -itU D j)

d © Z-Dj

=

-

3.4

A n_ x(X )

-> A n_ ! ( T N ) =

0.

i= 1

Next, note that if f is a rational function on X w hose divisor is T - in v a r ia n t , then

f = X»%u for some

u 6 M

X; this follows by restricting to the torus preceding section and the fact that the d eterm in ed uniquely by If

£

indeed, w r it in g

span

Njr im p ly th at

u is

f. This shows the exactness of the second row. m ust

for some Cartier divisor supported on

£ = 0(E )

for some Cartier divisor

function w hose divisor agrees w ith

divisor.

vj

is an algebraic line bundle, its restriction to T n

be triv ia l, so £ = 0(D )

Hence

and com plex n u m b e r

T^j. The le m m a in the

UDj;

E, take a rational

E on T n , and set

D = E - div(f).

D is T -in v a r ia n t as a Weil divisor, and t h erefo re as a Cartier The exactness of the upper r o w follows easily.

Finally, the fact that that it is a subgroup of

Pic (X )

is torsion free follows fr o m the fact

0 M / M (c r),

and each

M/M(cr)

is a lattice, so

torsion free. C o r o lla ry .

I f all m a x i m a l cones of A

are

n - d im e n s io n a lf then

P i c ( X ( A ) ) s H2 (X ( A ),Z ). Proo f.

W e must m ap the group of T-C artier divisors onto

w ith kernel

M. W e h a v e seen the isomorphisms:

{T -C a rtier divisors)

=

Ker ( © M/M(cjj)

©

i

H2 ( X ( A ) ; Z )

=

©uj

M/M(

) ,

©

M ( a jH a jf ic r ^ )) .

i 1), and V 3 = ~ e 2 - Show that the Weil

LI NE

divisor

a^Di + a 2 D2 + £303

only if

a^

+ a2

A

=

X

= X ( A ) if and

Show that

Z«m D2

A i(X )

be the com plete

v^ = 2 e ^ - e 2 ,V 2 = -e^ + 2 e 2 , and a^Di + a 2 D2 +

65

is a Cartier divisor on

= 0 (m od m).

P ic (X ) Exercise. Let

BUNDLES

Z 2 w ith edges along

fan in

V3

= Z-D2 .

= - e i ~ e 2 . Show that

^3 0 3 is a Cartier divisor on X = X ( A ) if and only if

a i = a 2 = a 3 (m od 3). Show that Pic (X )

=

Z-3Di

A x(X )

=

(Z-Di + Z -D 2 )/ Z -3 (D !-D 2) =

In particular,

A n_^X

Exercise.

X = X (A ),

Let

=

Z , Z © Z/3Z .

can h a ve torsion. w here

A

is the fan in

Z 3 o v e r the faces of

the co n v e x hull of the points e^, e 2 , e 3 , e^ - e 2 + 63 , and P ic (X ) s Z, A 2 (X ) £ Z 2, and

Show that E xercise.

Let

X = X (A ),

cube w ith ve rtic es by the vertices.

w here

(± 1 ,± 1 ,± 1 ),

Show that

-e^ - 63 .

A 2 ( X )/ P i c (X ) s Z.

A is the fan o v e r the faces of the Z 3 generated

and N the sublattice of

Pic(X ) = Z, generated by a divisor that is

the sum of the four irreducible divisors corresponding to the ve rtic es of a face. Show that

A 2 (X ) s Z5, and

A 2 (X ) / P ic (X ) s Z 4.

In §1.5 w e constructed a com plete fan

A

that cannot be

constructed as the faces o v e r a n y subdivided polytope.

In fact, the

exercise provin g that this fan is not a fan o v e r the faces of a co n ve x polytope a ctu a lly showed that e v e r y line bundle on

X (A )

is trivial;

i.e., P i c ( X ( A ) ) = 0. Exercise.

For this fan, show that

E xercise.

Let

A

n-dim ensional.

A 2( X ( A ) ) s Z5 0 Z / 2 Z .

be a fan such that all of its m a x im a l cones are

Show that the following are equivalent:

(i)

A

(ii)

E v e r y Weil divisor on

is simplicial; X (A )

is a (Q-Cartier divisor;

(iii)

P ic (X (A ))0 < Q —> A n_ i(X (A ))< 8 >Q is an isomorphism ;

(iv)

r a n k ( P i c ( X ( A ) ) ) = d - n.

The data

(u (a ) € M / M (a ))

for a Cartier divisor

D defines a

66

SECTION

c o n tin u o u s piecewise lin e a r fu n ctio n restriction of u (a );

t|jp to the cone

3.4

i(jp on the support

|A|: the

ct is defined to be the linear function

i.e., ^ p (v )

=

< u ( a ), v >

for

v €

ct .

The co m p atibility of the data makes this function

well defined and

continuous. Conversely, a n y continuous function on

|A|

and integral (i.e., given by an elem ent of the lattice comes fr o m a unique T-C artier divisor. is d eterm in ed by the p ro p erty that [D]

If

M)

D = SajDi,

that is linear on each cone,

the function

^D^v i^ = ” a i> e q u iv a le n tly

= I - ^ D( v i ) D i .

These functions beh a ve nicely w ith respect to operations on divisors. For exam ple,

iJjd + e

the linear function it follows that

= ipp + + E, so i|/mD = m +D. -u.

i|jp and

If D and iJje

A T-C artier divisor

differ by a linear function

D = Z ajD j

rational co n ve x polyhedron in PD = = Lem m a.

ipdiv(xu)

on X ( A )

u in

M.

also de te rm in es a

M(r defined by

{ u € M|r : > - a j { u € Mg* : u > i^p on

for

all i )

|A|) .

The global sections of the line bundle r(X,0(D)) =

Pro o f.

Note that

0(D)

a re

® C- Xu . u e PDnM

It follows fr o m the le m m a of the preceding section that r(UCT,0(D)) =

© c-xu. u € PD(a)nM

where Pp(a-)

=

{ u € M|r : > -aj

V

v j € cr} .

These identifications are compatible w ith restrictions to sm a ller open sets. It follows that

is

E are linearly eq u iva len t divisors,

T (X ,0 (D )) = n r ( U cr,0 (D ))

direct sum o v e r the intersection of the

is the corresponding

P p (c j)n M ,

as required.

LI NE

Exercise. (iii)

Show that

(i)

BUNDLES

P mD = m P D; (ii)

67

P D + div(xu) = P D " u i

P D + P E c P D+ E* W hen

|A| = N(r, the v a r i e t y

X (A )

is complete, and it is a

general fact that cohom ology groups of a coherent sheaf are finite dimensional on a n y complete v a r ie t y . that the polyhedron P ro p o sitio n .

Pp

In the toric case this m eans

is bounded:

I f the cones in

A

span

Njr

as a co n e , then

H X ,0 (D ))

is fin ite dimensional.

Proof.

If

Pp

in

there would be a sequence of vectors

M(r

nu m bers

tj

is finite.

w e r e unbounded, by using the compactness of a sphere converging to zero, such that

nonzero v e c t o r implies that have

In p a r t i c u la r , Pp O M

u in

M|r. The fact that

uj in

tpUj
0 for all j. Since the

,v

j

Pp,

and positive

converges to some >

> - aj

v* span

for all j N(r, w e must

u = 0, a contradiction. The next proposition answers the question of w h en a line bundle

is generated by its sections, i.e., w h en th ere are global sections of the bundle such that at e v e r y point at least one is nonzero. rea l-v a lu e d function

i|) on a v e c to r space is (u pper) c o n v e x if

ip(t-v + ( l - t ) - w )

for all vectors

v

Z, if Di

and

> tip(v) + ( l - t ) i p ( w )

and

of the toric v a r i e t y

w,

and all

0 < t < 1. For the simplest exam ple

IP* corresponding to the unique com p lete fan in

D2 are the divisors corresponding to the positive and

n e g a tive edges, the divisor function

Recall that a

ipp on

D = a^Di + a 2 D2 corresponds to the

IR defined by

This function is co n ve x ex a c tly w h e n 0(D) = 0 ( a i + a2)

on

a^ + a 2 is non negative.

IP*, this is ex a c tly the criterion for

0(D )

Since to be

generated by its sections. W e are concerned w ith continuous functions that the restriction

ip on

N jr such

ip|CT to each cone is given by a linear function

68

SECTION

u(cr) 6 M.

In this case c o n v e x ity m eans that the graph of

under the graph of graph of

u(cr)

the c o m p lem en t of dimensional cones

ct,

ip lies that the

so

ip is called s t r ic t l y c o n v e x if the graph of ct lies strictly under the graph of

u(cr),

on

for all n-

cr; eq u iva len tly , for a n y n -dim ensional cones

the linear functions

ct',

P ro p o sitio n . Let

for all n-dim ensional cones

\\) is "tent-shaped":

The co n ve x function

and

3.4

u

(

ct)

and

u(cr')

A s s u m e all m a x i m a l cones in

D be a T - C a r t i e r divisor on

X (A ).

A

Then

cr

are different. are n -d im e n s io n a l. 0(D )

is g e n e ra te d

b y its sections if and o n ly if ipp is convex. Pro o f.

On a n y toric

and only

v a r i e t y X,

if, for a n y cone

(i)

> - a*

(ii)

=

ct,

0(D)

for all i,

- aj for those

Indeed, (i) is the condition for

is generated by its sections if

there is a u(cr) € M

u

and i for w h ich v j €

ct

( ct) to be in the polyh edron

determ in es global sections, and (ii) says that UCT. The function

such that

X u^

. Pp

generates

that 0(D )

on

tpD *s determ in ed by its restrictions to the n-

dimensional cones, w h e r e its values are given by (ii). The c o n v e x i t y of ijjp is then eq u iva len t to (i). If

0(D)

is generated by its sections, and all m a x im a l cones of the

fan are n-dimensional, w e can reconstruct function

i|jp, fr o m the polytope ^p(v)

=

D, or e q u iv a le n t ly its

Pp:

m in

u e PDn M

=

m in ,

LI NE

w h e r e the

69

are the vertices of Pp.

Exercise. that

BUNDLES

If

0(D )

and

0(E)

are generated by their sections, show

P d + E = P d + P E-

Exercise. P DflM ,

If

0(D)

is generated by its sections, and

show that

( X u : u € S)

generates

0(D)

S is a subset of

if and only if S

contains the ve rtic es of Pp. W hen basis

0(D)

is generated by its sections, choosing (and ordering) a

{% u • u € PpOM } «p -

0 and

that the corresponding p r o je c t iv e toric v a r i e t y fold Veron ese em bedding of

P n in

X x j < m ).

Show

Xp C P r ~1 is the i n ­

P r - 1 , r = ( nJnm )'

S h °w th at the

construction of the preceding exercise gives the cone o v e r this em bedding as the affine toric v a r i e t y described in § 2 .2 . M ore generally, let W e call a com plete fan

P be the co n ve x hull of a n y finite set in M. A

c o m p a tib le w ith

P

if the function

ijjp

defined by

ipp(v) = m in is linear on each cone a in A . Since U €P vpp is convex, it determ in es a T-C artier divisor D = Dp on X - X ( A )

whose line bundle is generated by its sections. As before, these sections are linear combinations of the functions Exercise.

X u> as u va ries o v e r

Show that the im age of the corresponding m o rp h is m

(pD: X -* P r_1 , is a v a r i e t y of dimension

3.5

P flM .

k, w h e r e

k = d im (P ). ^18^

C o h o m o lo g y of lin e bu nd les

Let

D be a T-C artier divisor on a toric

\\t =

ijjj)

sections of

va riety

be the corresponding function on 0(D)

are a graded module:

X =X ( A ) ,

|A|. W e

and

let

kn ow th at the

H °(X ,0 (D )) = © H ° ( X , 0 ( D ) ) u,

w here n H ° (X ,0 (D ))u =

w ith

P D the polyhedron

f C -X u \ [ 0

if u € P n O M ° otherw ise

{u € M jr : u > ^ on

|A|). This can be

described in fancier w ords by defining a closed conical subset IA| for each

of

u € M: Z(u)

Then

Z(u)

u belongs to Pp

=

{ v € |A| : > i p ( v ) ) .

ex a c tly w h en

w h en the cohom ology group

Z(u) = |A|, or eq u iv a le n tly ,

H°(|A| ' Z (u))

vanishes, w h e r e this H°

74

SECTION

3.5

denotes the 0 th o r d in a r y or sheaf cohom ology of the topological space w ith com plex coefficients. E quivalently, if

H° (u)(IA|) = H°(|A|, =

|A|

'

Z (u))

K er(H °(|A|) -* H°(|A| s Z (u ) ) )

is the 0 th local c o h o m o lo g y group (or r e la tiv e group of the pair consisting of

|A| and the co m p lem en t of

ex a c tly w h en

Z(u)),

w e have

u € PD

H Z (U)(|A|) is not zero. Th erefore

H °(X ,0 (D ))

=

© H ° (X , 0 (D )) u

, H ° (X ,0 (D ))u =

H z (u )(|A|) .

This is the sta te m e n t that generalizes to the higher sheaf co hom ology groups

Hp(X ,0 (D ))

and to the higher local cohom ology groups

HZ(u)(IAD = HP(|A|, |A| N Z(u);C). P r o p o s itio n .

F o r all

Hp(X ,0 (D ))

s

p > 0

t h e r e a re ca n o n ica l is o m o rp h is m s :

© H p(X , 0 (D ) )u ,

Hp(X , 0 (D ))u a

H g (u )(|A|) .

These local cohom ology groups are often easy to calculate. exam ple, if X

is affine, so |A| is a cone and

and

are both convex, so all higher cohom ology vanishes —

|A| ' Z(u)

iJj is linear, then

For

w hich is one of the basic facts about general affin e varieties.

|A|

For toric

varieties, a sim ilar a rg u m e n t gives a stronger result than is usually true: all h ig h e r co h o m o lo g y groups of an a m p le line bun d le on a c o m p le te to ric v a r ie t y vanish. C o r o lla ry .

I f IAI

is c o n v e x

In fact, m o r e is true: and

0(D ) is g e n e ra te d b y its sections,

then • Hp(X ,0 (D )) = 0 fo r all p > 0. P ro o f.

Since

ijj is a co n ve x function, it follows that |A| ' Z(u)

is a co n vex set, so both

=

( v € |A| : < i|j( v ) }

|A| and

|A| ' Z(u)

are convex.

This implies

the vanishing of the corresponding cohom ology groups. It follows that for sections,

X complete and

0(D)

generated b y its

COHOMOLOGY

X (X ,0 (D ))

OF

LI NE

BUNDLES

=

S ( - 1 ) P d i m HP(X,0(D))

=

dim H °(X ,0(D ))

=

75

C ard (P Dn M ) .

W ith R ie m a n n -R o c h form ulas available to calculate the Euler characteristic, this gives an approach to counting lattice points in a co n ve x polytope.

We will com e back to this in Chapter 5. Note in

particular the fo rm u la for the a r ith m e tic genus: X (X ,0 x ) = d im H °(X ,0 x ) = Proof of the proposition. Cech com plex

Hp(0 (D ))

1.

is the p **1 cohom ology of the

C\ w ith

CP =

®

H ° ( u CTon . . . n u CT

®

®

U € M

CT0 f . . . , C T p

the sum o v e r all cones

c tq

0 (D ))

H Z(u)n a0n . . . na_( a o n • • • n a p) * U

. . . , 0. The proposition then follows from a standard spectral

sequence a rgu m ent: Lem m a.

L et

Z be a closed subspace of a space Y

of a fin ite n u m b e r of closed subspaces Y j, such th a t

H z n Y ^ Y '.T ) = 0

f o r all i > 0

and

T

th a t is a union a sheaf on

Y

and all Y* = Y j()n ... O Y jp .

Then h ‘z ( Y , T ) w h e re C "((Y j),7 ) c p ( { Y j) . T )

=

H* ( C ' ( { Y j ) , f )) ,

is th e c o m p le x whose pth t e r m is =

.

©

.

r ZnY

n

nY

J0 Proof.

Take an in je c t iv e resolution

com plex

C‘ ((Yj},3*).

(Y j n ... n Y j Jp

5’ of

7,

f ). F

and look at the double

The hypothesis implies that the columns are

resolutions of the com plex

C *({Y j),T ).

resolutions of the com plex

F z(Y ,D .

W e claim that the row s are Then calculating the co hom ology

76

SECTION

3.5

of the total com plex tw o w a y s (or appealing to the spectral sequence of a double com p lex) gives the assertion of the lem m a . The exactness of the row s will follow fro m the fact that the are in je ctive.

To see this, note that if

3 is in je c tiv e , the sequence

r z n w ( w , 3) -» n w , 3) -> n w n z n w , 3)

o is exact for a n y

W.

3q

o

This reduces the assertion to the absolute case, i.e.,

to showing that o -» h y , 3) -> ® r < Y Jf« is exact.

Since

e r ( Y j l n Y j 2 , 3) -> . . .

3 is a direct s u m m an d of its sheaf

("discontinuous") sections, w e can replace calculation is local at a point the simplex Exercise.

( j : y € Y j),

Let

usual, and let

in Y,

and the cohom ology is that of

w ith coefficients in the stalk

D = mDo,

For

so 0(D) = 0 (m ).

m > 0, v e r i f y that

u > ijjp ex a c tly w h e n the corresponding (b)

For

v q

,

m < 0, show that

v j , . . . , v n,

. . . , v ^ . . . , v n.

is convex.

X u give a basis for

Show d ir e c tly that

m, > 0

and

Z m j < m;

H°(lPn,0 (m D )). is concave, so that the sets Z(u)

and

H^^dslm ) = 0 unless

H1(lPn,0 (m D )) = 0 for all

Use this to v e r i f y that

HndPn,0 (m D )) = ® C * X Ui the sum o v e r those and

Dq, . . . ,D n as

Compute the cohom ology as

u = ( m j , . . . , m n) w ith

are co n ve x and unequal to N r,

mj < 0

3y .

is zero on the cone generated b y

is m e* on the cone generated by (a )

3. But then the

X = IPn, a toric v a r i e t y w ith its divisors

follows. Show that and

y

3 by

3 of a r b i t r a r y

i * n,

Z(u) = (0).

and that

u = ( m j , . . . , m n) w ith

Z m , > m.

Using the sam e techniques, w e h a v e the following im p o r t a n t result, w h ich is also special to toric varieties: P r o p o s itio n . proper m a p

Let

A'

be a r e f in e m e n t of A ,

f: X' = X ( A ') -* X = X (A ).

f * ( 0 X')

=

In particular, taking

Sx

and

R ^ tO x ')

g iv in g a b ir a t io n a l

Then =

0

fo r all

i >0 .

X ‘ to be a resolution of singularities, this says

that e v e r y toric v a r i e t y

X

has ra tio n a l singularities.

COHOMOLOGY

P ro o f.

The assertion is local, so

together w ith all of its faces, so

OF

(ii)

HKX'.OxO = 0

to be a cone

a

i >0. X is n orm a l

it is obvious since both spaces of sections are

and

f is birational; here

© C * X U, the sum o v e r

u € cr^PlM. The second follows from the fact that its sections, since the support

77

X = UCT. The claims are that

n x '.O x ') = n x .O x ) = A a ;

The first is a general fact, since

BUNDLES

w e can take A

(i)

for all

LI NE

IA 'I = |a| is convex.

C?x« is generated by

CHAPTER MOMENT M A PS

4.1

4

AND THE TAN GEN T BUNDLE

Th e m a n if o l d w i t h s in g u la r c o r n e r s

Although w e are w orkin g m a in ly w ith com plex toric varieties, it is w o r th noticing that th e y are all defined n a tu r a lly o v e r the integers, sim ply by replacing For a field

K, the

C by

1L in the algebras: Ua = Spec(Z[cj'‘'n M ]).

K -v alu ed points of Ua can be described as the

semigroup hom om orp hism s Horn sg( a v n M , K ) , w here

K is the m u ltip lic a tive semigroup

K *U {0 }.

For exam ple, for

K = 1R c C, w e h a v e the real points of the toric v a r ie t y . In fact, the sam e holds w h e n

K is just a sub-sem igroup of

C.

The im p o rta n t case is the semigroup of non n ega tive real nu m b ers IR> =

flR+ U{0},

w hich is a m u ltip lic ative sub-sem igroup of

case th ere is a retrac tion given by the absolute value, IR> C For a n y cone (U

a,

C ->

.

this determ in es a closed topological subspace

= H o m sg( a v n M , IR>)

together w ith a retraction

C

UCT =

H o m sg( a v n M , C)

UCT -» (U CT)>. For a n y fan

together to fo rm a closed subspace X (A )>

of X ( A )

A,

these fit

tog eth er w it h a

retraction X (A )> For exam ple, if

a

k copies of

IR> and

n-k

N,

then

X ( A )> .

can be a little worse.

e*, . . . ,e^

that

(U a)> is isomorphic to a product of

copies of

is a m anifold w ith corners.

of X>

X (A )

is generated b y vectors

fo rm part of a basis for X>

c

IR. Thus if X is nonsingular,

W hen

X

is singular, the singularities

For the toric v a r i e t y

78

X = IPn, w it h its

THE

MANIFOLD

WITH

SINGULAR

CORNERS

usual co verin g b y affine open sets U, = UCT., (Uj)> (tg:

... :1:

... :tn) w ith

P";> =

tj > 0.

consists of points

Hence

IR>n + 1 " (0) / IR+

=

{(to, . . . ,tn) € IRn + 1 :

w hich is a standard n-simplex. (x 0 : . . . : x n)

The fiber o v e r a point

tj > 0 and

to + . . . + t n = 1} ,

The retrac tion fr o m

*->

1

(to, . . .

( |x0 l, . . •, lxn|)

,tn)is

CPn to

DPn> is

.

acom pact torus of dimension

equal to C a r d {i: tj * 0} - 1.

The algebraic torus SN =

Tn

contains the c o m p a c t torus

H om (M ,S*)

c

Horn (M ,€ * )

where

S1 = U ( l )

circles.

From the isomorphism of C* w ith

isomorphism of Tn

=

is the unit circle in C*. So Sn

IR+ w ith

Sn

=

x

Sjsj:

TN ,

is a product of n

S1.* IR+ = S 1 x [R (v ia the

IR given by the lo g a rith m ), w e h a v e

Hom(M,lR + ) =

S n x Hom(M,IR)

=

Sn

x

N(r ,

a product of a com pact torus and a v e c t o r space. P ro p o sitio n .

The r e t r a c t i o n

the q u o t ie n t space of X ( A ) P roo f.

X ( A ) —» X (A )>

=

From w h a t w e just saw,

X (A )>

b y the action of the c o m p a c t torus

Look at the action on the orbits ( 0 T)> =

identifies

X (A )> f lO T =

Sn-

Ox: Hom(T-LDM,IR + )

H o m ( T _Ln M , D R )

=

N ( t ) jr .

S n (t ) acts fa ith fu lly on

quotient space (Ox )> = N ( t )|r. Since

'with

Sn

Ox = T n (t ) w ith

acts on Ox b y w a y of its

80

SECTION

4.1

projection to Sjsj(T), the conclusion follows. Note that the fiber of X —» X>

( 0 T)> can be identified w ith

w h ich is a com pact torus of dimension

Sn(t)>

( 0 T)> fit together in X> corresponding orbits X

n - d im (x ).

The spaces

in the same com b inatorial w a y as the

0 T in X. If one can get a good picture of the

m anifold w ith singular corners how

over

X>, this can help in understanding

is put together topologically/1^

The m anifold w ith corners sort of "dual polyhedron" to

A,

X>

can be described a b s tr a c tly as a

at least if X

v e r t e x for each n-dim ensional cone in

A;

is complete:

X>

has a

t w o v e rtic es are joined by

an edge if the corresponding cones h a v e a co m m o n ( n - l ) - f a c e , and so on for sm aller cones. then

X>

If

A = Ap

arises fro m a c o n ve x polytope in

M( r,

is h om eom orph ic to P. In the next section w e w ill see an

explicit realization of this hom eom orphism .

W e list some oth er simple

properties of this construction, leaving the verifications as exercises: (1)

If

r

is a n y positive n u m b er, the m apping

determ in es an a utom orphism of

t »-> t r

IR>. This determ in es an

a u tom orph ism of the spaces (U CT)> = Homsg( a v n M ,R > ), together to d e te r m in e a h om e om orp h is m from positive integer,

z »-> z r

(2)

If

r

is a

X,

com patible w ith

X> C X —» X>. The quotient

w ith the action of

Tjq/Sjsj = N|r acts on

Tjvj on

X. The inclusion

w ith respect to the inclusion (3)

to itself.

is an endom orphism of €, w h ic h induces

sim ila rly an endom orphism of a n y toric v a r i e t y the m aps

X>

w h ich fit

X/S n = X>, c o m p a tib ly X> c X

is e q u iv a ria n t

N r = Hom(M,IR + ) c H om (M ,C *) = T^.

There is a canonical m apping

Sjsj * X>

-* X, w h ich realizes

X as a quotient space. (4)

For a n y cone

t

, the inclusion

(0 T)> C (U T)> is a

deform a tion retract. Exercise.

Use the L e r a y spectral sequence for the m ap ping

to r e p ro v e the result that the Euler ch aracteristic of X of n-dim ensional cones.

X —> X>

is the n u m b e r

MOMENT

4.2

MAP

81

M om ent m ap

M o m e n t m aps occur fr e q u e n tly w h en Lie groups act on v a r ie t ie s / 2^ Toric va rieties provide a large class of concrete examples.

In this

section w e construct these maps explicitly, and then sketch the relation to general m o m e n t maps. Let

P

be a co n ve x polytope in

rise to a toric v a r i e t y the sections

X u for

X = X (A p ) u € PnM p:

w ith ve rtic es in

and a m orph ism

M,

giving

cp: X —» IP* - 1 via

(see §3.4). Define a m o m e n t m a p X

-»

Mr

by ^ (x )

Note that

=

^

V; V

SIX

Z

(x)l

U € PnM

|jl is S ^ - in v a r ia n t , since, for

IX u(t*x)| = IX u( t ) M X u(x)l = l%u(x)|. on the quotient space

X>

t in Sjsj and

It follows that

x in

X,

\i induces a m a p

X/S^ = X>: p:

Pro p o sitio n .

I X u( x ) l u .

X^

-*

M

r

.

The m o m e n t m a p defines a h o m e o m o r p h i s m f r o m

o nto the p o ly tope P. In fact, one gets such a h om eom orphism using a n y subset of the

sections

X u as long as P

is the co n ve x hull of the points, i.e., the

subset contains the vertices of P. P ro o f.

Let

the fan.

Q be a face of

P,

W e claim that in fact

and let

a

be the corresponding cone of

p. maps the subset

( 0 CT)> b ije c t iv e ly

onto the r e la tiv e in terior of Q: ( 0 ff)>

Int(Q) ,

as a real a n aly tic isomorphism. Let

pu(x ) = I X u(x )l/ X I X U (x)|, w h e r e the sum in the

den om in a to r is o v e r all ve rtices of that a point

u' in P n M

P). Th erefore

(or in a subset containing the

0 < pu(x ) < 1 and

x of (Oa)> is in

p.(x) = X p u(x )u .

Note

82

SECTION

H o m ( a i n M , IR +) c

4.2

H o rn sg( a v n M , IR>) ,

the inclusion by extending by zero outside

crJ\ It follows that for

x in

(Oa)>, pu(x ) > 0

if u € Q ;

pu(x) = 0 if

u t Q .

W ritin g this out, one is reduced to proving the following assertion: Lem m a.

L et V

be a fin ite -d im e n s io n a l real v e c t o r space, and let

be the c o n v e x h u ll of a fin ite set of v e cto rs

uj_f . . . , u r

K

in the dual

space V*.

A ssum e th a t

K is n o t con tained in a hyperplan e.

L et

El, . . . , e r

be a n y p ositive n u m b e rs , and define pp V —> IR b y the

fo r m u la p,(x)

=

Then the m a p p in g

Eie ui( x ) / ( e i G ul (x) + . . . + er e ur (x >) .

\i: V -» V * ,

p (x ) = p ^ (x )u i + . . . +pr ( x ) u r , defines

a real a n a ly tic is o m o rp h is m of V

onto the in t e r io r of K.

This is proved in the appendix to this section. Exercise.

Show that the vertices of the im age of the m o m e n t m a p

are the im ages of the points of The m ap fro m torus

Tn on

Xp

Xp

=

H o m (M , £ * )

dete rm in ed by the m a p

fixed by the action of the torus

Tn -

IPr ”1 is com patible w ith the actions of the

and the torus

Tn

of P flM .

to

X

Z r —> M

T = ( C * )r

on

P r_1 , w ith the m a p

-> H o m (Z r ,C*)

=

T

taking the basic ve ctors to the points

The action of the Lie group

S = (S 1) 1" on

IP1""1 dete rm in es

a m om en t map

m: If

IP1" 1 -»

x € P r - 1is represented b y

Lie(S)* = (IRr )* v =

=

IRr .

( x j , . . . , x r ) € Cr , then, up to a

scalar factor, this m o m e n t m ap has the fo rm u la

m (x )

=

— Z W 2 e* . Zlxjl 2 i = l

The following exercise shows that this agrees w ith a general construction of m o m e n t maps.

MOMENT

Exercise.

Define

r v at the origin

MAP

83

r v (t ) = >2 IIt*v ||2. The d e r iv a t iv e of

r v : T/S -» IR by

e of the torus is a linear m ap de( r v ) : IRr = Te(T/S)

i.e., de( r v ) is in (lRr )*.

->

IR

!Hl(x) = ||v||"2 *de( r v ).

Show that

The composite (n Xp ------

is then a m a p from Exercise. x

Xp

to

(IRr )*

-> M r

M r.

Show that this composite takes

x

to

jjl(x2), w h e r e

x 2 is the m ap defined in ( 1 ) of the preceding section.

A p p e n d ix

on c o n v e x i t y

The object is to p ro v e the following e le m e n t a r y fact. P r o p o s itio n .

L et

u*, . . . , ur

be points in

an y af f i ne h ype rpla n e, and let

IRn, n o t co n ta in e d in

K be th e ir c o n v e x hull.

L et

s i,...,e r

be a n y p ositive real n u m b e rs , and define H: IRn —> !Rn b y

H(x) -

w h e re f(x ) = p r o d u c t on

e

IRn. Then

-j~

£

f(x ) k = l

t ke ( “ " ’x ) uk , k

+ ... + er e ^U r’ x \

and ( , ) is the usual i n n e r

H defines a real a n a ly tic is o m o rp h is m of IRn

onto the i n t e r i o r of K. W e will deduce this from the following t w o related statem ents: ( A n)

L e t u i, . . . , u r

be v e cto rs in

IRn th a t span

DRn, and le t

be the cone (w ith v e r t e x at the o rig in ) th a t th e y span. L e t be a n y p ositive re a l n u m b e rs . F -

Then the m a p

C . . . ,er

F: IRn —> IRn defined b y

£ t k . d X n

Q ^(logD )

(1)

Dj,

-»

0

,

extended b y ze ro to X.

is trivial.

For (2), consider the canonical m ap of sheaves * ^ y (lo g D )

that takes

u € M

to d (% u)/%u. To see that this m a p is an iso m o r­

phism, it suffices to look locally on affine open sets

Ua, w h e r e the

assertion follows rea dily fro m the a bove description of

Q^(logD).

The second m apping in (1) is the residue m apping, w h ich takes co = X fjd X j/ X i fi is divisible by

4.4

to

©flip).. The residue is zero precisely w h e n each Xj,

i.e., w h en

co is a section of

Q^. ^

S erre d u a lity

For a v e c t o r bundle

E on a nonsingular com plete v a r i e t y

X,

S e rre

d u a lity gives isomorphisms H n _ i( X , E v ® Q ^ ) If X is a toric v a r i e t y and

E = 0(D )

isomorphism respects the grading b y consists of isomorphisms

s

H !( X , E ) * .

for a T -d ivis o r D, this M;

w ith

= 0 (- Z D j),

it

88

SECTION

4.4

H n -i( X , e ( - D - Z D i))_u for each

=

( H i ( X , Q ( D ) ) u) "

u € M. It is interestin g to give a direct proof.

the piecewise linear function associated to D, and function for point

the canonical divisor

Vjcorresponding to h ‘( X , 0 ( D ) ) u

=

sok(v|) = 1

-ZDj,

each divisor

Let

i|> = ifjp be

k = for the lattice

Dj. By the description

H*z ( N r ) , HJ( X , 0 ( - D - I D , ) ) _ u =

h

in §3.5,

£ .(N r ) .

where Z

=

{ v € N r : i|i(v) < u ( v )} ;

Z‘ =

{ v € N r : -ip(v) + k (v) < - u ( v ) }

=

{ v € N r : u ( v ) < i}j(v) - k(\ )} .

Serre du a lity am ounts to isomorphisms (SD)

Hnz7‘ ( N R )

b

'H z (N r ) " .

The next t h ree exercises outline a proof. Note that the set S

=

{ v € N r : k(v) = 1}

is the b o u n d a r y of a polyhedral ball the com plem en t of

Exercise.

If

{0}

in

B, so is a d eform ation r e t r a c t of

N r.

C is a n o n e m p t y closed cone in

N r , show th at th e r e are

canonical isomorphisms H*c (N r )

£

H‘(N r , N R ' C )

3

H H B .S 'S n C )

s

Hi_1(s

s

H n .j.^ sn c),

n snc)

the last b y A lexa n d e r duality, w h e r e the

~

denotes reduced

cohom ology and hom ology groups. Exercise.

Show that the em bedding

SflZ

S ' (S f lZ 1) is a

deform a tion retract. ^ Exercise. or

P r o v e (SD), first in the cases w h e r e

Z‘ = N r and

Z = (0);

h z7 1( n r )

s

Z = Nr

and

otherw ise Hn-,-1(s ' s n z ' )

£

Hn. j . i ( s ' s n z r

Z‘ = (0),

SER R E

a

Exercise.

Suppose

DUALITY

H n- j - l ( S n Z ) *

X = X (A )

s

H ^ N r )* . (6)

is a nonsingular toric v a r i e t y , and

is a strongly co n ve x cone in Njr. Show that the sum o v e r all |A|; and

(ii)

u in

M

89

(i)

|A|

r ( X , Q x ) = © C * X U,

that are positive on all nonzero v e cto rs in

H K X jQ ^ ) = 0

i > 0. ^

for

Grothendieck extended the Serre du ality th eo re m to singular varieties. c o m p le x

For this, the sheaf oo’x

Q x must be replaced by a dualizing

in a derived category.

When

X

is C oh en -M a ca u la y,

h o w e v e r , this dualizing complex can be replaced by a single dualizing sheaf cox . For a v e c to r bundle C oh en -M a ca u la y v a r i e t y

E on a com plete n-dim ensional

X, Grothendieck du a lity gives isomorphisms

H n _ i(X , E " ® c o x ) For a singular toric v a r i e t y

s

H *(X ,E )*.

X, ED* m a y not be a Cartier divisor

(or e v e n a 0 .

is c o m p le t e , and L is a line bundle on H n i (X , L v ® c o x )

P roo f.

s

X, then

H i( X , L ) * .

P a r t (a ) is local, so w e m a y assume

X = U a for some cone

a.

In this case, (a) is precisely w h a t w as proved in the preceding exercise. Then (b) follows; in fact, if E is a n y v e c to r bundle on du ality for

f*(E)

H n i (X ,E ^ 0 cox ) =

X, using Serre

on X', w e h a ve =

H n _ i( X , E ^ ® R f* (Q x .))

H n-i ( X ‘ , f* (E )v ® Q x .)

s

H U X 'J ^ E ))*

s

H !( X , E ) * ,

the last isomorphism using the last proposition in Chapter 3. Exercise.

Let j: U -» X

in X. Show that

be the inclusion of the nonsingular locus

j*(fiy ) =

U

90

SECTION

4.4

There is a p r e t t y application of du ality to la ttice points in polytopes.

If

P

is an n-dim ensional polytope in

M r w ith ve r tic e s in

M, w e h a v e seen that th ere is a com plete toric v a r i e t y am ple T-C a rtier divisor

D on

X w hose line bundle

X

0(D )

and an is generated

by its sections, and these sections are linear com binations of in P flM .

By refining the fan, one m a y take

X

%u for

to be nonsingular, if

desired. Consider the exact sequence 0 E xercise,

(a )

O(D-IDi) Show that

-> 0(D)

D - ZDj

-> 0 (D)Iz d

-*

0.

is generated b y its sections, and

these sections are

©

C .% u .

u € Int(P)nM

(b)

Deduce that X ( X . 0 (D)l 5;Di) =

w here

h ° (X ,0 (D )lZDl) =

DP is the bo u n d a ry of

C a rd O P flM ) ,

P.

N ow b y Serre-Grothendieck duality, % (X,0(D - XDj))

=

( - l ) nX (X ,0 (-D )) .

Since the higher cohom ology vanishes, ( - l ) nX ( X , 0 ( - D ) )

= C ard d n t(P )O M ) .

It is a standard fact in algebraic g e o m e tr y th at for a n y Cartier divisor

D on a com plete v a r i e t y Hv)

is a polyn om ial in v

=

X, the function

Z,

X (X ,0 (u D )),

of degree at most

is n if D is a m p l e . ^

f: Z

n = d im (X ),

In this case, for

and this degree

v > 0, f ( v ) = C a r d ( ^ - P n M ) .

P u tting this all together, w e h a v e the C o r o lla ry .

If P

v e rtic es in

M,

is a c o n v e x n -d im e n s io n a l p o ly to p e in th e re is a p o ly n o m ia l fp Card ('U 'Pn M )

fo r all integers v > 0 , and

=

of degree n fp(u)

Mg*

w ith

such t h a t

u

BETTI

NUMBERS

C a rd U n t(u -P )n M )

=

91

( - l ) n fp(-^u)

fo r all v > 0 . This form u la, called the inv e rsio n f o r m u la , w as con jec tu red by Ehrhart, and first proved by Macdonald in 1971. The above proof is from [Dani]. Exercise. at

Compute

fp(t»)

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

(l,b ),

M = Z 2 w ith v e rtic es

for the polytope in for

b a positive integer, and v e r i f y the

inversion form u la d irectly in this case. This can be interpreted by m eans of the a d ju n c t io n f o r m u l a , w h ich is an isomorphism

Q y ® 0 (D)|D = coD, given by the residue.

From the exact sequence 0

—>

Q£0(D)

—»

coD

-»

0

and the long exact cohom ology sequence, w e see that r(D,ooD) =

©

ۥ X u ,

u e Int(P)nM

HKDjCOq) = 0 for

0 < i < n-1,

X (D ,O d) =

dim Hn_ 1 (D, 00 [)) = 1. Th erefore

and

( - l ) n_1 X(D,ood) =

1 - C a rd d n t(P )n M ) .

This m eans that the a rith m e tic genus of For exam ple, if P (d,0),

and

(0,d),

then

X = IP2, D is a c u r v e of degree

(0,0), (d,0), (0,e), and

c u r v e of bidegree

4.5

Z 2 w ith v e rtic es at

is the polytope in

is 1 + 2 + . . . + (d -2 ) = ( d - l ) ( d - l ) / 2 . ve rtic es at

D is Card( In t (P ) D M).

(d,e),

If P

(d,e),

is a rectangle in then

(0,0),

d, so its genus Z 2 w ith

X = IP1 * IP1, D is a

and the genus of D is ( d - l ) ( e - l ) .

B etti n u m b e rs

For a sm ooth com pact v a r i e t y j th betti num ber.

When

X = X (A )

n u m b e r of k-dim ensional cones in

X,

let

pj = ra n k (H J( X ))

is a toric v a r ie t y , let A.

d^

In fact, these nu m b ers

be its be the

SECTION

92

4.5

d e te rm in e each other: Proposition. then

If X = X (A )

pj = 0 if j

is a n o n sin g u la r p r o j e c t i v e t o r ic v a r i e t y ,

is odd, and

P2k =

Set

hk = P2k- ^

p x (t)

Z ( - l ) 1" k ( k ) d n_ i . i =k

P x (t) “

P o in ca re

= Z h k t2k =

Z dn_ j(t 2 - 1 ) ' i =0

=

polynom ial, then

z d k ( t 2 - l ) n- k . k =0

For exam ple, for the topological Euler characteristic, X (X )

=

X ( - l ) j fJj =

P x (- 1 )

=

dn ,

as w e h a v e seen. Exercise.

I n v e r t the above form ulas to express the

dk

in te r m s of

the betti numbers:

"I In fact, one can assign to a n y complex algebraic v a r i e t y necessarily smooth, com pact, or irreducib le) a polyn om ial

X

(not

Px(t)>

called its v ir t u a l P o in c a r e p o ly n o m ia l, w ith the properties: (1)

P x (t )

=

H r a n k (H H X ))t [ if X

is n o n s in g u la r and

p r o j e c t i v e ( o r c o m p le te ); (2)

P x (t )

= X

P y (t) +

Y

is a closed algebraic subset of

and U = X n Y.

For exam ple, if

U is the co m p lem en t of r

points in

IP1, then

P j j ( t ) = t 2 + 1 - r. (N ote in particular that the coefficients can be negative.)

It is an easy exercise, using resolution of singularities and

induction on the dimension, to see that the polynom ials are u n iqu ely d eterm in ed by properties (1) and (2). Other properties follow easily from (1) and (2): (3)

If X

is a disjoint union of a fin ite n u m b e r of locally closed

sub varieties O(i), (4)

then

I f X = Y x Z,

P X(t) = X pQ(j)(t); then

P x (t ) = P Y (t ) - P z (t).

BETTI

Exercise.

If

NUMBERS

X -* Z is a fiber bundle w ith fiber

triv ia l in the Zariski topology, show that If

X

93

Y

that is locally

P x (t) = P y ^ ^ P z ^ )-

is nonsingular and complete, (1) says that

Euler ch aracteristic

X(X ).

W ith

X, Y,

and

P x (-l)

U as in (2),

is the

th e r e is a

long exact sequence

... w here

-» h|.u

H*

coefficients.

-

-> h[.y -

H^X

h|.+1u -» h|.+1x

denotes cohom ology w ith com pact supports, and rational It follows from this that

P x (-l)

m ust a lw a y s be the

Euler c h a r a c t e r is t ic w ith c o m p a c t support, i.e., P x (-l)

=

X c (X )

=

Z ( - l ) i d im (H j.X ) .

The existence of such polynomials follows fro m the existence of a m ixed Hodge stru ctu re on these cohom ology groups/11^ This gives a w eight filtration on these v e cto r spaces, com patible w ith the m aps in the long exact sequence, such that the induced sequence of the m th graded pieces rem a in s exact for all ... -

m:

gr^J (H j. U) -* g r ™ (H ^ X ) -* g r $ ( H ^ Y ) -* g r ^ O f j ^ U )

This m eans that the corresponding Euler ch aracteristic X ” (X )

=

Z ( - l ) i d i m ( g r ^ ( H 1c X ) )

is also additive in the sense of (2). If X then

g r $ ( H ™ X ) = H™X = Hm(X ),

is nonsingular and p r o je c t iv e ,

so % ™ (X ) = ( - l ) m dirn(Hm (X)).

Hence w e (m u s t) set P X(t )

=

Z ( - l ) m X ^ (X )tm m

Z ( - l ) i + m d irn(gr™ (H |: X ) ) t m . i , rn

One need not know a n y th in g about the m ixed Hodge structu res or the w eigh t filtration to use these virtu a l polynom ials to calculate betti num bers; one has only to use the basic properties that d e te r m in e them .

For exam ple, for a torus

Hence if X = X ( A )

T = (C*)^,

we have

P j ( t ) = ( t 2 - l ) k.

is an a r b it r a r y toric v a r ie t y , since it is a disjoint

SECTION

94

union of its orbits

0 T = TM(T)» by p r o p e r t y (4) w e h a v e

^XCA)^)

w here

4.5

21 P qt M

~

Z d n_ k ( t 2 - l ) k

=

dp denotes the n u m b e r of cones of dimension

true for a n y toric v a r ie t y .

In case

X (A )

p in

A.

This is

is nonsingular and com plete,

h o w e v e r , this is the ord in a ry Poincare polynom ial by p r o p e r t y (1), and this proves the proposition. In fact, the proposition is also tru e w h e n complete.

For this one needs to know that

A

is only simplicial and

gr^J (H ™ X) = Hm(X )

this case. This follows from the combination of t w o facts: intersection (co)hom ology groups

IHm(X )

n u m b e r of

m;

(ii)

if X

is

Hm(X ) = IHm( X ) . (12)

A toric v a r i e t y modulo all primes.

also in

the

of an a r b it r a r y co m p act

v a r i e t y h a v e a m ixed Hodge stru ctu re of pure w eig h t a V -m a n ifo ld , then

(i)

X

is defined o v e r the integers, so can be reduced

Since X

is a disjoint union of orbits

F q-va lu ed points of the torus

0 T, and the

T n (t ) = (Gm ) k is ( q - l ) k , it

follows that n

Card ( X ( F q) )

W hen

X

=

Z d _ .k ( q - l ) k k=0

1= 0 k = i

is nonsingular and p ro je ctive , Deligne’s solution of the Weil

conjectu res implies that

C a r d ( X ( F Dr ) ) p w h e r e the

Exercise.

=

2n Pj Z ( - l ) j Z X,r, , j =0 i = 0 J1

Xjj are uniquely d eterm in ed com plex n u m b ers w it h

Use the preceding tw o form u las to give a n o th er proof of the

proposition. W e will give a third proof of the proposition in Chapter 5. F orm ula ( * ) implies that for an a r b it r a r y toric v a r i e t y the Euler ch aracteristic w ith com pact support X C(X ) = P x ( “ l ) to the n u m b e r of n-dim ensional cones in

A.

X = X (A ), is equal

W e h a v e seen earlier

BETTI

NUMBERS

that the o r d in a r y Euler ch aracteristic

%(X)

is also equal to the

n u m b e r of these cones. This raises the question of w h e t h e r this is special to toric varieties, or is tru e for all varieties. Exercise. v a r ie t y .

Show that

X ( X ) = X C(X )

E q u ivalently, show that

X ( X ) = X ( Y ) + X (U )

a closed algebraic subset in a v a r i e t y a classical neighborhood of Y that

X (N n Y ) = 0 . (13)

for e v e r y com p lex algebraic whenever

X w ith co m p lem en t

in X such that

Y

U. If N

% ( Y ) = X (N ),

show

CHAPTER IN TER SEC TIO N

5.1

5 THE ORY

C h o w g rou p s

In this ch ap ter w e will w o rk out some of the basic facts about intersections on a toric v a r ie t y . A ^ (X )

On a n y v a r i e t y

X,

the Chow group

is defined to be the free abelian group on the k-dim ensional

irreducible closed subvarieties of X, by the cycles of the fo rm

modulo the subgroup generated

[ d i v ( f )], w h e r e

f is a nonzero rational

function on a (k + l)- d im e n s io n a l s u b v a r ie ty of on an a r b it r a r y toric v a r i e t y A n- i ( X )

X

X.

W e h a v e seen that

the toric divisors gen e rate the group

of Weil divisors modulo rational equivalence.

The obvious

generalization is valid as w e l l : ^ P r o p o s itio n . X = X (A )

The Chow g ro u p

Let

at least

of an a r b i t r a r y t o r ic v a r i e t y

is g e nera ted b y the classes of the o rb it closures V(

©

dim a = n - i

A k(Off)

Oa is an open subset of affin e space

the sam e principle that

Ai(Oa) = Z-[Oa] and

Since the restriction from

-»

0 .

A 1, w e see b y

A^(O a) = 0 for

A^CX*) to A k (0 CT) m aps

[V(cr)]

k * i.

to [0 CT],

a simple induction shows that the classes [V(a)J, d i m ( a ) = n-k, generate

A^(Xj).

For a Cartier divisor

D on a v a r i e t y

96

X,

the s u p p o rt of

D is

CHOW

GROUPS

97

the union of the codimension one subvarieties is not zero.

W e say that

p r o p e r ly if V

W

such that

ordvy(D)

D m eets an irreducible s u b v a r ie ty

is not contained in the support of

V

D. In this case one

can define an in te rse ctio n cycle D*V by restricting

D to V

(i.e., by

restricting local defining equations), determ in in g a Cartier divisor on V,

and taking the Weil divisor of this Cartier divisor:

Let us w o r k this out w h e n Cartier divisor, and

X

is a toric v a r ie t y ,

V = V (a ).

divisor on the toric v a r i e t y

In this case,

D«V = [Dly]-

D = Xa jD j

D lv (a)

is a Ta T-C artier

V ( ct), so w e will h a v e

D.V(cr) the sum o v e r all cones

=

Z b y V (y ) ,

y containing

a

w ith

d im (y ) = d i m ( a ) + 1 ,

and the

by are certain integers. To com p ute the m u ltip lic ity

suppose

y is spanned by

v^

Let

let

by,

cr and a finite set of m in im a l edge vectors

i € Iy. Here is an exam ple w h e r e

Iy has th ree vectors:

e be the g enerator of the one-dim ensional lattice

that the im age of each

D|y

v* in Ny/Ng

sj be the integers such that

vj

N y / N a

is a positive m u ltiple of

m aps to

s^e

in

N y / N a .

such e,

and

Then

by

is given by the form u la by

To see this, for a n y

=

cone

ai si

for all i in

y containing

linear function ony corresponding to the that

V(cr)

ct let u(y)

.

€ M / M (y )

divisor

be the

D. The assumption

is not contained in the support of D translates to the

condition that M(cr)/M(y).

Iy

As

u(y)

vanishes on

d e te rm in e the divisor the m u ltip lic ity

ct,

w hich m eans that

y varies o v e r cones in the star of

by

D|v( .

Therefore

u(y)

cr, these ct

is in u(y)

is a facet of

y,

a* = -

=

- = Sj ( - ) = spby ,

as asserted. When

X is nonsingular, there is only one i = i(y) in

Iy,

and

Sj = 1, so by = aj(y) is the coefficient of D^y) in D. In this case, each Dk is a Cartier divisor, and v(y)

a and v k span a cone

if

y

Dk-V(cr) 0 In fact, if X (A ) v(y)

if

ct and v k do not span a cone in A

is nonsingular, Dk and V(