Annals of Mathematics Studies Number 131home.ustc.edu.cn/~hanzr/pdf/Introduction to Toric Varieties-Fulton.pdf · relations with commutative algebra and lattice points in polyhedra

Annals o f Mathematics Studies

Number 131

Brought to you by | University of Science and Technology of ChinaAuthenticated

Download Date | 2/26/19 11:53 AM

T H E WILLIAM H. ROEVER LECTURES IN GEOMETRY

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 Mathematical 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 sh in g t o n U n iv ersity S t . L o uis

PRINCETON UNIVERSITY PRESS

PRINCETON, NEW JERSEY

1993



Copyright © 1993 by Princeton University Press

A LL RIGHTS RESERVED

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



http://pup.princeton.edu

William H. Roever 1874-1951



Dedicated to the memory of

Jean-Louis and Yvonne Verdier



CONTENTS

Chapter 1 Definitions and examples

1.1 Introduction 31.2 Convex polyhedral cones 81.3 Affine toric varieties 151.4 Fans and toric varieties 201.5 Toric varieties from polytopes 23

Chapter 2 Singularities and compactness

2.1 Local properties of toric varieties 282.2 Surfaces; quotient singularities 312.3 One-parameter subgroups; limit points 362.4 Compactness and properness 392.5 Nonsingular surfaces 422.6 Resolution of singularities 45

Chapter 3 Orbits, topology, and line bundles3.1 O rb its 513.2 Fundamental groups and Euler characteristics 563.3 Divisors 603.4 Line bundles 633.5 Cohomology of line bundles 73

Chapter 4 Moment maps and the tangent bundle

4.1 The manifold w ith singular corners 784.2 Moment map 814.3 Differentials and the tangent bundle 854.4 Serre duality 874.5 Betti numbers 91

Chapter 5 Intersection theory

5.1 Chow groups 965.2 Cohomology of nonsingular toric varieties 1015.3 Riemann-Roch theorem 1085.4 Mixed volumes 1145.5 Bezout theorem 1215.6 Stanley's theorem 124

Notes 131

References 149

Index of Notation 151

Index 155

v i iBrought to you by | University of Science and Technology of ChinaAuthenticated




PREFACE

Algebraic geom etry has developed a great deal of m achinery for

studying higher dimensional nonsingular and singular varieties; for

example, all sorts of cohomology theories, resolution of singularities,

Hodge theory, intersection theory, Riemann-Roch theorems, and

vanishing theorems. There has been real progress recently toward at

least a rough classification of higher dimensional varieties, particularly

by Mori and his school. For all this — and for anyone learning

algebraic geom etry — it is important to have a good source of

examples.

In introductory courses this can be done in several ways. One

can study algebraic curves, where much of the story of their linear

systems (line bundles, projective embeddings, etc.) can be worked out

explicitly for low genus/D For surfaces one can work out some of the classification, and work out some of the corresponding facts for the

special surfaces one finds/2) Another approach is to study varieties

that arise in "classical" projective geometry: Grassmannians, flag

varieties, Veronese embeddings, scrolls, quadrics, cubic surfaces, etc/3)

Toric varieties provide a quite different yet e lem entary w a y to

see m an y examples and phenomena in algebraic geometry. In the

general classification scheme, these varieties are v e r y special. For

example, they are all rational, and, although they m ay be singular,

the singularities are also rational. Nevertheless, toric varieties have

provided a rem arkab ly fertile testing ground for general theories.

Toric varieties correspond to objects much like the simplicial complexes

studied in e lem entary topology, and all the basic conceptss on toric

varieties — maps between them, line bundles, cycles, etc. (at least

those compatible w ith the torus action) — correspond to simple

"simplicial" notions. This makes everyth ing much more computable

and concrete than usual. For this reason, we believe it provides a good

companion for an introduction to algebraic geom etry (but certainly not

1The num bers refer to the notes at the back of the book.



X PR EF A C E

a substitute for the study of curves, surfaces, and projective

geometry!).

In addition, there are applications the other w ay , and interesting relations w ith com m utative algebra and lattice points in polyhedra.

The geom etry of toric varieties also provides a good model for how

some of the compactifications of sym m etr ic varieties look; indeed, this

was the origin of their study. Although we won't study compac

tifications in this book, knowing about toric varieties makes them

easier to understand.

The goal of this mini-course is to develop the foundational

material, w ith m any examples, and then to concentrate on the

topology, intersection theory, and Riemann-Roch problem on toric

varieties. These are applied to count lattice points in polytopes, and

study volumes of convex bodies. The notes conclude w ith Stanley's

application of toric varieties to the geometry of simplicial polytopes.

Relations between algebraic geometry and other subjects are

emphasized, even when proofs without algebraic geom etry are possible.

When this course was first planned there was no accessible text

containing foundational results about toric varieties, although there

was the excellent introductory survey by Danilov, as well as articles

by Brylinski, Jurkiewitz, and Teissier, and more technical monographs

by Demazure, Kempf - Knudsen - Mumford - Saint-Donat, Ash -

Mumford - Rapoport -Tai, and Oda, where most of the results about

toric varieties appeared for the first t im e .^ Since then the excellent

book of Oda [Oda] has appeared. This allows us to choose topics based

on their suitability for an introductory course, and to present them in

less than their m ax im um generality, since one can find complete

arguments in [Oda]. Oda's book also contains a wealth of references

and attributions, which frees us from attempting to give complete

references or to assign credits. In no sense are we try ing to survey the

subject. Almost all of the material, including solutions to m an y of the

exercises, can be found in the references. We make no claims for

originality, beyond hoping that an occasional proof m a y be simpler

than the original; and some of the intersection theory on singular toric

varieties has not appeared before.

These notes were prepared in connection w ith the 1989 William H.



PREFACE xi

Roever Lectures in Geometry at Washington University in St. Louis. I

thank D. Wright for organizing those lectures. They are based on

courses taught at Brown and the University of Chicago. I am grateful

to C. H. Clemens, D. Cox, D. Eisenbud, N. Fakhruddin, A. Grassi, M.

Goresky, P. Hall, S. Kimura, A. Landman, R. Lazarsfeld, M. McConnell, K.

Matsuki, R. Morelli, D. Morrison, J. Pommersheim, R. Stanley, and B.

Totaro for useful suggestions in response to these lectures, courses, and

prelim inary versions of these notes. Readers can also thank H. Kley

and other avid proofreaders at Chicago for finding m an y errors.

Particular thanks are due to R. MacPherson, J. Harris, and B.

Sturmfels w ith whom I have learned about toric varieties, and V.

Danilov, whose survey provided the model for these courses. The

author has been supported by the NSF.

In rewriting these notes several years later, w e have not

attempted to include or survey recent work in the subject.^ ) In the

notes in the back, however, we Have pointed out a few results which have appeared since 1989 and are closely related to the text. These

notes also contain references for more complete proofs, or for needed

facts from algebraic geometry. We hope this will make the text more

accessible both to those using toric varieties to learn algebraic

geom etry and to those interested in the geom etry and applications of

toric varieties. In addition, these notes include hints, solutions, or

references for some of the exercises.

March, 1993 William FultonThe University of Chicago



E R R A T A

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

Page For

12, line 12 Gordon

Read

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| a cone in A

63, line 23 abelian abelian if A contains a coneof maximal dimension

64 Replace lines 13-15 with;

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 - & 2 X a2x

70, line 28 ma^ ma2

79, lines mD D19, 23, 24

September, 1996 William Fulton



Introduction to Toric Varieties





CHAPTER 1

DEFINITIONS AND EXAMPLES

1.1 Introduction

Toric varieties as a subject came more or less independently from the

work of several people, primarily in connection w ith the study of

compactification problems/1 This compactification description gives a

simple w a y to say what a toric va r ie ty is: it is a normal va r ie ty X

that contains a torus T as a dense open subset, together w ith an

action T x X —» X of T on X that extends the natural action of T

on itself. The torus T is the torus C*x . . . x C* of algebraic groups,

not the torus of topology, although the latter will play a role here as

well. The simplest compact example is projective space P n, regarded

as the compactification of Cn as usual:

(C *)n C Cn C lPn .

Similarly, any product of affine and projective spaces can be realized

as a toric variety .

Besides its brev ity , this definition has the v irtue that it explains

the original name of toric varieties as "torus embeddings." Unfortunately, this name and description m ay lead one to think that one

would be interested in such varieties only if one has a torus one wants

to compactify; indeed, one m ay wonder if there wouldn’t have been

more general interest in this subject, at least in the West, if this name

had been avoided. The action of the torus on a toric va r ie ty will be

important, as well as the fact that it contains the torus as a dense

open orbit, but the problem with this description is that it completely

ignores the relation w ith the simplicial geometry that makes their

study so interesting. At any rate, we far prefer the name "toric

varieties," which is becoming more common.

In this introductory section we give a brief definition of toric

varieties as we will study them; in the following sections these notions

3Brought to you by | University of Science and Technology of ChinaAuthenticated


4 SECTION 1.1

will be made more complete and precise, and the basic facts assumed

here will be proved. A toric va r ie ty will be constructed from a lattice

N (which is isomorphic to Zn for some n), and a fan A in N,

which is a collection of “strongly convex rational polyhedral cones" a in the real vector space Njr = N<8>zlR, satisfying the conditions

analogous to those for a simplicial complex: every face of a cone in A

is also a cone in A , and the intersection of two cones in A is a face

of each. A strongly convex rational polyhedral cone a in Njr is a

cone w ith apex at the origin, generated by a finite number of vectors;

"rational” means that it is generated by vectors in the lattice, and

"strong" convex ity that it contains no line through the origin. We often

abuse notation by calling such a cone simply a "cone in N".

Let M = Hom(N,Z) denote the dual lattice, w ith dual pairing

denoted ( , >. If cr is a cone in N, the dual cone is the set of

vectors in Mg* that are nonnegative on a. This determines a

com m utative semigroup

Sq- = a v n M = { u € M : (u ,v ) > 0 for all v € a } .

This semigroup is finitely generated, so its corresponding "group

algebra" C[SCT] is a finitely generated comm utative (C-algebra. Such

an algebra corresponds to an affine variety: set

Ua = Spec(C[S(J]) .

If t is a face of a, then SCT is contained in ST, so CfS^] is a

subalgebra of C[ST], which gives a map UT Uff. In fact, UT is a

principal open subset of UCT: if we choose u € Sa so that t = a f l u x,

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

varieties fit together to form an algebraic variety , which we denote by

X (A ). (The "embedding" notation for this is TNem b (A ), but w e w on ’t

follow this convention.) Note that smaller cones correspond to smaller

open sets, which explains w h y the geometry in N is preferred to the

equivalent geom etry in the dual space M.

We turn to some simple examples. For these, the lattice N is

taken w ith a fixed basis e^, . . . , en, w ith Xj., . . . ,X n the elements in

C[M] corresponding to the dual basis. For n < 3, we usually w rite X,

Y, and Z for the first three of these. We first consider some affine



I NTRODUCTI ON 5

examples, where A consists of a cone a together with all of its faces,

and X (A ) is the affine va r ie ty UCT.

The origin {0} is a cone, and a face of every other cone. The dual

semigroup is all of M, w ith generators ±e£, . . . , ±e * , so the

corresponding group algebra is

CtM] = C[Xi, x r 1, X2, X2_1 Xn> X n-1] ,

which is the affine ring of the torus: U{q} = T = (C *)n. So eve ry toric

va r ie ty contains the torus as an open subset.

If a is the cone generated by ej, . . . ,en, then Sa is generated

by the dual basis, so

c [s CTi = m lf X2, . . . , Xn] ,

which is the affine ring of affine space: UCT = Cn.For another example take n = 2, and take a generated by e2

and 2ei - e2.

^ ^ ^ X XSemigroup generators for Sa are e^, e + e2 and e + 2e2, so

C[Sa ] = C[X, XY , X Y 2] = C [U ,V ,W ]/ (V 2 - U W ) .

Hence, is a quadric cone, i.e., a cone over a conic:



6 SECTION 1.1

Next we look at a few basic examples which are not affine. For

n = 1, the only non-affine example has A consisting of the cones

IR>0, iR<o> and {0}, which correspond to the affine toric varieties €,

C, and C*. These three cones form a fan, and the corresponding toric

va r ie ty is constructed from the gluing:

^ — ----- •------------ ►

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

<C <-> C* €

with the patching isomorphism given b y x h x " 1 on the overlap.

This is of course the projective line IP1.

Consider, for n = 2, the fan pictured:

This t ime UaQ = S p e c M X .X ^ Y ] ) = C2 and UCTl = SpecCClY.XY"1]) = C2.

The resulting toric va r ie ty is the blow-up of C2 at the origin. To see

this, realize the blow-up as the subvariety of C2 * IP1 defined by the

equation XT* = Y T q, where To and T^ are homogeneous

coordinates on IP1. This has an open cover by the two varieties Uq

and U i where Tq and T^ are nonzero, each isomorphic to C2; on

Uq coordinates are X and T i /Tq = X -1Y, and on coordinates are

Y and Tq/T^ = X Y -" 1, which coincides w ith the toric construction.

Next, for n = 2, consider the fan



I NTRODUCTI ON 7

The dual cones in M = 22 are

Each UCT. is isomorphic to C , with coordinates (X ,Y ) for erg,

(X “ 1,X“ 1Y ) for CTj_, and (Y '^ X Y * " * ) for 0 2 - These glue together to

form the projective plane IP2 in the usual way: if (Tq :T i:T2 ) are the

homogeneous coordinates on IP , X = T^/To <X = Ti/To and Y = T2/T0. (Note

again how the geometry in N is more agreeable than that in M.)

For a more interesting example, consider a fan as drawn, where

the slanting arrow passes through the point ( - l ,a ) , for some positive

integer a.c t 4

a 3

The four affine varieties are UCTl = Spec(C[X,Y]), = Spec(C[X,Y 1]),

UCT3 = Spec(C[X- * ,X -aY~*]), and = Spec(C[X_1,XaY]). We have the

patching

U c t 4

Ur

(x 1,xay )

I(x _1, x -ay -1)

(x ,y )

I( x ,y -1)

Ur

Ur



8 SECTION 1.2

which projects to the patching x “ * «— > x of € with C. The

varieties UCTl and Ua2 (and Ua3 and Ua4) patch together to the

va r ie ty C * IP*, so all together we have a P*-bundle over IP*. These

rational ruled surfaces are sometimes denoted Fa, and called

Hirzebruch surfaces.

Exercise. Identify the bundle Fa P* w ith the bundle P (0 (a )® l l )

of lines in the vector bundle that is the sum of a trivial line bundle and

the bundle 0 (a ) on P * . ^ )

Each of the four rays t determines a curve DT in the surface.

Such a curve will be contained in the union of the two open sets Ua

for the two cones a of which t is a face, meeting each of them in a

curve isomorphic to C, glued together as usual to form P*. The

equation for DxnU a in s C2 is % u = 0, where u is the

generator of SCT that does not vanish on t . For example, if t is

the ray through e2 , the curve DT is defined by the equation Y = 0

on U„ = Spec(C[X,Y]), and XaY = 0 on UCT4 = Spec(£[XaY ,X "1]).

Exercise. V er i fy that DT s IP1. Show that the normal bundle to DT

in Fa is the line bundle 0 (-a ), so the self-intersection number (D*D)

is -a. Find the corresponding numbers for the other three r a y s .^

Beginners are encouraged to experiment before going on. See if

you can find fans to construct the following varieties as toric varieties:

P n, the blow-up of Cn at a point, € * P*, P* x p*, <Ca x IPt>, and

pa x pb vVhat ar? all the one-dimensional toric varieties? Construct

some other two-dimensional toric varieties.

1.2 Convex polyhedral cones

We include here the basic facts about convex polyhedral cones that will

be needed. These results can be found in their natural generality in

any book on convex ity/4 but the proofs in the polyhedral case are so

simple that it is nearly as easy to prove them as to quote texts. We

include proofs also because they show how to find generators of the

semigroups, which is what we need for actual computations.



CONVEX POLYHEDRAL CONES 9

Let V be the vector space N(r, with dual space V * = M|r. A

convex polyhedral cone is a set

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

generated by any finite set of vectors vj_,... , v s in V. Such vectors,

or sometimes the corresponding rays consisting of positive multiples of some vj, are called generators for the cone a.

We will soon see a dual description of cones as intersections of half

spaces. The dimension d im (a ) of a is the dimension of the linear

space lR*a = a + ( - a ) spanned by a. The dual a v of any set a is

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

ctv = (u € V* : <u,v> > 0 for all v € a } .

Everyth ing is based on the following fundamental fact from the theory

of convex sets/5^

( * ) I f o is a convex polyhedral cone and vq t cr, then there is

some uq € with <uq,vq> < 0.

We list some consequences of (* ) . A direct translation of ( * ) is the

duality theorem :

( 1 ) ( c r v ) v = cr.

A face t of a is the intersection of a with any supporting

hyperplane: t = a f lu 1 = { v € a : <u,v> = 0} for some u in a v. A

cone is regarded as a face of itself, while others are called proper faces.

Note that any l inear subspace of a cone is contained in every face of

the cone.

(2) A n y face is also a convex polyhedral cone.



10 SECTION 1.2

The face a f lu 1 is generated by those vectors v j in a generating set

for a such that <u,V}> = 0. In particular, we see that a cone has

only fin ite ly m a n y faces.

(3) A n y intersection of faces is also a face.

This is seen from the equation P K a n u ^ ) = a f l d u j h for uj € <jv.

(4) A n y face of a face is a face.

In fact, if t = a f lu 1 and y = T fltu1)1 for u € ctv and u' € t v , then

for large positive p, u' + pu is in crv and y = a f l fu ' + pu )1.

A facet is a face of codimension one.

(5) A n y proper face is contained in some facet.

To see this, it suffices to show that if t = orflu^ has codimension

greater than one, it is contained in a larger face. We m ay assume that a spans V (or replace V by the space spanned by a); let W be

the linear span of t . The images Vj in V/W of the generators of a

are contained in a half-space determined by u. By moving this half

space in the sphere of half-spaces in V/W, one can find one that

contains these vectors vj but with at least one such nonzero vector in

the boundary hyperplane. In other words, there is a uq in cr so

that uox contains t and at least one of the vectors v\ not in W;

this means that afTuox is a larger face. When the codimension of t

in a is two, so V/W is a plane, there are exactly two such support

ing lines, which proves that any face of codimension two is the

intersection of exactly two facets.

From this we deduce by induction on the codimension:

(6) A ny proper face is the intersection of all facets containing it.

Indeed, if t is any face of codimension larger than two, from ( 5 ) we

can find a facet y containing it; by induction t is the intersection of

facets in y, and each of these is the intersection of two facets in a,

so their intersection t is an intersection of facets.

(7) The topological boundary of a cone that spans V is the union of

its proper faces (or facets).




Since a face is the intersection with a supporting hyperplane, points in

proper faces of a have points arbitrarily near which are not in cr.

Since a has interior points, by looking at the line segment from a

point in a face to a point in the interior, we see likewise that points in

faces have points arb itrarily near which are interior points. Con

versely, if v is in the boundary of the cone a, let Wj —> v, w j ^ a.

By ( * ) , there are vectors uj € a v with <Uj,Wj> < 0. By taking the uj

in a sphere, we find a converging subsequence, so we m ay assume uj

has a limit u q . Then u q € C7V and v is in the face a f lu ox.

When cr spans V and t is a facet of a, there is a u € ctv,

unique up to multiplication by a positive scalar, w ith t = afiu^. Such a vector, which we denote by uT, is an equation for the hyperplane spanned by t .

(8) I f a spans V and a * V, then a is the intersection of

the half-spaces HT = { v € V : <uT,v> > 0 } , as t ranges over the

facets of a.

If v w ere in the intersection of the half-spaces but not in a, take

any v 1 in the interior of cr. Let w be the last point in a on the

line segment from v' to v. Then w is in the boundary of a, and so

is in some facet t . Then <ux ,v ‘> > 0 and <uT,w> = 0, so <uT,v> < 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 among

the generators of cr, solve for a vector u annihilating the set; if

neither u or -u is nonnegative on all generators of a it is dis

carded; otherwise either u or -u is taken as a generator for a v; if

the n-1 vectors are in a facet t , this vector will be the one denoted

uT above. From (8) we deduce the fact known as Farkas' Theorem:

(9) The dual of a convex polyhedral cone is a convex polyhedral

cone.

If cr spans V, the vectors uT generate crv'; indeed, if u in

were not in the cone generated by the uT, applying ( * ) to this cone,

there is a vector v in V with <uT ,v> > 0 for all facets t and



12 SECTION 1.2

<u,v> < 0, and this contradicts (8). If a spans a smaller linear space

W = IR’ cr, then a '' is generated by lifts of generators of the dual cone

in W * = V*/W-L, together w ith vectors u and -u as u ranges over a basis for W x.

This shows that polyhedral cones can also be given a dual definition as the in te rsec t ion of h a l f - sp ac e s : for g e n e ra to r s u l t . , u t

of o'*,cr * ( v € V : <uj[, v > > 0 , . . . , <ut,v> > 0 } .

If we now suppose cr is rational, meaning that its generators can be taken from N, then crv is also rational; indeed, the above

procedure shows how to construct generators uj in cr^nM.

Proposition 1. (Gordon’s Lem m a) I f cr is a rational convex

polyhedral cone, then Sa = a v HM is a fin ite ly generated semigroup.

Proof. Take u*, . . . ,u s in crÔM that generate as a cone. Let

K = {S t jU j : 0 < tj < 1). Since K is compact and M is discrete, the

intersection K f iM is finite. Then K n M generates the semigroup.

Indeed, if u is in a vOM, write u = Z r ^ i , rj > 0, so ri = mj + tj

w ith mj a nonnegative integer and 0 < tj < 1. Then u = Zm jU j + u1,

w ith each u* and u' = StjUj in KnM.

It is often necessary to find a point in the re la tive in te r io r of

a cone a, i.e., in the topological interior of a in the space {R»cr

spanned by a. This is achieved by taking any positive combination of

dim(cr) linearly independent vectors among the generators of a. In

particular, if a is rational, we can find such points in the lattice.

(10) I f t is a face of cr, then crÔT-1- is a face of o'*, with

d im (x ) + d i m ( a v' n T x) = n = dim(V). This sets up a one -to -one

order-revers ing correspondence between the faces of a and the

faces of ctv. The smallest face of o is aO (-a ) .

To see this, note first that the faces of o ” are exactly the cones

a v f l v x for v € a = ( a v)v. If t is the cone containing v in its

relative interior, then a v f l v x = a v f l (T v/O vx) = a v f lx x, so every

face of o'* has the asserted form. The map t »-» t * = a v f l x x is

clearly order-reversing, and from the obvious inclusion t C ( t * ) * it




follows form ally that t * = ( ( t * ) * ) * , and hence that the map is

one-to-one and onto. It follows from this that the smallest face of

o is ( a ' T f l t a T = ( a v')J- = a n ( -a ) . In particular, we see that

d im (an (-cr ) ) + d im (a v) = n. The corresponding equation for a general

face t can be deduced by putting t in a m aximal chain of faces of

a, and comparing w ith the dual chain of faces in o'*.

(11) I f u € crN/, and x = a f l u x, then x v = + IR>q« ( -u ).

Since both sides of this equation are convex 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 crn (-u )v = a D u A, as required.

Proposition 2. Let o be a rational convex polyhedral cone, and let

u be in Sa = a v f)M. Then x = crflux is a rational convex

polyhedral cone. All faces of a have this fo rm , and

ST = SCT + .

Proof. If t is a face, then t = a f l u x for any u in the relative

interior of a vn T A, and u can be taken in M since a vDTA is

rational. Given w € ST, then w + p-u is in for large positive p,

and taking p to be an integer shows that w is in Sa + Z>q*(-u ).

Finally, we need the following strengthening of ( * ) , known as a

Separation Lem m a, that separates convex sets by a hyperplane:

(12) I f o and cr* are convex polyhedral cones whose intersection

t is a face of each, then there is a u in a N/n ( - a ,)v' with

t = a f lu x = cr'flu1 .

This is proved by looking at the cone y = a - a ‘ = a + (-cr*). We know

that for any u in the relative interior of yv, yflux is the smallest

face of y:

ynuA = yn(-y) = ( a - a ,) n ( a ' - a ) .

The claim is that this u works. Since o is contained in y, u is in

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

Conversely, if v € a(1uA, then v is in o ' - a, so there is an equation

v = w ' - w, w ' € a', w € (j. Then v + w is in the intersection x of



14 SECTION 1.2

cr and cr', and the sum of two elements of a cone can be in a face

only if the summands are in the face, so v is in t . This shows that

oTlux = t , and the same argument for -u shows that a 'flu-1* = t .

Proposition 3. I f a and a 1 are rational convex polyhedral cones whose intersection t is a face of each, then

ST = SCT + Sa» .

Proof. One inclusion ST 3 SCT + is obvious. For the other inclu

sion, by the proof of (12) we can take u in a vO(-cj,)vnM so that

t = cjn u x = a ' f lu x. By Proposition 2 and the fact that -u is in Sa',

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

(13) For a convex polyhedral cone a, the following conditions are

equivalent:

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

(ii) a contains no nonzero linear subspace;

(iii) there is a u in a'"' with a f lu x = (0);

( iv ) crv spans V*.

The first two are equivalent since CTfl(-a) is the largest subspace in

a; the second two are equivalent since a f l ( - a ) is the smallest face of

cr. The first and last are equivalent since dim(<7 0 ( - a ) ) + d im (a v) = n.

A cone is called strongly convex if it satisfies the conditions of

(13). Any cone is generated by some minimal set of generators. If the

cone is strongly convex, then the rays generated by a minimal set of

generators are exactly the one-dimensional faces of a (as seen by

applying ( * ) to any generator that is not in the cone generated by the

others); in particular, these minimal generators are unique up to

multiplication by positive scalars.

Exercise. If t is a face of a, with W = IR-t , show that o ’ =

(cr + W)/W is a convex polyhedral cone in V/W (rational if a is

rational), and the faces of cj are exactly the cones of the form

Y = (y + W)/W as y ranges over the cones of a that contain t .

Exercise. For a cone ct and v € a, show that the following are

equivalent: (i) v is in the relative interior of a; (ii) <u,v> > 0



AFFINE TORIC VARIETIES 15

for all u in a v ' a x; (iii) a vf l v x = a x; ( iv ) a + IR>q*(-v ) = IR-cr;

(v ) for all x € a there is a positive number p and a y in cr

w ith p-v = x + y. (6)

Exercise. If t is a face of a cone cr, show that the sum of two

vectors in cr can be in t only if both of the summands are in t .

Show conversely that any convex subset of a cone cr satisfying this

condition is a face.

Since we are main ly concerned w ith these cones, we will often

say "a is a cone in N" to mean that cr is a strongly convex rational

polyhedral cone in Njr. We will sometimes write "t •< cr" or "cr > t "

to mean that t is a face of cr. A cone is called simplicial, or a

simplex, if it is generated by linearly independent generators.

Exercise. If cr spans Njr, must cr and cr'' have the same minimal

number of generators? ^

1.3 A f f in e to r ic v a r ie t ie s

When a is a strongly convex rational polyhedral cone, we have seen

that SCT = a v flM is a finitely generated semigroup. Any additive

semigroup S determines a "group ring" C[S], which is a com m utative

C-algebra. As a complex vector space it has a basis %u, as u varies

over S, w ith multiplication determined by the addition in S:

__ û + u'

The unit 1 is X°* Generators {uj} for the semigroup S determine

generators (X Ui) f ° r the C-algebra C[S].Any finitely generated com m utative C-algebra A determines a

complex affine var ie ty , which we denote by Spec(A). We rev iew this

construction and its related notation/8 If generators of A are

chosen, this presents A as C[X*, . . . ,Xm]/I, where I is an ideal;

then Spec(A) can be identified with the subvariety V (I ) of affine

space Cm of common zeros of the polynomials in I, but as usual for

modern mathematicians, it is convenient to use descriptions that are



16 SECTION 1.3

independent of coordinates. In our applications, A will be a domain,

so Spec(A) will be an irreducible variety. Although Spec(A) officially

includes all prime ideals of A (corresponding to subvarieties of V (I)),

when w e speak of a point of Spec(A) we will mean an ordinary closed

point, corresponding to a maximal ideal, unless we specify otherwise.

These closed points are denoted Specm(A). Any homomorphism

A B of C-algebras determines a morphism Spec(B) -> Spec(A) of

varieties. In particular, closed points correspond to C-algebra

homomorphisms from A to €. If X = Spec(A), for each nonzero

element f € A the principal open subset

Xf = Spec(Af) c X = Spec(A)

corresponds to the localization homomorphism A »-» Af.

For A = C[S] constructed from a semigroup, the points are easy

to describe: they correspond to homomorphisms of semigroups from S

to C, w here C = C* U {0} is regarded as an abelian semigroup via

multiplication:

Specm(C[S]) = Homsg( S , C ) .

For a semigroup homomorphism x from S to C and u in S, the

value of the corresponding function %u at the corresponding point of

Specm(C[S]) is the image of u by the map x: %u(x) = x(u).

When S = SCT arises from a strongly convex rational polyhedral

cone, w e set A c = C[Sa], and

UCT - SpectClS*]) = Spec(Aa) ,

the corresponding affine toric variety. All of these semigroups will be

sub-semigroups of the group M = S{q}. If e*, . . . ,en is a basis for N,

and e£, . . . , e* is the dual basis of M, write

Xj = € C[M] .

As a semigroup, M has generators ±e£, . . . , ±e* , so

C[M] = C[X1, X i - 1.X2.X2- 1, . . . .X n .X n "1]

= C[X i , . . . , X n]X r .Xn ,

which is the ring of Lauren t polynomials in n variables. So




U{0} = Spec(C[M]) s C* x . . . x C* = (C *)n

is an affine algebraic torus. All of our semigroups S will be sub

semigroups of a lattice M, so C[S] will be a subalgebra of C[M]; in

particular, C[S] will be a domain. When a basis for M is chosen as

above, w e usually w rite elements of C[S] as Laurent polynomials in

the corresponding variables Xj. Note that all of these algebras are

generated by monomials in the variables X\.

The torus T = Tjsj corresponding to M or N can be written

intrinsically:

Tn = Spec(C[M]) = Hom(M,C*) = N<E>Z<D* .

For a basic example, let cr be the cone with generators

e l> • • • » ek f ° r some k, 1 < k < n. Then

S<t = 2>om i + %>0'e2 + • • • + 2 >o*e^ + 2 -e£ +1 + . . . + 2 -e* .

Hence A „ = <C[X1 ,X 2, . . . ,X k ,X k + 1,X k + 1~1, . . . . X ^ X ^ 1], and

Ua = C x . . . X C x C* x . . . x C = Ck x (C *)n- k .

It follows from this example that if u is generated by k

elements that can be completed to a basis for N, then UCT is a

product of affine k-space and an (n-k)-dimensional torus. In

particular, such affine toric varieties are nonsingular.

Next we look at a singular example. Let N be a lattice of rank

3 , and let a be the cone generated by four vectors v^, V2 , V3 , and

V4 that generate N and satisfy v^ + V3 = V2 + V4 . The va r ie ty UCT

is a "cone over a quadric surface", a va r ie ty met frequently when

singularities are studied. If we take N = Z and v, = ej for i = 1 , 2 ,

3, so v 4 = e i + e3 - e2,



18 SECTION 1.3

then Sa is generated by e£, e^, e£ + e^, and e^ + e^, so

A a = C[Xlf X3, X iX 2, X2X3 ] = C [W ,X ,Y ,Z ]/ (W Z - X Y ) .

Therefore Ua is the hypersurface defined by W Z = X Y in C4.

A homomorphism of semigroups S —> S' determines a

homomorphism C[S] —> €[S'] of algebras, hence a morphism

Spec((C[S']) - > Spec(C[S]) of affine varieties. In particular, if t is

contained in a, then SCT is a sub-semigroup of ST, corresponding to

a morphism Ux -* UCT. For example, the torus Tjsj = U{o) maps to all

of the affine toric varieties UCT that come from cones a in N.

Lem m a. If t i s a face of a , then the m ap UT —> Ua embeds UT as a principal open subset of U^.

Proof. By Proposition 2 in §1.2, there is a u € SG w ith t = o f l u 1

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

This implies immediately that each basis element for C[ST] can be

w ritten in the form x w _pu = X W / (X U)P for w € SCT. Hence

A-j- - (A a)û f

which is the algebraic version of the required assertion.

Exercise. Show that if t c a and the mapping UT -» UCT is an open

embedding, then t must be a face of a. ^

More generally, if tp: N' -> N is a homomorphism of lattices such

that cpm maps a (rational strongly convex polyhedral) cone a' in N'

into a cone a in N, then the dual q)v: M -> M' maps SCT to SCT»,

determining a homomorphism A CT A a n d hence a morphism

Uff' - Uff.

Exercise. Show that if S C S' C M are sub-semigroups, the

corresponding map Spec(C[S']) -» Spec(€[S]) is birational if and only if

S and S' generate the same subgroup of M.

The semigroups Sa arising from cones are special in several

respects. First, it follows from the definition that SCT is saturated,

i.e., if p*u is in Sa for some positive integer p, then u is in Sa. In




addition, the fact that cr is strongly convex implies that a v spans

M|r, s o Sct generates M as a group, i.e.,

M = Sa + (-Sa) .

Exercise. Show conversely that any finitely generated sub-semigroup

of M that generates M as a group and is saturated has the form

crv nM for a unique strongly convex rational polyhedral cone a in N.

The following exercise shows that affine toric varieties are defined

by m on om ia l equations.

Exercise. If Sa is generated by uj, . . . , u*, so

A ct = C[%ul, . . . , %ut] = C [ Y Y t] /1 ,

show that I is generated by polynomials of the form

Y i al . Y 2a2- . . . -Ytat - Y ! b l-Y2b2- . . . -Ytbt ,

where a^, . . . , a*, bj, . . . , bt are nonnegative integers satisfying the

equation

a i ui + . . . + a tut = b ju i + . . . + btut .

If a is a cone in N, the torus Tj acts on Uff,

TN x Uct “♦ ,

as follows. A point t € Tfyj can be identified with a map M —» (Cw of

groups, and a point x € UCT with a map SCT -» € of semigroups; the

product t-x is the map of semigroups SCT -» C given by

u t (u )x (u ) .

The dual map on algebras, C[SCT] C[Sff]®C[M], is given by mapping

X u to X U<8>XU for u € Sa. When a = (0), this is the usual product

in the algebraic group Tjsj. These maps are compatible with inclusions

of open subsets corresponding to faces of a. In particular, they extend

the action of Tn on itself.

Exercise. If a 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 ~’ ^<j x a' •

Except for the following exercises, we will not be concerned with

varieties arising from more general semigroups, although these are of

considerable interest to algebraists/12^

Exercise, (a ) Let S c 1L be the sub-semigroup generated by 2 and

3. Then

CIS] = C[X2,X3] = C[Y,Z]/(Z2 - Y 3) ,

so Spec(C[S]) is a rational curve with a cusp.

(b) Find C[S] when S is the sub-semigroup of Z>q generated by a

pair of re la tive ly prime positive integers.

(c) Find corresponding generators and relations for S generated by

3, 5, and 7. (13)

Exercise. If a c 1R2 is the cone generated by e2 and X-e^ - e2 ,

w ith X an irrational positive number, show that a vn (Z 2) is not

finitely generated, and that (C[av f l (2 2)] is not noetherian.

1.4 Fans and toric varieties

By a fan A in N is meant a set of rational strongly convex

polyhedral cones a in N(r such that

(1) Each face of a cone in A is also a cone in A ;

(2) The intersection of two cones in A is a face of each.

For simplicity here we assume unless otherwise stated that fans are

f in ite , i.e., that they consist of a finite number of cones. This will

assure that our toric varieties are of finite type, not just locally of

finite type. From now on a cone in N, or a cone, will be assumed to

be a rational strongly convex polyhedral cone, unless otherwise stated.

From a fan A the toric var ie ty X (A ) is constructed by taking

the disjoint union of the affine toric varieties U^, one for each a in

A , and gluing as follows: for cones a and t , the intersection a f lx is a face of each, so UCTnT is identified as a principal open subvariety of



FANS AND TORIC VARIETIES

Ua and of UT; glue Ua to UT by this identification on these open

subvarieties. The fact that these identifications are compatible comes

im m ediate ly from the order-preserving nature of the correspondence

from cones to affine varieties. The fact that the resulting complex

va r ie ty is Hausdorff (or the algebraic va r ie ty is separated — or the

resulting "prescheme" is a "scheme") comes from the following

lem m a.^^

Lemm a. If a a n d t a r e cones t h a t in te rs e c t in a co m m o n face,

th e n th e d iag o n al m a p Ug-nx -> UCT * UT is a closed em bedding.

Proof. This is equivalent to the assertion that the natural mapping

A ct® A t —> A anT surjective. And this follows from the fact that

SctHt = Scr + ST, as we saw in Proposition 3 of §1.2, which was

(appropriately!) a consequence of the Separation Lem m a for cones.

In particular, for any two cones a and t of A , we have the

identity UanUx = UanT. If a is a cone in N, and A consists of a

together w ith all of its faces, then A is a fan and X (A ) is the affine

toric va r ie ty UCT. We will see later that these are the only toric

varieties that are affine.

Let us work out a few simple examples. In dimension one, with

N = Z, the only possible cones are the right or left half-lines and the

origin. We have seen all possible fans, which give C, C*, and IP1.

For some simple two-dimensional examples, we draw some fans

in N = Z2. For the fan

take two copies of C2, corresponding to the algebras C[X,Y] and

C[X_1,Y]; gluing gives (P1 x €. Similarly, for the fan



22 SECTION 1.4

take four copies of <C2, and glue to get IP1 x IP1. The preceding two

examples are special cases of a general fact:

Exercise. If A is a fan in N, and A ' is a fan in N', show that the

set of products a x a', a € A , a ' € A', forms a fan A x A ' in N © N ‘,

and show that

X ( A x A ' ) = X (A ) x X (A ' ) .

The generalization of the construction of IP2 is:

Exercise. Suppose vectors vg, v , . . . ,vn generate a lattice N of

rank n, w ith vq + + . . . + v n = 0. Let A be the fan whose cones

are generated by any proper subsets of the vectors vg, . . . ,vn.

Construct an isomorphism of X (A ) w ith projective n-space IPn.

The vectors v^, . . . , v n in the preceding exercise can be taken to

be the standard basis e j, . . . ,en for N = Zn, w ith vg = -e^ - . . . ~en.

This corresponds to constructing IPn as the closure of Cn. A more

sym m etr ic description of IPn can be given by taking N to be the

lattice Zn + 1/Z-(1,1,... ,1), w ith vj the image of the ith basic vector,

0 < i < n. With this description, Tn = (C * )n+1/(C* is embedded

naturally in Cn + 1 ' {0} /C* = IPn.

Exercise. Find the toric varieties corresponding to the following three

fans in Z2:

Suppose (p: N1 —» N is a homomorphism of lattices, and A is a

fan in N, A 1 a fan in N ‘, satisfying the following condition:

for each cone ct* in A ', there is some

cone (j in A such that (p(a') C cr.

As w e saw in the preceding section, this determines a morphism



TORIC VARIETIES FROM POLYTOPES 23

U(j' u a c X (A ). The morphism from U^' to X (A ) is easily seen to

be independent of choice of cr, and these morphisms UCT» X (A )

patch together to give a morphism

<p*: X (A ' ) -> X (A ) .

At the end of §1.1 we constructed the Hirzebruch surface Fa as a

toric var ie ty . The projection Fa -> P 1 is determined by the

projection Z2 -> Z taking (x ,y ) to x. (Note that the second

projection does not satisfy the required condition, if a * 0.)

Exercise. Show that for any integer m, the map Z -> Z2 given by

z ( z ,m z ) determines a section P 1 -* Fa of this projection.

The actions of the torus Tjsj on the varieties UCT described in the

preceding section are compatible with the patching isomorphisms,

giving an action of Tn on X (A ). This extends the product in Tjsj:

Tn x X (A ) -» X (A )

II J fTn x t n TN

The converse is also true: any (separated, normal) va r ie ty X

containing a torus Tn as a dense open subvariety, w ith compatible

action as above, can be realized as a toric va r ie ty X (A ) for a unique

fan A in N. We will have no use for this, so we don't discuss the

proof/15^

Although w e will deal w ith complex toric varieties in these

lectures, the interested reader should have no difficulty carrying out

the same constructions over any other base field (or ground ring).

1.5 Toric varieties from polytopes

A con vex polytope K in a finite dimensional vector space E is the

convex hull of a finite set of points. A (proper) face F of K is the

intersection w ith a supporting affine hyperplane, i.e.,

F = { v € K : <u,v> = r ) ,



24 SECTION 1.5

w here u € E* is a function w ith <u,v> > r for all v in K; K is

usually included as an improper face. We assume for simplicity that

K is n-dimensional, and that K contains the origin in its interior. A

facet of K is a face of dimension n-1. The results of §1.2 can be

used to deduce the corresponding basic facts about faces of convex

polytopes. For this, let cr be the cone over K x 1 in the vector space

E x IR. The faces of cr are easily seen to be exactly the cones over the

faces of K (w ith the cone {0} corresponding to the em pty face of K);

from this it follows that the faces satisfy the analogues of properties

(2 )- (7 ) of §1.2.

As for the duality theory of polytopes, the polar set (or polar ) of

K is defined to be the set

(Often {u € E* : <u,v> < 1 V v € K } = -K° is taken to be the polar

set, but this does not change the results.) For example, the polar of the

octahedron in IR3 w ith vertices at the points (±1,0,0), (0,±1,0), and

(0,0,±1) is the cube w ith vertices (±1,±1,±1).

Proposition. The polar set K° is a convex polytope, and K is the

polar of K°. I f F is a face of K, then

is a face of K°, and the correspondence F F* is a one-to -one,

order-revers ing correspondence between the faces of K and the

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

i.e., its vertices lie in a lattice in E, then K° is also rational, with

its vertices in the dual lattice.

K° = (u € E* : > -1 for all v € K ) .

F* = {u € K° : = -1 V v € F }

Proof. With a the cone over K x 1, the dual cone a '/ consists of



TORIC VARIETIES FROM POLYTOFES 25

those u x r in E* x (R such that <u,v> + r > 0 for all v in K. It

follows that is the cone over K° x 1 in E* x (R. The assertions of

the proposition are now easy consequences of the results in §1.2 forcones. For example, the duality (K°)° = K follows from the duality (crN' )N' = a. For a face F of K, if t is the cone over F x l f then the dual a v f lT x is the cone over F* x 1, from which the duality between

faces of K and K° follows.

Exercise. Let K be a convex polyhedron in E, i.e.,

K = ( v € E : > -a*, . . . , <ur ,v> > -a r )

for some ui, . . . ,u r in E* and real numbers a*, . . . ,a r . Show that

K is bounded if and only if K is the convex hull of a finite set/16^

A rational convex polytope K in N|r determines a fan A whose

cones are the cones over the proper faces of K. Since we assume that

K contains the origin in its interior, the union of the cones in A will

be all of Njr. All of the fans we have seen so far whose cones cover Nr

have this form.

More generally, if K1 is a subdivision of the boundary of K, i.e.,

K1 is a collection of convex polytopes whose union is the boundary of

K, and the intersection of any two polytopes in K1 is a polytope in K1,

then the cones over the polytopes in K* form a fan. Here are some

examples of such K1:

Note that the second can be "pushed out", so that the cone over it is

the cone over a convex polytope, but the third cannot.

There are m an y fans, however, that do not come from any

convex polytope, however subdivided. To see one, start w ith the fan

over the faces of the cube w ith vertices at (±1 ,±1,±1) in Z3. Let A

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



26 SECTION 1.5

that the vertex (1,1,1) is replaced by (1,2,3). It is impossible to find

eight points, one on each of the eight positive rays through the

vertices, such that for each of the six cones generated by four of these

vertices, the corresponding four points lie on the same affine plane:

Exercise. Suppose for each of the eight vertices v there is a real

number r v , and for each of the six large cones cr there is a vector

ua in M|r = IR3, such that <ucr,v> = r v whenever v is one of the

four vertices in a. Show that there is then one vector u in M r

w ith <u,v> = r v for all v. In particular, the points pv = ( l / r v )«v

cannot have each quadruple corresponding to a cone lying in a plane

unless all eight points are coplanar/17^

A particularly important construction of toric varieties starts

w ith a rational polytope P in the dual space M[r. We assume that P

is n-dimensional, but it is not necessary that it contain the origin.From P a fan denoted Ap is constructed as follows. There is a cone

c t q of Ap for each face Q of P, defined by

crq = { v € N|r : <u,v> < <u',v> for all u € Q and u' € P } .

In other words, c t q is dual to the “angle" at Q consisting of all vectors pointing from points of Q to points of P; this dual cone ctq'' is

generated by vectors u‘ - u, where u and u‘ v a r y among vertices

of Q and P respectively. It is not hard to ve r i fy directly that these

cones form a fan. It is more instructive, however, to realize the fan as

a fan over a "dual polytope" in N j r .

Proposition. The cones c t q , as Q varies over the faces of P,

fo rm a fan Ap. I f P contains the origin as an in te r io r point, then

Ap consists of cones over the faces of the polar polytope P°.

Proof. If the origin is an interior point, it is immediate from the

definition that c t q is the cone over the dual face Q* of P°, and the

second assertion follows. It follows from the definition that Ap is

unchanged when P is translated by some element u of M, or when

P is multiplied by a positive integer m: A mp + U = Ap. Since any P

spanning M r* can be changed to one containing the origin as an

interior point by such translation and expansion, the first assertion

also follows.



TORIC VARIETIES FROM POLYTOPES 27

Conversely, by the duality of polytopes, it follows that for a

convex rational polytope K in Nr (containing the origin in its

interior) the fan of cones over the faces of K is the same as Ap,

where P = K° is its polar polytope. The toric va r ie ty X (A p ) will

sometimes be denoted Xp. We look at some examples:

(1) If P is the simplex in IRn with vertices at the origin and

the points ei, . . . , e n, then Ap is the fan used to construct (Pn as a

toric variety: X (A p ) s IPn.

(2) If P is the cube in IR3 with vertices at ± e^ ± e^ ± e^ , then

Ap is the fan over faces of the octahedron with vertices ±ej, and

X (A P) = P 1 x P 1 x P 1.

(3) Let P be the octahedron in IR with vertices at the points

(±1,0,0), (0,±1,0), and (0,0,±1), so the fan Ap is the fan over faces

of the cube with vertices (±1,±1,±1). If N is taken to be the lattice

of points (x,y,z) € Z3 such that x = y = z (mod 2), then any three

of the four vertices of any face a of the cube generate N, and the

sums of opposite vertices of a are equal. Each of the six corres

ponding open subvarieties UCT has a singular point isomorphic to the

cone over a quadric surface described in §1.3. We will come back to

this example later.



CHAPTER 2

SINGULARITIES AND COMPACTNESS

2.1 Local propert ies of tor ic va r ie t ie s

For any cone a in a lattice N, the corresponding affine va r ie ty UCT

has a distinguished point, which we denote by xCT. This point in UCT is

given by a map of semigroups

Sa = ct^HM -» {1,0} C C*U{0} = € ,

defined by the rule

f 1 if u € a x u <

[ 0 otherwise

Note that this is well defined since a x is a face of crv, which implies

that the sum of two elements in cannot be in a x unless both are

in a x.

Exercise. If a spans Njr, show that xa is the unique fixed point of

the action of the torus Tjsj on Ua. If cj does not span N|r, show

that there are no fixed points in UCT. ^

We first find the singular points of toric varieties. Suppose to

start that ct spans Njr, s o a x = (0). Let A = A a, TTt the m aximal

ideal of A corresponding to the point xa, so m is generated by all

X u for nonzero u in SCT. The square m 2 is generated by all X u f ° r

those u that are sums of two elements of Sa ' {0}. The cotangent

space m/m2 therefore has a basis of images of elements X u f ° r

those u in SCT ' {0} that are not the sums of two such vectors. For

example, the first elements in M lying along the edges of are

vectors of this kind. Suppose Ua is nonsingular at the point xa. One

characterization of nonsingularity is that the cotangent space m/m2

is n-dimensional, since dim (UCT) = d im d ^ ) = n. This implies in

particular that ctv cannot have more than n edges, and that the



LOCAL PROPERTIES OF TORIC VARIETIES

minimal generators along these edges must generate SCT. Since SCT

generates M as a group, the minimal generators for Sa must be a

basis for M. The dual a must therefore be generated by a basis for

N. Hence UCT is isomorphic to affine space Cn.

A general a has smaller dimension k. Let

No- = <rnN + (-CTnN)

be the sublattice of N generated (as a subgroup) by a ON. Since cr

saturated, Na is also saturated, so the quotient group N (a ) = N/Na

also a lattice. We m ay choose a splitting and w rite

N = Nct ® N" , ex = a ‘ ® (0) ,

where a ' is a cone in Na. Decomposing M = M' 0 M" dually, we

have SCT = ( (a -T O M 1) 0 M", so

UCT S UCT. x Tn„ £ U„i x (C*)n- k .

More intrinsically, the maps NCT —» N —> N (a ), ct‘ —» ct —» {0},

determine a fiber bundle

U<r‘ Ucr TN(ff)

that splits: Ua s Ua-i * Tm (cr), n° f canonically. If isnonsingular, UCT» must also be nonsingular, and the preceding

discussion applies: a 1 must be generated by a basis for NCT. This

proves:

Proposition. An affin e to ric v a r ie ty UCT is n o n s in g u la r if a n d

o n ly if a is g e n e ra te d b y p a r t of a basis fo r th e la tt ic e N, in

w h ich case

U q. = x (C * )n“k k = d im (c j) .

We therefore call a cone n o n s in g u la r if it is generated by part of

a basis for the lattice, and we call a fan nonsingular if all of its cones

are nonsingular, i.e., if the corresponding toric va r ie ty is nonsingular.

Although a toric va r ie ty m ay be singular, e ve ry toric va r ie ty is

n o rm al:

Proposition. E ach rin g A CT = C[SCT] is in te g ra lly closed.



30 SECTION 2.1

Proof. If a is generated by v* , . . . , v r , then ctv = w here

T| is the ray generated by v* € ct, s o A ct = n A T.. We have seen that

each At . is isomorphic to CIX*, X2 , • . . , Xn, Xn-1], which is

integrally closed, and the proposition follows from the fact that the

intersection of integrally closed domains is integrally closed.

Exercise, (a) If S is any sub-semigroup of M, show that CIS] is

integrally closed if and only if S is saturated.

(b) If S is a sub-semigroup of M, let S be its saturation, i.e.,

S = {u € M : p*u € S for some positive integer p). Show that C[S] is

the integral closure of C[S], i.e., Spec(C[S]) —» Spec(C[S]) is the

normalization map. Carry this out for S generated by e^ + 2e2 and„ * * . _ -7, * -n *2e1 + e2 in M = Ze^ + Ze2.

(c) Deduce Gordon's lemma from the com m utative algebra fact

that the integral closure of a finitely generated domain over a field is

finitely generated as a module over the domain.

Another important fact about toric varieties (but one w e won't

need here) is that they are all Cohen-Macaulay varieties: each of

their local rings has depth n, i.e., contains a regular sequence of n

elements, where n is the dimension of the local ring. We sketch the

proof (assuming fam iliarity w ith properties of depth), referring to

[Dani] for details. As above, it is enough to consider the case w here

dim(cr) = n. In addition, we m ay replace the lattice N by any sub

lattice N' of finite index; this follows from the fact that, if <j‘ is the

corresponding cone in N', the inclusion A CT —> A CT« splits as a map of

Ajj-modules, the splitting given by projecting a v flM ' onto a v HM.

If d im (a ) = n, set A = A CT, and I = ® C % U, the sum over all

u in Int(cxv)nM. Orienting and all its faces, one has an exact

sequence

0 —* A/I —> Cn_i —> Cn_2 . . . —* Ci —» Cq —* 0 ,

where Cn_^ is the direct sum of rings C I ^ D t ^ M ] , for all k-

dimensional faces t of a ; the boundary maps are given by

projections of faces in ct onto smaller faces, w ith signs determined

by the orientation. One knows by induction that Cn_j has depth

n-k, and it follows that A/I has depth n-1. If I were a principal



SURFACES; QUOTIENT SINGULARIT IES 31

ideal, it would follow that A has depth n. If there is a uq in

In t (a v')HM such that u - uo is in aÔM for all u € In t (a v)HM,

then I = A «X U°- Although this need not be true, it can be achieved by replacing N by a sublattice N ‘ of finite index/2^

The following exercise states a useful fact, although in situations

where it comes up the conclusion can usually be seen by hand.

Exercise. If a torus Tn acts algebraically on a finite-dimensional

vector space W, i.e., the map Tn GL(W) is a morphism of affine algebraic groups, show that W decomposes into a direct sum of the

spaces

W u = ( w € W : t-w = %u( t )w for all t € Tn )

as u varies over M. ^

Exercise. Deduce that for any algebraic action of a torus T on a

finite dimensional vector space V, the ring of invariants Sym (V )^ is

C ohen-M acau lay .^

Exercise. Use the preceding exercise to show that the homogeneous

coordinate ring of the Segre variety:

€ [ X xY lt X i Y 2, • • • , X iYq, X2Y i , . . . , X2Y q, . . . , XpY lf . . . , XpY q]

is Cohen-Macaulay.

More recently, Gubeladze has proved a conjecture of Anderson

that the affine ring A CT of a toric va r ie ty satisfies the generalization of

a conjecture of Serre: all projective modules are free, or, in geometric

language, all vector bundles on affine toric varieties are tr iv ia l/5^

2.2 Surfaces; quotient singularities

Consider the case where N = Z2 and a is generated by e2 and

m e* - e2, generalizing the case m = 2 that we looked at earlier.



32 SECTION 2.2

Then A a = C[Sff] = ClX, XY, X Y 2 , . . . , X Y m ]. Setting X = Um and

Y = V/U, we have

A ct = C lU m,U m“ 1V, . . . ,U V m" 1, V m ] C C [U , V ] ,

so Ua = Spec(ACT) is the cone over the rational n o rm a l cu rv e of

degree m. The inclusion of A CT in C[U,V] corresponds to a mapping

C2 - Uff.The group G = = { m th roots of unity} s Z/mZ acts on <D2

by c (u , v ) = (cu,Cv), and Ua is the quotient va r ie ty C2/G, i.e.,

Ua is a cyclic quotient singularity. Algebraically, G acts on the

coordinate ring C[U,V] b y F h F ( cU , cV ) , and, via this action,

A ct = C[U,V]G

is the ring of invariants.

The toric structure can be used to see this more naturally. Let

N' C N be the lattice generated by the generators e2 and me^ - e2

of a, and let a 1 be the same cone as ct, but regarded in N' (since

N 'r = N r ). Since a' is generated by two generators for N', UCT» = C2,

and the inclusion of N ‘ in N gives a map <C2 = Ua» Ua. The claim

is that this is the same as the map constructed by hand above. To see

this, note that N ‘ is generated by me^ and e2 , so M ‘ D M is

generated by and e^, corresponding to monomials U and Y,

with Um = X. The generators for Sa' are e£ and + e^, so

A a* = C[U,UY] = C[U,V], with V = UY, and we recover the previous

description.

A similar procedure applies to an arb itrary singular two-

dimensional affine toric variety. First note that, w ith an appropriate

choice of basis for N, we m ay assume the cone a has the form




for some 0 < k < m, with k and m relatively prime integers. Toprove this, note that any minimal generator along an edge of a is part of a basis for N = Z^, so we can take one to be (0,1), and the

other (m,x), for m a positive integer. Applying an automorphism of

the lattice,

f 1 0 \ ( m 0 \ _ ( m 0 AV c 1 i ‘ l x 1 J ~ Vcm+x 1 J ’

we see that x can be changed arb itrarily modulo m, so w e can take

x = -k, 0 < k < m. Of course, if x = 0 (mod m), then cr is generated

by a basis for N, and we are in the nonsingular case. That k and m

are rela tively prime comes from the fact that (m ,-k ) is taken to be a

m inimal generator along the edge.

With ex as above, A CT = the sum over (i,j ) w ith

j < ~-i. Let N' be generated by m e i - k e 2 and e2 , i.e. by

m e i and e2- Then as before, M' is generated by ^ e ^ and e^,

corresponding to monomials U and Y, and the corresponding cone

a' has SCT« generated by — e£ and k* — e£ + e^. Therefore A G* is

C[U,UkY] = C[U,V], w ith V = UkY, so = C2.

The group G = um acts on UCT' = C2 by C-(u,v) = (cu ,ekv), and

UCT = Ua./G = C2 /G .

Equivalently, Aa = (A a')G. In fact, G acts on the larger ring

<C[M'] = a u . l T ^ V . V - 1] = a U . i r ^ Y . Y -1], by U h CU, Y h Y,

and the ring of invariants is <C[X, X -1, Y, Y "* ] = C[M]. Therefore

Aa = Aa.nc[M ] = a ct. n «D[M'])G - ( a ct.)g .

We will describe A a more explicitly in §2.6.

More generally still, and more intrinsically, for a lattice N of



34 SECTION 2.2

any rank, if N' C N is a sublattice of finite index, let M C M' be the

dual lattices. There is the canonical duality pairing

M'/M x N/N' — > Q/Z 01— > C* ,

the first map by the pairing < , >, the second by q ^ exp(2iriq). Now

G = N/N' acts on C[M'] by

v * X u = exp(2Tu<u,,v > ) 'X ul

for v € N, u' € M'. And, via this natural action,

( * ) C[M']G = C[M] .

Hence G = N/N' acts on the torus Tn» and Tn»/G = T j- To prove (* ) ,

it suffices to take a basis e*, . . . ,en for N so that mêi, . . . ,m nen

are generators for N', for some positive integers mj. Then C[M'] is the Laurent polynomial ring in generators Xj, and C[M] is the

Laurent polynomial ring in generators Uj, w ith (U|)mi = X*. An

element (a*, . . . ,a n) in 0Z/m jZ = N/N' acts on monomials by

multiplying Ui*l* . . . 'U ^ n by exp( 2'n:i(2ai£i/mi)), from which ( * )

follows at once. In the special case when N has rank 2, w ith basis

e i and e2 , and N' is generated by me^ and e2 , N/N' is isomor

phic to |JLm , with the image of e^ corresponding to C = exp(2iri/m),

and one checks that the general action of N/N' specializes to the

above action of |JLm .

Now suppose cr is an n-simplex in N, i.e., a is generated by n

independent vectors. Let N' C N be the sublattice generated by the

minimal elements in a f lN along its n edges. As before, this gives a

cone a' in N', w ith Cn = Ua* Ua. The abelian group G = N/N'

acts on UCT', with

UCT = Ua./G = Cn/G .

This follows as before from the case of the torus, by intersecting the

ring A a. w ith £ [M ‘]G = C[M].

If cr is any simplex, i.e., generated by independent vectors, then

Ua is a product of a quotient as above and a torus. In particular, if A

is a simplicial fan, — all the cones in A are simplices — then X (A ) is

an orbifold or V-manifo ld , i.e., it has only quotient s ingu larit ies .^




Exercise. Let a be the cone generated by

Gl> G2> • • • > Gn-1> -e i - e2 - . . . -en_i + m e n ,

where ej_, . . . ,en is a basis for N. Show that

(i) = Cn/p.m, where the m th roots of unity jJLm act by

C-(xlf . . . ,xn) = (C-Xi, . . . ,C-xn);

(ii) UG is the cone over the m-tuple Veronese embedding of

IP11' 1.

Exercise. Show that if m and a*, . . . ,a n are positive integers, the

quotient of Cn by the cyclic group [JLm acting by

c * ( z i , . . . ,zn) = ( c a iz i , . . . ,eanzn)

can be constructed as an affine toric var ie ty UCT, by taking n

N ’ = T. z-d/a^-e, c N = N' + I - ( l / m ) - ( e 1 + . . . +en) ,i = 1

and the cone ct generated by e^, . . . ,en. When a^ = , . . = an = 1,

show that this agrees with the construction of the preceding exercise.

One can sometimes extend this construction to non-affine toric

varieties by making the group actions compatible on affine open

subvarieties. An interesting example is the twisted or weighted

p ro je c t iv e space (P(do, . . . ,dn), where do, . . . ,dn are any positive

integers. To construct this as a toric variety , start w ith the same fan

used in the construction of projective space, i.e., its cones generated by

proper subsets of {vq, vj_, . . . , v n}, where any n of these vectors are

linearly independent, and their sum is zero; however, the lattice N is

taken to be generated by the vectors ( l/ d ^ V j , 0 < i < n. The

resulting toric va r ie ty is in fact the var ie ty

IP(d0 ,dn ) = Cn+1 n { 0 } / C* ,

where C* acts by C*(x(), • • • .xn) = (Cd°XQ £dnxn).

Exercise, (a) With aj the cone generated by the complement of ex,

use the preceding exercise to identify UCT. with a quotient of Cn, and

use the standard map (x*, . . . ,xn) (x^: . . . :x j_ i : l :x i , . . . :xn) to

identify this quotient with the open set Ui in P(do, • . . ,dn) consisting



36 SECTION 2.3

of points whose ith coordinate is nonzero.

(b) Write this twisted projective space as a quotient of a nonsingular

toric va r ie ty by a finite abelian g rou p .^

For any cone in a simplicial fan, one can find a sublattice to write

the corresponding open subvariety of the toric va r ie ty as a quotient by

a finite abelian group. Sometimes, as in the example of twisted

projective spaces, one can find one sublattice that works for all these

open sets at once, but this is not always possible:

Exercise. Let A be the fan in N = Z2, covering IR2, w ith edges

generated by the vectors - e j , ~e2> anc* 2ei + 3e2- Showthat for eve ry sublattice N ‘ of finite index in N, w ith A ' the

corresponding fan in N1, the var ie ty X (A ') is singular.

2.3 O n e -p a ram ete r subgroups; limit points

In this section, we use one-parameter subgroups of the torus, and their

limit points in toric varieties, to see how to recover the fan from the

torus action.

For each integer k we have a homomorphism of algebraic groups

-* Gm , z ■"» z k ■

Here we write <Gm for the multiplicative algebraic group, i.e., for C*.

Exercise. Show that these are all of them:

Homalg. gp.(Gm.Gm) = Z . (8)

Given a lattice N, with dual M, we have the corresponding

torus

= Hom(M,Gm) .

From the preceding exercise, it follows (by taking a basis for N) that

Horn«Gm,TN) = Hom(Z,N) = N .

This means that every one -pa ram ete r subgroup X: Gm —» Tn is given



ONE - PAR AM ETER SUBGROUPS; L IMIT POINTS 37

by a unique v in N. Let Xv denote the one-parameter subgroup

corresponding to v. For z € €*, Av (z) is in Tn , so is given by a

group homomorphism from M to <C*. Explicitly, for u in M,

Xv(z ) (u ) = X u(Xv( z » = z<UlV> ,

where < , > is the dual pairing M ® N —> Z.

Dually,

Horn(TN,(Gm) = Hom(N,Z) = M .

Every characte r %: Tn -» Gm is given by a unique u in M. The

character corresponding to u can be identified w ith the function

X u in the coordinate ring C[M] = F(Tn,0*)-

Exercise, (a) Show that for lattices N and N', the mapping

Horn z (N ',N ) -» Homalg gp (TN-,TN) , (p >-* (p* ,

is an isomorphism. (A mapping between tori induces a corresponding

mapping on one-parameter 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 duality pairing

< , > : M x N -* Z .

Note in particular that the above prescription shows how to

recover the lattice N from the torus Tn - Given a cone a, we next

want to see how to recover a from the torus embedding Tn c LfCT.

The key is to look at lim its lim Av(z) for various v € N, asz~*°the complex variable z approaches the origin. For example,

suppose cr is generated by part of a basis e^, . . . ,e^ for N, so Ua

is x (C * )n~k. por v = ( m i , ... ,m n) € Zn, Av (z) = (zm i , . . . , z mn).

Then Av (z) has a limit in if and only if all mj are nonnegative

and m, = 0 for i > k. In other words, the limit exists exactly when v

is in a. In this case, the limit is (8j_, ... ,8n), where 8j = 1 if lrq = 0

and 8j = 0 if rrtj > 0. Each of these limit points is one of the

distinguished points xT for some face t of cr.



38 SECTION 2.3

In general, for each cone t in a fan A , we have defined the distinguished point xT in UT. If t is a face of cr, then Ux is contained in UCT, so we must be able to realize xT as a hom om or

phism of semigroups from Sa to C. This homomorphism is

which is well defined since x An a v is a face of cjv. From this it follows

that the resulting point xT of X (A ) is independent of a; that is, if

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

UCT to the point defined in Uy. We note also that these points are all

distinct; this follows from the fact that xT is not in UCT if cr is a

proper face of t . As we will see later, there is exactly one such point

in each orbit of Tn on X (A ).

Claim 1. I f v is in |A|, and t is the cone of A that contains

v in its re la t ive in te r io r , then lim Av (z) = xT .z - » 0

For the proof, look in for any ct containing t as a face, and

identify Av (z) w ith the homomorphism from M to C* that takes

u to zÛiV . For u in Sa, we have <u,v> > 0, w ith equality

exactly when u belongs to t x . It follows that the limiting

homomorphism from Sa C M to £ is precisely that which defines

xT. (One should check that this is the topological limit, say by choosing

m generators %u for SCT to embed UCT in Cm.)

Claim 2. I f v is not in any cone of A , then lim Av (z) does notz 0

exist in X (A ).

In fact, if v is not in a, the points Av (z) have no limit points in UCT

as z approaches 0. To see this, take u in ctv w ith <u,v> < 0

(possible since a = (crv)v'). Then %u(Av (z )) = zû ,v -> » as z 0.

With these claims, a f lN is characterized as the set of v for

which Av (z) has a limit in UCT as z -* 0, and the limit is xa if

v is in the relative interior of a. For those v not in the support

|A| of A , i.e., the union of the cones in A , there is no limit (or

converging subsequence).



COMPACTNESS AND PROPERNESS 39

Exercise. For v € N, show that Xv : C* —» Tjsj extends to a

morphism from C to X (A ) if and only if v belongs to |A|,

and Xv extends to a morphism from IP1 to X (A ) if and only

if v and -v belong to |A|.

2.4 Compactness and properness

Recall that a complex var ie ty is compact in its classical topology

exactly when it is complete (proper) as an algebraic variety . For a

toric var iety , we can see this in terms of the fan:

A toric var ie ty X (A ) is com pact if and only if its support |A|

is the whole space Njr.

Because of this we say that a fan A is com plete if |A| = Njr.

One implication is easy: if the support were not all of N r , since A is

finite, there would be a lattice point v not in any cone; the fact that

Xv (z) has no limit point as z -* 0 contradicts compactness.

Before proving the converse, we state the appropriate

generalization. Let (jr. N‘ -» N be a homomorphism of lattices that

maps a fan A ' into a fan A as in §1.4, so defining a morphism

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

Proposition. The m ap <p*: X (A ') —► X (A ) is p roper i f and only if

cp-1(|A|) = |A'|.

A va r ie ty is compact when the map to a point is proper, so the

preceding case is recovered by taking the second lattice to be {0}.

Proof of the proposition. =*: If v' is in N ‘ but in no cone of A ',

and v = cp(v') is in a cone of A , then (p*(Xv .(z)) = Xv (z) has a limit

in X (A ), but Xv .(z) has no converging subsequences as z 0,

which contradicts properness.

We use the valuative cr iterion of properness: a morphism

f: X —» Y of varieties (or a separated morphism of schemes of finite

type) is proper if and only if for any discrete valuation ring R, with

quotient field K, any comm utative diagram



40 SECTION 2.4

Spec(K) > X

r I

Spec(R) » Y

can be filled in (uniquely) as shown. When X is irreducible, one m ay

assume the image of the map Spec(K) —» X is in a given open subset

U of X . (10)

Apply this w ith X = X (A ' ) D U = T^p, Y = X (A ), f = tp*; assume

Spec(R) maps to U^. The map from Spec(K) to U is given by a

homomorphism a: M ‘ —> K* of groups. We want to find cr* mapping

to cr so we 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 is the order

function of the discrete valuation, then ord°oc°(p* is nonnegative on

cr^nM; equivalently,

ip(ord°oc) = ord°ot°cp* € ( a v )v = a .

By the assumption, there is a cone a 1 so that cp(cr‘) c cr and

ord°cx € cr'. This says that ord°oc is nonnegative on a ,v, which

is precisely the condition needed to fill in the diagram.

A fundamental example of a proper map is blowing up. We

saw the first example of this in the construction of the blow-up of

Uq- = C2 at the origin xa = (0,0) in Chapter 1. More generally,

suppose a cone a in A is generated by a basis v^, . . . , v n for N.

Set vq = v i + . . . + v n, and replace a by the cones that are gener

ated by those subsets of (vo, v* , . . . , v n) not containing ( v j , . . . , v n).

This gives a fan A 1, and the resulting proper map X (A ')-> X (A ) is

the blow-up of X (A ) at the point xa. To see this, since nothing is

changed except over UCT, we m ay assume A consists of a and all



COMPACTNESS AND PROPERNESS 41

of its faces, so X (A ) = UCT = € n, with xCT the origin, and v* = e* for

1 < i < n. Then X (A ') is covered by the open affine varieties U^.,

where a 4 is the cone generated by eg, e i, . . . , e if . . . ,en, 1 < i < n, and is generated by e i* ,e** - e * , . . . , en* - e^. The corres

ponding coordinate ring is

A CT. = CtXi( X i X f 1, X n X f 1] .

On the other hand, the blow-up of the origin in Cn is the subvariety

of £ n x (Pn-1 defined by the equations XjTj = XjTj, where Ti, . . . , Tn

are homogeneous coordinates on P n“ *. The set Uj where Tj =* 0 has

Xj = Xi«(Tj/Tj), so Uj is Cn, with coordinates X* and Tj/Tj = Xj/Xj

for j * i. So Uj = UCT. with ctj as above, and w ith the same gluing.

Exercise. For a nonsingular affine toric va r ie ty of the form

x (C * )n“ k, show how to construct the blow-up along {0} x (£ * )n~k

as a toric variety.

Exercise. If A is an infinite fan, show that X (A ) is never compact,

even if |A| = Njr and all Av(z) have limits in X (A ). W hy does the

criterion for properness not apply? Give an example of an infinite fan

A in Z2 whose support is all of IR2. Show that, for infinite fans, the

proposition remains true with the added condition that there are only

finitely m any cones in A ' mapping to a given cone in A.

Exercise. (F iber bundles) Suppose 0 -» N1 -» N -» N" 0 is an

exact sequence of lattices, and suppose A', A , and A " are fans in N',

N, and N“ that are compatible with these mappings as in §1.4, giving

rise to maps

X (A ' ) -> X (A ) -> X (A " ) .

Suppose there is a fan A M in N that lifts A ", i.e., each cone in A "

is the isomorphic image of a unique cone in A ", such that the cones

a in A are of the form

a = + a " = {v* + v " : v ' € a', v " € a " }

for a' a cone in A ' and cr" a cone in A". Show that the above sequence is a locally trivial fibration.



42 SECTION 2.5

Exercise. Construct the projective bundle

P (0 (a 1)© e ( a 2) e . . • ® 0 (a r )) -> IPn

as a toric v a r ie t y / 12^

Exercise. Let cp*: X (A ') -» X (A ) be the map arising from a

homomorphism of lattices cp: N* —* N mapping A 1 to A as in §1.4.

Show that for a cone a' in A 1, cp* maps the point xCT« of X (A ' ) to

the point xCT of X (A ), where cr is the smallest cone of A that

contains ct‘.

Exercise. Find a subdivision of the cone generated by (1,0,0), (0,1,0),

and (0,0,1) in Z3, including the ray through (1,1,1), such that:

(i) the resulting toric va r ie ty X' is nonsingular; (ii) the resulting

proper map X' C3 is an isomorphism over <C3 ' (0); but (iii) this

map does not factor through the blow-up of C3 at the origin. In

particular, this map does not factor into a composite of blow-ups along

smooth centers/14^

2.5 Nonsingular surfaces

Let us see what two-dimensional nonsingular com plete toric varieties

look like. They are given by specifying a sequence of lattice points

vo, Vi, . . . , v d_i, v d = v 0

in counterclockwise order, in N = Z , such that successive pairs

generate the lattice.

From the fact that vq and v* are a basis for the lattice, and v^

and v 2 are also a basis, we know that v 2 = - vq + a^v^ for some



NONSINGULAR SURFACES 43

integer a^. In general, we must have

aiv i = v i - l + Vi + i , 1 < i < d ,

for some integers aj. We will discuss later the conditions these integers

must satisfy.

The possible configurations are topologically constrained. For

example, two of the cones cannot be arranged with v j in the angle

strictly between v i+i and -v j and v j + i in the angle strictly between -v j and - v i+i:

Exercise. Prove this.

We want to classify all of these surfaces. The cases when the

number d of edges is small are easy to do by hand:

Exercise. Show that for d = 3, the resulting toric va r ie ty must be

P 2, and for d = 4, one gets a Hirzebruch surface Fa, both as

constructed in §1.1.

Given one of these toric surfaces, we know how to construct

another that is the blow-up of the first at a T^-fixed point: simply

insert the sum of two adjacent vectors. In fact:

Proposition. All com plete nonsingular toric surfaces are obtained

from P 2 or P a by a succession of blow-ups at Tjsj-f ixed points.

This follows from the preceding exercise and the following:

Claim. I f d > 5, there m us t be some j, 1 < j < d, such that

v j _ i and V j+i generate a strongly convex cone, and

v j - v j _ ! + v J + 1 .



44 SECTION 2.5

The proof of this claim is outlined in the following exercise.

Exercise, (a) Show that if d > 4 there must be two opposite vectors

in the sequence, i.e., v j = - v j for some i, j.

(b) Suppose Vi = - vq and i > 3. Show that v j = V j_ i + V j +i for

Exercise. Show that the integers a^, . . . , a^ must satisfy the

equation

(*> ( U h ?;1,)-••••(?;*) = u ? )- (i7)

Exercise. Show that inserting v* = + v^ + i between ar

v k +1 changes the sequence of integers a*, . . . ,a j by adding 1

Exercise. Show that inserting v* = v^ + v^ + i between v^ and

v k +1 changes the sequence of integers a*, . . . ,a j by adding 1 to a^

and a^ + i and inserting a 1 between them. Deduce from this and

the preceding discussion that the integers a*, . . . , a<j must satisfy the

equation

( * * ) a i + a2 + • ■ • + a^ = 3d - 12 .

Exercise. Show conversely that any integers a*, . . . , a<j satisfying

( * ) and ( * * ) arise from a nonsingular two-dimensional toric variety.

Show that the sequence 0, 2, 1, 3, 1, 3, 1, 1 satisfies these conditions,

and describe the corresponding surface/18^

Exercise. Each vj determines a curve D4 = IP1 in X (as in §1.1).

Show that the normal bundle to this embedding is the line bundle

GC-a^) on IP1. Show that successive curves meet transversally, but

are otherwise disjoint:

some 0 < j < i.

d2

(D i • D i) = -ai

The classification of smooth projective toric varieties of higher

dimension is an active problem/20^



RESOLUTION OF SINGULARITIES 45

2.6 Resolution of singularities

In §2.2 we fixed the cones and refined the lattice. Now we fix the

lattice and subdivide the cones. Suppose A ' is a re f inem ent of A ,

i.e., each cone of A is a union of cones in A'. The morphism

X (A ') —> X (A ) induced by the identity map of N is birational and

proper; indeed, it is an isomorphism on the open torus Tjsj contained

in each, and it is proper by the proposition in §2.4.

This construction can be used on singular toric varieties to

resolve singularities. Consider the example where cx is the cone in

Z2 generated by 3 e i - 2e2 and e2 , and insert the edges through the

points e i and 2e±- e2 - The indicated subdivision

gives a nonsingular X (A ') mapping birationaily and properly to UCT.

This can be generalized to any two-dimensional toric singularity.

Given a cone ct that is not generated by a basis for N, we have seen

that w e can choose generators and e2 for N so that cr is

generated by v = e2 and v' = m e i - k e 2 , 0 < k < m, with k and m

rela tive ly prime. Insert the line through e*:

The cone generated by e and e2 corresponds to a nonsingular open

set, while the other cone generated by e and m e^ -k e2 corresponds

(m , -k )



46 SECTION 2.6

to a va r ie ty whose singular point is less singular than the original one.

To see this, rotate the picture by 90°, moving e* to e2 , and then

translate the other basic vector vertically (by a m atr ix i ) as

§2.2) to put it in the position (m ^ - k i ) , with m i = k, 0 < k^ < m i,

and ki = aj_k - m for some integer a^ > 2:

This corresponds to a smooth cone when k^ = 0. Otherwise

m/k = a^ - k^/mi = a^ - l/ (m j/ k i ) , and the process can be repeated.

The process continues as in the Euclidean algorithm, or in the

construction of the continued fraction, but w ith alternating signs:

a 2 "

JL_a r

with integers a* > 2. This is called the H irzebruch-Jung continued

fraction of m/k.

Exercise, (a) Show that the edges drawn in the above process are

exactly those through the vertices on the edge of the polygon that is

the convex hull of the nonzero points in ctDN:




(b ) S h o w that there are r added vertices v^, . . . , v r be tween the

given vert ices v = vq and v' = v r + i, and ajVj = v ^ i + v i+i.

(c) S h o w that these added rays correspond to exceptional divisors

Ej = IP*, fo rm ing a chain

with self-intersection numbers (Ej-Ej) = -aj.

(d) Show that {v q , . . • , v r + i ) is a minimal set of generators of the

semigroup a ON.

Exercise. Show that the algebra A a = C[Sa] has a minimal set of

generators {U^V**, 1 < i < e), where the embedding dimension e and

the exponents are determined as follows. Let b2 , . . . , be_i be the

integers (each at least 2) arising in the Hirzebruch-Jung continued

fraction of m/(m-k). Then

si = m , S2 = m - k , sj+i = bpSj - Si_i for 2 < i < e-1 ;

t i = 0 , t2 = 1 , tj+i = bj*11 - t j_i for 2 < i < e -1 S 21^

Exercise. Let a be the cone generated by e2 and (k + l ) e i - ke2-

Show that Sa is generated by u* = e£, U2 = ke£ + (k + De^, and

u3 = e^ + e^, w ith (k + l ) u 3 = u* + U2- Deduce that

A a = C [ Y i , Y 2,Y3] / ( Y 3k + 1 - Y i Y 2) .

which is the rational double point 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 IP* and with self

intersection - 2 . 22^

Exercise. Let a be generated by e2 and m e j - ke2 as above, and

let a ' be generated by e2 and m'e^ - k 'e2 , with 0 < k' < m'

re la tive ly prime. Show that UCT» is isomorphic to UCT if and only if

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

Given a fan A in any lattice N, and any lattice point v in N,

one can subdivide A to a fan A ' as follows: each cone that contains



48 SECTION 2.6

v is replaced by the joins (sums) of its faces w ith the ray through v;

each cone not containing v is left unchanged.

Since A 1 has the same support as A , the induced mapping from

X (A ') to X (A ) is proper and birational. The goal is to choose a

succession of such subdivisions to get to a nonsingular toric variety.

Exercise. Show that one can subdivide any fan, by successively

adding vectors in larger and larger cones, until it becomes simplicial.

Now if cr is a k-dimensional simplicial cone, and vj_, . . . , v^ are

the first lattice points along the edges of cr, the m ult ip lic ity of ct is

defined to be the index of the lattice generated by the Vj in the lattice

generated by a:

m u lt (a ) = [Na : Zv^ + . . . + Zv^] .

Note that UCT is nonsingular precisely when the multiplicity of cr is

one.

Exercise. Show that if mult(cr) > 1 there is a lattice point of the

form v = EtjVi, 0 < < 1. For such v, taken minimal along its ray,

show that the multiplicities of the subdivided k-dimensional cones are

tpm u lt (a ) , w ith one such cone for each nonzero tj.

From the preceding two exercises, one has a procedure for

resolving the singularities of any toric va r ie ty — never leaving the

world of toric varieties:

Proposition. For any toric var ie ty X (A ) , there is a re f in em en t A

of A so that X (A ) -» X (A ) is a resolution of singularities.

In particular, the resulting resolution is equivariant, i.e., the map

commutes w ith the action of the torus.




Exercise. For surfaces, show that this procedure is the same as that

described at the beginning of this section. Show that the integer a*

found there is the multiplicity of the cone generated by v j_ i and

v i+i.

Let N be a lattice of rank 3, with a the cone generated by

vectors vj_, V2 , V3 , and V4 that generate N as a lattice and satisfy

v i + v 3 = v 2 + V4 , as we considered in §1.3:

There are three obvious ways to resolve the singularity by subdividing:

A i Draw the plane through v^ and V3 (take v = v^ or V3 );

A 2 Draw the plane through v 2 and V4 (take v = v 2 or V4 );

A 3 Add a line through v = v^ + V3 = v 2 + V4 .

The first two of these replace a by two simplices, the third by four.

Since the third refines each of the first two, the corresponding

resolution maps to each of them:

X (A 3)

^ \X (A i ) X (A 2)

\ ^X



50 SECTION 2.6

This description of two different minimal resolutions is well known for

this cone over a quadric surface: each of X (A i ) X and X (A 2 ) X

has fiber IP1 over the singular point, corresponding to the two rulings

of the quadric, while X (A $ ) —> X has fiber IP^xlP1, the quadric itself.

The transformation from X (A i ) to X (A 2 ) is an example of a "flop",

which is a basic transformation in higher-dimensional birational

geometry. In fact, toric varieties have provided useful models for

Mori’s program/25^

Let us consider a global example. Let A be fan in IR3 over the

faces of the cube w ith vertices at (±1 ,±1 ,±1 ) ; take N to be the

sublattice of 1? generated by the vertices of the cube, as in Example

(3) at the end of Chapter 1. The six singularities of X (A ) can be resolved by doing any of the above subdivisions to each of the face of

the cube. The following is a particularly pleasant w a y to do it:

These added lines determine a tetrahedron; the fan A of cones over

faces of this tetrahedron determines the toric va r ie ty X( A ) = (P3.

Since A is also a refinement of A, we have morphisms

X (A ) « - X ( 2 ) -* X( A ) = IP3 .

Exercise. Show that the morphism X( A ) -* X( A ) is the blow-up of

IP3 along the four fixed points of the torus. In X( A ) the proper

transforms of the six lines joining these points are disjoint. Show that

the morphism X( A ) -* X (A ) contracts these lines to the six singular

points of X (A ).



CHAPTER 3

ORBITS, TOPOLOGY, AND LINE BUNDLES

3.1 Orbits

As w ith a n y set on wh ich a group acts, a toric va r i e ty X = X ( A ) is a

disjoint union of its orbits by the action of the torus T = Tn - W e will

see that there is one such orbit 0 T for each cone t in A ; it is the

orbit containing the distinguished point xT that w a s described in §2.1.

M oreover ,

0 T = «C * )n~k if d i m ( T ) = k .

If t is n-dimensiona l , then 0 T is the point xT. If t = (0), then

0 T = Tn - W e will see that 0 T is an open subvar i e ty of its closure,

which is denoted V ( t ) . The va r iety V ( t ) is a closed subvar ie ty of X

that is again a toric var iety. In fact, V ( t ) will be a disjoint union of

those orbits for wh ich y contains t as a face.

Before work ing this out, let us look at the simplest example:

T = ( C * ) n acting as usual on X = Cn. The orbits are the sets

{ ( z i , . . . , z n ) € Cn : zj = 0 for i € I, zj # 0 for i £ 1} ,

as I ranges over all subsets of {1, . . . ,n). This is the orbit containing

xT, w h e r e t is generated by the basic vectors e* for i € I. All of the

above assertions are evident in this example. Note also that

Ox = Horn ( t xD M , (C*) ,

which is a fo rmu la that will be true general ly as well. The general

case of a nonsingular affine va r ie ty is obtained by crossing this

example w ith a torus (€*)*.

For a compact example , consider the projective space lPn

corresponding to the fan of cones generated by proper subsets of

{ v q , . . • , v n), w h e r e the vectors generate the lattice and add to zero.

If t is generated b y the subset { v j : i € I), then V ( t ) is the



52 SECTION 3.1

intersection of the hyperp lanes zj = 0 for i € I, and 0 T consists of

the points of V ( t ) whose other coordinates a re nonzero.

In the general case we will first describe the orbits 0T and their

closures V ( t ) abstractly, and then show how to embed them in X (A ).

For each x we defined Nx to be the sublattice of N generated (as a

group) by t flN, and

N (x ) = N/NT , M ( t ) = t -'-d m

the quotient lattice and its dual. Define 0T to be the torus

corresponding to these lattices:

° x = Tn (t ) - Hom (M (x ),C *) = Spec(€[M(T)]) = N(x)<g>zC* .

This is a torus of dimension n -k , where k = dim (x), on which Tn

acts transitively via the projection Tn -* Tn (t )*The star of a cone t can be defined abstractly as the set of

cones a in A that contain t as a face. Such cones cr are

determined by their images in N ( t ), i.e. by

o7 = a + (N t )|r / ( N t )r C N|r/ (N x )|r = N (t )|r .

These cones {ct : x < a ) form a fan in N(x), and we denote this fan

by Star(x). (W e think of S tar(x ) as the cones containing t , but

realized as a fan in the quotient lattice N ( t ).)

Set

V (t ) = X (S tar (x ) ) ,

the corresponding (n-k)-dimensional toric variety. Note that the torus

embedding Ox = Tn (t ) c V (x ) corresponds to the cone (0 ) = T in



ORBITS 53

N(t ).

This toric va r ie ty V ( t ) has an affine open covering (Uct( t )}, as

cr varies over all cones in A that contain t as a face:

U(j(t ) = Spec(C[avnM(T)l) = Spec(C[avriTxn M]) .

Note that a v n T x is a face of crv (the face corresponding to t by

duality). For a = t , Ut (t ) = 0T.

To embed V (x ) as a closed subvariety of X (A ) , we construct a

closed embedding of Ua(T ) in Ua for each cr > t . Regarding the

points as semigroup homomorphisms, the embedding

Uct( t ) = HomSg (a v flT'LnM ,C ) Homsg( a vn M ,C ) = UCT

is given by extension by zero; again, the fact that a vOT1 is a face of

ctv implies that the extension by zero of a semigroup homomorphism

is a semigroup homomorphism. The corresponding surjection of rings

dcr^H M]---- > C[avn T xn M] ,

is the obvious pro jection : it takes X u to %u if u is in a v f lT xnM ,

and it takes %u to 0 otherwise.These maps are compatible: if t is a face of a, and cr a face

of cr', the diagram

Uct(t ) ^ Uct»(t )

£ £UCT ^ Uff.

commutes, since it comes from the com m utative diagram

HomSg(avriT1nM ,C) Homsg(alvn T xnM,(C)

1 1Homsg(<7vn M , C) Homsg( a lv nM,(C)

where the horizontal maps are restrictions and the vertical maps are

extensions by zero. These maps therefore glue together to give a closed

embedding

V (t ) X (A ) .

If x is a face of t ', we have closed embeddings



54 SECTION 3.1

V ( t ' ) V ( t )

given, on the open set U CT for a € StarCx'), by extension by zero f rom

HomSg(CTv riTl -LnM, (C ) to Homsg( a v n T An M , C ) ; a l te rnat ively , regard

V ( t ) as a toric va r ie ty and apply the preceding construction. In

s u m m a r y , w e have an order-revers ing correspondence f r om cones t

in A to orbit closures V ( t ) in X(A ).

It follows from this description that the ideal of V ( T ) f i U a in A CT

is ® C *X U> the sum over all u in Sa such that <u,v> > 0 for v in

the relative interior of t . For a nonsingular surface, this agrees w ith

the description in §1.1 of the curve defined by an edge of a fan.

Exercise. For a cone or, show that eve ry M-graded prime ideal

in A a has this form for some face t of cr. Show that e ve ry

Tfvj-invariant closed subscheme of Ua is defined by a graded ideal

in A ct. ^

Exercise. Show similarly that if X (A ) is an affine var ie ty , then A

consists of all faces of some cone a, so X (A ) = UCT.

Consider the singular example X = UCT, where a is the cone in

Z2 generated by v^ = 2e^ - e2 and V2 = &2- We saw in §1.1 that UCT

is a cone over a conic, defined in C3 by an equation V 2 = UW. Then

V (a ) is the vertex of the cone, which is the origin in C3, and, if

and T 2 are the edges through v* and V2 , then V ( t ^ ) is the line

U = V = 0, and V ( t 2 ) is the line V = W = 0. The orbits 0T. are the

complements of the origin in these lines, and 0 {q) = Tjvj is the

complement of these lines in X.

Proposition. There are the following relations am ong orbits 0 T,

orbit closures V ( t ), and the affine open sets U a :

(a) UCT = 11 0T ;t < a

(b) V ( t ) = Ji 0* ;V >T

(c) 0 T = V ( t ) V u v (y )y > T



ORBITS 55

Proof. For (a), note that a point of UCT is given by a semigroup

homomorphism x: cr^nM -» C. This point is in Tn = 0{q} exactly

when x does not take on the value 0 , for then it extends to a

homomorphism from M to C*. In general,

x-1(C*) = a vflTxnM

for some face x of a. This follows from the characterization of faces

given in an exercise in §1 .2 : the sum of two elements of crv cannot be

in x _1(C*) unless both are in x_1(C*). This means precisely that x

corresponds to a point of 0T C U ^x ).

For (c), passing to N (x), i.e., working in the toric va r ie ty V (x ),

we m ay assume x = { 0 }, in which case we must show that

Tn = X (A ) - U V(y) .Y*(0)

By intersecting w ith open sets of the form UCT, this follows from (a).

Then (b) follows from (c) by induction on the dimension.

It follows from the proposition that X (A ) is a disjoint union of

the Ox, which are the orbits of the T r a c t io n . In addition, an orbit

0 T is contained in the closure of Oa exactly when ct is a face of x.

In particular, the closed orbits are exactly those 0CT for ct a maximal

cone in A. It also follows that if A 0 is the fan obtained from a fan A

by removing some maximal cones ct (while retaining their faces),

then X (A ° ) is obtained from X (A ) by removing the closed orbits Oa.

If A = Ap is constructed from a polytope P as in §1.5, there is a cone ctq for each face Q of P, so an invariant subvariety

V q = V ( c t q ) for each face. Note that V q is contained in V q « exactly when Q is a face of Q', and the (complex) dimension of V q is equal

to the (real) dimension of Q.

If X (A ) is nonsingular, it follows from the definitions that each orbit closure V (x ) is nonsingular. It is a good general exercise now to

compute the orbits and orbit closures in all of the toric varieties we

have seen, including those that are singular.

Exercise. Show that V (x ) f lU a is the disjoint union of the orbits Oy as y varies over all cones such that x < Y < cr. In particular, V (x ) is

disjoint from UCT if x is not a face of cr.



56 SECTION 3.2

Exercise. Show that the distinguished point xx is the origin (identity)

of the torus 0T. Show that 0T is the unique closed Tfyj-orbit in UT.

Exercise. Let cp*: X (A ' ) -» X (A ) be the map arising from a

homomorphism of lattices mapping a fan A ' in N' to a fan A in N,

as in §1.4. If (p maps N ‘ onto N, show that cp* maps the orbit 0T«

of X (A ') onto the orbit 0T of X (A ) , where t is the smallest cone of

A that contains t ‘. If cp“ ^(lA|) = IA'1, deduce that cp* maps V ( t ' )

onto V ( t ). ^

Exercise. Any v in N determines a mapping Av : C* —> Tjsj, so an

action of C* on X (A ). Show that the set of fixed points of this action

is the union of those V (y ) for which v is in Ny.

3.2 Fu n dam en ta l groups and Euler ch a ra c te r is t ic s

First we look at the fundamental group tti(X (A ) ) of a toric var iety .

Base points will be omitted in the notation for fundamental groups;

they m ay be taken to be the origins of the embedded tori. The main

fact is that complete toric varieties are simply connected. In fact,

Proposition. Let A be a fan that contains an n -d im ensiona l cone.

Then X (A ) is simply connected.

Proof. The first observation is that the inclusion Tn X (A ) gives a

sur jection

tti(Tn ) ----~ Tri(X (A )) .

This is a general fact for the inclusion of any open subvariety of a

n orm a l variety ; the point is that a normal va r ie ty is locally

irreducible as an analytic space, so that its universal covering space

cannot be disconnected by throwing aw ay the inverse image of a closed

subvariety/3^

Now for any torus Tn , there is a canonical isomorphism

N n i (T N) >



FUNDA MEN TAL GROUPS AND EULER CHARACTERISTICS 57

defined by taking v € N to the loop S1 c (C* —» Tjsj, where C* —> T j

is the map Av defined in §2.3. If v is in ctHN for some cone cr,

the loop can be contracted in UCT, since lim Av (z) = xa exists in UCT;z —»0

in fact, we have seen that Av extends to a map from C to Ua. The

contraction is given by

f Xv (tz ) z € S1, 0 < t < 1

X y ’t ( z ) = 1 c l[ xa z € S , t = 0

If a is n-dimensional, then a f lN generates N as a group, so the

fact that such loops are trivial in UCT implies that all loops are trivial.

Corollary. I f o is a k -d imensional cone, then TTi(Ua) = Zn” k.

Proof. This follows from the fact that = U^' x (C^)n_k, and

u 1(C*‘) - uiCS1) = 2.

More intrinsically, if a' is the cone in the lattice NCT generated

by a, the fibration Ua' -» -> Tn (ct) induces a canonical

isomorphism

TTi(Ua) T t id N ^ ) ) = N (a ) .

Exercise. Let A be the fan in IR2 consisting of three cones: the

origin, and the two rays through 2e^+e2 and e i+ 2e2- Show that

t t i (X (A ) ) s Z/3Z.

In complete generality, if N ‘ is the subgroup of N generated by

all crON, as cr varies over A , then

tt1( (X (A ) ) = N/N' .

To see this, note that for each a, TrÛtf) = N/NCT. By the general van

Kampen theorem,

t t i ( (X (A ) ) ) = t i1(U U (T) = l im u i t U j = lim N/Nq. = N/XNa = N/N' .

For affine toric varieties, a similar argument shows more:

Proposition. I f a is an n -d im ensional cone, then Ua is

con tractible.



58 SECTION 3.2

Proof. We want to define a homotopy

H : UCT x [0,1] -4 UCT

between the retraction r: UCT —» xCT and the identity map. Choose a

lattice point v in the interior of cj. Regarding the points of Ua as

semigroup homomorphisms from SCT to C, define H by

H(x x t ) (u ) = t <"u ,v ">• x(u) for t > 0 ,

and H(x x 0) = xa. It is easy to see that H(x x t) is a semigroup

homomorphism whenever x is. For u = 0, H(x * t )(u ) = x(0) = 1 for

all t. For u € S CT x [o], <u,v> > 0, so H(x x t ) (u ) -> 0 as t -» 0. It

follows that H(cp x t) approaches xCT as t —> 0, and, since generators

of Sa determine an embedding of UCT in some Cm, the resulting

mapping is continuous.

Corollary. If dim(cr) = k, then Oa c UCT is a deformation re trac t.

Proof. One can use the same proof as in the proposition, or an

isomorphism Ua = UCT' x 0CT.

Corollary. There is a canonical isomorphism HHUq-jZ) = A ^ M ta ) ) ,

where M (a ) = a^flM.

Proof. Since 0 CT is the torus Ti\|(ff), its cohomology is the exterior

algebra on the dual M (a ) of its first homology group N(a).

Knowing the cohomology of the basic open sets Ua can give some

information about the cohomology of X (A ). When one has an open

covering X = UiU . . . UU$ of a space, if all intersections of the open

sets are simply connected, the cohomology of X can be computed as

the Cech cohomology of the covering. In general one has a spectral

sequence

= ® Hq (Ui n . - . n u , ) =* Hp+q( x ) . (4)io < ■ --< ip p

Apply this to the covering by open sets Uj = UCT. , where the aj are

the maximal cones of A:



FUNDAMEN TAL GROUPS AND EULER CHARACTERISTICS 59

E1p-q = © A qM(CTj n . . .fieri ) =* Hp+q(X (A ) ) .i0 < - - < ip p

In particular, this gives a calculation of the tcpological Euler

characterist ic %( X(A) ) . For every cone t that has dimension less

than n, the alternating sum X ( - l ) q rank ( A qM (x ) ) vanishes, while if

the dimension is n, this alternating sum is one. Therefore

X (X (A ) ) = X ( - l ) p+q rank q = # n-d imensional cones in A.

Assume all maximal cones in A are n-dimensional, as is the case

if A is complete. Since each U^. is contractible,

E1° ' q = 0 for q > 1 .

In addition, the complex E^’0 is

0 —> © ZCT —> © ZCT.flCT: “ * ® • • • »i 1 i < J i<j<k 1 J K

which is a Koszul complex (or the cochain complex of a simplex). This

implies that

E2p'° = 0 for p > 1 .

From the spectral sequence one therefore has

H2(X (A ) ) = E^ 1 = Eg’1 = K e r (E *' 1 - E^’ 1)

= K e r ( © M(ajflCTj) © M(crj 0 0^0 0^)) . i< j i < j < k

Note that any element u of M(or) = a xnM gives a nowhere

vanishing function X u on Ua. An element in the above kernel gives

a cocycle defining a line bundle on X (A ).

Exercise. V er ify that the torus Tjsj acts on such a line bundle,

compatibly w ith its action on X (A ). Ver ify that the above

isomorphism with H2(X ) takes the cocycle for a line bundle to the

first Chern class of the line bu nd le .^



60 SECTION 3.3

In particular, e very element of H2(X ) is the first Chern class of a line bundle of this type.

3.3 Divisors

On any va r ie ty X, a Weil divisor is a finite formal sum X a jV i of

irreducible closed subvarieties Vj of codimension one in X. A Cartier

divisor D is given by the data of a covering of X by affine open sets

Ua , and nonzero rational functions fa called local equations, such

that the ratios fa /fp are nowhere zero regular functions on Ua nUp.

The ideal sheaf 0 (-D ) of D is the subsheaf of the sheaf of rational

functions generated by fa on Ua ; the inverse sheaf 0(D) is the

subsheaf of the sheaf of rational functions generated by l/ fa on Ua .

Regarded as a line bundle, its transition functions from Ua to Up are

fa /fp. A Cartier divisor D determines a Weil divisor, denoted [D], by

[D] - Z ordv (D)-V ,c o d( V , X) = l

where ordy(D ) is the order of vanishing of an equation for D in the

local ring along the subvariety V. When X is normal these local rings

are discrete valuation rings, so the notion of "order" is the naive one.

For normal varieties, the map D [D] embeds the group of Cartier

divisors in the group of Weil divisors. A nonzero rational function f

determines a principal divisor d iv (f ) whose local equation in each

open set is f. ^

We will be prim arily concerned w ith divisors on a toric va r ie ty

X = X (A ) that are mapped to themselves by the torus T = T^. The

irreducible subvarieties of codimension one that are T-stable

correspond to edges (or rays) of the fan. Number the edges t * , . . . ,t<i,

and let v* be the first lattice point met along the edge t *. These

divisors are the orbit closures:

Di - V ( t j ) .

The T -W e i l divisors are the sums X ajDj for integers a*.

We want to describe the Cartier divisors that are equivariant



DI V IS OR S 61

by T, which we call T-C a r t ie r divisors. First, consider the affine case

X = Uq-, w ith dim(cr) = n. Let D be a divisor that is preserved by T,

corresponding to a fractional ideal I = r (X ,0(D )). We claim that I is

generated by a function X u for a unique u € a v flM. This can be

seen algebraically as follows. The fact that the divisor is T-invariant

implies that I is graded by M, i.e., I is a direct sum of spaces C*XU

over some set of u in M. Since I is principal at the distinguished

point Xq, I/ml must be one-dimensional, where m is the sum of all

C*XU for u * 0 . It follows that there is a unique u w ith I = A a*X u-

Consequently a general T-Cartier divisor on UCT has the form d iv (X u)

for some unique u € M.

Lem m a. Let u € M, and let v be the first latt ice point along an

edge t . Then o r d y (T) (d iv (X u)) = <u,v>, so

[ d iv ( X u)l = Xi Di .

Proof. The order can be calculated on the open set UT = C x (C * )n_1,

on which V ( t ) corresponds to {0} * (C * )n-1. This reduces the

calculation to the one-dimensional case, i.e., to the case where N = Z,

t is generated by v = 1 , and u € M = Z. Then X u is the monomial

X u, whose order of vanishing at the origin is u.

nFor example, look at the cone cr in Z^ generated by the vectors

V1 = 2ej[-e2 and V2 = e2 , so UCT is the cone over a conic, w ith two T-Weil divisors D* and D2 (which are straight lines on the cone). If

u = (p,q) € M = Z2, then d iv (X u) = (2p-q)D i + qD2- In particular, we see that D* and D2 are not Cartier divisors, although 2Di and 2 D2

are.

Exercise. Let a be the cone in Z2 generated by v* = 2e i ~ e 2 and

v 2 ~ -e i + 2e2, corresponding to divisors and D2 . Show that a^Di + a2D2 is a Cartier divisor on Ua if and only if a* = a2 (mod 3).

7Exercise. Let cr be the cone in Z generated by v^ = ej_, V2 = &2>

V3 = e3 , and V4 = e i - e2 + 63. Show that a^Di + a2D2 + a3D3 + a4D4

is a Cartier divisor on UCT if and only if a^ + a3 = a2 + a4 .

Exercise. Take a as in the preceding exercise, but replace the lattice

by N = Z-(l/2b )e i + Z*(l/b)e2 + Z* (l/a )e3 + Z «( l/2b )(e i + e2 + 63 ), where



62 SECTION 3.3

a and b are positive integers, with a > 1 and gcd(a,2b) = 1. Find

the first lattice points along the four edges, and find which HajDj are

Cartier divisors on UCT. In particular, show that no positive multiple of

ZDj is a Cartier divisor.

Exercise. Show that for each irreducible Weil divisor DT on an affine

toric va r ie ty Ua, there is an effective Cartier divisor that contains DT

with multiplicity

For a cone a of dimension less than n, a T-Cartier divisor on

Ua is of the form d iv (X u) for some u € M, but

d iv (X u) = d iv (X u') <=* u - u' € a xnM = M (a ) .

To see this write UCT = U^' x Tn (ct) as usual. Note that a and a'

have the "same" edges, so corresponding Weil divisors. Therefore

T-Cartier divisors on UCT correspond to elements of M /M (a ).

On a general toric va r ie ty X (A ), it follows that a T-Cartier

divisor is defined by specifying an element u ( cj) in M /M (a ) for each

cone a in A , defining divisors d iv (X _u^ ) on Ua (the minus sign is

taken to conform to the literature). Equivalently, X u^ generates

the fractional ideal of 0(D) on UCT:

r ( u ff, 0 (D)) = a ct- x u(ct).

These must agree on overlaps; i.e., when t is a face of a, u(cr) must

map to u ( t ) under the canonical map from M /M (a ) to M/M(x). In

short,

{ T-Cartier divisors) = lim M/M (a )

= Ker ( 0 M/M (aj) —> ® M / M ta j f la j ) ) ,i i < J

with a, the maximal cones, as in the preceding section.

Exercise. Show that a Weil divisor ZajDj is a Cartier divisor if and

only if for each (m ax im al) cone cr there is a u (a ) € M such that for

all Vj € a, <u(a),Vi> = -aj. ^

Exercise. If A is simplicial, show that any Weil divisor D is a

(Q-Cartier divisor, i.e., some positive multiple of D is a Cartier divisor.



LINE BUNDLES 63

3.4 Line bundles

Let Pic(X) be the group of all line bundles, modulo isomorphism. For

any irreducible va r ie ty X, the map D (3(D) gives a hom om or

phism from the group of Cartier divisors on X onto Pic(X), w ith

kernel the group of principal divisors. Let A n_ i (X ) denote the group

of all Weil divisors modulo the subgroup of divisors [div(f)] of rational

functions. The map D »-> [D] determines a homomorphism from

Pic(X) to A n_i(X ), which is an embedding when X is normal:

Pic(X) c_> A n- i (X ) .

For X a toric variety , any u € M determines a principal Cartier

divisor d iv (X u)> giving a homomorphism from M to the group

D iv jX of T-Cartier divisors. The following proposition shows that

Pic(X) can be computed by using only T-Cartier divisors and functions,

and similarly for A n_^(X) with T-Weil divisors. In addition, it gives a

recipe for calculating these groups for a complete toric va r ie ty X.

Proposition. Let X = X (A ), where A is a fan not contained in

any p roper subspace of N|r. Then there is a com m u ta t ive diagram

with exact rows:

0 -> M -> DivTX -> Pic(X) -> 0

II £ sd

0 M —> © Z-Dj A n_ i (X ) -* 0i = l

In particu lar, rank (P ic (X )) < ran k (A n_ i (X ) ) = d - n, where d is

the n u m b e r of edges in the fan. In addition, Pic(X) is free abelian.

Proof. First note that X ' U Dj = Tn is affine, w ith coordinate ring

the unique factorization ring C [X i ,X i_1, . . . ,X n,X n_1], so all Cartier

divisors and Weil divisors on Tn are principal. This gives an exact (9)sequence' '



64 SECTION 3.4

dA n - i tU D j ) = © Z-Dj - A n_x(X) -> A n_ ! (T N) = 0 .

i= 1

Next, note that if f is a rational function on X whose divisor is

T-invariant, then f = X»%u for some u 6 M and complex number

X; this follows by restricting to the torus T j. The lemma in the

preceding section and the fact that the vj span Njr imply that u is

determined uniquely by f. This shows the exactness of the second row.

If £ is an algebraic line bundle, its restriction to Tn must

be trivial, so £ = 0(D) for some Cartier divisor supported on UDj;

indeed, writing £ = 0(E) for some Cartier divisor E, take a rational

function whose divisor agrees w ith E on Tn , and set D = E - div(f).

Hence D is T-invariant as a Weil divisor, and therefore as a Cartier

divisor. The exactness of the upper row follows easily.

Finally, the fact that Pic(X) is torsion free follows from the fact

that it is a subgroup of 0M/M(cr), and each M/M(cr) is a lattice, so

torsion free.

Corollary. I f all m a x im a l cones of A are n -d im ensiona lf then

P ic (X (A ) ) s H2(X (A ),Z ).

Proof. We must map the group of T-Cartier divisors onto H2 (X (A ) ,Z ) ,

w ith kernel M. We have seen the isomorphisms:

{T-Cartier divisors) = Ker ( © M/M(cjj) © M/M(<jjfl ctj) ) ,i i< j

H2(X (A ) ;Z ) = K e r ( © M (a j f la j) —> © M (a jH a jf icr^ )) .J i<j<k

An element © u j in the first kernel is mapped to the sum 0 ( u j - u j )

in the second (noting that IVKaj) = 0 since ctj is n-dimensional). It

is an easy exercise to show that this map is surjective, w ith kernel

isomorphic to M.

Exercise. Give an example with m aximal cones not n-dimensional

where the map from P ic (X (A )) to H2 (X (A ) ;Z ) is not su r jec t iv e/10^

Exercise. Let A be the complete fan in Z2 w ith edges along

V1 = e l> v 2 ~ “ Gl + n ie2 , (m > 1), and V3 = ~e2- Show that the Weil



LINE BUNDLES 65

divisor a^Di + a2D2 + £303 is a Cartier divisor on X = X (A ) if and

only if a^ + a2 = 0 (mod m). Show that

Pic(X) = Z«m D2 A i (X ) = Z-D2 .

Exercise. Let A be the complete fan in Z2 w ith edges along

v^ = 2e ^ - e 2 , V2 = -e^ + 2e2 , and V3 = - e i ~ e 2 . Show thata^Di + a2D2 + ^3 03 is a Cartier divisor on X = X (A ) if and only if

a i = a2 = a3 (mod 3). Show that

Pic(X) = Z-3Di = Z ,

A x(X) = (Z-Di + Z-D2)/Z-3(D !-D2) = Z © Z/3Z .

In particular, A n_^X can have torsion.

Exercise. Let X = X (A ), where A is the fan in Z3 over the faces of

the convex hull of the points e^, e2 , e3 , e^ - e2 + 63 , and -e^ - 63.

Show that Pic(X) s Z, A 2 (X) £ Z2, and A 2(X )/P ic (X ) s Z.

Exercise. Let X = X (A ), where A is the fan over the faces of the

cube w ith vertices (±1 ,±1 ,±1 ), and N the sublattice of Z3 generated

by the vertices. Show that Pic(X) = Z, generated by a divisor that is

the sum of the four irreducible divisors corresponding to the vertices of

a face. Show that A 2(X ) s Z5, and A 2(X )/P ic (X ) s Z4.

In §1.5 we constructed a complete fan A that cannot be

constructed as the faces over any subdivided polytope. In fact, the

exercise proving that this fan is not a fan over the faces of a convex

polytope actually showed that every line bundle on X (A ) is trivial;

i.e., P ic (X (A ) ) = 0.

Exercise. For this fan, show that A 2 (X (A ) ) s Z5 0Z/2Z.

Exercise. Let A be a fan such that all of its m axim al cones are

n-dimensional. Show that the following are equivalent:

(i) A is simplicial;

(ii) Every Weil divisor on 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 ) rank (P ic (X (A )) ) = d - n.

The data (u (a ) € M /M (a )) for a Cartier divisor D defines a



66 SECTION 3.4

continuous piecewise linear function i(jp on the support |A|: the

restriction of t|jp to the cone ct is defined to be the linear function

u(a); i.e.,

^p (v ) = for v € ct .

The compatibility of the data makes this function well defined and

continuous. Conversely, any continuous function on |A| that is linear

and integral (i.e., given by an element of the lattice M) on each cone,

comes from a unique T-Cartier divisor. If D = SajDi, the function

is determined by the property that ^D^v i = ” a i> equivalently

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

These functions behave nicely with respect to operations on divisors.

For example, iJjd + e = ipp + +E, so i|/mD = m +D. Note that ipdiv(xu) is the linear function -u. If D and E are linearly equivalent divisors,

it follows that i|jp and iJje differ by a linear function u in M.

A T-Cartier divisor D = ZajDj on X (A ) also determines a

rational convex polyhedron in M(r defined by

PD = {u € M|r : <u,vj> > -a j for all i )

= {u € Mg* : u > i p on |A|) .

Lem m a. The global sections of the line bundle 0(D) are

r(X,0(D)) = ® C-Xu .u e PDnM

Proof. It follows from the lemma of the preceding section that

r(UCT,0(D)) = © c - x u .u € PD(a)nM

where

Pp(a-) = {u € M|r : <u,vj> > -aj V vj € cr} .

These identifications are compatible w ith restrictions to smaller open

sets. It follows that T(X ,0(D)) = n r ( U cr,0(D)) is the corresponding

direct sum over the intersection of the Pp(cj)nM, as required.



LINE BUNDLES 67

Exercise. Show that (i) PmD = m P D; (ii) PD + div(xu) = P D " ui

When |A| = N(r, the var ie ty X (A ) is complete, and it is a

general fact that cohomology groups of a coherent sheaf are finite

dimensional on any complete variety. In the toric case this means

that the polyhedron Pp is bounded:

Proposition. I f the cones in A span Njr as a cone , then

H X ,0 (D )) is finite dimensional. In part icu la r , PpOM is finite.

Proof. If Pp were unbounded, by using the compactness of a sphere

in M(r there would be a sequence of vectors uj in Pp, and positive numbers tj converging to zero, such that tpUj converges to some

nonzero vector u in M|r. The fact that > - a j for all j

implies that <u,vj> > 0 for all j. Since the v* span N(r, we must

have u = 0, a contradiction.

The next proposition answers the question of when a line bundle

is generated by its sections, i.e., when there are global sections of the

bundle such that at every point at least one is nonzero. Recall that a

real-valued function i|) on a vector space is (upper) convex if

for all vectors v and w, and all 0 < t < 1. For the simplest example

Z, if Di and D2 are the divisors corresponding to the positive and

negative edges, the divisor D = a^Di + a2 D2 corresponds to the

function ipp on IR defined by

This function is convex exactly when a^ + a2 is nonnegative. Since

generated by its sections.

We are concerned with continuous functions ip on Njr such

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

(iii) PD + PE c PD + E*

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

of the toric va r ie ty IP* corresponding to the unique complete fan in

0(D) = 0 (a i + a2) on IP*, this is exactly the criterion for 0(D) to be



68 SECTION 3.4

u(cr) 6 M. In this case convexity means that the graph of ip lies

under the graph of u(cr) for all n-dimensional cones ct, s o that the

graph of \\) is "tent-shaped":

The convex function ip is called str ic t ly convex if the graph of on

the complement of ct lies strictly under the graph of u(cr), for all n-

dimensional cones cr; equivalently, for any n-dimensional cones cr

and ct', the linear functions u ( ct) and u(cr') are different.

Proposition. Assume all m ax im a l cones in A are n-d im ensional.

Let D be a T -C art ie r divisor on X (A ). Then 0(D) is generated

by its sections if and only if ipp is convex.

Proof. On any toric va r ie ty X, 0(D) is generated by its sections if

and only if, for any cone ct, there is a u(cr) € M such that

(i) <u(cr),Vi> > - a* for all i, and

(ii) <u(cr) ,v j > = - aj for those i for which vj € ct .

Indeed, (i) is the condition for u (ct) to be in the polyhedron Pp that

determines global sections, and (ii) says that X u^ generates 0(D) on

UCT. The function tpD *s determined by its restrictions to the n-

dimensional cones, where its values are given by (ii). The convex ity of

ijjp is then equivalent to (i).

If 0(D) is generated by its sections, and all m axim al cones of the

fan are n-dimensional, we can reconstruct D, or equivalently its

function i|jp, from the polytope Pp:

^p (v ) = min <u,v> = min <u*,v> , u e PD n M



LINE BUNDLES 69

where the are the vertices of Pp.

Exercise. If 0(D) and 0(E) are generated by their sections, show

that P d + E = P d + P E-

Exercise. If 0(D) is generated by its sections, and S is a subset of

PDflM, show that ( X u : u € S) generates 0(D) if and only if S

contains the vertices of Pp.

When 0(D) is generated by its sections, choosing (and ordering) a

basis {% u • u € PpOM} for the sections gives a mapping

«p - <PD : X (A ) -» IP1" 1 , x ^ ( X U1(x): ... :X Ur(x )) ,

to projective space P r _ *, r = Card (PpnM ).

Lem m a. I f |A| = Nir, the mapping (pp is an embedding, i.e., D is

v e ry ample, if and only if ipp is s tr ic t ly convex and for every

n -d imensional cone cr the semigroup Sa is generated by

{u - u (a ) : u € PDnM ) .

Proof. Take corresponding homogeneous coordinates Tu on

indexed by the lattice points u in Pp. Let a be an n-

dimensional cone in the fan, and let u(or) be the corresponding

element of PpflM , so X u^ generates 0(D) on UCT. It follows

easily from the strict convexity of ijjp, that the inverse image by

cpD of the set C*""1 C where Tu(ct) * 0 is the open set UCT.

The restriction UCT —> Cr_ * is then given by the functions X u_u^ »

and the fact that they generate SCT means that the corresponding

map of rings is surjective, so the mapping is a closed embedding. The

proof of the implication =* is similar.

Exercise. Let X = X (A ), with A the fan over the faces of the cube

with vertices (±1 ,±1 ,±1 ), N the lattice generated by the vertices. Let

D be the sum of the divisors corresponding to the four vertices of the

bottom of the cube. Show that PD is the octahedron with vertices

(0,0,0), CA,0,^2), (O,1/*,1/*), (0 ,-V / 2), ( - 1/2 ,0 ,1/2), and (0,0,1). Show

that cpp embeds X in IP5, with image defined by the equations

^0^5 = T1T4 = T2T3 . With X ( A ) = IP3 as at the end of Chapter 2, show



70 SECTION 3.4

that this identifies X with the image of the rational map from IP3 to

IP5 defined by the linear system of quadrics through the four T^-fixed

points: (x0: x j: x2: x3) >-* (x 0xi: xox2: xox3: x *x2: x j x 3: x2x3). (12)

Proposition. On a complete toric varie ty , a T -C a rt ie r divisor D is

ample, i.e., some positive m ult ip le of D is ve ry ample, if and only if

its function is s tr ic t ly convex.

Proof. Since +mD = m +D» im Plicat i °n ^ follows from thelemma. For the converse, note that replacing D by m*D replaces

the polytope PD by m *PD = {u € M r : <u,Vi> > -m*a| for all i}. For

any u € SCT it follows from the fact that > -a* if Vj £ ct

that u + m-u(cr) is in nri'Pp for large m. Since SCT is a finitely

generated semigroup, it follows that it is generated by elements

u - m«u(cr) as u runs through m -PDf lM , for m sufficiently large.

By the lemma, it follows that mD is v e ry ample.

Note in particular that for complete toric varieties, unlike for

general complete varieties, every ample line bundle is generated by

its sections.

Exercise. When D is ample, show that the polytope P^ is

n-dimensional and the elements u (a), as a varies over the

n-dimensional cones of A , are exactly the vertices of PD.

Exercise. For the toric va r ie ty X = P n w ith its divisors Dq, . . . >Dn>

ve r i fy directly that D = ZajD| is generated by its sections <=* ipj) is

convex <=» ao + . • . + an > 0, and that D is ample «=» is strictly

convex <=* ag + . . . + an > 0 *

Exercise. For X = Fm the Hirzebruch surface, w ith edges generated

by v i = e j, V2 = &2> v 3 = - e i + m e 2 , V4 = - e 2 , show that D = Za^D

is generated by its sections if and only if a2 + a4 > 0 . and a^ + a3 >

m a j . Show that Pic(X) is free on generators A = D* and B = D4 ,

and that aA + bB is ample a and b are positive. V e r i fy that A

is a fiber of the mapping from Fm to P 1, and B is the first Chern

class of the universal quotient bundle of 0 ® 0 (m ) on Fm.

Exercise. Show that any two-dimensional complete toric va r ie ty is projective. Show that any ample divisor on such a va r ie ty is v e ry



LINE BUNDLES 71

ample.(13)

Exercise. (Demazure) If X (A ) is complete and nonsingular, show

that a T-divisor is ample if and only if it is v e ry am ple/14)

These ideas can be used to give simple examples of complete

varieties that are not projective. Form the three-dimensional fan A

whose edges in Z pass through v^ = -e^, V2 = - e 2 , V3 = - e 3 , V4 =

e* + e2 + e3 , V5 = V3 + V4 , V5 = v* + v 4, and V7 = V2 + v 4, and with

cones through the faces of the triangulated tetrahedron shown:

It is easy to ve r i fy that X = X (A ) is complete and nonsingular. The

claim is that X is not projective. Since Pic(X) consists of line

bundles of the form 0(D) for D a T-Cartier divisor, it suffices to show

that no function can be strictly convex.

Exercise. If ip were such a strictly convex function, show that

^ (v * ) + ip(v5 ) > ip(v3) + ip(ve)

i|>(v2 ) + +(vfc) > ijXvi) + 4KV7 )

4Kv3 ) + 4XV7 ) > 4j(v2) + 1MV5 )

which add to a contradiction.

7Exercise, (a) Describe the birational map from this va r ie ty X to IP

determined by this subdivision of the pyramid. In particular, show

that the blowing up occurs over a plane triangle in IP .

(b) Show that the toric va r ie ty obtained by truncating the pyram id

and omitting V4 has a singular point of multiplicity 2 .



72 SECTION 3.4

Exercise. Prove the toric version of Chow's Lemma, that any

complete toric va r ie ty can be dominated birationally by a projective

toric va r ie ty/ 15) Carry this out for the preceding example.

Any complete nonsingular toric var ie ty , such as this example, has

m any nontrivial line bundles, since Pic(X) s Zd-n. We have seen an

example of a singular complete toric va r ie ty that is even farther from

being projective, having no nontrivial line bundles at all/16)

We saw in §1.5 that a convex n-dimensional polytope P w ith

vertices in M determines a fan Ap and a complete toric va r ie ty

Xp = X (Ap). This var ie ty comes equipped with a Cartier divisor D = Dp, whose corresponding convex function iJj = i)j[) is defined by

ip(v) = min <u,v> = min <u,v> = min <Uj,v> ,u € P u € PflM

where the uj are the vertices of P. Equivalently, if aj is the (m a x

imal) cone corresponding to ui# then X u* generates 0(D) on Ua..

Exercise. Show that P q = P, and ve r i fy that D is ample. Show

conversely that if D is an ample T-Cartier divisor on a toric va r ie ty

X (A ), then the fan constructed from PD is A.

Exercise. Given P as above, show that Dp is v e ry ample if and only

if for each vertex u of P, the semigroup generated by the vectors

u' - u, as u1 varies over POM, is saturated.

Exercise. Let P be the polytope w ith vertices at ( 0 ,0 ,0 ), (1,0,0),

( 0 ,0 ,1 ), ( 1 ,1 ,0 ), and ( 0 ,1 ,1 ) in Z3. Find the fan Ap, and show that

D =* Dp is v e ry ample, embedding Xp in P 4 as the cone over a

quadric surface, defined by an equation T1T4 = T2T3 .

Exercise. Let M = Z3, P the polytope w ith vertices ( 0 ,0 ,0 ), ( 0 ,1 ,1 ),

(1,0,1), and (1,1,0). Show that D = Dp is ample but not v e r y ample on X - Xp. Show in fact that (pp: X —» P 3 realizes X as the double

cover of P 3 branched along the four coordinate planes.

Exercise, Let P be an n-dimensional convex polytope w ith vertices

in M, and assume that Dp is v e ry ample, so w e have Xp c P r-1 ,

r = Card(PHM). Let a be the cone in N x Z whose dual a v is the



COHOMOLOGY OF LINE BUNDLES 73

cone over P * 1 in M * Z. Identify the affine toric va r ie ty Ua with

the cone over Xp in Cr . Deduce that Xp C P *"”1 is arithm etica lly

norm al and Cohen-Macaulay; i.e., the homogeneous coordinate ring of

Xp in P r_1 is normal and Cohen-Macaulay/17^

Exercise. Let P = { ( x i , . . . , x n) € IRn : xj > 0 and Xxj < m ). Show

that the corresponding projective toric va r ie ty Xp C P r ~1 is the in

fold Veronese embedding of P n in P r-1 , r = ( nJnm )' Sh°w that the

construction of the preceding exercise gives the cone over this

embedding as the affine toric va r ie ty described in §2 .2 .

More generally, let P be the convex hull of any finite set in M.

We call a complete fan A compatible with P if the function ijjpdefined by ipp(v) = min <u,v> is linear on each cone a in A. Since

U € Pvpp is convex, it determines a T-Cartier 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 X u> as u varies over PflM.

Exercise. Show that the image of the corresponding morphism

(pD: X -* P r_1, is a va r ie ty of dimension k, where k = dim(P). 18^

3.5 Cohom ology of line bundles

Let D be a T-Cartier divisor on a toric va r ie ty X = X (A ), and let

\\t = ijjj) be the corresponding function on |A|. We know that the

sections of 0(D) are a graded module: H°(X,0(D)) = © H °(X ,0 (D ))u,

where

n f C-Xu if u € P n OMH°(X,0(D ))u = \ °

[ 0 otherwise

with PD the polyhedron {u € M jr : u > ^ on |A|). This can be

described in fancier words by defining a closed conical subset Z(u) of

IA| for each u € M:

Z(u) = { v € |A| : <u,v> > ip (v ) ) .

Then u belongs to Pp exactly when Z(u) = |A|, or equivalently,

when the cohomology group H°(|A| ' Z(u)) vanishes, where this H°



74 SECTION 3.5

denotes the 0 th ordinary or sheaf cohomology of the topological space

with complex coefficients. Equivalently, if

H° (u)(IA|) = H°(|A|, |A| ' Z(u))

= Ker(H°(|A|) -* H°(|A| s Z (u )) )

is the 0 th local cohomology group (or relative group of the pair

consisting of |A| and the complement of Z(u)), we have u € PD

exactly when HZ(U)(|A|) is not zero. Therefore

H°(X,0(D)) = ©H°(X ,0 (D ))u , H°(X,0(D ))u = H z (u )(|A|) .

This is the statement that generalizes to the higher sheaf cohomology

groups Hp(X,0(D)) and to the higher local cohomology groups

HZ(u)(IAD = HP(|A|, |A| N Z(u);C).

Proposition. For all p > 0 there are canonical isomorphisms:

Hp(X,0(D)) s ©H p(X,0 (D ))u , Hp(X ,0 (D ))u a H g(u )(|A|) .

These local cohomology groups are often easy to calculate. For

example, if X is affine, so |A| is a cone and iJj is linear, then |A|

and |A| ' Z(u) are both convex, so all higher cohomology vanishes —

which is one of the basic facts about general affine varieties. For toric

varieties, a similar argument gives a stronger result than is usually

true: all h igher cohomology groups of an ample line bundle on a

complete toric varie ty vanish. In fact, more is true:

Corollary. I f IAI is convex and 0(D) is generated by its sections,

then • Hp(X,0(D)) = 0 for all p > 0.

Proof. Since ijj is a convex function, it follows that

|A| ' Z(u) = ( v € |A| : <u,v> < i|j(v ) }

is a convex set, so both |A| and |A| ' Z(u) are convex. This implies

the vanishing of the corresponding cohomology groups.

It follows that for X complete and 0(D) generated by its

sections,




X(X,0 (D )) = S (-1 )Pd im HP(X,0(D))

= dim H°(X,0(D)) = Card(PDnM ) .

With Riemann-Roch formulas available to calculate the Euler

characteristic, this gives an approach to counting lattice points in a

convex polytope. We will come back to this in Chapter 5. Note in

particular the formula for the arithmetic genus:

X (X ,0 x ) = d im H °(X ,0x ) = 1 .

Proof of the proposition. Hp(0(D )) is the p**1 cohomology of the

Cech complex C\ with

CP = ® H ° (u CTon . . . n u CT 0(D))

® ® HZ(u)na0n . . . na_( a on • • • n a p) *U € M CT0 f . . . , CTp U F

the sum over all cones c t q , . . . ,<Jp in A. The boundary maps in the

complex C’ preserve the M-grading, which gives its cohomology the

grading. As before, we have H^(u)n|T|(|T|) = 0 for all cones t and all

u and all i > 0. The proposition then follows from a standard spectral

sequence argument:

Lem m a. Let Z be a closed subspace of a space Y that is a union

of a fin ite n u m b e r of closed subspaces Yj, and T a sheaf on Y

such that H zn Y ^Y '.T ) = 0 for all i > 0 and all Y* = Y j()n ... OYjp .

Then

h ‘z (Y ,T ) = H* (C ' ( { Y j ) , f )) ,

where C "((Y j),7 ) is the complex whose pth te rm is

cp ( {Y j) .T ) = . © . r ZnY n nY (Y j n ... n Y j f ) .J0 Jp F

Proof. Take an in jective resolution 5’ of 7, and look at the double

complex C‘((Yj},3*). The hypothesis implies that the columns are

resolutions of the complex C*({Yj),T). We claim that the rows are

resolutions of the complex F z (Y ,D . Then calculating the cohomology



76 SECTION 3.5

of the total complex two ways (or appealing to the spectral sequence of

a double complex) gives the assertion of the lemma.The exactness of the rows will follow from the fact that the 3q

are injective. To see this, note that if 3 is injective, the sequence

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

is exact for any W. This reduces the assertion to the absolute case, i.e.,

to showing that

o -» h y , 3) -> ® r<Y Jf« e r ( Y j l n Y j2 ,3) -> . . .

is exact. Since 3 is a direct summand of its sheaf 3 of a rb itra ry

("discontinuous") sections, we can replace 3 by 3. But then the

calculation is local at a point y in Y, and the cohomology is that of

the simplex ( j : y € Y j) , with coefficients in the stalk 3y .

Exercise. Let X = IPn, a toric var ie ty w ith its divisors Dq, . . . ,Dn as

usual, and let D = mDo, so 0(D) = 0 (m ). Compute the cohomology as

follows. Show that is zero on the cone generated by v j , . . . , v n,

and is m e* on the cone generated by v q , . . . , v^ . . . , v n.

(a) For m > 0, ve r i fy that is convex. Show directly that

u > ijjp exactly when u = (m j , . . . ,m n) w ith m, > 0 and Z m j < m;

the corresponding X u give a basis for H°(lPn,0(mD)).

(b) For m < 0, show that is concave, so that the sets Z(u)

are convex and unequal to Nr, and H^^dslm) = 0 unless Z(u) = (0).

Use this to ve r i fy that H1(lPn,0(mD)) = 0 for all i * n, and that

HndPn,0 (mD)) = ® C *X Ui the sum over those u = (m j , . . . ,m n) with

mj < 0 and Zm , > m.

Using the same techniques, we have the following important

result, which is also special to toric varieties:

Proposition. Let A ' be a re f in em en t of A , giving a b irational

proper m ap f: X' = X (A ') -* X = X (A ). Then

f * (0 X') = Sx and R ^ tO x ' ) = 0 for all i > 0 .

In particular, taking X ‘ to be a resolution of singularities, this says

that e ve ry toric va r ie ty X has rational singularities.




Proof. The assertion is local, so we can take A to be a cone a

together w ith all of its faces, so X = UCT. The claims are that

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

(ii) HKX'.OxO = 0 for all i > 0 .

The first is a general fact, since X is normal and f is birational; here

it is obvious since both spaces of sections are © C *X U, the sum over

u € cr^PlM. The second follows from the fact that C?x« is generated by

its sections, since the support I A 'I = |a| is convex.



CHAPTER 4

MOMENT M APS AND THE TANGENT BUNDLE

4.1 The m an ifo ld w ith singular corners

Although we are working mainly with complex toric varieties, it is

worth noticing that they are all defined naturally over the integers,

simply by replacing C by 1L in the algebras: Ua = Spec(Z[cj'‘'nM]).

For a field K, the K-valued points of Ua can be described as the

semigroup homomorphisms

Horn sg( a vn M ,K ) ,

where K is the multiplicative semigroup K *U {0 }. For example, for

K = 1R c C, we have the real points of the toric variety .

In fact, the same holds when K is just a sub-semigroup of C.

The important case is the semigroup of nonnegative real numbers

IR> = flR + U{0}, which is a multiplicative sub-semigroup of <L. In this

case there is a retraction given by the absolute value, z •-» |z|:

IR> C C -> IR> .

For any cone a, this determines a closed topological subspace

(U<j)> = Homsg( a vn M , IR>) C UCT = Homsg( a v n M ,C)

together w ith a retraction UCT -» (UCT)>. For any fan A , these fit

together to form a closed subspace X (A )> of X (A ) together w ith a

retraction

X(A )> c X (A ) X (A )> .

For example, if a is generated by vectors e*, . . . ,e^ that

form part of a basis for N, then (Ua)> is isomorphic to a product of

k copies of IR> and n-k copies of IR. Thus if X is nonsingular,

X> is a manifold w ith corners. When X is singular, the singularities

of X> can be a little worse. For the toric va r ie ty X = IPn, w ith its



THE MANIFOLD WITH SINGULAR CORNERS

usual covering by affine open sets U, = UCT., (Uj)> consists of points

(tg: ... :1: ... :tn) w ith tj > 0. Hence

P";> = IR>n + 1 " (0) / IR +

= {(to, . . . ,tn) € IRn + 1 : tj > 0 and to + . . . + tn = 1} ,

which is a standard n-simplex. The retraction from CPn to DPn> is

1(x 0 : . . . : xn ) *-> ( |x0 l, . . • , lxn|) .

The fiber over a point (to, . . . ,tn) is a compact torus of dimension

equal to Card {i: tj * 0} - 1.

The algebraic torus Tn contains the com pact torus Sjsj:

SN = Hom(M ,S*) c Horn (M ,€*) = TN ,

where S1 = U ( l ) is the unit circle in C*. So Sn is a product of n

circles. From the isomorphism of C* w ith S1.* IR+ = S1 x [R (via the

isomorphism of IR+ w ith IR given by the logarithm), we have

Tn = Sn x Hom(M,lR + ) = Sn x Hom(M,IR) = Sn x N(r ,

a product of a compact torus and a vector space.

Proposition. The re trac tion X (A ) —» X (A )> identifies X (A )> 'with the quotient space of X (A ) by the action of the com pact torus Sn-

Proof. Look at the action on the orbits Ox:

(0T)> = X (A )> flOT = Hom(T-LDM,IR + )

= H o m ( T _LnM,DR) = N (t )jr .

From what we just saw, Sn (t ) acts faithfully on Ox = Tn (t ) with

quotient space (Ox)> = N (t )|r. Since Sn acts on Ox by w a y of its



80 SECTION 4.1

projection to Sjsj(T), the conclusion follows.

Note that the fiber of X —» X> over (0T)> can be identified with

S n ( t ) > which is a compact torus of dimension n - d im (x). The spaces

(0T)> fit together in X> in the same combinatorial w a y as the

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

manifold w ith singular corners X>, this can help in understanding

how X is put together topologically/1^

The manifold with corners X> can be described abstractly as a

sort of "dual polyhedron" to A , at least if X is complete: X> has a

vertex for each n-dimensional cone in A; two vertices are joined by

an edge if the corresponding cones have a common (n - l ) - fa ce , and so

on for smaller cones. If A = Ap arises from a convex polytope in M(r,

then X> is homeomorphic to P. In the next section we will see an

explicit realization of this homeomorphism. We list some other simple

properties of this construction, leaving the verifications as exercises:

(1) If r is any positive number, the mapping t »-> t r

determines an automorphism of IR>. This determines an

automorphism of the spaces (UCT)> = Homsg( a vnM,R>), which fit

together to determine a homeomorphism from X> to itself. If r is a

positive integer, z »-> zr is an endomorphism of €, which induces

similarly an endomorphism of any toric va r ie ty X, compatible with the maps X> C X —» X>.

(2) The quotient Tjq/Sjsj = N|r acts on X/Sn = X>, compatibly

with the action of Tjvj on X. The inclusion X> c X is equivariant

with respect to the inclusion Nr = Hom(M,IR + ) c Hom(M,C*) = T^.

(3) There is a canonical mapping Sjsj * X> -* X, which realizes

X as a quotient space.

(4) For any cone t , the inclusion (0T)> C (UT)> is a

deformation retract.

Exercise. Use the Leray spectral sequence for the mapping X —> X>

to reprove the result that the Euler characteristic of X is the number

of n-dimensional cones.



MOMENT M A P 81

4.2 M om en t m ap

Moment maps occur frequently when Lie groups act on varieties/2

Toric varieties provide a large class of concrete examples. In this

section we construct these maps explicitly, and then sketch the

relation to general moment maps.

Let P be a convex polytope in with vertices in M, giving

rise to a toric va r ie ty X = X (A p ) and a morphism cp: X —» IP* - 1 via

the sections X u for u € PnM (see §3.4). Define a m o m e n t m ap

p: X -» M r

by

^ (x ) = ^ V ; V Z IXu(x ) lu .SIX (x)l U€PnM

Note that |jl is S^-invariant, since, for t in Sjsj and x in X,

IXu(t*x)| = IXu( t )M X u(x)l = l%u(x)|. It follows that \i induces a map

on the quotient space X/S^ = X>:

p : X^ - * M r .

Proposition. The m o m e n t m ap defines a hom eom orph ism from

X> onto the poly tope P.

In fact, one gets such a homeomorphism using any subset of the sections X u as long as P is the convex hull of the points, i.e., the

subset contains the vertices of P.

Proof. Let Q be a face of P, and let a be the corresponding cone of

the fan. We claim that in fact p. maps the subset (0 CT)> b ijective ly

onto the relative interior of Q:

(0 ff)> Int(Q) ,

as a real analytic isomorphism.

Let pu(x) = IXu(x )l/X IXU (x)|, where the sum in the denominator is over all u' in PnM (or in a subset containing the

vertices of P). Therefore 0 < pu(x) < 1 and p.(x) = Xpu(x)u. Note

that a point x of (Oa)> is in



82 SECTION 4.2

Hom (ai n M , IR +) c Hornsg(a vnM , 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 .

Writing this out, one is reduced to proving the following assertion:

Lem m a. Let V be a fin ite-dimensional real vec to r space, and let K

be the convex hull of a finite set of vectors uj_f . . . ,u r in the dual

space V*. Assume that K is not contained in a hyperplane. Let

El, . . . ,e r be any positive numbers, and define pp V —> IR by the

formula

p,(x) = Eie ui(x )/ (e iG ul (x) + . . . + er e ur (x>) .

Then the mapping \i: V -» V*, p (x ) = p^(x)ui + . . . +pr (x )u r , defines

a real analytic isomorphism of V onto the in te r io r of K.

This is proved in the appendix to this section.

Exercise. Show that the vertices of the image of the m om ent map

are the images of the points of X fixed by the action of the torus Tn -

The map from Xp to IPr ”1 is compatible w ith the actions of the

torus Tn on Xp and the torus T = (C *)r on P r_1, w ith the map

Tn = Hom (M,£*) -> Hom (Zr ,C*) = T

determined by the map Zr —> M taking the basic vectors to the points

of PflM. The action of the Lie group S = (S 1)1" on IP1""1 determines

a m o m e n t m ap

m : IP1" 1 -» Lie(S)* = (IRr )* = IRr .

If x € P r - 1 is represented by v = (x j , . . . ,x r ) € Cr , then, up to a

scalar factor, this moment map has the formula

m (x ) = — Z W 2 e* .Zlxjl2 i= l

The following exercise shows that this agrees w ith a general

construction of moment maps.



MOMENT M A P 83

Exercise. Define r v : T/S -» IR by r v (t ) = >2 IIt*v ||2. The der ivative of

r v at the origin e of the torus is a linear map

de( r v ) : IRr = Te(T/S) -> IR

i.e., de( r v ) is in (lRr )*. Show that !Hl(x) = ||v||"2 *de( r v ).

The composite

(n <rriXp ------ JPr ~1 -----> (IRr )* -> M r

is then a map from Xp to M r.

Exercise. Show that this composite takes x to jjl(x2), where

x x2 is the map defined in ( 1 ) of the preceding section.

Append ix on c o n v e x i ty

The object is to prove the following elementary fact.

Proposition. Let u*, . . . , ur be points in IRn, not contained in

any aff ine hyperplane, and let K be their convex hull. Let s i , . . . , e r

be any positive real numbers, and define H: IRn —> !Rn by

H(x) - - j ~ £ t ke ( “ " ’x ) uk ,f(x ) k = l k

where f(x ) = e + ... + er e Ur’ x\ and ( , ) is the usual inner

product on IRn. Then H defines a real analytic isomorphism of IRn

onto the in te r io r of K.

We will deduce this from the following two related statements:

(A n) Let u i, . . . ,u r be vectors in IRn that span DRn, and let C

be the cone (w ith vertex at the origin) that they span. Let . . . ,e r

be any positive real numbers. Then the m ap F: IRn —> IRn defined by

F<*> - £ tk . <Uk' ,‘ , ukk=l

determines a real analytic isomorphism of IRn onto the in te r io r of C.



84 SECTION 4.2

(Bn,m) With C c (Rn and F: IRn -> IRn as in (A n), let

tu: IRn -> IRm be a l inear surjection. Then ir°F maps IRn onto

the in te r io r of ti(C), and all of the fibers are connected manifolds

isomorphic to IRn~m.

The statement (A^) is easy, since F is a mapping w ith positive

derivative, and it is easy to compute the limit of F(x) as x -» ±<».

More generally, for arb itrary n, and for m < n, the m atr ix

v aXj 1 < i,j < m

is a positive definite sym m etr ic matrix. Indeed, the value of this

quadratic form on a vector t = (t*, . . . , t m) is

Z ek e (uk>x)- (w k , t ) 2 , k = l

where w^ is the projection of u^ on the first m coordinates. In

particular, the Jacobian determinant of F is nowhere zero, so F is

a local isomorphism.

With this we can ve r i fy the implication (A m) =* (Bn m). A fter

changing coordinates, we m ay assume that it: IRn -» (Rm is the

projection to the first m coordinates. Let p: (Rn IRn“ m be the

projection to the last n -m coordinates. Let G = tt°F, and for each

y € lRn" m, let Gy : p-1(y ) = IRm IRm be the map induced by G.

Since Gy(z) = e^w k’2^Wfc, w ith the positive, and the

projections w^ span IRm, the assertion (A m) implies that each Gy

is a one-to-one map onto the interior of tt(C). It follows that for each

q € Int(iT(C)), the projection from G~*(q) to 0Rn_m induced by p is

one-to-one and onto. By the above Jacobian calculation, it is an

isomorphism of manifolds. This verifies (Bn>m).

We next ve r i fy that F is one-to-one. It suffices to show that its

restriction to a given line is one-to-one. A fter change of coordinates,

this line m a y be taken to be the line with Xj fixed for 2 IR. Since g

is one-to-one, the restriction of F to the line is one-to-one.



DIFFERENTIALS AND THE TANGENT BUNDLE 85

We show next that the image of F contains points arb itrarily

close to any point on an edge of C. If several Uj lie on the same ray

through the origin, they can be grouped in the formula for F, so one

m ay assume no two uj are on the same ray. Suppose the edge passes

through uj_. Since the ray through u^ is an edge, one m ay find a

vector v so that (u i ,v ) = 0 but (u j,v ) < 0 for j > 1. For any vector

w, the limit of F(Av + w ) as A —> +<» is eûl ,w êiui. For appro

priate choice of w, this limit can be placed arbitrarily along the edge.

It follows that once the image of F or 7t°F is known to be

convex, it must contain the whole interior of the cone.

Now we show the implication (Bn n_ i ) =* (A n). By what we have

already proved, it suffices to show that the image of F is convex.

Equivalently, we must show that the intersection of the image of F

with any line is connected or empty. Any line is of the form TT- 1 ( q )

for some projection tt from IRn to 0Rn-1, and some q € DR11" 1. But

F(IRn) fl TT_ 1( q ) = F ((7T°F)- 1(q )) ,

and (TC°F)~*(q) is connected or em pty by (Bn n _i).

This, with an evident induction, completes the proofs of (A n) and

(Bn,m).Finally, we show how the proposition follows from (A n + i ) , by

forming a cone over K in IRn + *. Let

uk = uk x 1 € IRn x IR = (Rn + 1 ,

and define F : IRn+1 -* IRn+1 by F(x,t) = Z e ^ e ^ k ^ e 1 uk. By (A n+i )

F maps IR11* 1 isomorphically onto the interior of the cone C

generated by the uk. The part mapping to Int(K) x 1 consists of those

(x,t) with S ekeûk'xêt - -1, i.e., with t = - log f(x ). So

H(x) = F(x,-log f(x )) maps IRn isomorphically onto the interior of K.

4.3 D iffe ren t ia ls and the tangent bundle

Proposition. I f X is a nonsingular toric va r ie ty , and Di, . . . ,Da

are the irreducible T -divisors on X, then -ZD* is a canonical

divisor; i.e.,



86 SECTION 4.3

Qx s ° x < - Z d,) .i= 1*

Proof. If e i, . . . ,en are a basis for N, let Xj = X e i be the

coordinates corresponding to a dual basis for M, and set

dXx dXn0 0 = /V . . . XV----------------

X i Xn

a rational section of . Another choice of basis for N gives the

same differential form, up to multiplication by ±1. We must show

that the divisor of oo is -XDj. On an open set UCT, we m ay assume

ct is generated by part of a basis, say e*, . . . , e^, so we have

U a = Spec (€ [X 1 XK.XK + i .X k + r 1. . . . . X ^ X ^ 1])

and

±1co = -------------- dXi /v . . . ^ d X n .

x r ....xn

This shows that div(co) and -ZD^ have the same restriction to Ua.

For example, let X be a nonsingular complete surface, w ith

notation as in §2.5, so (Dj-Dj) = -a^ The canonical divisor K = -ZDj

has self-intersection number

(K-K) = KDj-Di) + 2d = -Sa j + 2d = - (3d - 12) + 2d = 12 - d .

Since there are d two-dimensional cones, we also know that

X (X ) = d. Hence

(K-K) + X (X ) (12 - d) + d

----------------------------------------1 5 ---------------------------1 ■ x ( x - 3 x ) •

which is Noether's formula for the surface X. We will use the

Riemann-Roch formula in the next chapter to generalize this to higher

dimensions.

We also need to know the whole sheaf of differential forms

(the cotangent bundle ) not just its top exterior power — at least up to

exact sequences — so we can compute its Chern classes. For this we



SERRE DUALITY

use the locally free sheaf Q^(logD) of differentials with logarithm ic

poles along D = ZD*. At a point x in D^n . . . HD^, w ith x not in

the other divisors, if X*, . . . ,X n are local parameters such that

Xj = 0 is a local equation for Dj, 1 dXnX i x 2 x k

give a basis for Q^(logD) at x . ^

Proposition. (1) There is an exact sequence of sheaves

d0 -» -> Q^(logD) -» ®^0 Di -» 0 ,

where 0Q. is the sheaf of functions on Dj extended by zero to X.

(2) The sheaf Q^(logD) is trivial.

Proof. For (2), consider the canonical map of sheaves

* ^ y ( logD )

that takes u € M to d(%u)/%u. To see that this map is an isomor

phism, it suffices to look locally on affine open sets Ua, where the

assertion follows readily from the above description of Q^(logD).

The second mapping in (1) is the residue mapping, which takes

co = XfjdXj/Xi to ©flip).. The residue is zero precisely when each

fi is divisible by Xj, i.e., when co is a section of Q^. ^

4.4 Serre du a l ity

For a vector bundle E on a nonsingular complete va r ie ty X, Serre

duality gives isomorphisms

Hn_i(X ,E v® Q ^ ) s H !(X , E ) * .

If X is a toric va r ie ty and E = 0(D) for a T-divisor D, this isomorphism respects the grading by M; w ith = 0(-ZDj), it

consists of isomorphisms



88 SECTION 4.4

H n - i (X , e ( - D - ZD i) )_u = (H i (X ,Q (D ))u) "

for each u € M. It is interesting to give a direct proof. Let i|> = ifjp be

the piecewise linear function associated to D, and k =

function for the canonical divisor -ZDj, so k(v|) = 1 for the lattice

point Vj corresponding to each divisor Dj. By the description in §3.5,

h ‘ (X ,0 (D ))u = H*z (N r ) , HJ(X ,0 (-D - ID , ) )_u = h £.(Nr ) .

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 duality amounts to isomorphisms

(SD) Hnz7‘ (N R) b 'H z (N r ) " .

The next three exercises outline a proof. Note that the set

S = { v € N r : k(v) = 1}

is the boundary of a polyhedral ball B, so is a deformation retract of the complement of {0} in N r .

Exercise. If C is a nonempty closed cone in N r , show that there 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 n s n c ) s H n . j . ^ s n c ) ,

the last by Alexander duality, where the ~ denotes reduced

cohomology and homology groups.

Exercise. Show that the embedding SflZ S ' (S flZ1) is a

deformation retract. ^

Exercise. Prove (SD), first in the cases where Z = N r and Z‘ = (0),

or Z‘ = N r and Z = (0); otherwise

h z71(n r ) s Hn-,-1(s ' sn z ' ) £ Hn. j . i ( s ' s n z r



SERRE DU ALITY 89

a H n- j - l ( S n Z ) * s H ^ N r )* . (6)

Exercise. Suppose X = X (A ) is a nonsingular toric va r ie ty , and |A|

is a strongly convex cone in Njr. Show that (i) r ( X , Q x ) = © C *X U,

the sum over all u in M that are positive on all nonzero vectors in

|A|; and (ii) HKXjQ^) = 0 for i > 0 . ^

Grothendieck extended the Serre duality theorem to singular

varieties. For this, the sheaf Qx must be replaced by a dualizing complex oo’x in a derived category. When X is Cohen-Macaulay,

however, this dualizing complex can be replaced by a single dualizing

sheaf cox . For a vector bundle E on a complete n-dimensional

Cohen-Macaulay var ie ty X, Grothendieck duality gives isomorphisms

Hn_i(X, E"®cox ) s H * (X ,E ) * .

For a singular toric va r ie ty X, ED* m ay not be a Cartier divisor

(or even a <Q-Cartier divisor). Nevertheless, it defines a coherent sheaf

0 x ( - 2 Dj), whose local sections are rational functions w ith at least

simple zeros along the divisors Dj. In fact, this is the dualizing sheaf:

Proposition. Let oox = O ^ -Z D j ) .

(a) I f f: X 1 —» X is a resolution of singularities obtained by refin ing

the fan of X, then f * (Q x .) = oox and R !f * (Q Xi) = 0 for i > 0 .

(b) I f X is com plete , and L is a line bundle on X, then

H n i (X , Lv®cox ) s H i(X , L ) * .

Proof. Part (a) is local, so we m ay assume X = Ua for some cone a.

In this case, (a) is precisely what was proved in the preceding exercise.

Then (b) follows; in fact, if E is any vector bundle on X, using Serre

duality for f*(E) on X', we have

Hn i (X ,E ^0 cox ) = Hn_i( X ,E^®Rf*(Qx .))

= H n-i (X ‘ , f*(E )v® Qx .) 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 be the inclusion of the nonsingular locus U

in X. Show that j * ( f i y ) =



90 SECTION 4.4

There is a pretty application of duality to lattice points in

polytopes. If P is an n-dimensional polytope in M r w ith vertices in

M, we have seen that there is a complete toric va r ie ty X and an

ample T-Cartier divisor D on X whose line bundle 0(D) is generated

by its sections, and these sections are linear combinations of %u for u

in PflM. By refining the fan, one m ay take X to be nonsingular, if

desired. Consider the exact sequence

0 O(D-IDi) -> 0(D) -> 0(D)Izd -* 0.

Exercise, (a ) Show that D - ZDj is generated by its sections, and

these sections are

© C.%u . u € Int(P)nM

(b) Deduce that

X (X .0 (D)l5;Di) = h°(X ,0(D )lZDl) = C a rd O P f lM ) ,

where DP is the boundary of P.

Now by Serre-Grothendieck duality,

%(X,0(D - XDj)) = ( - l ) nX (X ,0 (-D )) .

Since the higher cohomology vanishes,

( - l ) nX (X ,0 (-D )) = C arddn t (P )O M ) .

It is a standard fact in algebraic geometry that for any Cartier

divisor D on a complete var ie ty X, the function f: Z Z,

Hv) = X (X ,0 (u D ) ) ,

is a polynomial in v of degree at most n = dim(X), and this degree

is n if D is a m p le .^ In this case, for v > 0, f ( v ) = Card (^ -PnM ).

Putting this all together, we have the

Corollary. I f P is a convex n -d imensional polytope in Mg* with

vertices in M, there is a polynomial fp of degree n such that

Card('U'PnM) = fp(u)

for all integers v > 0 , and



BETTI NUMBERS 91

C ard U n t (u -P )n M ) = ( - l ) n fp(-û)for all v > 0 .

This formula, called the inversion fo rm u la , was conjectured by

Ehrhart, and first proved by Macdonald in 1971. The above proof is

from [Dani].

Exercise. Compute fp(t») for the polytope in M = Z2 w ith vertices

at ( 0 ,0 ), ( 1 ,0 ), and ( l ,b ), for b a positive integer, and ve r i fy the

inversion formula directly in this case.

This can be interpreted by means of the adjunction fo rm u la ,

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

From the exact sequence

0 —> —» Q£<g>0(D) coD - » 0

and the long exact cohomology sequence, we see that

r(D,ooD) = © €• X u ,u e Int(P)nM

HKDjCOq) = 0 for 0 < i < n-1, and dim Hn_1(D,00[)) = 1. Therefore

X(D,Od) = ( - l ) n_1 X(D,ood) = 1 - C a rd d n t (P )n M ) .

This means that the arithmetic genus of D is Card( Int(P ) D M).

For example, if P is the polytope in Z2 w ith vertices at (0,0),

(d,0), and (0,d), then X = IP2, D is a curve of degree d, so its genus

is 1 + 2 + . . . + (d-2) = (d - l ) (d - l )/ 2 . If P is a rectangle in Z2 w ith

vertices at (0,0), (d,0), (0,e), and (d,e), then X = IP1 * IP1, D is a

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

4.5 Betti nu m bers

For a smooth compact var ie ty X, let pj = rank(HJ(X )) be its

j th betti number. When X = X (A ) is a toric var iety , let d^ be the

number of k-dimensional cones in A. In fact, these numbers



92 SECTION 4.5

determine each other:

Proposition. I f X = X (A ) is a nonsingular p ro je c t iv e tor ic v a r ie ty ,

then pj = 0 i f j is odd, and

P2k = Z ( - l ) 1" k ( k ) d n_i .i = k

Set hk = P2k- ^ P x (t ) “ Poincare polynomial, then

p x (t ) = Z h k t2k = Z dn_ j(t2 - 1 ) ' = z dk( t2 - l ) n- k .i = 0 k =0

For example, for the topological Euler characteristic,

X (X ) = X ( - l ) j fJj = Px (-1 ) = dn ,

as we have seen.

Exercise. Invert the above formulas to express the dk in terms of

the betti numbers:

"I -

In fact, one can assign to any complex algebraic va r ie ty X (not

necessarily smooth, compact, or irreducible) a polynomial Px(t)>

called its v ir tua l Po incare polynomial, with the properties:

(1) P x (t) = H rank(HHX))t [ if X is nonsingular and

p ro je c t iv e (o r complete);

(2) P x (t) = P y ( t ) + Y is a closed algebraic subset of

X and U = X n Y.

For example, if U is the complement of r points in IP1, then

P jj (t ) = t2 + 1 - r. (Note 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 polynomials are uniquely

determined by properties (1) and (2). Other properties follow easily

from (1) and (2):

(3) I f X is a disjoint union of a finite n u m b e r of locally closed

subvarieties O(i), then PX(t) = X pQ(j)(t);

(4) I f X = Y x Z, then Px (t ) = P Y (t )-Pz (t).



BETTI NUMBERS 93

Exercise. If X -* Z is a fiber bundle w ith fiber Y that is locally

trivial in the Zariski topology, show that Px (t ) = P y ^ ^ P z ^ ) -

If X is nonsingular and complete, (1) says that P x ( - l ) is the

Euler characteristic X(X). With X, Y, and U as in (2), there is a

long exact sequence

. . . -» h|.u - H^X -> h[.y - h|.+1u -» h|.+1x

where H* denotes cohomology with compact supports, and rational

coefficients. It follows from this that P x ( - l ) must a lways be the

Euler characteristic with com pact support, i.e.,

P x ( - l ) = Xc(X) = Z ( - l ) i dim(Hj.X) .

The existence of such polynomials follows from the existence of a

mixed Hodge structure on these cohomology groups/11 This gives a

weight filtration on these vector spaces, compatible with the maps in

the long exact sequence, such that the induced sequence of the m th

graded pieces remains exact for all m:

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

This means that the corresponding Euler characteristic

X ” (X ) = Z ( - l ) i d im (g r ^ (H 1cX ))

is also additive in the sense of (2). If X is nonsingular and projective,

then g r$ (H ™ X ) = H™X = Hm(X), so % ™ (X) = ( - l ) mdirn(Hm(X)).

Hence we (must) set

P X(t ) = Z ( - l ) m X ^ ( X ) t m - Z ( - l ) i + m dirn(gr™(H|: X ) ) t m .m i , rn

One need not know anything about the mixed Hodge structures or

the weight filtration to use these virtual polynomials to calculate betti

numbers; one has only to use the basic properties that determine

them. For example, for a torus T = (C*)^, we have P j ( t ) = ( t 2 - l ) k.

Hence if X = X (A ) is an arb itrary toric var iety , since it is a disjoint



94 SECTION 4.5

union of its orbits 0T = TM(T)» by property (4) we have

^XCA)^) ~ 21 P qt M = Z d n_k ( t2 - l ) k

where dp denotes the number of cones of dimension p in A. This is

true for any toric variety. In case X (A ) is nonsingular and complete,

however, this is the ordinary Poincare polynomial by property (1), and

this proves the proposition.

In fact, the proposition is also true when A is only simplicial and

complete. For this one needs to know that gr^J (H™X) = Hm(X) also in

this case. This follows from the combination of two facts: (i) the

intersection (co)homology groups IHm(X) of an arb itrary compact

va r ie ty have a mixed Hodge structure of pure weight m; (ii) if X is

a V-manifold, then Hm(X) = IHm(X ) . (12)

A toric va r ie ty X is defined over the integers, so can be reduced

modulo all primes. Since X is a disjoint union of orbits 0T, and the

number of Fq-valued points of the torus Tn (t ) = (Gm)k is ( q - l ) k, it

follows that

When X is nonsingular and projective, Deligne’s solution of the Weil

conjectures implies that

where the Xjj are uniquely determined complex numbers w ith

Exercise. Use the preceding two formulas to give another proof of the

proposition.

We will give a third proof of the proposition in Chapter 5.Formula ( * ) implies that for an arb itrary toric va r ie ty X = X (A ),

the Euler characteristic with compact support X C(X) = P x ( “ l ) is equal

to the number of n-dimensional cones in A. We have seen earlier

n

1 = 0 k = iCard (X (F q)) = Z d _ . k ( q - l ) k

k = 0

2n PjCard (X (FDr ) ) = Z ( - l ) j Z X,r, ,

p j =0 i = 0 J1



BETTI NUMBERS

that the ord inary Euler characteristic %(X) is also equal to the

number of these cones. This raises the question of w hether this is

special to toric varieties, or is true for all varieties.

Exercise. Show that X (X ) = X C(X) for eve ry complex algebraic

variety . Equivalently, show that X (X ) = X (Y ) + X (U ) w henever Y

a closed algebraic subset in a va r ie ty X w ith complement U. If N

a classical neighborhood of Y in X such that %(Y) = X (N ), show

that X(N n Y ) = 0 . (13)



CHAPTER 5

INTERSECTION THEORY

5.1 Chow groups

In this chapter we will work out some of the basic facts about

intersections on a toric variety. On any va r ie ty X, the Chow group

A^(X) is defined to be the free abelian group on the k-dimensional

irreducible closed subvarieties of X, modulo the subgroup generated

by the cycles of the form [d iv ( f )], where f is a nonzero rational

function on a (k+l)-d imensional subvariety of X. We have seen that

on an arb itrary toric va r ie ty X the toric divisors generate the group

A n- i (X ) of Weil divisors modulo rational equivalence. The obvious

generalization is valid as w e l l : ^

Proposition. The Chow group A^(X) of an a rb i t ra ry toric va r ie ty

X = X (A ) is generated by the classes of the orbit closures V(<j) of

the cones o of dimension n-k of A.

Proof. Let X* C X be the union of all V(cr) for all cr of dimension

at least n-i. This gives a filtration X = Xn D Xn_i D . . . D X_i = 0

by closed subschemes (say w ith reduced structure). The complement

of X j_ i in Xi is the disjoint union of orbits 0 CT, as a varies over

the cones of dimension n-i. From the exact sequence relating a closed

subscheme and its complement, we have

A k(X i_ ! ) -» A k(Xj) -> © A k(Off) -» 0 .dim a = n - i

Since a torus Oa is an open subset of affine space A 1, we see by

the same principle that Ai(Oa) = Z-[Oa] and A^(Oa) = 0 for k * i.

Since the restriction from A^CX*) to Ak (0CT) maps [V(cr)] to [0CT],

a simple induction shows that the classes [V(a)J, d im (a ) = n-k,

generate A^(Xj).

For a Cartier divisor D on a va r ie ty X, the support of D is



CHOW GROUPS 97

the union of the codimension one subvarieties W such that ordvy(D)

is not zero. We say that D meets an irreducible subvariety V

properly if V is not contained in the support of D. In this case one can define an intersection cycle D*V by restricting D to V (i.e., by

restricting local defining equations), determining a Cartier divisor D|y

on V, and taking the Weil divisor of this Cartier divisor: D«V = [Dly]-

Let us work this out when X is a toric var iety , D = XajDj is a T-

Cartier divisor, and V = V (a ). In this case, D lv (a) a T-Cartier

divisor on the toric va r ie ty V ( ct), so we will have

D.V(cr) = Z byV(y) ,

the sum over all cones y containing a w ith dim(y) = d im (a ) + 1 ,

and the by are certain integers. To compute the multiplicity by,

suppose y is spanned by cr and a finite set of m inimal edge vectors

v^ i € Iy. Here is an example where Iy has three vectors:

Let e be the generator of the one-dimensional lattice N y / N a such

that the image of each v* in Ny/Ng is a positive multiple of e, and

let sj be the integers such that vj maps to sê in N y / N a . Then b y

is given by the formula

a ib y = for all i in I y .

si

To see this, for any cone y containing ct let u(y) € M /M (y ) be thelinear function on y corresponding to the divisor D. The assumptionthat V(cr) is not contained in the support of D translates to the condition that u(y) vanishes on ct, which means that u(y) is in M(cr)/M(y). As y varies over cones in the star of cr, these u(y)

determine the divisor D|v(<j)« In particular, when ct is a facet of y, the multiplicity b y is -<u(y),e>. Therefore



a* = -<u(y),Vj> = - <u(y), Sj-e> = Sj ( - <u(y) ,e>) = spby ,

as asserted.

When X is nonsingular, there is only one i = i(y) in I y , 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

Dk-V(cr)v ( y ) if a and v k span a cone y

0 if ct and v k do not span a cone in A

In fact, if X (A ) is nonsingular, Dk and V(<j) meet transversally in

v ( y ) in the first case, and they are disjoint in the second case.

Exercise. I f y is simplicial, so there is one i = i(y) in I y , show that

mult(cr) (2)b v a i (v ) m u l t ( V ) ■

In general, if a subvariety V is contained in the support of a Cartier divisor D, the intersection D-V is defined only up to rational equivalence on V; i.e., D*V is in Am_i(V), where m = dim(V). This intersection class can be defined by finding a Cartier divisor E on V whose line bundle *s isomorphic to the restriction of 0x(D) toV, and setting D-V to be the rational equivalence class of [E]. If f is a rational function on X such that V is not contained in the support of D' = D + div(f), then D-V is represented by the cycle D'*V defined as above in the case of proper intersection/3 For toric varieties, with D and V(ct) as before, but with V (a ) contained in the support of D, we can take some u in M so that D' = D + div(%u) meets V (a )

properly. In fact, if u(a) is the linear function in M/M(a) defining

D on a, any u in M that maps to u ( ct) in M /M(a) will do. Then D«V(a) is rationally equivalent to D '-V Xcj ) , which is calculated as above. In the simplicial case, we get

D-V(ct) ~ D'-V(ar) = Z ( a . ^ x ----------\ + ) V(y) ,m u l t ( y ) u r i

where the sum is over all y spanned by a and one of the vectors

Vi(v)*



CHOW GROUPS 99

If V is a complete curve, the cycle or cycle class D*V has a well

defined degree, which is denoted (D*V). The following exercise is the

generalization to arb itrary dimension of the facts about intersecting

divisors on a nonsingular surface that were seen in Chapters 1 and 2.

Exercise. Suppose an (n-l)-d im ensional cone a is the common face

of two nonsingular n-dimensional cones y' and y". Let vj_, . . . , v n_i

be the minimal lattice points on the edges of a, ai d let v ‘ and v "

be the m inimal lattice points on the other edges of y' and y"

respectively.

(a) Show that there are (unique) integers a*, . . . , an_i such that

v 1 + v " = aj/Vi + a2*V2 + . . . + &n- l ' v n - l •

(b) Show that, for 1 < k < n-1,

Dk -V (a ) ~ ck‘ V(y ') + ck" V (y ") ,

where ck‘ and ck“ are some integers whose sum is - a k.

(c) Deduce that the intersection number is

(Dk-V (a ) ) = (Dr D2- • ■ • -Dk2- . . . -Dn-!) = - a k .

(d) Show that, in fact, V(cj) = IP1, and <?(Dk)|y (CT) = 0 ( - a k).

Exercise. If X is nonsingular and complete, show that a T-divisor

D is ample if and only if (D*V (a )) > 0 for all cones cr of dimension

n - l . ( 4>

On a nonsingular n-dimensional va r ie ty X, one sets A P(X) =

A n_p(X). There is an intersection product A p(X )® A q(X) A p+q(X),

making A*(X ) = 0 A p(X) into a commutative, graded ring. For

subvarieties V and W that meet properly, i.e., each component of

their intersection has codimension equal to the sum of the codimen

sions of V and W, one can define an intersection cycle V*W by

putting appropriate multiplicities in front of each component of the

intersection; these multiplicities are one when the intersections are

transversal. In general, one has only a rational equivalence class/5^

For a general toric va r ie ty X (A ), if a and t are cones in A , then



1 0 0 SECTION 5 . 1

V ( ct) O V ( t ) =V(y) if a and t span the cone y

0 if a and t do not span a cone in A

This follows, even sch em e-th eore tica lly , from the description of these varieties in §3.1. This intersection is proper exactly w hen the

dimension of Y is the sum of the dimensions of a and t . If A is

nonsingular and the intersection is proper (or em p ty), then V (a ) and

V ( t ) m eet transversa lly in V(y) (or 0 ), so V ( c t ) ' V ( t ) = V(y) (or 0). A lternatively , if a has (m inim al) generators v ^ , . . . , v ^ , then V(cr) is the tran sv ersa l intersection of D^, . . . and these form ulasfollow from the case of divisors considered above.

If A is only simplicial, one has in tersections of cycles or cycle classes only with rational coefficients. The Chow group

A * ( X ) q = 0 A p (X ) q = ® A n_p(X)®<Q

has the s tru c tu re of a graded <Q-algebra. This is a general fact for any v a r ie ty th a t is locally a quotient of a manifold by a finite g r o u p .^ In case a and t span y, with dim(y) = d im (a) + d im (x), then

m u l t ( a ) - m u l t ( T )V ( c t ) - V ( t ) - ------------------------------- v(y) .

m u l t ( y )

If a and t are contained in no cone of A, then V ( ct)*V(t ) = 0. As

in the nonsingular case, this follows from the description of the intersection of the divisors, using the fact th a t some positive multiple of each Dir is a Cartier divisor.

One can often m ake calculations on singular varieties by resolving singularities. Any proper morphism f: X' -* X d eterm ines a push-

fo rward m ap f*: A^(X') —» A^(X), th a t takes the class of a v a r ie ty

V to the class of deg(V/f(V)M(V) if f(V) has the sam e dimension

as V, and to 0 otherwise. In the toric case, if A' is a re f in e m en t of A, we h ave a proper birational m ap f: X(A') -> X(A). We h ave seen th a t if a 1 is a cone in A' th a t is contained in a cone a of A of the sam e dimension, then f m aps V(<j‘) b irationally onto V (a ) , so on cycles we h ave f*[V(a')] = [V(a)]. If a ' is not contained in a n y cone of A of the sam e dimension, then d im (f(V (a ') ) ) < d im (V (a ')) , and f jV ( a ' ) ] = 0.



COHOMOLOGY OF NONSINGULAR VARIETIES 101

5.2 Cohom ology of nonsingu lar va r ie t ie s

Let X = X (A ) be a nonsingular (or at least simplicial) complete toric

variety. We have seen three proofs that

%(X) = m ,

where m = dn is the number of n-dimensional cones of X; in fact,

we know formulas for the betti numbers of X in terms of the

numbers of cones of various dimensions. On the other hand, w e have

potential generators for the homology groups, but w ith lots of relations,

by using the orbit closures corresponding to cones of all dimensions.

What we need is a w ay to associate to each n-dimensional cone a

a subcone t , s o that the resulting m varieties V ( t ) give a basis for

H*(X). ^ We will start with the assumption that X is nonsingular,

and then point out the modifications needed when A is only

simplicial.

For any ordering o^, . . . , a m of the top-dimensional cones,

define a sequence of subcones i j C ctj, 1 i) and that meet cr* in a cone of dimension n-1. In other

words, Tj is the intersection of those walls of cr* that are in ter

sections w ith n-cones larger in the ordering. In particular, t^ = (0),

and T m = crm. The key assumption that will make this work is:

( * ) I f Tj is contained in cij, then i < j.

For the following two-dimensional fan, the first two orderings satisfy

( * ) , while the third does not:

Lem m a. I f X is p ro je c t iv e and A simplicial, then the n-

dimensional cones of A can be ordered so that ( * ) holds.



102 SECTION 5.2

Proof. Take a v e ry ample divisor, corresponding to a strictly convex

function and for each n-dimensional cone cr let u(cr) € M be the

element defining the restriction of c|> to a. These u (a ) are the

vertices of a polytope P in M jr that describes the sections of the

corresponding line bundle, and the idea is to choose a height function

on M so that these vertices have different heights, and to order the

a by the value of the height function. That is, choose some v € N so

that the m values are distinct, and order the ex's so that

<u(cr i) ,V> < <u(<T2 ) ,V> < . . . < <u(<7m ),V> .

Let uj = u(cji). Consider a fixed CTj. Using the correspondence between

cones in A and faces of P, t * is the cone corresponding to the

smallest face of P that contains uj and all edges ef P that connect

Ui to vertices uj with j > i. The claim ( * ) follows from the fact that

this face contains no uj with j < i.

Exercise. For the non-projective complete toric varieties considered

in §3.4, find an ordering of the top-dimensional cones that verifies

assumption (* ) . Is there an example where ( * ) is impossible to

achieve? ^

Theorem. If ( * ) holds and X is nonsingular, then the classes

[V(Ti)l fo rm a basis for A *(X ) = H*(X;Z).

The proof depends on a simple consequence of (* ) :

Lem m a, (a) For each cone y in A there is a unique i = i(y)

such that Tj C Y C In fact, i(Y) is the smallest in teger i such

that <7 i contains y.

(b) I f Y is a face of y‘, then i(y) < i(y').

Proof. For the uniqueness in (a), if tj C y C <jj and Tj C y C cjj,

then Tj C cr j and Tj C ct*. Then ( * ) implies that i = j. For the

existence, given Y, let i be minimal such that y C cr*. Write y as

the intersection of some of the (n -l)-d im ensional faces of aj; since

these are all intersections of cri w ith some other n-dimensional cones,

and by assumption these must come later in the sequence, y must

contain t j , as desired. Assertion (b) follows from the second




statement in (a).

Now, for 1 < i < m, set

Y, = U o* = v(Tj) n ua. ,T jc y cC T j 1

and

Z\ = Yj U Yj+ i U . . . U Y m .

Lem m a. (1) Each Zx is closed, Z\ - X, and Zx ' Zi+i = Y [.

(2) I f X is nonsingular, then Y { s Cn_ki, where k* = dimCxj).

Proof. It follows from (a) of the preceding lemma that X is a

disjoint union of Y^, . . . , Y m. Since the closure of Oy is the union of

all Oy« for y' containing y, the fact that Z x is closed follows from

(b) of the lemma, and the rest of assertion (1) is clear. Since

V (T j )n Uff. is an affine open set in the toric va r ie ty V (T j) corres

ponding to a maximal (n-kj)-dimensional cone in NCtj), ( 2 ) follows

from what we saw in §2 .1 .

We can now prove the theorem, showing by descending induction

on i that the canonical map A*(Zj) —» H^(Zj) is an isomorphism, and

that the classes of the closures Yj = V (t j ), j > i, form a basis. We

have a com m utative diagram with exact rows:

A p(Zi+1) -> Ap(Zj) - A p(Yj) -> 0

i i- H2p.l<Yi> - H2p«Zi* 1) - H2p<Z,) - H2p(Y,) - H2p_1(Z,»i) ->

where the second row is the long exact sequence of B ore l-M oore homology, i.e., homology w ith locally finite chains. Since Yj is an

affine space, A^CYj) = H2* (Y i) s Z, generated by the class of Yj. By

induction, the first vertical map is an isomorphism and Hq(Zi+i ) = 0

for q odd. A diagram chase shows that the middle vertical a rrow is

an isomorphism, and that A*(Zj) is free on the classes fV (x j) ], for

j * i.

If A is only simplicial, all of the above argument is valid, provided rational coefficients are used in place of integers. The



104 SECTION 5.2

n — kdifference is that now each Yj is a quotient € i/G* of affine space

by a finite group, and for such a quotient, the Chow groups and Borel-

Moore homology are the subspaces invariant by the group:

A*(Cr /G)q = (A * (C r )Q)G , H*«Cr /G;<Q) = (H*(Cr ;<Q))G .

Theorem . //(* ) holds and A is simplicial, then the classes [V (x j) ]

fo rm a basis for the vector space A *(X )q = H*(X;Q).

Corollary. With the same assumptions,

d P = n _ h n ' k ’

where dp is the n u m b er of p-d imensional cones in A and hj is

the rank of A*(X) (or H2 i(X)). Equivalently,

hk = Zi =k

Proof. The dimension of A n_k(X)Q is the number of cones Tj of dimension k. The number of p-dimensional cones y w ith Tj C y C cr j

is ( n - p ) » an^, by (a) of the second lemma, each cone occurs exactly

once in this way.

It follows from the fact that X is locally a quotient of a manifold

by a finite group that Poincare duality is valid:

H‘(X;Q) ® H2n~‘(X;C) -> H2n(X;Q) = <Q

is a perfect pairing. Since HKXjQ) is dual to Hj(X,(Q), we deduce

Corollary 1. For A simplicial, h^ = hn_k fo r all 0 < k < n.

In our situation we can see this explicitly as follows. With

a l> • • ■ » a m anc corresponding t j , . . . ,T m as above, let T j ‘ be the intersection of <7 j with all a j such that j is less than i and

dim(ajfiCTj) = n - 1 .

Exercise. S h o w t h a t V ( t i ' ) , . . . , V ( T m ‘) g i v e a b a s i s f o r A*(X )q.

F r o m t h e f a c t t h a t d i m ( T j ' ) + d i m ( T j ) = n f o r al l i, d e d u c e t h a t

h k = h n _k. U s e t h e i n t e r s e c t i o n s of t h e v a r i e t i e s V ( t j ) a n d V ( T j ' ) to




show directly that the intersection pairings A^ (X )q ® A n_^(X)(Q -> (Q

are perfect pairings.

The fact that X is an orbifold (or V-manifold), i.e., locally a

quotient of a manifold by a finite group, has another important

consequence: both the cohomology and homology (w ith rational

coefficients) coincide with the intersection homology IH*(X;<Q). That

is, the canonical maps

Hi(X;(Q) -> H2n-i(X;<Q)

are isomorphisms. It has been established — first using Deligne's proof

of the Weil conjectures, then by Saito's putting a pure Hodge structure on the intersection homology groups — that for any projective var ie ty

X, the intersection homology satisfies the hard Lefschetz theorem : if

co in H2(X ) is the class of a hyperplane section, the maps

IHn_i(X;<Q) IHn + i(X;Q)

are all isomorphisms/9 This implies in particular that multiplication

by oo is in jective from IH* to IHi+2 for i < n. For a V-manifold,

this implies that the rank of H2p-2(X) is no more than the rank of

H2p(X) for 2p < n, which gives

Corollary 2. For A simplicialf hp_i < hp for 1 < p < [ / y ] .

It would be interesting to prove hard Lefschetz for projective

orbifolds without appealing to the deeper theorem for intersection

homology. Even for toric varieties, proving this fact has been a

challenge. If A is not simplicial, the homology and cohomology of

X (A ) can be much more complicated, and in fact need not be

combinatorial invariants of the fan. However, the intersection

homology of an arb itrary toric var ie ty does have a such a description,

which gives relations between the ranks of the intersection homology

and the numbers of cones of each dimension. Then hard Lefschetz gives

inequalities among these numbers/10^

Exercise. Let X = X(A), where A is the cone over the faces of the

cube w ith vertices at ( ± 1 , ± 1 , ± 1 ), and lattice generated by the

vertices. Show that A0(X) = H0(X) = Z, A3(X) = H6(X) = Z, and



106 SECTION 5.2

A i (X ) = H2(X ) s Z, A 2(X ) = H4 (X) £ Z®5, H3(X ) £ Z®2 , (11)

Finally, we want to describe the intersection ring A*X = H*X of

a nonsingular projective toric va r ie ty X = X (A ). Let Dj_, . . . ,D j be

the irreducible T-divisors, corresponding to the m inimal lattice points

v l> • • • »v d a l°ng the edges. By what we saw in the last section, if a

cone ct is generated by V j , . . . » v ^ , then V(ct) is the transversal

intersection of D^, . . . »Dik- Since the varieties V(ct) span the

cohomology, this means that A*X is generated as ^ Z-algebra by the

classes of D*, . . . ,D^. We have seen that = 0 in A^X if

v i ^, . . . , V i do not generate a cone, since the intersection of the

divisors is empty. In addition, if u is any element of M, the divisor

of the rational function X u on X is 21 <u,Vj>Dp and this must

vanish in A*X by definition.

Proposition. For a nonsingular p ro je c t ive toric va r ie ty X,

A*X = H*X = Z[Dj_, . . . ,Dd]/I, Where I is the ideal generated by all

(i) Dj^* ... for v ^ , . . . , Vj^ not in a cone of A ;

d(ii) 21 <u,Vj>Di for u in M .

i = l

In fact in (i) it suffices to include only the sets of Vj without repeats,

and in (ii) one needs only those u from a basis of M.

Proof. Let A ’ = ZtD*, . . . ,Dd]/I, w ith D*, . . . ,D<j regarded as

indeterminates and I generated by the elements in (i) and (ii). By the

discussion before the proposition, there is a canonical surjection from

the ring A # to A*X that takes Dj to the class of the corresponding

divisor. For each k-dimensional cone a, let p (a ) be the monomial

D^- . . . *Dj^ in A*, where v ^ , . . . , are *he generators of ct.

Choose an ordering of the n-dimensional cones ct , . . . , crm satisfying

( * ) , w ith t i , . . . , T m as before. To complete the pôof, i.e., to show

that A ’ -> A*X is injective, it suffices to show that p (x i ) , . . . ,p (T m)

generate A* as a Z-module. For this we need an "algebraic moving

lemma:"




Lem m a . Let oc < Y < p be cones in A , and let k = dim(y). Then

there is an equation p(y) = in A\ with Yj cones of

dimension k in A with a < Yj but Yi £ P, and mj integers.

Geometrically, V (p) C V (y ) C V (a ) , and this says we can m ove V (y )

off V (p ) by a rational equivalence, while staying inside V(oc).

Proof. Renumbering v^, . . . fv$, we m ay assume that oc is

generated by v*, . . . , v p, Y by v*, . . . , v k, and p by v 1? . . . , v q,

and that v^, . . . , v n form a basis for N, w ith l = 1 and <u,Vj> = 0 for i < n, i * k.

Relation (ii) then gives an equation Dk = an + l^n + l + • • • + ad^d» multiplying by D i- . . .*D k_i, we have

dDr . . . -Dk = Y. ajCDi- . . . -DK-ihDi .

i= n + 1

Using (i) to throw away any terms for which v^, . . . , v k_i, vj do not

generate a cone, this is the required equation.

Next we show that A ’ is additively generated by monomials in

Dj_, . . . ,Dd without multiple factors, i.e., by the elements p(y) as y

varies over all cones in A. By induction it suffices, if Vj € y, to write

Dj-p(y) as a linear combination of such monomials. Use the lem m a to

write Dj = SajDj, a sum over i such that Vj i y. Then Dj*p(y) =

ZaiDppCy) has the required form.

Finally, to see that the p(xj) generate A', by descending

induction on i we show that if y lies between t j and dj, then p(y) is in the submodule generated by those p ( t j ) w ith j > i. If Y = t j

we are done, and if not we use the lemma to write p(y) = X m tp(Yt), a

sum over cones Yt w ith t* c yt <£ dj. By ( * ) , such Yt must lie

between Tj and d j for some j > i, and the inductive hypothesis

concludes the proof of the proposition.

If A is only simplicial, the same holds, w ith essentially the same

proof, but w ith rational coefficients:

(Q[Di, . . . ,D d] / I A*(X)<q H*(X;<Q) ,



108 SECTION 5.3

with I generated by (i) and (ii). The difference in this case is that if

t is generated by v* , . . . ,Vi , then Dj • . . . = V (t ).1 K i k m ult (T )

The proposition remains true for arb itrary complete and non

singular (or simplicial) toric varieties. The proof given here works

if the maximal cones can be ordered to satisfy the assumption (* ) .

Danilov proved it in general by showing that the ring IXDj., . . . ,Dd]/J,

where J is the ideal generated by the relations (i), is a Gorenstein

ring/12) If X arises from a polytope, this ring is called the S tanley -

Reisner ring of the polytope.

5.3 R iem an n -R och theorem

We begin by summarizing some general intersection theory. The

operation of intersecting w ith divisors determines, for any line bundle

L on any var ie ty X, first Chern class homomorphisms

Ak/X) A k_ i(X ) , a c i (L ) ^ a ,

defined by the formula c i (L ) ^ [V] = [E], for V a k-dimensional

subvariety, where E is a Cartier divisor on V whose line bundle

G y ® is isomorphic to the restriction of L to V. From this one can

construct, for an arb itrary vector bundle E on X, Chern class

homomorphisms a Ci(E) a from A^(X) to A^_i(X), satisfying

formulas analogous to those in topology. From these one can define

the Chern character ch(E) and the Todd class td(E). These charac

teristic classes are contravariant for arb itrary maps of varieties. For

a line bundle L, these are given by the formulas

ch(L) = exp (c i (L ) ) = 1 + c i (L ) + . . . + ( l/ n ! )c i (L )n ,

n = dim(X); and td(L) = c i (L )/ ( l - exp (-c i (L ) ) ) = 1 + L) + . . . .

For general bundles, they are determined by requiring them to be

multiplicative, i.e., for any short exact sequence 0 —» E' -* E E" 0

of bundles, ch(E) = ch(E')-ch(E"), and td(E) = td(E ')-td (E").(13)

Every va r ie ty X has a "homology Todd class" Td(X) in A*(X)<q,

(or in Borel-Moore homology H*(X;(Q)), which has the form



RIEM AN N- ROCH THEOREM 109

Td(X) = TdnX + Tdn_iX + . . . + Td0X ,

with Td^CX) € A^(X )q. The top class TdnX = [X] is the fundamental

class of X. If X is nonsingular, then

Td(X) = td(Tx ) - [X] ,

where td(Tx) is the Todd class of the tangent bundle Tx of X. If

f: X' —> X is a proper birational morphism, with f* (Ox') = Ox anc*

R^COx') = 0 for i > 0, then f* (Td(X ‘)) = Td(X). The H irzebruch -

R iem ann-R och formula says that for any vector bundle E on a

complete var ie ty X,

X (X ,E ) = J c h (E ) -T d (X ) ,

where the integral sign means to take the degree of the zero

dimensional piece of the term following. For a line bundle L, it says

X (X ,L) = Z - j - d e g ree (c i (L )k ^ Td^(X)) . k =0 k!

For L trivial, this says that %(X,Ox) = degree(Tdo(X)). If X is

nonsingular, X (X ,L ) is the degree of the top-codimensional piece of

ch (L )^ td(Tx)-When X is a toric va r ie ty with divisors Dj_, . . . ,D j as before,

the proposition at the end of §4.3 shows that the Chern classes of the

cotangent bundle Q x and the structure sheaves Op. are related by

, dc (Q y ) • TT c (0 p.) = 1 .

i = 1 1

Here c = 1 + c + C2 + . . . + cn is the total Chern class. From the

sequence 0 ©(-Dj) -» Ox 0 , we have c (0 p.) = (1 - D i)"1.

Combining, and using the fact that cj(Ev) = (- lVcÊ ), we get

Lem m a. The total Chern class of a nonsingular tor ic va r ie ty is

dc(Tx) = TT (1 + Di) = Z [V(ct)I .

i -1 a € A

By the definition of the Todd class of a bundle, we have similarly



110 SECTION 5.3

d Djtd(TX) = TT ---------i = l 1 - exp(-Dj)

= 1 + ^ -C1(X) + ^ ( C1(X )2 + C2(X )) + ^ Cl(X )-c2(X ) + . . . .

Here Cj(X) = c^Ty)- (Note, however, that the Dj are not Chern roots

of Ty> since there are more than n of them.) If X (A ) is singular,

we can compute Td(X) by subdividing to find a proper birational

morphism f: X (A ') —> X (A ) from a nonsingular toric var iety . From

the last proposition in §3.5 it follows that Td (X (A )) = f * (T d (X (A 1))),

which gives a method for calculating the Todd class in any given

example.

Exercise. For any toric va r ie ty X, show that Tdn_ i (X ) =

and Tdo(X) = [x] for any point x in X.

We will apply the Hirzebruch-Riemann-Roch formula to the line

bundle 0(D) of a T-Cartier divisor D on X (A ) that is generated by

its sections. As we saw in §3.4, the higher cohomology groups of such a

line bundle vanish, and the dimension of the space of sections is the

number of lattice points in a certain convex polytope P = Pp w ith

vertices in the lattice M. Denote the number of lattice points that are

in P by # (P ), i.e., # (P ) = Card(POM). By Riemann-Roch, we

therefore have a formula for the number of such lattice points:

n 1(RR) # (P ) = Z — degree(Dk ^ Td^(X)) .

k = 0 k!

Note that any bounded convex polytope P w ith vertices in M

arises in this way. Given P, we m ay find a complete fan A that is

compatible w ith P, so there is a T-Cartier divisor D on X = X (A )

such that 0(D) is generated by its sections, and so that these sections

are precisely the linear combinations of the functions X u for u in

PHM. Moreover, by subdividing A if necessary, we m ay assume that

X (A ) is smooth and projective, although this is rare ly necessary.For any nonnegative integer v, let v*P = {^*u : u € P). From



R I E M A N N - R O C H THEOREM 111

(RR) and the fact that v P is the polytope corresponding to the divisor

t»*D we deduce that v is a polynomial function of v of

degree at most n:

Let fp(u) be this polynomial. Note that fp(t») = X(X,0 (t»-D )) for all

v € Z. Since Tdn(X) = [X], the leading coefficient is an = (Dn)/n!,

where (Dn) denotes the self-intersection number obtained by

intersecting D with itself n times.

The lattice M determines a volume element on M[r, by

requiring that the volume of the unit cube determined by a basis

of M is 1. Denote the volume of a polytope P by Vol(P). The only

fact we will need is the basic and elem entary identity that relates

the volume to the number of lattice points:

# ( v - P )( * ) Vol(P ) = lim ----- — .

V - » OO v

This follows from the fact that the error term in estimating the

volume by using unit cubes centered at lattice points of a polytope

is bounded by the (n-l)-d imensional area of the boundary of the

This follows from the fact that the error term in estimating the

volume by using unit cubes centered at lattice points of a polytope

is bounded by the (n-l)-d imensional area of the boundary of the polytope, and this vanishes in the limit.

Applying (RR), we see that # (v *P )/ v n —» (Dn)/n! as u —» oo,

where (Dn) is the self-intersection number. Therefore,

Corollary. If the polytope P corresponds to the divisor D, then

The divisor D is described by a collection of elements u (a ) in

M/M(a), one for each cone a. For any cone a in A , let P a be the

intersection of P with the corresponding translation of the subspace

perpendicular to cr:

* ( v - P ) = X ( X , 0 ( v D ) ) = Z ak v kk =0

ak = Y j degree(Dk - T d k(X ) ) .

Vol(P )( D-■ . . -D)

n!(Dn)

n!

P = P f l f o H u ( ct) ) ;



112 SECTION 5.3

i.e., PCT = P H ( a x + u) for any u in M that maps to u(cr) in

M /M (ct). The lattice M (a ), translated by u (a ), determines a volume

element on the space a x + u (a), whose dimension is the codimension

of a, or the (complex) dimension of V (a ). We have

Corollary. Vo l(Pa) = deg( ^ [V(cr)]) .

Proof. Suppose first that V(cr) does not belong to the support of D,

i.e., u (a ) = 0. In this case, D restricts to the divisor on the toric

va r ie ty V (a ) defined by the collection of elements u ( t ) € M ( ct) / M ( t )

as t varies over the star of a. This is a divisor whose line bundle

0(D|y(cr)) is generated by its sections, and these sections are the linear

combinations of characters X u for u in CT^nPnM = P fff lM (a ) , so we

are reduced to the situation of the preceding corollary. In the general

case D can be replaced by D + d iv (X u)> where u is any element of

M that maps to u(cr) in M/M(a). We have seen that the poiytope

corresponding to D + d iv (X u) is P - u. The first case applies to this,

and noting that rationally equivalent divisors have the same

intersection numbers, the corollary follows.

In particular, if X (A p ) is constructed from the polytope P as in

§1.5, then the cones cr correspond to the faces of P, and in this case

PCT is the face of P corresponding to cr.

We know that the classes of the varieties V(cr) span A*(X )q , s o

the Todd class can be written as a <Q-linear combination of these classes:

Td(X) = Z rCT[V (a ) ] ,<T € A

for some rational numbers rCT. As we pointed out, in any example one

can calculate such coefficients, but because of the relations among

these generators for the Chow groups, the coefficients are not unique.

Having such an expression for the Todd class, however, gives a

marvelous formula for the number of lattice points/*6 Combining

(RR) w ith the preceding corollary, we have

* (P ) = Z i W o K P J ,a e A



RIEM ANN-R OCH THEOREM 113

and

n* (v - P ) = 21 a k v k . with a k = Z r a*Vol(Pa) .

k=0 codim (a) = k

For example, in the two-dimensional case, we m ay take

Td(X) = [X] + H(Z[Dil) + [x] ,

where x is any point of X. Starting with a convex rational polytope

P in the plane, this leads to

Pick's Formula: # (P ) = Area(P ) + Vi • Per im eter (P ) + 1 .

Note that the lengths of the edges of P are measured from the

restricted lattice, so the length of an edge between two lattice points is

one more than the number of lattice points lying strictly between

them.

Area(P ) = ^

Per im eter (P ) = 7

* ( P ) = 1 3 = ^ + ^ - + l 2 2*(-u-P) = 1 + -^ v + ^T"1’2

As we have seen, the value of the polynomial v »-* ^ (v -P ) at v = -1

is the number 6 of interior lattice points.

There is great interest in finding useful formulas for the number

of lattice points in a convex polytope, which has sparked efforts in

finding explicit formulas for the Todd classes of toric varieties. One

wants coefficients ra that depend only on the geom etry around the

cone a. It is also interesting to look for combinatorial formulas for other characteristic classes of a possibly singular toric var iety . For example, F. Ehlers has shown that MacPherson's Chern class is a lways

the sum of the classes of all orbit closures, each w ith coefficient 1 ,

whether the toric va r ie ty is singular or not/18^

Exercise. Let N = {(xq, . . . ,x n) € Zn + 1 : X xj = 0), and define vectors v 0, • • • , v n in N whose sum is zero by setting vq = (1 ,0 , . . . , - ! ) ,



114 SECTION 5.4

v i = (-1,1,0,... ,0 ), . . . , v n = ( 0 , . . . ,-1,1). Let A be the fan w ith cones

generated by proper subsets of {vq , . . . , v n), so X (A ) = [Pn, and each

V (a ) is an intersection of coordinate hyperplanes in (Pn. Show that

the coefficient rCT of V (a ) can be taken to be the fraction of a in

the space spanned by a (by volume obtained by intersecting w ith the

unit ball and using the induced metric from the Euclidean m etr ic on Rn + 1) (19)

Exercise. Let X be the toric va r ie ty whose fan consists of the cones

over faces of our standard cube. Compute Td(X), and use this to

compute #(i^«P), where P is the dual octahedron/20^

Exercise. (R. Morelli) Let M = Z3, and let P be the polytope w ith

vertices at ( 0 ,0 ,0 ), ( 1 ,0 ,0 ), (0 ,1 ,0 ), and ( 1 ,1 ,m), where m is a

positive integer. Show that the polynomial #(u*P) corresponding to

P is 1 + ■ - v + v 2 + ~ - v 3. In particular, if m > 13, the one-6 6

dimensional Todd class T d i (X (A p ) ) cannot be written as a n on

negative linear combination of the one-dimensional cycles V (a ) .

5.4 M ixed vo lum es

Intersection theory on toric varieties can be used to prove interesting

facts about convex bodies in Euclidean spaces, by exploiting the

interpretation of the volume of a polytope as a self-intersection number of a divisor on a toric variety. The basic notions involved here

are Minkowski sums and mixed volumes. If P and Q are any convex

compact sets, their Minkowski sum is the convex set

P + Q = (u + u‘ : u € P, u' € Q) ;

note that for a positive integer v, v>P is the Minkowski sum of v

copies of P. We work in an n-dimensional Euclidean space Mjr

coming from a lattice M, which determines a vo lume Vol(P ) for any

compact convex set P. This volume is positive when P has a

nonempty interior. For n = 2, the mixed volume V(P,Q) of two

convex compact sets can be defined by the equation

2-V(P,Q) = VoKP + Q) - Vo l(P ) - Vol(Q) ,



MIXED VOLUMES 115

where of course Vol denotes ordinary area. In particular, V (P ,P ) is

the area of P. For example, if Q = B(e) is the disk of radius e, then

P + Q is a closed e-neighborhood of P, and the right side of this

equation, when divided by e, approaches the circumference of P as

e approaches 0 .

The mixed volume is multilinear, so V (P ,B (e)) = e*V(P,B), where B

is the unit disk. It follows that 2»V(P,B) is the circumference of P.

The isoperimetric inequality that the area of P is ct most the square of the circumference divided by 4 t t translates to the inequality

V (P,Q )2 > V(P,P)*V(Q,Q).

Our goal in this section is to prove generalizations of these facts.

To start, we will consider only n-dimensional convex polytopes whose

vertices are in M. All of the assertions, in fact, will extend to general

convex compact sets, by approximating them with such rational

polytopes (for a finer lattice). We have seen that if a complete fan A is compatible with a polytope P, i.e., the corresponding function i|vp is

linear on the cones of A , then P corresponds to a T-Cartier divisor

D on the toric va r ie ty X = X (A ). The line bundle 0(D ) is generated

by its sections, which have a basis of characters X u for u in PflM.

As we saw at the end of §3.4, the assumption that P is n-dimensional

implies that the map from X to projective space given by these

sections has an image that is an n-dimensional variety.

If any finite number of such polyhedra P i , . . . ,P S are given, we

m ay find one such A that is compatible with each of them; if desired,

by subdividing the fan, we m ay even make X = X (A ) nonsingular and

projective. If Ej is the divisor on X corresponding to the polytope

Pj, then, for any nonnegative integers v\ t . . . , v s, the polytope

t i f P l + . . . + ^s'Ps corresponds to the divisor V y E i + . . . +

From the corollary in the preceding section we deduce a fundamental

result of Minkowski that the volume of a nonnegative linear



116 SECTION 5.4

combination of polytopes is a polynomial of degree n in the

coefficients v±, . . . , v s. In fact, we have the formula

( 'V fE i + ... + Vg-Es) n(1) V o K i^ .P i + . . . + i;s .P s ) = —— ----------— — •

For n polytopes P i , . . . , P n, n! times the coefficient of V i > . . . ' V n in this polynomial is defined to be the m ixed vo lum e of Pj_, . . . , P n, and

is denoted VCPj, . . . ,Pn)- We will give a closed formula for this in a

minute, but what will be useful is this intersection-theoretic

characterization:

( E i •... • En )(2) V ( P i , . . . , P n) = -

n!

The mixed volume has a simple expression as an alternating sum

of ordinary volumes:

n(3) n !*V (Pl l . . . ,P n) = V o K P i+ . - .+ P n ) - Z VoK P i + ... + P , + ... + P n)

i = 1

n+ z Vol(P1 + ...+ P i + ...+ P i+...+ Pn) - . . . + ( - l ) " -1 Z Vol(Pi) .

i<j »=1

This follows from (2) and the algebraic identity

nn!-(Er ...-En) = (E! + ...+En)n - Z (Ex+ . . .+ £ (+ . . . + En)n

i = 1

n+ X! (Ei + ...+ Ej + ...+ Ej + ...+ En)n - . . . + ( - l ) n X (Ej)n .

i< j i = 1

(Here and in the following, to avoid a flood of parentheses, w e often

write En in place of (En) for a divisor E; it is a lways the intersection

number that is meant.) Many other properties of mixed vo lum e are

easy to prove either from (2), or directly from the definition. For

example, we see immediately that V (P i , . . * ,Pn) *s a sym m etr ic and

multilinear function of its variables:

(4) V(a-P + b-Q,P2 , . . . , P n ) = a-V(P, P2, . . . , Pn) + b-V(Q, P2, . •., P n)



MIXED VOLUMES 117

for nonnegative integers a and b. It also follows from this definition

that

(5) V (P , . . . ,P ) = Vol(P) .

Exercise. Prove, for two polytopes P and Q, the Steiner

decomposition:

( 6 ) Vol(P + Q) = Z ( " )V (P , i ;Q ,n - i ) ,i = 0

where V (P , i ;Q ,n - i ) denotes the mixed volume of i copies of P and

n-i copies of Q.

The mixed volume is invariant under arb itrary translations of the polytopes:

(7) V ( P 1 + U1, . . . ,P n + u n) = V ( P lf . . . , P n )

for any uj_, . . . ,un in M. This follows from the fact that Pj + uj

corresponds to the divisor Ej - d iv (X Ui), because intersection numbers

of rationally equivalent divisors are the same.

1Exercise. If M is replaced by — M, show that the mixed volumes

are all multiplied by m n. Show that

(8 ) V (A (P i ) , . . . ,A (P n)) = |det(A)|*V(Pi, . . . ,P n)

for any linear transformation A of M.

An important fact that is far less obvious is the nonnega tiv ity of

mixed volumes. This follows from the fact that the intersection

number of n divisors Ej, when each O(Ej) is generated by its sections, is nonnegative. The following is a stronger result:

(9) V (Q lf . . . ,Qn) < V (P i , . . . ,Pn) if Q jC Pj for 1 < i < n .

It suffices to prove this when Qj = P[ for i £ 2. If Ej are the divisors

corresponding to Pj, and Fi corresponds to Qi, the fact that Qi is

contained in P i implies that Ei = Fi + Y w ith Y an effective

divisor. Changing Ej to rationally equivalent divisors if necessary, we

m ay assume each E* restricts to a Cartier divisor on Y. Then

(Er E2 . . . . -E n) - (F r E2 - . . . -E n) = (Y .E2 . . . . -E n) - (E2|y - . . . - E J y ) •



118 SECTION 5.4

This is nonnegative since it is the intersection number of n-1 divisors

on Y whose line bundles 0(Ej|y) are generated by their sections/21 In particular, if P is the smallest convex set containing Pj_, . . . ,P n>(9) gives the inequality

(10) V (P i , . . . ,Pn) < Vol(P ) .

We also derive a stronger version of nonnegativity:

(11) V (P t , . . . ,Pn) > 0 if Int(Pj) * 0 for 1 < i < n .

ATo see this, replace M by — M and translate the polytopes if

necessary so that the intersection Q = O P j has nonempty interior,

and then (9) and (5) imply that the mixed volume is at least the volume of Q.

For a simple example, let P^ and P 2 be the polyhedra in IR2o

with vertices in H as shown:

Then V o l (P i ) = 2, Vo l(P2) = lV2, and VoKPj. + P 2) = l l H , so

V (P 1,P2) = ^ (V o l (P 1 + P2) - VoK Pi) - Vo l(P2) ) * 4 .

The volume of the convex hull P is 4H.

Exercise. Prove the additiv ity form ula : given P and Q such that

PUQ is convex, and Pk + i, . . . , P n» then

(12) V(PUQ,k; Pk + 1, ... ,P n) + V(PflQ,k; Pk + 1, ... ,P n )

= V (P ,k ;P k+1, . . . , P n ) + V (Q ,k ;P k+1. . . . . P n ) .

where the notation means that the first polytope is repeated k times.



MIXED VOLUMES 119

Finally, we have a deeper result, known as the A lexandrov - Fenchel inequality:

(13) V ( P lf . . . ,P n ) 2 > V t P i . P j . P a P n ) -V (P 2 ,P 2 ,P3, . . . ,P n ) •

Following Teissier and Khovanskii, this can be deduced from the Hodge

index theorem. If Ej_, . . . ,En are the corresponding divisors on the

var ie ty X, which we can take to be nonsingular, (13) is equivalent to

the inequality

(Er . . . -En ) 2 > (E i-E i-Ev . . . .En)(E 2 -E2 -E3. . . . -En ) .

The maps cpj: X —> IPri determined by the divisors Ej have n-

dimensional images. The Bertini theorenr/22 implies that if Hj is a

generic hyperplane in IPri, then Y = (p3_1(H3 )n . . . n ( fn- 1(Hn) is a

nonsingular irreducible surface. Since c p 1 (Hj) is rationally equivalent

to Ej, if we define Di and D2 to be the restrictions of E and E2

to the surface Y, this inequality becomes

(Dr D2 )2 > (D1-D1) (D 2 .D2) .

This is a consequence of the Hodge index theorem, which says that

if A and B are divisors on a nonsingular projective surface, with

(A «A ) > 0 and (A*B) = 0, then (B*B) < 0. 23 To deduce the displayed

inequality, we know that (Dj/Di) > 0 , and the inequality is obvious

if (D i 'D i ) = 0. If (Di*Di) > 0, take A = D* and B = aD2 - bD^, w ith

a = (D i-D j) and b = (Dj_'D2 ); the index theorem gives the inequality

(aD2 - bD^ )2 < 0 , which is immediately seen to be equivalent to the

displayed equation. Note that if n = 2, the Bertini argument is

unnecessary.

Exercise, (a) Given P, Q, Rk+1> • • • anc* 0 .. . ,R n)k > V(P,k; Rk + 1>... ,Rn) ‘ .V(Q,k; Rk + i , ... ,Rn)k_i.

k(b) Show that V (P t P n) k TT V (P i ,k ;Pk + i Pn) .

i = 1(c) Deduce the inequality

(14) V (P lf . . . , Pn) n 2 V 0KP1). . . . "Vol(Pn) . (24)



120 SECTION 5.4

The inequality (14) is equivalent to the inequality

(Er . . . *En) n > ( E ! n)* . . . *(Enn) .

This is also a general fact, for divisors Ej such that each 0(Ej ) is

generated by its sections, on any complete irreducible n-dimensional

variety. This appealing generalization of the Hodge index theorem was

apparently only recently discovered by Demailly.

Exercise. Prove the B runn-M inkow ski inequa lity for two convex

polytopes P and Q:

7 v o Kp + q ) * 7 v o i ( P ) + 7 v o i ( Q ) . (26)

Now let M - Zn, so M jr = (Rn, w ith its usual metric. Let B

denote the ball of radius one, and B(e) the ball of radius e. All of the

preceding results extend to arb itrary compact convex sets, by the

following exercise.

Exercise. Define a distance d(P,Q) between compact convex sets to

be the m in im um e such that P c Q + B(e) and Q C P + B(e). Show

that d defines a metric (w ith triangle inequality). Show that the

operations P, Q ^ P + Q; P »-* Vol(P); P*, . . . ,P n »-* V (P i , . . . ,P n)

are continuous. Show that for any compact convex Q and any

positive e there is a positive integer m and an n-dimensional convex

polytope P w ith vertices in M such that d(P,Q) < e.

The mixed volumes are particularly interesting when taken with

several copies of one compact convex set and the other copies a ball.

For example, the expansion

Vol(P + B(e)) = £ (?)V(P, i ;B(e),n-i) = Z ( ^ ) v ( P , i ;B ,n - i ) e n- ii = 0 i - 0

interprets the mixed volumes of P and the unit ball B in term s of

the rate of growth of the volume of an e-neighborhood of P:

- ^ V o l ( P + B (t)) |t = 0 = n (n - l ) ' . . . • (n -k + l) *V (P ,n -k ;B ,k )



BEZOUT THEOREM 121

In particular, the first derivative of this function, which we denote by

S(P), is given by the formula

S(P) = n-V(P, . . . , P, B) .

So S(P) is the limit of (Vo l(P + B(e)) - Vol(P))/e as e goes to 0.

When the boundary of P is smooth, S(P) is its (n-l)-d im ensional

area; if not, this is a reasonable measure for it. Inequality (14) then

implies a classical isoperimetric inequality

S (P )n > nn•vn*Vol(P)n~l ,

where v n is the volume of B.

It is also interesting to interchange the roles of P and B, since

the limit of (Vol(B + e-P) - Vol(B))/e as e goes to 0 can be written as

S(B)*d(P)/2, where d(P) is the mean d iameter of P. In other words,

d(P) = 2‘ n«V(P,B, . . . ,B)/S(B), and (14) gives the inequality

d (P )n > 2n-nn-Vol(P)-Vol(B)n' 1/S(B)n = 2n-Vol(P)/Vol(B) .

In particular, this gives the inequality Vol(P ) < (d/2)n*Vol(B), where

d is the maximal diameter of P.

Exercise. Show that f(t ) = V (t«P + ( l- t )Q ,k ; Rk + l, • • • ,Rn)1/k has the

upper convexity property: f(t ) > ( l - t ) f (O ) + t f ( l ) . In particular,

S(P + Q)1/(n_1) > S (P )1/(n_1) + S(Q)1/(n_1) . (27)

5.5 B ezou t th eo rem

A regular function on the torus T = Hom(M,C*) has the form

F = Z a u X u, the sum over a finite number of points u in M. With

M = Zn, this is a Laurent polynomial, i.e., a finite linear combination

of monomials in variables X*, . . . ,X n with arb itrary integer

exponents. The convex hull of the points u for which au is not zero is

called the Newton polytope or polyhedron of F. We need the

following simple fact.

Lem m a. Let A be a complete fan compatible with the Newton



122 SECTION 5.5

polytope P of F, and let E be the T -C art ie r divisor on X (A )

corresponding to P. Then E + div(F) is an effective divisor on X (A ).

Proof. The assertion is that the order of E + div(F) along any

codimension one subvariety D of X (A ) is nonnegative. This is clear

if D meets T, since F is regular on T. Otherwise D is the

subvariety corresponding to an edge of A. With v the generator of

this edge, we have

ordD(F) > min ordD( X u) = min <u,v> = ipp(v) = - o r d D(E ) , au^ 0 uePHM

as required.

Suppose Fi, . . . ,Fn are Laurent polynomials. We want to

estimate the number of their common zeros in T = (C * )n. Let

Z = { z € T : Fx(z) - . . . = Fn(z) = 0 } .

If z is an isolated point of Z, let i ( z ;F i , ... ,Fn) denote the in ter

section multiplicity of the hypersurfaces defined by the ¥[ at the

point z; this intersection number is a positive integer, and it is 1

exactly when the hypersurfaces meet transversally at z. Our Bezout formula, generalizing slightly results of D. N. Bernstein, A. G. Kouchnirenko, and A. G. Khovanskii/^^ is

Proposition. Let Pj be the Newton polytope of Fj. Then

X ! i ( z ;F 1, . . . , F n) < n ! - V (P lf . . . ,Pn) ,z i so la t e d in Z

where V(P±, . . . ,Pn) is the m ixed volume.

Proof. Take A compatible w ith all of the polytopes, so each Pj

corresponds to a T-Cartier divisor Ei on X (A ). We have seen that

the right side of the inequality is the intersection number (E i* . . . «E n).

Since Dj = div(Fj) + Ej is rationally equivalent to Ej, this intersection

number is the same as (Dj_* ...*Dn). Each divisor Dj meets the torus

T in the hypersurface defined there by Fj = 0, so the left side of the

inequality is at most the sum of the intersection multiplicities of the

divisors D*, . . . ,Dn at their isolated points of intersection. The line



BEZOUT THEOREM 123

bundles CKD*) = ©(E*) are generated by their sections, so we are

reduced to a general fact of intersection theory: if D*, . . . ,Dn are

effective Cartier divisors on a complete irreducible va r ie ty , whose line

bundles 0(Dj) are generated by their sections, then the sum of their

intersection multiplicities at their isolated points of intersection is at

most the intersection number (Dj/ ... 'Dn).

Note that it is not necessary to assume that Z is finite; and even

if Z is finite, the divisors Dj, which represent the closures of the

hypersurfaces Fj = 0 , can have non-isoiated intersections on the

complement of the torus.

If the polytopes Pj are fixed, it is also true that, for generic F*

with these P\ as Newton polygons, the intersections are all isolated

and transversal, and equality holds in the preceding display. This

follows from the fact that, in this case, the divisors D* will be the

inverse images of generic hyperplanes for the corresponding maps from

X (A ) to projective spaces, and the Bertini theorem guarantees that

these will have only isolated transversal intersections.

For an example with n = 2, consider the polynomials

Ft = X2 + Y 2 + X3Y and F2 = X2Y + X Y 2 + 1 .

The Newton polygons P i and P2 are those sketched in the preceding

section. Since the mixed volume V (P i , P 2 ) is 4, the number of

common isolated zeros must be at most 8 . Early estimates bounded

the number of zeros by n!*Vol(P), where P is the convex hull of

. . . ,Pn. As we saw in the preceding section, the mixed volume

gives a sharper estimate. In the above example, the vo lume of this

convex hull is 4 1/2 , so that estimate would allow 9 solutions.

Exercise. Show that there are exactly 8 common solutions to these

equations in (C*)^, so all of the intersection numbers must be 1.

Consider the case of n polynomials Fj in C [X i , ... ,Xn], w ith

deg(Fj) < d|. The Newton polytope of F[ is contained in dpP, where

p = { ( 11, . . . , tn) € IRn : tj > 0 and Zt* < 1 ). The standard fan for IPn,

regarded as the compactification for <Cn as usual, is compatible w ith

these polytopes, so the sum of the intersection numbers at isolated

zeros in (C * )n — or Cn or lPn — is at most



124 SECTION 5.6

n !-V (d r P , . . . , d n.P) = n !-dr . . . -dn • V ( P , ... ,P)

= n !*d i ' ... *dn * Vol(P ) = dr . . . -dn ,

since Vol(P ) = 1/nl, so the usual Bezout theorem is recovered.

5.6 S tan ley 's th eo rem

Stanley has given a beautiful application of toric varieties to the

problem of characterizing the numbers of vertices, edges, and faces of

all dimensions of a convex simplicial polytope K in Euclidean n-space.

Let K be a convex polytope in 3-space, w ith fo vertices, f*

edges, and f2 faces. To begin, we have E uler ’s formula :

( 1 ) f0 - h + *2 * 2 .

If the polytope is simplicial, i.e., all of its faces are triangles, the facts

that each face has three edges and each edge is on two faces give

( 2 ) 3 f2 = 2 f ! .

To bound a solid takes more than three vertices, i.e.,

(3) f0 > 4 .

Note that the triple is determined once the number of vertices is

specified. It is an easy exercise to show that any triple of integers

( f0* f 1 > f2 ) satisfying these three conditions can be realized from a

convex simplicial polytope in 3-space. Starting w ith a solid simplex

(w ith fo = 4), it suffices to show how to construct a new polytope

with one more vertex than a given one. This is achieved by putting a

new vertex just outside the middle of a face, and taking the convex

hull of the new set of vertices.

For n = 4, the numbers fo, f i , f2> f3 satisfy Euler's equation:

( 1 ) fo " h + h ‘ *3 = 0 .

Since 3-simplices have four 2-faces, each on two 3-simplices,

( 2 ) f 2 = 2 f 3



STANLEY 'S THEOREM 125

To bound a 4-dimensional solid, as before,

(3) fo * 5 .

There is also a quadratic inequality, valid in all dimensions, which

comes from the fact that two vertices can be joined by at most one

edge:

This time there is also a less obvious lower bound on the number of

edges:

For example, if fo ,= 5, these conditions determine the other

numbers: f\ = 10, f2 = 10, and f3 = 5. This example is achieved by

(the boundary of) a 4-simplex.

Exercise. The conditions ( l ) - ( 5 ) allow two possibilities for (fo» f l» f2 * 3 w ith six vertices: (6,14,16,8) and (6,15,18,9). Construct simplicial

polytopes to realize these two possibilities.

The claim is that these five conditions are necessary and

sufficient for the existence of a simplicial 4-polytope. In general, the

problem is to characterize the n-tuples (fo , f i » . . . , fn- l ) of integers that can be realized as the numbers of vertices, edges, . . . , (n - l ) - fa ces

of an n-dimensional convex simplicial polytope. Although it is easy to

generalize some of the equations listed above for polytopes of dimension

three and four — at least for equations (1) - (4) — it is far from obvious

what the complete answer should be, or how to prove it.

There is a convenient w ay to rewrite the equations, which

suggests generalizations, by looking at successive differences. This

replaces the sequence (fo ,fi ,* • • » fn- l ) an equivalent sequence of integers (ho ,h i,. . . , hn). Write the integers f* down the right side of

a triangle, and the integer 1 down the left, and then fill in the spaces

from the top down so that each integer inside is the difference of the

integer above it to the right and that above and to the left:

(4) f l < Vi f0 ( f0 - 1 ) .

(5) f l > 4 f0 - 10 .



126 SECTION 5.6

1 fo

1 f0- i f l

1 f0-2 f i - f o +1 f2

1 f0-3 fi~2fo+3 f2“ f

Denote the bottom row, from right to left, by (ho ,h i, . . . ,h n). In

formulas, the relations are simply

(*> hP = £i = p

where we set f_ i = 1 .

Euler’s equation is equivalent to the equation

h0 = hn .

For n = 3 and n = 4, equations (1) and (2) are equivalent to the

equations

(A ) ho = hn and h* = hn_ i .

The generalization fo £ n+1 of equation (3) is equivalent to hn_ i > 1,

or, modulo the above, to

(B i ) h i > h0 = 1 .

Equation (5) for n = 4 is equivalent, modulo the equations (A ), to

(B2 ) h2 £ h i ,

while equation (4) is equivalent to

(C) h2 - h i < ( h l h° + .

The D ehn -Som m erv il le equations are the generalizations of the

equations (A): hp = hn_p. They were expressed in this form by

Sommerville in the 1920's. The generalizations of the equations (B) are

not hard to guess, but (C) is more subtle. The complete answer is given

in the following theorem, which was conjectured by P. McMullen.



S T A N L E Y ' S THEO REM 127

Given a sequence fg, f i , . . . ,fn- l of integers, set f_ i = 1 and

m = [ i f ] - F °r 0 < p < n, define hp by ( * ) , and, for 1 < p < m, set

Sp = h p - h p_ i .

Theorem . A sequence of integers fo, f i , . . . , fn- 1 occurs as the

n u m b e r of vertices, edges, . . . , (n -1 ) - faces of an n -d imensional

convex simplicial poly tope if and only if the following conditions hold:

(I) (D ehn -S om m erv il le ) hp = hn_p for 0 0 for 1 np_i > . . . > nr > r > 1, then

for 1 < p < m - 1.

Note that for any positive integer p, any positive integer gp has

a unique expression in the form (a), by taking np m aximal such that

( pP) < gp, and then continuing with the difference. In words, (b) says

that the bound for gp+i is obtained from the expression for gp by

increasing each en try in each binomial number by one. If gp= 0, the

condition is interpreted to mean that gp+i = 0. The existence of a

convex polytope w ith such face numbers was established by Billera and

Lee by a direct ingenious argument; the necessity was proved by

Stanley/31 We give Stanley's argument.

The main idea is to produce a toric va r ie ty X = X (A ) for some

complete simplicial fan A , so that the number d^ of k-dimensional

cones in A is the same as the number f^ - i of (k- l)-d im ensional

faces of the given simplicial polytope. To do this, note that, since the

polytope is simplicial, moving all of its vertices slightly gives a polytope



128 SECTION 5.6

with the same numbers of faces in each dimension. By such a

perturbation we can assume the vertices are in the rational points Nq of a given lattice N, or by refining N, that the vertices are all in N.

We m ay also translate so that the origin is in the interior of the

polytope. Now define A to be the set of cones over the faces of the

polytope (w ith vertex at the origin), together w ith the cone (0). The

equation d^ = f^ - i (w ith do = f - i = 1) is then clear. The va r ie ty

X = X (A ) is complete, and locally a quotient of Cn by a finite abelian

group, since A is simplicial. In addition, X is projective. In fact, if

P in M(r is the polar polytope to the given polytope, then A = Ap,

and X = X (A p ) comes equipped with an ample line bundle.

We can now apply the theorems about the cohomology of the

toric va r ie ty X from the first part of this chapter. The rational

cohomology H*(X) = H*(X;Q) vanishes in odd dimensions, and

hp = d im H 2p(X ) = I ( - l > l" P ( J ) d n_, = X ( - D i_P ( p ) fn- i - i •1=P F 1= P

We saw that (I) is a corollary of Poincare duality, and that (II ) (a )

follows from the hard Lefschetz theorem for intersection homology. To

prove (II)(b), we use the other general fact we know about the

cohomology of X: it is generated by classes of divisors. Set

R* = H2 i (X;<Q)/Go.H2 i" 2 (X;<Q) ,

where, as before, go is the class of a hyperplane. Then R* = ©R* =

H*(X)/(co) is a comm utative graded Q-algebra, w ith R° = <Q, that is

generated by the part R* of degree one. Macaulay characterized the

sequences (g^, g2 , . . . ) of integers that can be the dimensions of such

an algebra: they must form a Macaulay vector as in (II). 32 Noting

that

gp = d im R p = dim H2p(X) - dim H2p“ 2 (X ) = hp - hp_i

for 1 < p < m, the proof of the theorem is complete.

Exercise, (a) Show that the relation between the fp's and the hp*s

can be expressed by the polynomial identity

Z hp t p = Z f n - i - i ( t - l ) 1 •p =0 P i =0



STANLEY 'S THEOREM 129

(b) Show that

fP - l = q? 0 ( n - j ) h n - q for 0 < p < n .

(c) Show that the Dehn-Sommerville equations (I) are equivalent to

the equations:

fp = ( - 1 ) " - 1 z V l ) k ( £ ; l ) f k for -1 < p < n .k = p v ^ '

Exercise. For n = 5, use (I) to solve for f2 , f3 , and f4 in terms of

fo and f i , and write (II) as inequalities for fo and f*.

Exercise. Prove that

f h ± + p + l \ / v - n + p - l Ahp < [ p J = [ p J for 0 < p < n ,

where v = fo is the number of vertices. (Here the convention is that

(^ ) takes its usual values for 0 < b < a, and is otherwise 0 , except

that ( q ) = 1.) Deduce the Motzkin-McMullen upper bound

con jecture :

fp S k| o T ^ k ( V k k ) ( p + l - k ) w h e n n = 2 m ;

fp ~ kf 0 ^ r ( k + l ) ( P ^ r - k ) w h e n n = 2 m + l .

Prove that these inequalities are valid for any convex n-dimensional

polytope, simplicial or not. Show that these inequalities become

equalities for a (simplicial) cyclic polytope: the convex hull of v

points on the rational normal curve { ( t ,t2 ,t3, . . . ,tn) )/ 33^

Exercise. Deduce the lower bound con jec ture : for a simplicial

polytope,

fp * ( ; ) f o - ( p t l ) p for 1 - P - n * 2 ;

fn_ l > ( n - l ) f o - (n + l ) ( n - 2) .

Exercise. A convex n-dimensional polytope is called simple if each



130 SECTION 5.6

vertex lies on exactly n (n -l)- faces . Characterize the sequences

(fo, . . . , fn- l ) ° f integers that arise from a simple polytope/34^

These links between toric varieties and polytopes have spurred

renewed interest in both subjects.



NOTES

P re fa c e

1. See for example,

E. Arbarello, M. Cornalba, P. A. Griffiths, and J. Harris, Geometry of Algebraic Curves, Vol. I, Springer-Verlag, 1985.

2. See [Beau], and

W. Barth, C. Peters, and A. Van de Ven, Compact Complex Surfaces, Springer-Verlag, 1984.

3. See

J. Harris, Algebraic Geometry: A First Course, Springer-Verlag, 1992.

4. The Danilov article is [Dani]. The others are [Bryl], [Jurk], [Dema],[KKMS], [AMRT], and [Oda'].

5. Again Oda has come to the rescue, with a survey:

T. Oda, "Geometry of toric varieties," pp. 407-440, in Proc. of the Hyderabad Conf. on Algebraic Groups, 1989 (S. Ramanan, ed.), Manoj Prakashan, Madras, 1991.

A new text emphasizes convex geometry:

G. Ewald, Combinatorial Geometry and Algebraic Geometry, to appear.

Chapter 1

1. See [Dema], [KKMS], and

I. Satake, "On the arithmetic of tube domains," Bull. Amer. Math. Soc. 79 (1973), 1076-1094.

Y. Namikawa, Toroidal Compactification of Siegel Spaces,Springer Lecture Notes 812, 1980.

2. The first assertion is seen by writing down the transition functionsfor the bundle. See [Shaf, Ch. VI] for the basic facts about vectorbundles, including 0 (a).

3. Let 1 be the ideal sheaf of DT in Fa. On U(T1 the conormal sheaf



132 NOTES TO CHAPTER 1

V32 is trivial and generated by Y; and on UCT4 it is trivial and generated by XaY, so the change of coordinates is by Xa, which identifies 1/32 with 0(a). The self-intersection number is the degree of the normal bundle 0(-a). The self-intersection numbers for the other three curves are 0, a, and 0. We will describe the divisors corresponding to rays more fully in Chapter 3, and intersection numbers in Chapter 5. See [Beau] for basic facts about the surfaces Fa.

4. Some of these references are:

A. Brjrfndsted, An Introduction to Convex Polytopes,Springer-Verlag, 1983.

B. Griinbaum, Convex Polytopes, J. Wiley and Sons, 1967.

R. T. Rockafellar, Convex Analysis, Princeton Univ. Press, 1970.

The Appendix in [Oda] has a more complete list of facts relevant to toric varieties, with precise references.

5. See for example Lemma 4.4 in the book by Br^ndsted, §2.2 in that by Grunbaum, or Theorem 11.1 in that by Rockafellar.

6 . See [Oda, Lemma A.4], and the article

R. Swan, "Gubeladze's proof of Anderson's conjecture," pp. 215-250 in Azumaya Algebras, Actions, and Modules, Contemp.Math. 124, Amer. Math. Soc., 1992.

7. Not in dimensions larger than three. For example, if cr is the cone in Z4 generated by the eight vectors ej and - e* + 2 X ej, then cr'"' has twelve minimal generators 2 e* + e * , i * j. J 1

8 . See [Shaf, §V.l], [Hart, Ch. I].

9. If this is difficult now, try it after you have read about orbits in Chapter 3.

10. The quotient field of C[S] is the same as that of C[S + (-S)], so one can assume S and S' are sublattices; choose a basis for S' so that multiples of these elements form a basis for S.

11. If J is the ideal generated by such relations, the fact that the canonical map from C[Yi, . . . , YJ/J to Aa is an isomorphism follows from the fact that the X u f° r u in SCT form a basis for A^, and the monomials mapping to a given X u differ by elements of J. Finding a minimal generating set for this ideal is more subtle, cf.

B. Sturmfels, "Grobner bases of toric varieties," Tohoku Math. J.43 (1991), 249-261.

12. Two of the many references are



NOTES TO CHAPTER 2 133

J. Herzog, “Generators and relations of abelian semigroups and semigroup rings," Manuscripta Math. 3 (1970), 175-193.

M. Hochster, "Rings of invariants of tori, Cohen-Macaulay rings generated by monomials and polytopes," Ann. Math. 96 (1972), 318-337.

Many of the results about toric varieties extend to such rings, and to the "nonnormal toric varieties" one gets by gluing their affine varieties together.

13. See

J. Herzog and E. Kunz, "Die Wertehalbgruppe eines lokalen Ringsder Dimension 1," S. B. Heidelberger Akad. Wiss. Math. Natur. Kl. (1971), 27-67.

14. See [Shaf, Ch. V, §4.3].

15. In §2.3 we will see how to recover A from X(A). For the general proof see [Oda1, Theorem 4.1].

16. Both are equivalent to the cone over K * 1 meeting E x 0 only at the origin.

17. Given the eight real numbers r v , choosing three generators of each cr determines the vector uCT, and the equation <ua,v> = r v for the fourth generator is then a linear equation among the eight numbers. These six equations have a solution space of dimension three.

Chapter 2

1. A point given by a homomorphism x: SG —> C of semigroups is fixed exactly when x(u) = 0 for all u * 0 , since if u * 0 there is a point in the torus t: M -> C* with t(u) # 1, and (t*x)(u) = t(u)x(u). For ct not to span Njr means that Sa contains some nonzero u together with -u, so x(u)x(-u) = 1 .

2. See [Oda, §3.2] for Ishida’s more intrinsic and more general construction of these complexes.

3. Use the simultaneous diagonalizability of commuting matrices; see §15 of

J. Humphreys, Linear Algebraic Groups, Springer-Verlag, 1975.

4. Diagonalizing, one may assume T = (C*)r acts on V = (Ln by

(Zi, . . . ,zr )-Cj = (z1mU. . . . •zrmrj ) e j .

The ring of invariants is C[S], where




S = { (pi, . • ■ ,pn) € (Z>0)n : ZmyPj = 0 V i ) .

5. The reference is

J. Gubeladze, "Anderson's conjecture and the maximal monoidclass over which projective modules are free," Math, of the USSR - Sbornik 63 (1989), 165-180.

For more on this see the article by Swan mentioned in Note 6 to Chapter 1.

6 . For more on quotient singularities, see

F. Hirzebruch, "Uber vierdimensionale Riemannsche Flachenmehrdeutiger analytischer Funktionen von zwei komplexen Veranderlichen," Math. Ann. 126 (1953), 1-22.

D. Prill, "Local classification of quotients of complex manifolds bydiscontinuous groups," Duke Math. J. 34 (1967), 375-386.

E. Brieskorn, "Rationale Singularit aten komplexer Flachen,"Invent. Math. 4, (1968), 336-358.

0. Riemenschneider, "Deformationen von Quotientensingularitaten (nach zyklischen Gruppen)," Math. Ann. 209 (1974), 211-248.

F. Ehlers, "Eine Klasse komplexer Mannigfaltigkeiten und dieAuflosung einiger isolierter Singularit aten," Math. Ann. 218 (1975), 127-156.

7. (a) For UCT., the identification with the preceding exercise is by

m = dj and (a*, . . . ,an) = (do, . . . ,d j_ i,d i+i, . . . ,dn). (b) Take N‘

generated by the v*. This realizes lP(do, . . . ,dn) as Pn/G, where G = N/N' = pdq x . . . x Pdn / pc , where c is the greatest common

divisor of the d*. For more on twisted projective spaces see

S. Mori, "On a generalization of complete intersections," J. Math. Kyoto Univ. 15 (1975), 619-646.

8 . If cp is an algebraic group homomorphism, and the corresponding map ip*: C lTJ ’ 1] -» <C[T,T_1) takes T to F(T), show that F(TiT2) = F(T1)F(T2). and deduce that F(T) = for some k.

9. Such limits play an important role in invariant theory:

D. Mumford and J. Fogarty, Geometric Invariant Theory, Springer-Verlag, 1982.

For constructions of quotients of toric varieties by subtori of the given torus, see

M. M. Kapranov, B. Sturmfels, and A. V. Zelevinsky, "Quotients of




toric varieties," Math. Ann. 290 (1991), 643-655.

10. When f is not proper, and X is embedded in a variety Xso that the morphism f extends to a proper morphism from X to Y — which is in fact always possible — it is easy to construct such a discrete valuation ring and maps. In fact, one can take R to be the ring €{t) or C[[t]] of convergent or formal power series, with the maps corresponding to an analytic map of a small disk, with center mapping to a point of the closure of X that is not in X. For a formal proof, see [Hart, Ch. II, Exer. 4.11].

11. Look at the covering by the affine open UCT and consider the distinguished points xT. For more on this, see [Oda, §1.5].

12. Let N be the lattice of rank r + n -1 generated by vectors w j, . . . ,w r and v q , . . . , v n, with relations

wi + . . . + w r = 0 , v q + . . . + v n = a jw j + . . . + ar w r .

Let A be fan consisting of cones generated by subsets of these vectors that do not contain all of the Wj's or all of the vj's. See [Oda, §1.7] for more on toric bundles.

13. Given the descriptions of the points in terms of semigroup homomorphisms, this is equivalent to the fact that an element u in cr'' is in ctx exactly when cp*(u) is in ( a 1)1, which follows from the fact that cp(a') contains points in the relative interior of ct.

14. The simplest example has rays also through (2,1,1), (2,1,2), and (3,1,2), with cones as indicated:

(0 ,0 ,1 )

For more on this see

M. Teicher, "On toroidal embeddings of 3-folds," Israel J. Math. 57 (1987), 49-67.

Recently it has been shown that any two nonsingular complete toric varieties of the same dimension can be obtained from each other by a sequence of maps that are blow-ups along smooth toric centers or the inverses of such maps. In fact, there are two independent proofs:




J. Wlodarczyk, "Decompositions of birational toric maps in blowups and blow-downs. A proof of the weak Oda conjecture," preprint.

R. Morelli, "The birational geometry of toric varieties," preprint.

15. Write v j = -av j + bvj + i and vj + = -cv j - d v i + i, with a, b, c, and d positive integers. The determinant of ( _* _bd) should be 1, but ad + be > 2 .

16. For (a), take the longest possible sequence of consecutive vectors in the same half-plane, and apply the topological constraint (see Note 15) to see where adjacent vectors can lie. For (b), write Vj = -bjVQ + b j 'v i,and set cj = bj + bj1; since C2 £ 3 and Cj = 1, there is a j withcj > cj+1 and cj > Cj_i; show that aj = 1 .

17. The matrices give changes of bases from one pair of adjacentvectors to the next.

18. The condition ( * * ) insures that the vectors go around the origin only once.

19. See the calculations of Note 3 to Chapter 1.

20. For example, see [Oda] and

V. E. Voskresenskii and A. A. Klyachko, "Toroidal Fano varieties and root systems," Math. USSR Izv. 24 (1985), 221-244.

V. Batyrev, "On the classification of smooth projective toric varieties," Tohoku Math. J. 43 (1991), 569-585.

21. For the relations among these generators — as well as the relation between the integers bj and the integers aj — see the article by Riemenschneider in Note 6 .

22. The Hirzebruch-Jung continued fraction for (k+l)/k produces a string of k 2 's.

23. If k 'k1 = 1 (mod m), write k*k‘ = 1 - s-mb; the matrix ( )maps N = Z2 isomorphically to itself and maps ct onto cr', giving an isomorphism between Ua and UCT». The converse follows from the factthat the configuration obtained by the resolution process describedhere is the minimal resolution (none of the self-intersection numbers of exceptional rational curves is - 1 ), which is uniquely determined by the singularity. This means that the singularity determines the sequence a^, . . . , ar , up to replacing it by the reverse sequencear , . . . ,a i , which corresponds to the pair (m,k‘) with kk‘ s 1

(mod m). Note that the singularity of UCT is a cone over a lens space, and two such lens spaces are homeomorphic exactly when m ‘ = m




and k' = k or kk1 s ±1 (mod m). For a reference, see the Hirzebruch article in Note 6 .

24. The existence of such v is a simple case of a theorem of Minkowski, which can be seen by comparing the approximate number of points to the volume of large solids {X t ^ , -m i < t* < mj). For the general case, see §24.1 of

G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, fourth edition, Oxford Univ. Press, 1960.

25. The cone over a quadric was described in

M. F. Atiyah, "On analytic surfaces with double points," Proc. Royal Soc. A 247 (1958), 237-244.

For flips, flops, and the relation to Mori's program see:

J. Kollar, "The structure of algebraic threefolds — an introduction to Mori's program," Bull. Amer. Math. Soc. 17 (1987), 211- 273.

M. Reid, "Decomposition of toric morphisms," pp. 395-418 in Arithm etic and Geometry II, Progress in Math. 36, Birkhauser, 1983.

M. Reid, "What is a flip?", preprint, 1992.

T. Oda and H. S. Park, "Linear Gale transforms and Gelfand-Kapranov-Zelevinskij decompositions," Tohoku Math. J. 43 (1991), 375-399.

Recently, Batyrev has used toric varieties as a testing ground for the "mirror symmetry" conjectures coming from physics:

V. V. Batyrev, "Dual polyhedra and mirror sym m etry for Calabi- Yau hypersurfaces in toric varieties," preprint, 1992.

V. V. Batyrev, "Variations of the mixed Hodge structures of affine hypersurfaces in algebraic tori," Duke Math. J. 69 (1993), 349-409.

Chapter 3

1. Any M-graded ideal (or submodule of the field of rational functions) has the form 0 C X u> the sum over some subset of M; see Note 3 to Chapter 2. For complete proofs of these facts, including the following exercise, see [Oda1, §5].

2. This follows from the fact that cp*(xT«) = xT, because cp* takes




TN'-orbits to Tisporbits. The last assertion uses the properness of (p*.

3. Note that normality is crucial; the fact that the closed complement is small is not enough. The complement of a point in an irreducible variety can be simply connected without the variety being simply connected — for example if the variety is constructed by identifying two points in a simply connected variety. For an amplification of these points, see

W. Fulton and R. Lazarsfeld, “Connectivity and its applications in algebraic geometry," Springer Lecture Notes 862 (1981), 26-92.

4. This is true with any coefficient group or sheaf: if C* is an injective resolution of the sheaf, this is the spectral sequence of the double complex 0 C q(UjQn . . . OUj ), with the vertical maps coming from the complex C* and the horizontal maps the "Cech" maps of alternating sums of restrictions. For details, see [Hirz, §1.2] and

R. Godement, Topologie algebrique et theorie des faisceaux,Act. Sci. et Ind., Hermann, Paris, 1958.

5. See [Hirz, §1.4] for basic facts about Chern classes.

6 . If the data { fa } is used to define Cartier divisors, the data [focgcx) defines the same Cartier divisor whenever g^ is a nowhere zero regular function on Ua ; in addition, one identifies data with equivalent restrictions to a common refinement of the open covering.If the variety V meets the affine open set Ua , the ideal p of V flU^ is a prime ideal in the affine ring A of Ua , and the local ring of V is the localization Ap. If X is normal and V has codimension one, Ap is integrally closed and one-dimensional, so a discrete valuation ring. The order of D at V is the order of fa with respect to this discrete valuation. That a Cartier divisor is determined by its Weil divisor follows from the fact that A is the intersection of these rings Ap. For more on this see [Shaf, §111.1], [Hart, §11.6], and [Fult, §2.2].

7. There is a u € Sa such that <u,v> = 1, where v is the first lattice point along t .

8 . This is local, so it is enough to do it on an affine UCT, in which case it follows from the preceding lemma.

9. This exact sequence follows readily from the definitions, see [Fult, §1.8].

10. For example, X = Tn, with n > 2. All algebraic line bundles are trivial, but H2(Tn) = A 2M =* 0 ; the torus has analytic line bundles that are not algebraic.

11. We have seen that (i) =» (ii), and (ii) =* (iii) =» (iv ) are clear from




the proposition. If a is a maximal cone that is not simplicial, with generators v j , . . . , v r , one can find positive integers a* so that thepoints ( l/a jW i are not in a hyperplane, so there is no u (a ) with<u(a),Vi> = ~kaj for 1 < i < r and any positive k.

12. The graph of this map is the resolution X (A ) .

13. Find a strictly convex function if; with ^ (vj) € Z. For the second assertion, use the generators of semigroups found in §2.6. In fact, if D is an ample divisor on an n-dimensional toric variety, then (n - l )D is always very ample. This is proved in

G. Ewald and U. Wessels, "On the ampleness of invertible sheaves in complete projective toric varieties," Results in Math. 19(1991), 275-278.

14. If the corresponding function is given by u(a) on the maximal cone cr, both are equivalent to the condition that > -aj whenever vj £ cr.

15. Use all of the hyperplanes spanned by (n-l)-dimensional cones.

16. For more on the projectivity of nonsingular toric varieties see

P. Kleinschmidt and B. Sturmfels, "Smooth toric varieties with small Picard number are projective," Topology 30 (1991), 289-299.

More examples like those in the text can be found in

M. Eikelberg, "The Picard group of a compact toric variety,"Results in Math. 22 (1992), 509-527.

17. If P = {ui, . . . ,ur ), the affine ring of UCT is generated by variables Y*, . . . , Y r (with Y 4 = Xûi x ^ ), and relations generated by TTYjai - TTY^* if ZajUj = Ib jU i and Xaj = Zbp This is the homogeneous coordinate ring of Xp in P r_1.

18. It suffices to look at the restriction of cp to the torus Tjsj, where the map is a map of algebraic tori Tn -* (<C*)r /C* determined by a homomorphism of lattices from N to Zr/ Z * ( l , ... ,1). The dimension of cp(X) is the rank of the image of N, which is the dimension of P.

Chapter 4

1. For the topology of moment maps for more general actions of tori on algebraic varieties, see

M. Goresky and R. MacPherson, "On the topology of algebraic torus




actions," Springer Lecture Notes 1271 (1987), 73-90.

2. For more on moment maps, including generalizations of these statements and proofs, see [Jurk] and

M. F. Atiyah, "Convexity and commuting Hamiltonians," Bull. London Math. Soc. 14 (1982), 1-15.

M. F. Atiyah, "Angular momentum, convex polyhedra and

algebraic geometry," Proc. Edinburgh Math. Soc. 26 (1983), 121-138.

V. Guillemin and S. Sternberg, "Convexity properties of the moment mapping," Invent. Math. 67 (1982), 491-513.

F. C. Kirwan, Cohomology of Quotients in Symplectic and Algebraic Geometry, Princeton Univ. Press, 1984.

L. Ness, "A stratification of the null cone via the moment map," Amer. J. Math. 106 (1984), 1281-1325; appendix by D. Mumford, "Proof of the convexity theorem," 1326-1329.

3. This is a general construction for divisors with normal crossings. There is a spectral sequence Hq(X,Q^(log D)) =* Hp + q(X ' D ,C), with Q^(logD) = A p(Q^(logD)). Reference:

P. Deligne, Equations Differentielles a Points Singuliers Reguliers, Springer Lecture Notes 163, 1970.

4. For details and generalizations to singular toric varieties, see [Oda, Ch. 3].

5. Construct this inductively over the simplices of S. Note that, on S, Z is defined by the equation 41 < u and Z' by the equation u + 1 <so there is a band between them on each simplex.

6 . For the four isomorphisms use: (i) the first exercise; (ii) duality of cohomology and homology; (iii) the second exercise; ( iv ) the first exercise.

7. The sections of ©x(“ ^Di) are computed as before. For i > 0, u € M, setting Z = (v € |A|: u (v) > 0 }, Z' = [v € |A|: u (v ) + k(v ) < 0 ), andS = (v € |A|: k(v ) = 1), the same argument shows that

Hkx .Q^ -u s H'z,(|A|) S Hi-1(S x z'ns) = Hi_1(zns) ,

which vanishes since ZflS is the intersection of a strongly convex cone with a polyhedral sphere.

8 . For more on the dualizing complex on toric varieties, see [Oda, §3.2].

9. One reason for this is the general Riemann-Roch theorem, which




gives an interpretation for the coefficients of the polynomial. We will discuss this in Chapter 5.

10. For this and generalizations to several divisors, see

A. G. Khovanskii, “Newton polyhedra and the genus of complete intersections," Funct. Anal. Appl. 12 (1978), 38-46.

11. Serre realized in the 1960's, based on the Weil conjectures, that there should be such virtual betti numbers. This motivated the search for weights, mixed Hodge structures, and motives. See pp. 185-191 of

A. Grothendieck, "Recoltes et Semailles: reflexions et temoignage sur un passe de mathematicien," Montpellier, 1985.

For the construction of the mixed Hodge structures, see

P. Deligne, "Theorie de Hodge, II, III" Publ. Math. I.H.E.S. 40 (1971), 5-57, 44 (1974), 5-77.

Although these papers do not explicitly do the construction for cohomology with compact supports, the methods required to do this are similar. This has been carried out, on the dual Borel-Moore homology groups, in

U. Jannsen, "Deligne homology, Hodge-D-conjecture, and motives," in Beilinson's Conjectures on Special values of L-Functions, (M. Rapoport, N. Schappacher, and P. Schneider, eds.), pp. 305-372, Academic Press, 1988.

These virtual polynomials are discussed in

V. I. Danilov and A. G. Khovanskii, "Newton polyhedra and an algorithm for computing Hodge-Deligne numbers," Math. USSR Izv. 29 (1987), 279-298.

A. H. Durfee, "Algebraic varieties which are a disjoint union of subvarieties," Lecture Notes in Pure and Appl. Math. 105 (1987), 99-102.

12. This is proved in [Dani, §14], based on

J. H. M. Steenbrink, "Mixed Hodge structure on the vanishingcohomology," in Real and Complex Singularities, Oslo, 1976, (P. Holm, ed.), pp. 565-678, Sitjthoff & Noordhoff, 1977.

It can also be deduced from the fact that the intersection homology always has a pure Hodge structure, as in the discussion in §5.2.

13. Here is one way to prove this. Note that the equality %(X) =X C(X) is true whenever X is an even-dimensional oriented manifold, since H^(X) and Hdim^ - 1(X) are dual vector spaces. For a general X, take a covering by a finite number of affine open sets Xa ; Mayer-




Vietoris sequences imply that

X(x) = Z ( - i ) r+1x ( x 0Cln . .. n x a r ) ;

Xc(x) = Z ( - D r+1x c( x0Cln . .. n x a r ) .

It therefore suffices to show that %(X) = X C(X) if X is affine. Let Y be the singular locus of X, and U = X ' Y. It suffices by induction on the dimension to show that X(X) = X (Y ) + X(U). Let tt: X X be a resolution of singularities, with Y = t t '^ Y ) a divisor with (strong) normal crossings. It follows by induction on the number of components of Y that X (Y ) = X C(Y ), so X (X ) = X (Y ) +X(U ). There is a neighborhood N of Y in X such that Y is a deformation retract of N and Y is a deformation retract of tt“ *(N). (For example, embed X as a closed subvariety of <Lm so that Y = MfiX for a linear subspace M of (Cm, and take N to be the intersection of X with an e-neighborhood of M; the fact that tt is proper implies that tt-1Y is a deformation retract of tt_1(N).) By Mayer-Vietoris, the equation X (X ) = X (Y )+ X(U) is equivalent to the equation X(tt_1(N) ' Y) = 0. Since tt_1(N) ' Y = N ' Y, it follows that X(N N Y) = 0, and this is equivalent to the equation X(X) = X (Y ) + X(U)-

When Y is a point and N is a small neighborhood of Y, the equation X(N ' Y ) = 0 says that the link of the singularity has zero Euler characteristic. This was noticed by Sullivan:

D. Sullivan, "Combinatorial invariants of analytic spaces,"Springer Lecture Motes 192 (1971), 165-168.

Sullivan has shown that stratified spaces with odd-dimensional strata have vanishing Euler characteristic. S. Weinberger, and M. Goresky and R. MacPherson have verified that this can be used to extend the results of this exercise to arbitrary spaces that can be stratified with even-dimensional strata. For relevant techniques, see

M. Goresky and R. MacPherson, Stratified Morse Theory, Springer-Verlag, 1988.

Chapter 5

1. For general results about intersection theory, we refer to [Fult]. The facts used in the proof of the proposition are proved in [Fult, §1.8-1.9]. Recently the author, R. MacPherson, and B. Sturmfels have shown that the relations among the generators [V (a )l for A^(X) are generated by those of the form [d iv (Xu)l for u in M (t ) , and t a cone of A of dimension n - k - 1 .




2. Assume that a has independent (minimal) generators w j, . . . ,w r , and y has one more generator v = v it and let e be a lift of e to Np so Ny = NCT®Z*e. Take a positive integer integer p so that

P’V = m i*w i + . . . + m r • w r + p*s*e ,

for some integers mj and s = sj. Then

p-mult(y) = [Ny: Z Z*wj + Z-p.v]= [Nct: Z Z’Wj]*[Z*e : 2*p*s*e] = mult(a)-p*s ,

which shows that s = mult(y)/mult((j).

3. For basic facts about intersecting with divisors, in particular the fact that rationally equivalent divisors on X determine rationally equivalent cycles on V, see [Fult, Ch. 2].

4. All the extremal rays in Mori's sense are determined by curves V (cj), and one can see directly whether V (ct) is numerically effective or not. See [Oda, §2.5] and the article by Reid in Note 25 of Chapter 2 for toric constructions of the corresponding contractions of such curves.

5. In general, when X is nonsingular, V*W can be constructed as a rational equivalence class of the expected dimension on VflW . For the construction of intersections on a nonsingular variety, see [Fult, Ch. 8 ].

6 . When X is globally a quotient of a manifold M by a finite group G, then A*(X)q can be identified with the ring of invariants in A*(M)q. When X is only locally such a quotient, the construction is harder, but there are now several ways to extend intersection theory to these varieties (and beyond):

H. Gillet, "Intersection theory on algebraic stacks and Q-varieties," J. of Pure and Appl. Algebra 34 (1984), 193-240.

A. Vistoli, "Alexander duality in intersection theory," Compositio Math. 70 (1989), 199-225.

A. Vistoli, "Intersection theory on algebraic stacks and on the moduli spaces," Invent. Math. 97 (1989), 613-670.

S. Kimura, "On varieties whose Chow groups have intersection products with <Q-coefficients," University of Chicago thesis, 1990.

7. For this we follow [Dani], based on the article by Ehlers in Note 6 of Chapter 2, and

J. Jurkiewicz, "Chow rings of projective non-singular torus embedding," Colloquium Math. 43 (1980), 261-270.

8 . This problem is related to the "shellability" problem for cones. See

A. Bjoerner, M. LasVergnas, B. Sturmfels, N. White, and G. Ziegler,




Oriented Matroids, Cambridge Univ. Press, 1993.

9. To see that Hm(X) = IHm(X) = Hdim(x ) -m(X) when X is an n- dimensionai V-manifold, the point is that intersection homology is calculated by putting local conditions on cycles. There are now quite a number of surveys about intersection homology. The original paper is

M. Goresky and R. MacPherson, "Intersection homology theory," Topology 19 (1983), 135-162.

A recent survey, also emphasizing geometric intuition, is

R. MacPherson, Intersection Homology and Perverse Sheaves,AMS Lecture Notes, 1991.

For Hodge theory and intersection homology, see

M. Saito, "Mixed Hodge modules," Publ. Res. Inst. Math. Sci. (Kyoto Univ.) 26 (1990), 221-333.

10. For the relations between intersection homology betti numbers and the numbers of cones, including statements, proof, and some history, see

K.-H. Fieseler, "Rational intersection cohomology of projective toric varieties," J. Reine Angew. Math. 413 (1991), 88-98.

For more on these questions, see Note 35.

11. Let X —» X be the resolution of singularities considered in §2.6, and let Y be the union of the six lines in X that is mapped to the set Y of six singular points in X. The facts that Hi(X,Y) = Hj(X,Y) = Hj(X)

— 7for i > 2, and that X is the blow-up of P at four points give most of the answer, together with an exact sequence

0 - H3(X) -* H2(Y ) -> H2(X ) — H2(X) 0 .

To finish, the 6 by 5 matrix in the middle is calculated. Note in particular that A*(X) -* H*(X) need not be surjective; replacing X by X x X, one sees that the cycle map need not be surjective in even degrees. For a recipe for calculating the rational homology of a three- dimensional complete toric variety, see

M. McConnell, "The rational homology of toric varieties is not acombinatorial invariant," Proc. Amer. Math. Soc. 105 (1989), 986-991.

12. See [Dani, §10].

13. For Chern classes, Chern character, and Todd class in topology, see [Hirz]. For Chow theory, see [Fult, Ch. 3].

14. For the Hirzebruch-Riemann-Roch formula, as well as a discussion




of Grothendieck's extension to morphisms between smooth projective varieties, see [Hirz], and [Fult, Ch. 15]. For the extension to the singular case of Baum-Fulton-MacPherson, as well as the extension to the non- projective case by Fulton-Gillet, see [Fult, Ch. 18].

15. In the nonsingular case, td d y ) = 1 + Dj + . . . . In the singular case, take a toric resolution of singularities, and note that divisors corresponding to any new edges are mapped to zero. Note in particular that Tdn_i(X ) need not be in the image of Pic(X), since we have seen that Z Di need not be a Q-Cartier divisor. The expression for Tdo(X) follows from the fact that X(X,©x) = 1 when X is complete.

16. In the nonsingular case this can be found in [Dani, §11], based on

A. G. Khovanskii, "Newton polyhedra and toric varieties," Funct. Anal. Appl. 11 (1977), 289-296.

17. Morelli has given general formulas for the coefficients, showing that it is possible to assign the numbers rCT in a way that depends only on ct:

R. Morelli, "Pick's theorem and the Todd class of a toric variety," Advances in Math., to appear.

Pommersheim has given some explicit formulas, relating some of these coefficients to Dedekind sums:

J. E. Pommersheim, "Toric varieties, lattice points and Dedekind sums," Math. Ann. 296 (1993), 1-24.

Generalizations have recently been announced by S. Cappell and J. Shaneson. For an approach using equivariant K-theory and Lefschetz- Riemann-Roch, see

M. Brion, "Points entiers dans les polyedres convexes," Ann. sci.E. N. S. 21 (1988), 653-663.

18. A proof has recently appeared:

G. Barthel, J.-P. Brasseiet, and K.-H. Fieseler, "Classes de Chern des varietes toriques singulieres," C. R. Acad. Sci. Paris 315(1992), 187-192.

19. For a combinatorial description of these numbers and a proof, see

P. Diaconis and W. Fulton, "A growth model, a game, an algebra, Lagrange inversion, and characteristic classes," Rend. Sem. Mat. Torino, in press.

Another proof has been given by Morelli in the paper of Note 17.

20. The coefficients rCT can be taken to be 1 for dim(cr) = 0, 1/2 for dim(cr) = 1, 7/36 for dim(cr) = 2, and 1/6 for dim(cr) = 3. This




gives the formula tfC'U-P) = 1 + (7/Z)v + 2 v2 + (2/3)u3.

21. This follows from the fact that the intersection number can be realized by counting the number of intersections of inverse images of generic hyperplanes in the corresponding projective spaces. For a stronger result, proved jointly with R. Lazarsfeld, see [Fult, §12.2].

22. A modern treatment of the Bertini theorems is given in

J.-P. Jouanolou, Theoremes de Bertini et Applications,Birkhauser Boston, 1983.

23. For a proof and discussion, see [Fult, Ex. 5.2.4]. Note that there is no need to take X to be nonsingular; the surface Y will then only be irreducible and complete, but the inequality (D1/D2)2 £ (Di*Di)(D2*D2 ) is still valid for arbitrary Cartier divisors and D2 , provided at least one has nonnegative self-intersection; for example, one can apply the usual index theorem on a resolution of singularities of Y.

24. The inequalities follow formally from the case k = 2. For example, for k = n = 3,

V (P 1,P2>P3)6 = V (P 1>P2 ,P3)2 'V (P2,P3.Pl)2-V(P3,Pi,P2 )2

* TT V(P,.P,.Pj) .* * j

Taking two of these equal, one sees that

V (P 1,P1,P2) 3 £ V (P 1,P1,P1) 2 .V (P2,P2 .P2 ) = Vo l (P i )2-Vol(P2) ,

which is (b). This gives

V (P 1>P2 (P3)9 £ Vol(P1)2-Vol(P2 )-Vol(P2 )2-Voi(P3 )-Vol(P3 )2-Vol(P1) ,

and taking cube roots gives (c).

25. This can be deduced from the case n = 2, by the same formal manipulations as in the preceding note. For a direct proof for ample divisors, see §5 in

J.-P. Demailly, "A numerical criterion for very ample line bundles,” preprint.

The general case can be deduced from this by replacing E* by Ej + eH* with Hj ample, and letting e go to zero — which is analogous to approximating bounded convex sets by n-dimensional convex polytopes. (Thanks to L. Ein for pointing this out.)

26. Use ( 6 ) with the preceding exercise.

27. For more on mixed volumes, including references, history, and many more inequalities that can be deduced from these formulas, see [BZ, Ch. 4].




28. New results on Newton polyhedra have been found in a series of papers by Gelfand, Kapranov, and Zelevinsky; see, e.g.,

I. M. Gelfand, M. M. Kapranov, and A. V. Zelevinsky, "Newton polytopes of the classical resultant and discriminant," Advances in Math. 84 (1990), 237-254.

29. See

D. N. Bernstein, "The number of roots of a system of equations," Funct. Anal. Appl. 9 (1975), 183-185,

A. G. Kouchnirenko, "Polyedres de Newton et nombres de Milnor,"Invent. Math. 32 (1976), 1-31.

30. This is a result of the author and R. Lazarsfeld, the point being that any non-isolated components must give nonnegative contributions to the total intersection number. For a proof, see [Fult, §12.2].

For a generalization in another direction — to other groups — see

B. Ya. Kazarnovskii, "Newton polyhedra and the Bezout formulafor matrix-valued functions of finite-dimensional representations," Funct. Anal. Appl. 21 (1987), 319-321.

31. The references are

L. J. Billera and C. W. Lee, "A proof of the sufficiency ofMcMullen's condition for f-vectors of simplicial convex polytopes," J. Combin. Theory (A) 31 (1981), 237-255.

R. Stanley, "The number of faces of a simplicial convex polytope," Advances in Math. 35 (1980), 236-238.

32. For a proof of Macaulay's result, see

G. F. Clements and B. Lindstrom, "A generalization of acombinatorial theorem of Macaulay," J. Combin. Th. 7 (1969), 230-238.

33. For a discussion of these problems, with references, see

R. Stanley, "The number of faces of simplicial polytopes andspheres," in Discrete Geometry and Convexity (J. Goodman et al., eds.), Ann. New York Acad. Sci. 440 (1985), 212-223.

C. W. Lee, "Some recent results on convex polytopes,"Contemporary Math. 114 (1990), 3-19.

34. The dual of a simple polytope is simplicial.

35. For more on the relations between toric varieties and face numbers, see

R. Stanley, "Generalized h-vectors, intersection cohomology of toric varieties, and related results," in Commutative




Algebra and Combinatorics, pp. 187-213, Advanced Studies in Pure Math. 11, 1987.

R. Stanley, "Subdivisions and local h-vectors," J. Amer. Math. Soc. 5 (1992), 805-851.

This intrusion of algebraic geometry into the world of polytopes has provided a challenge to combinatorialists. Recently G. Kalai and P. McMullen have announced proofs of Stanley’s theorem that do not depend on toric varieties. McMullen's recent preprint, "On simple polytopes," provides a challenge back to algebraic geometers: to understand better intersection theory on singular toric varieties.



INDEX OF NOTATION

T = Tn the torus associated to N (or M) 3, 17

N, M dual lattices, of rank n 4

N(r N® IR 4

< , > pairing between a space or lattice and its dual 4, 37

ct, t , y (strongly convex rational polyhedral) cones 4, 9, 15

A fan 4, 20

ctv dual of a cone ct 4, 9

Sa semigroup defined by ct, equal to a v f l M 4, 12

(C[S] "group algebra" of semigroup S 4, 16

Ua affine toric va r ie ty of the cone a 4, 16, 54

X (A ) toric va r ie ty of the fan A 4, 20

IR> nonnegative real numbers 6 , 13, 78, 98

Fa Hirzebruch surface 7-8

V, V * dual vector spaces 9

ux perpendicular to facet t 11

Ht half-space determined by t 11

1R«ct subspace spanned by ct 9, 12

dim(cr) dimension of ct, equal to dimdR*cr) 9

ctx vector space perpendicular to a 12

t * CTv f l T x , the dual face to t 12

Z> nonnegative integers 13

x •< ct t is a face of a 15

%u element of C[S] corresponding to u in S 15, 37

Spec(A) affine va r ie ty of A 15

Specm(A) closed points of Spec(A) 16

Homsg semigroup homomorphisms 16

A a group ring C[Sa]; the affine ring of UCT 16

(p# morphism induced by cp: N ‘ -» N 22-23

K° polar set or polar of K 24

Ap fan from polytope P c M|r 26

Xp X (A p ) the toric va r ie ty of Ap 27



xa distinguished point of UCT 28, 37-38

Nct lattice generated by aON 29

N(or) N/Na 29,52

[±m group of m th roots of unity 33

IP(do,... ,dn) twisted or weighted projective space 35

(Gm multiplicative group C* 36

Av one-parameter subgroup C* —> Tn from v € N 37

|A| support of fan A 38

m u lt (a ) multiplicity of a 48

0T orbit corresponding to t 51-56

V ( t ) closure of 0T, X (S tar (x )) 51-56

M (t ) T xfl M 52

Star(x ) Star of cone t 52

a a + ( N t ) r / ( N t ) r 52

Uct(t ) Spec(C[ cF^n M (t )]), open affine in V (t ) 53

tti(X ) fundamental group of X 56-57

H*(X) cohomology group of X 58

%(X) topological Euler characteristic of X 59, 80, 93-95

D* irreducible divisors corresponding to edges of fan 60

vj generators of edges of fan 60

19(D) line bundle (invertible sheaf) of divisor D 60

[D] Weil divisor of Cartier divisor D 60

ordy order along a codimension one subvariety 60

d iv (f ) divisor of a rational function f 60

u (a ) eit. of M/M(ct) X u^ generates 0(D) on Ua 62

Pic(X) Picard group of X 63-65

A kX kth Chow group of X 63, 96

piecewise linear function which is u (a ) on a 66, 68

P d( cj) {u € M r : u > ijjp on a ) 66

Pj) polytope (u € M r : u > ) giving sections of 0(D) 66

cpD map to projective space defined by divisor D 69

H^( ) ith local cohomology group with support in Z 74

HP(X ,T ) pth sheaf cohomology group with coefs. in IF 74

X(X,0(D) ) Z ( - l ) pdim Hp(X,0(D)) 75

X(A )> manifold w ith singular corners of X (A ) 78

152 INDEX OF NOTATION



INDEX OF NOTATION 153

Sfsj compact torus HomtM.S1) 79

[i moment map from X (A ) to M|r 81

p. induced map from X (A )> to polytope in M(r 81

sheaf of i-forms on X 86

Q^(logD) differentials w ith logarithmic poles along D 87

ooy dualizing sheaf of X 89

f p("U) Card(t>*PnM) 90,111

dk number of k-dimensional cones in A 91

Pj (X) j lh betti number of X 91-95

P y (t ) (v ir tua l) Poincare polynomial 92-93

hk p2k (X (A ) ) 92, 126

X C(X) Euler characteristic with compact support 93, 94

V*W intersection cycle or cycle class 97, 99

(D-V) degree of D-V 99

A*(X)q Chow group A n_j(X)®<Q with rational coefficients 99

a*, t j top dimensional cones and associated subcones 101

i(y) smallest i such that a j contains y 102

p(cr) monomial in Dj corresponding to generators of a 106

ch(E) Chern character of vector bundle E 108

td(E) Todd class of vector bundle E 108

Td(X) Todd class of va r ie ty X 108-109

c(E) total Chern class of vector bundle E 109

# (P ) Card(POM) 110

v P [ V ’U : u € P) 110

(Dn) or Dn self-intersection number of a divisor D 111

Vol(P) volume of a polytope P 111

PCT P fl ( crx + u(cr)) 111

P + Q {u + u': u € P, u’ € Q), Minkowski sum 114

V ( P i , . . . , P n) mixed volume of polytopes P^, . . . , Pn 114, 116

B, B(e) unit ball, ball of radius e 115, 120

i(z; F-i, ... ,Fn) intersection multiplicity of Fj at z 122

f^ number of k-dimensional faces of a polytope 125





INDEX

action of torus 19, 23, 31adjunction formula 91affine toric va r ie ty 4 , 16Alexandrov-Fenchel inequality

119ample divisor 70-72, 74, 99arithmetic genus 75, 91A n singularity 47

betti numbers 91-95Bezout theorem 121-124blowing up 6 , 40-43, 50, 71 Borel-Moore homology 103, 108 boundary of a cone 1 0-11

boundary of a polytope 25, 90,111

Brunn-Minkowski inequality120

canonical divisor 85-86, 88

Cartier divisor 60character 37Chern class 59-60, 108-109,

113Chow group 63, 96-101Chow’s Lem m a 72Cohen-Macaulay 30, 31, 73, 89 compatible fan 73complete fan 39

complete toric va r ie ty 39cohomology

betti 58, 93, 101-108, 128

of line bundle 67, 74, 110

convex function 67convex polyhedral cone 9convex polytope,

polyhedron 23, 25

cotangent bundle 8 6 , 108-109 cotangent space 28cube 24, 27, 50, 65, 69-70,

105-106, 114 cyclic quotient singularity 32,

35cyclic polytope 129Dehn-Somerville equations 126 differentials 86

dimension of a cone 9

distinguished point 28, 37-38, 42, 51, 55-56, 61

divisor 60-63dual of a cone 4, 9dual face 12

dualizing sheaf 89duality theorem for cones 9

equivariant Euler characteristic

Euler’s formula exceptional divisor

facefacetfan (in a lattice) Farkas’ theorem fiber bundle

48, 60, 80 59, 80,

93-95 124-126

47

9, 23 10, 24

20 11

29, 41, 70, 93 56-57fundamental group

generated by sections 67-68 generators of a cone 9Gordon’s lemma 12, 30Grothendieck duality 89Gubeladze's theorem 31



156 INDEX

half-space 11

hard Lefschetz theorem 105 Hirzebruch surface 7-8, 43, 70 Hirzebruch-Jung

continued fraction 46

Hirzebruch-Riemann-Roch 109 Hodge index theorem 119-120

interior of a cone 12

intersection homology 94, 105,128

intersection multiplicity 99, 122 intersection product 97-101 intersection ring 106-108inversion formula 91isoperimetric inequalities 121

lattice 4Laurent polynomial 16, 121 line bundle 8 , 44, 59-60, 63-77 local cohomology group 74logarithmic poles 87

lower bound conjecture 129

McMullen conjecture 126

Macaulay vector 127-128

manifold w ith corners 78-80 Minkowski sum 114mixed volume 114-121

moment map 81-85monomials 17, 19

Mori’s program 50

moving lemma (algebraic) 106-

107

multiplicity of cone 48

Newton polytope,

polyhedron 121

Noether’s formula 86

non-projective va r ie ty 71-72,102

nonsingular 28-29normal 29, 73

octahedron 24, 27. 69-70, 114 one-param eter subgroup 36-39 orbifold 34orbit 51-56

Pick’s formula 113piecewise linear function 6 6 , 68

Poincare duality 104polar 24polyhedron 66

polytope 23toric va r ie ty of 23-27

principal divisor 60projective bundle 8 , 42projective space 6-7, 22, 70,

76, 113-114, 123 proper morphism 39-40proper intersection 97-98, 100

quadric cone 5, 17, 27, 49, 72 quotient singularity 31-36,

100,104<Q-Cartier divisor 62, 65, 89

rational cone 20

rational normal curve 32rational ruled surface 8

rational polytope 24rational singularities 47, 76 refinement of fan 45relative interior 12

residue 87, 91resolution of singularities 45-50



INDEX 157

Riemann-Roch 108-114

saturated 18-19, 30self-intersection number 8 ,

4 7 ,111Separation Lemma 13Serre duality 87-91simple polytope 129-130

simplex, simplicial cone 15simplicial fan 34, 65Stanley-Reisner ring 108Stanley's theorem 124-130star of cone 52Steiner decomposition 117strictly convex function 68

strongly convex cone 4, 12, 14

support of Cartier divisor 96-97support of fan 38

surface (nonsingular) 42-44

weighted projective space

35-36Weil conjectures 94Weil divisor, T-Weil divisor 60

T-Cartier divisor T-Weil divisor Todd class toric va r ie ty torustorus action

61-64, 66

60, 63 108-114

4, 20 36, 79

23, 51-56twisted projective space 35-36

upper bound conjecture 129

Veronese embedding 35, 73

v e ry ample divisor 69

virtual Poincare polynomial

92-93

volume 111

V-manifold 34

Documents

Annals of Mathematics Studies Number 131home.ustc.edu.cn/~hanzr/pdf/Introduction to Toric Varieties-Fulton.pdf · relations with commutative algebra and lattice points in polyhedra