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Annals o f Mathematics Studies
Number 131
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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.
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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
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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
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William H. Roever 1874-1951
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Dedicated to the memory of
Jean-Louis and Yvonne Verdier
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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
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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.
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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.
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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
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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
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Introduction to Toric Varieties
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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
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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
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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:
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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
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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
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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.
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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.
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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).
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CONVEX POLYHEDRAL CONES 11
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
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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 cr^OM 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 cr^OT-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
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CONVEX POLYHEDRAL CONES 13
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
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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
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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:
__ ^u + 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
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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
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AFFINE TORIC VARIETIES 17
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,
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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)^u 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
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AFFINE TORIC VARIETIES 19
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
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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
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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
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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
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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 ) ,
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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* : <u ,v> > -1 for all v € K ) .
F* = {u € K° : <u ,v> = -1 V v € F }
Proof. With a the cone over K x 1, the dual cone a '/ consists of
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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 : <u i,v> > -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
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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.
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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.
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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
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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.
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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
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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 a^OM 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.
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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
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SURFACES; QUOTIENT SINGULARIT IES 33
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
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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 m^ei, . . . ,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 .^
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SURFACES; QUOTIENT SINGULARIT IES 35
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
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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
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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.
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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 z^UiV . 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 )) = z^u ,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).
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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
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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
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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.
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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
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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 .
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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^
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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 )
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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:
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RESOLUTION OF SINGULARITIES 47
(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
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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.
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RESOLUTION OF SINGULARITIES 49
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
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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 ).
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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
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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
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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
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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
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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.
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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) >
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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, Tr^Utf) = 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.
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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:
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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 .^
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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
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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 <u ,v i>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
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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.
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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' '
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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
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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
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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 ) = <u (a ),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.
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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 < u j , v j > > - 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
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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
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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
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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 <u (a ),v j> > -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
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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 .
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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
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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°
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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,
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COHOMOLOGY OF LINE BUNDLES 75
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
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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.
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COHOMOLOGY OF LINE BUNDLES 77
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.
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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
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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
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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.
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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
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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.
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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.
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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 < i < n. If
g(x^) denotes the first component of F(x), then (as in the case m = 1
above) the assumptions of case (A * ) are valid for g: IR —> IR. Since g
is one-to-one, the restriction of F to the line is one-to-one.
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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 e^ul ,w ^eiui. 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 eke^uk'x^et - -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.,
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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
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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 < i < k, then
dXt dX2 dXkw * v 1 * * ' * v * ^^k + l » • • • > 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
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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
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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 ) =
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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
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BETTI NUMBERS 91
C ard U n t (u -P )n M ) = ( - l ) n fp(-^u)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
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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).
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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
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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
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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)
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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
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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 s^e 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
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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 ----------\ + < u , V : ( y ) > ) 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)*
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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
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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.
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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 < m, by letting Tj be
the intersection of cr* w ith all those crj that come after cjj (i.e.,
with j > 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.
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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 <u (a),v> 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
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COHOMOLOGY OF NONSINGULAR VARIETIES 103
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
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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
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COHOMOLOGY OF NONSINGULAR VARIETIES 105
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
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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 p^oof, 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:"
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COHOMOLOGY OF NONSINGULAR VARIETIES 107
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 < p < k < q < n .
Take u in M so that < u ,vk > = 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) ,
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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
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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) = (- lVc^E ), 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
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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
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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) ) ;
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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
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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 , . . . , - ! ) ,
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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) ,
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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
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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)
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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 ) •
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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.
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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 < i < k, prove theinequality
V (P , i ;Q ,k - i ;Rk + 1>.. . ,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)
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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 )
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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
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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
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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
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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
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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 .
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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.
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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 < p < m ;
(II) (g i, . . . ,gm) is a Macaulay vector, i.e.,
(a) gp > 0 for 1 < p < m ,
and, if one writes
- ("p’M p'D *-*("/)■with np > 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
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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
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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
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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.
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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
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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
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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
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134 NOTES TO CHAPTER 2
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
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NOTES TO CHAPTER 2 135
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:
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136 NOTES TO CHAPTER 2
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
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NOTES TO CHAPTER 3 137
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
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138 NOTES TO CHAPTER 3
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
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NOTES TO CHAPTER 4 139
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 <u ( ct),vj> > -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 = X^ui 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
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140 NOTES TO CHAPTER 4
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
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NOTES TO CHAPTER 4 141
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-
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142 NOTES TO CHAPTER 5
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 .
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NOTES TO CHAPTER 5 143
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,
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144 NOTES TO CHAPTER 5
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
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NOTES TO CHAPTER 5 145
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
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146 NOTES TO CHAPTER 5
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].
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NOTES TO CHAPTER 5 147
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
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148 NOTES TO CHAPTER 5
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.
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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
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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
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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
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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
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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
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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
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