Cox D., Little J., Schenck H. Toric Varieties (Draft, 2009)(530s)_MAg

Toric Varieties

David Cox

John Little

Hal Schenck

DEPARTMENT OF MATHEMATICS, AMHERST COLLEGE, AMHERST, MA01002

E-mail address: [email protected]

DEPARTMENT OF MATHEMATICS AND COMPUTER SCIENCE, COLLEGE OFTHE HOLY CROSS, WORCESTER, MA 01610


DEPARTMENT OF MATHEMATICS, UNIVERSITY OF ILLINOIS AT URBANA-CHAMPAIGN, URBANA, IL 61801


c

2009, David Cox, John Little and Hal Schenck

Preface

The study of toric varieties is a wonderful part of algebraic geometry that has deepconnections with polyhedral geometry. Our book is an introduction to this richsubject that assumes only a modest knowledge of algebraic geometry. There areelegant theorems, unexpected applications, and, as noted by Fulton [30], toricvarieties have provided a remarkably fertile testing ground for general theories.

The Current Version. The January 2009 version consists of seven chapters: Chapter 1: Affine Toric Varieties Chapter 2: Projective Toric Varieties Chapter 3: Normal Toric Varieties Chapter 4: Divisors on Toric Varieties Chapter 5: Homogeneous Coordinates Chapter 6: Line Bundles on Toric Varieties Chapter 7: Projective Toric Morphisms

These are the chapters included in the version you downloaded. The book also hasa list of notation, a bibliography, and an index, all of which will appear in morepolished form in the published version of the book. Two versions are availableon-line. We recommend using postscript version since it has superior quality.

Changes to the August 2008 Version. The new version fixes some typographicalerrors and has improved running heads. Other additions include:

Chapter 4 includes more sheaf theory. Chapter 4 has an exercise about support functions and tropical polynomials. Chapter 5 now discusses sheaves associated to a graded modules.

iii

iv Preface

The Rest of the Book. Five chapters are in various stages of completion: Chapter 8: The Canonical Divisor of a Toric Variety Chapter 9: Sheaf Cohomology of Toric Varieties Chapter 10: Toric Surfaces Chapter 11: Toric Singularities Chapter 12: The Topology of Toric Varieties

When the book is completed in August 2010, there will be three final chapters: Chapter 13: The Riemann-Roch Theorem Chapter 14: Geometric Invariant Theory Chapter 15: The Toric Minimal Model Program

Prerequisites. The text assumes the material covered in basic graduate courses inalgebra, topology, and complex analysis. In addition, we assume that the readerhas had some previous experience with algebraic geometry, at the level of any ofthe following texts:

Ideals, Varieties and Algorithms by Cox, Little and OShea [17] Introduction to Algebraic Geometry by Hassett [43] Elementary Algebraic Geometry by Hulek [52] Undergraduate Algebraic Geometry by Reid [84] Computational Algebraic Geometry by Schenck [90] An Invitation to Algebraic Geometry by Smith, Kahanpaa, Kekalainen and

Traves [94]Readers who have studied more sophisticated texts such as Harris [40], Hartshorne[41] or Shafarevich [89] certainly have the background needed to read our book.

We should also mention that Chapter 9 uses some basic facts from algebraictopology. The books by Hatcher [44] and Munkres [72] are useful references.

Background Sections. Since we do not assume a complete knowledge of algebraicgeometry, Chapters 19 each begin with a background section that introduces thedefinitions and theorems from algebraic geometry that are needed to understand thechapter. The remaining chapters do not have background sections. For some of thechapters, no further background is necessary, while for others, the material moresophisticated and the requisite background will be provided by careful referencesto the literature.

The Structure of the Text. We number theorems, propositions and equations basedon the chapter and the section. Thus 3.2 refers to section 2 of Chapter 3, andTheorem 3.2.6 and equation (3.2.6) appear in this section. The end (or absence) ofa proof is indicated by , and the end of an example is indicated by .

Preface v

For the Instructor. We do not yet have a clear idea of how many chapters canbe covered in a given course. This will depend on both the length of the courseand the level of the students. One reason for posting this preliminary version onthe internet is our hope that you will teach from the book and give us feedbackabout what worked, what didnt, how much you covered, and how much algebraicgeometry your students knew at the beginning of the course. Also let us know ifthe book works for students who know very little algebraic geometry. We lookforward to hearing from you!

For the Student. The book assumes that you will be an active reader. This meansin particular that you should do tons of exercisesthis is the best way to learnabout toric varieties. For students with a more modest background in algebraicgeometry, reading the book requires a commitment to learn both toric varieties andalgebraic geometry. It will be a lot of work, but its worth the effort. This is a greatsubject.Whats Missing. Right now, we do not discuss the history of toric varieties, nordo we give detailed notes about how results in the text relate to the literature. Wewould be interesting in hearing from readers about whether these items should beincluded.

Please Give Us Feedback. We urge all readers to let us know about: Typographical and mathematical errors. Unclear proofs. Omitted references. Topics not in the book that should be covered. Places where we do not give proper credit.

As we said above, we look forward to hearing from you!

January 2009 David CoxJohn LittleHal Schenck

Contents

Preface iii

Notation xi

Part I: Basic Theory of Toric Varieties 1

Chapter 1. Affine Toric Varieties 31.0. Background: Affine Varieties 31.1. Introduction to Affine Toric Varieties 101.2. Cones and Affine Toric Varieties 221.3. Properties of Affine Toric Varieties 34Appendix: Tensor Products of Coordinate Rings 48

Chapter 2. Projective Toric Varieties 492.0. Background: Projective Varieties 492.1. Lattice Points and Projective Toric Varieties 552.2. Lattice Points and Polytopes 632.3. Polytopes and Projective Toric Varieties 752.4. Properties of Projective Toric Varieties 86

Chapter 3. Normal Toric Varieties 933.0. Background: Abstract Varieties 933.1. Fans and Normal Toric Varieties 1063.2. The Orbit-Cone Correspondence 1153.3. Equivariant Maps of Toric Varieties 1243.4. Complete and Proper 138

vii

viii Contents

Appendix: Nonnormal Toric Varieties 148

Chapter 4. Divisors on Toric Varieties 1534.0. Background: Valuations, Divisors and Sheaves 1534.1. Weil Divisors on Toric Varieties 1694.2. Cartier Divisors on Toric Varieties 1744.3. The Sheaf of a Torus-Invariant Divisor 187

Chapter 5. Homogeneous Coordinates 1935.0. Background: Quotients in Algebraic Geometry 1935.1. Quotient Constructions of Toric Varieties 2025.2. The Total Coordinate Ring 2165.3. Sheaves on Toric Varieties 2245.4. Homogenization and Polytopes 229

Chapter 6. Line Bundles on Toric Varieties 2436.0. Background: Sheaves and Line Bundles 2436.1. Ample Divisors on Complete Toric Varieties 2606.2. The Nef and Mori Cones 2796.3. The Simplicial Case 289Appendix: Quasicoherent Sheaves on Toric Varieties 299

Chapter 7. Projective Toric Morphisms 3037.0. Background: Quasiprojective Varieties and Projective Morphisms 3037.1. Polyhedra and Toric Varieties 3077.2. Projective Morphisms and Toric Varieties 3157.3. Projective Bundles and Toric Varieties 321Appendix: More on Projective Morphisms 332

Chapter 8. The Canonical Divisor of a Toric Variety 3378.0. Background: Reflexive Sheaves and Differential Forms 3378.1. One-Forms on Toric Varieties 3478.2. p-Forms on Toric Varieties 3528.3. Fano Toric Varieties 352

Chapter 9. Sheaf Cohomology of Toric Varieties 3539.0. Background: Cohomology 3539.1. Cohomology of Toric Line Bundles 3659.2. Serre Duality 380

Contents ix

9.3. The Bott-Steenbrink-Danilov Vanishing Theorem 3859.4. Local Cohomology and the Total Coordinate Ring 385Appendix: Introduction to Spectral Sequences 395

Topics in Toric Geometry 401

Chapter 10. Toric Surfaces 40310.1. Singularities of Toric Surfaces and Their Resolutions 40310.2. Continued Fractions and Toric Surfaces 41210.3. Grobner Fans and McKay Correspondences 42310.4. Smooth Toric Surfaces 43310.5. Riemann-Roch and Lattice Polygons 441

Chapter 11. Toric Singularities 45311.1. Existence of Resolutions 45311.2. Projective Resolutions 46111.3. Blowing Up an Ideal Sheaf 46111.4. Some Important Toric Singularities 461

Chapter 12. The Topology of Toric Varieties 463

Chapter 13. The Riemann-Roch Theorem 465

Chapter 14. Geometric Invariant Theory 467

Chapter 15. The Toric Minimal Model Program 469

Bibliography 471

Index 477

Notation

Basic Notions

integers, rational numbers, real numbers, complex numbers

semigroup of nonnegative integers

image and kernel

!

"#

direct limit

!

$

"

inverse limit

Rings and Varieties

&% ')(*',+.-

polynomial ring in / variables&%0% '

(

*'

+

-0-

formal power series ring in / variables&% '21

(

(

*'

1

(

+

-

ring of Laurent polynomials3547698

affine or projective variety of an ideal:.4;-

coordinate ring of an affine or projective variety&%>;-?

graded piece in degree @ when ; is projective&4

xii Notation

E

completion of local ringE

local ring of a variety at a point and its maximal ideal

4

Notation xiii

Y 4.8

-dimensional cones of Y minimal generator of 9 ? , .

YS4 8

Y

maximal cones ofY

?

3

sublattice 4 6:9 ? 8 RBC+B

)4

6

8

9-?

?

4

6

8

quotient lattice ? ?3

;

4

6

8 dual lattice of ? 4 6 8 , equal to 6 I 9-;B

D

B

4

6

8

star of 6 , a fan in ?4

6

8

Y

M

4

6

8

star subdivision of Y along 6 4

6

8

index of a simplicial cone

Polyhedra

=

I> 4

xiv Notation

$

projective toric variety of a lattice polytope or polyhedron

toric variety of a basepoint free divisor

@

lattice homomorphism of a toric morphism 6 6

and its real extension

3

distinguished point of 43

4

6

8

orbit of 6 . Y; 4

H

8

R

4

6

8

closure of orbit of 6 . Y , toric variety ofB

D

B

4

6

8

R

4

8

torus-invariant prime divisor on 6 of . YS4 8

torus-invariant prime divisor on $ of facet ! 4 $

affine toric variety of recession cone of a polyhedron4 6

affine toric variety of a fan with convex support

Specific Varieties

+

+

affine and projective / -dimensional space 4+ 8

weighted projective space

M

multiplicative group of nonzero complex numbers S 47

M

8

+

standard / -dimensional torus

?.

?

rational normal cone and curve

47

+

8 blowup of +

at the origin

46 8 blowup of 6 along ; 4 H 8 , toric variety of Y M 4 H 8

Notation xv

G !D.4 8

Picard group of a normal variety B C C

48

support of a divisor

restriction of a divisor to an open set

4 4

'

O

'

8

local data of a Cartier divisor on

,

3

3

6 Cartier data of a torus-invariant Cartier divisor on 6

polyhedron of a torus-invariant divisorY

fan associated to a basepoint free divisor $ Cartier divisor of a polytope or polyhedron&

support function of a Cartier divisorB

4 Y

?

8

support functions integral with respect to ?

Intersection Products

24 8 degree of a divisor on a curve

intersection product of Cartier divisor and complete curve

Z

Z

numerically equivalent Cartier divisors and complete curves?

(

4 8

?

(

4 8 4

=

) )4 8

8N

*

and 4 ( 4 8 F8N*

T4 8

cone in ?(

4 8

generated by nef divisors 4 8

cone in ?( 4 8

generated by complete curves 4 8

Mori cone, equal to the closure of 4 8

G !D.4 8

@

G !D.4 8

*

4 A8

primitive relation of a primitive collection

Sheaves and Bundles

"

structure sheaf of a variety

M

"

sheaf of invertible elements of

"

#"

4 8

sheaf of a Weil divisor on

"

constant sheaf of rational functions when

is irreducible

restriction of a sheaf to an open set 4 4

8

sections of a sheaf over an open set

ideal sheaf of a subvariety !

; sheaf onBC

D47EF8

of anE

-module ;

sheaf on 6

of a graded L -module

"!

4#"8

sheaf on 6

of the graded L -module L 4#"8

stalk of a sheaf at a point

%$&('

tensor product of sheaves of

"

-modules

xvi Notation

Part I: Basic Theory ofToric Varieties

Chapters 1 to 9 introduce the theory of toric varieties. This part of thebook assumes only a minimal amount of algebraic geometry, at the levelof Ideals, Varieties and Algorithms [17]. Each chapter begins with a back-ground section that develops the necessary algebraic geometry.

1

Chapter 1

Affine Toric Varieties

1.0. Background: Affine VarietiesWe begin with the algebraic geometry needed for our study of affine toric varieties.Our discussion assumes Chapters 15 and 9 of [17].

Coordinate Rings. An ideal 6 ! L R F% '(*',+.- gives an affine variety354768

R

.

+

O

4

8

R

for all O .6

and an affine variety ; ! +

gives the ideal:.4;S-

R

L

:.4;-

can be interpreted as the -valued polynomial functions on ; .Note that

A%>;S-

is a

-algebra, meaning that its vector space structure is compatiblewith its ring structure. Here are some basic facts about coordinate rings:

A%>;S-

is an integral domain :.4; - &%>; ( - , where

M

498

R

for

.

&%>;-

.

Two affine varieties are isomorphic if and only if their coordinate rings areisomorphic

-algebras.

3

4 Chapter 1. Affine Toric Varieties

A point of an affine variety;

gives the maximal ideal

O

.

F%>; -

O

4

8

R

!

F%>; -; - arise this way.Coordinate rings of affine varieties can be characterized as follows (Exercise 1.0.1).Lemma 1.0.1. A -algebra E is isomorphic to the coordinate ring of an affinevariety if and only if E is a finitely generated -algebra with no nonzero nilpotents,i.e., if O . E satisfies O

P

R

for some , W , then O R .

To emphasize the close relation between ; and &%>;- , we sometimes write

(1.0.1) ; R BC D47&%>; - 8

This can be made canonical by identifying ; with the set of maximal ideals ofF%>; -

via the fourth bullet above. More generally, one can take any commutativering

E

and define the affine scheme BC D.47EF8 . The general definition of Spec usesall prime ideals of E , not just the maximal ideals as we have done. Thus someauthors would write (1.0.1) as ; R BC D 47A%>;S- 8 , the maximal spectrum of A%>;S- .Readers wishing to learn more about affine schemes should consult [26] and [41].

The Zariski Topology. An affine variety ; ! +

has two topologies we will use.The first is the classical topology, induced from the usual topology on

+

. Thesecond is the Zariski topology, where the Zariski closed sets are subvarieties of ;

(meaning affine varieties of +

contained in ; ) and the Zariski open sets are theircomplements. Since subvarieties are closed in the classical topology (polynomialsare continuous), Zariski open subsets are open in the classical topology.

Given a subset L ! ; , its closure L in the Zariski topology is the smallestsubvariety of ; containing L . We call L the Zariski closure of L . It is easy to giveexamples where this differs from the closure in the classical topology.

Affine Open Subsets and Localization. Some Zariski open subsets of an affinevariety

;

are themselves affine varieties. Given O . &%>;- SX , let;,N

R

.

;

O

4

8

P

R

!

;M

Then;,N

is Zariski open in;

and is also an affine variety, as we now explain.Let

;

!

+

have:.4

1.0. Background: Affine Varieties 5

give rational functions on;

. Then let(1.0.2) A%>;S- N R O

.

A4;S-;S- N 8

is the affine variety;2N

.

Example 1.0.2. The / -dimensional torus is the affine open subset47

M

8

+

R

+

S

354 ')( J',+98

!

+

with coordinate ringA% ' ( *' + -

R

A% ' 1

(

(

*' 1

(

+

-;S- N

from (1.0.2) is an example of localization. In Exercises 1.0.2and 1.0.3 you will show how to construct this ring for all affine varieties, not justirreducible ones. The general concept of localization is discussed in standard textsin commutative algebra such as [2, Ch. 3] and [25, Ch. 2].

Normal Affine Varieties. Let E be an integral domain with field of fractions .Then

E

is normal, or integrally closed, if every element of which is integralover

E (meaning that it is a root of a monic polynomial in E % ' - ) actually lies in E .For example, any UFD is normal (Exercise 1.0.5).Definition 1.0.3. An irreducible affine variety ; is normal if its coordinate ringF%>; -

is normal.

For example,

+

is normal since its coordinate ring &% '( *' +.- is a UFDand hence normal. Here is an example of a non-normal affine variety.

Example 1.0.4. Let

R

354 'R

8

!

. This is an irreducible plane curvewith a cusp at the origin. It is easy to see that

F%

-

R

A% '

-

K

'

R

L

. Nowlet

'

and be the cosets of ' and in &%

-

respectively. This gives '

.

F4

8

.

A computation shows that '

.

&%

-

and that 4 'V8

R

'

. ConsequentlyF%

-

and hence

are not normal.We will see below that

is an affine toric variety.

An irreducible affine variety ; has a normalization defined as follows. LetF%>; -

Z

R

"

.

F4


Example 1.0.5. We saw in Example 1.0.4 that the curve

!

defined by'

R

has elements '

.

&%

-

such that '

.

F%

-

is integral overF%

-

. InExercise 1.0.6 you will show that A% ' - ! F4

8

is the integral closure ofF%

-

and that the normalization map is the map

defined by 4

8

.

At first glance, the definition of normal does not seem very intuitive. Once weenter the world of toric varieties, however, we will see that normality has a verynice combinatorial interpretation and that the nicest toric varieties are the normalones. We will also see that normality leads to a nice theory of divisors.

In Exercise 1.0.7 you will prove some properties of normal domains that willbe used in 1.3 when we study normal affine toric varieties.

Smooth Points of Affine Varieties. In order to define a smooth point of an affinevariety

;

, we first need to define local rings and Zariski tangent spaces. When ;

is irreducible, the local ring of ; at is

R

O

.

&4; -

and 24 8 PR .

Thus

consists of all rational functions on;

that are defined at . Inside of

we have the maximal ideal

R

.

4

8

R

.

In fact,

is the unique maximal ideal ofX

, so that

is a local ring.Exercises 1.0.2 and 1.0.4 explain how to define

when;

is not irreducible.The Zariski tangent space of ; at is defined to be

4


Points lying in the intersection of two or more irreducible components of ;

are always singular ([17, Thm. 8 of Ch. 9, 6]).Since

!

47

+

8

R

/ for every .

+

, we see that

+

is smooth. For anirreducible affine variety ; !

+

of dimension @ , fix . ; and write :.4;S- isnormal once we prove that

X

is normal for all .;

. Hence it suffices to showthat

is normal whenever is smooth.This follows from some powerful results in commutative algebra:

is aregular local ring when is a smooth point of

; (see above), and every regularlocal ring is a UFD (see [25, Thm. 19.19]). Then we are done since every UFD isnormal. A direct proof that

X

is normal at a smooth point .;

is sketched inExercise 1.0.10.

The converse of Propostion 1.0.9 can fail. We will see in 1.3 that the affinevariety

354 '

RS8

!

is normal, yet354 '

R8

is singular at the origin.

Products of Affine Varieties. Given affine varieties ;( and ; , there are severalways to show that the cartesian product ; ( ; is an affine variety. The most directway is to proceed as follows. Let ; ( !

P

R

B9C

D.47A% ')(*' P - 8

and ; !

+

R

BC

D47A%

(

+

- 8

. Take:.4


A fancier method is to use the mapping properties of the product. This willalso give an intrinsic description of its coordinate ring. Given ; ( and ; as above,;(# ;

should be an affine variety with projections ' ;V( ; ; ' such thatwhenever we have a diagram

""

$$

$$;( ;

//

;(

;

where

'

;

' are morphisms from an affine variety , there should bea unique morphism $ (the dotted arrow) that makes the diagram commute, i.e.,

'

$

R

'

. For the coordinate rings, this means that whenever we have a diagram&%>; -

F%>;(*-

//

..

A%>;( ; -

%%F%

-

with

-algebra homomorphisms

M

'

&%>;

'

- A%

-

, there should be a unique -algebra homomorphism $ M (the dotted arrow) that makes the diagram commute. Bythe universal mapping property of the tensor product of -algebras, A%>; ( - 1 F%>; -

has the mapping properties we want. Since F%>; (*- =&%>; - is a finitely generated

-algebra with no nilpotents (see the appendix to this chapter), it is the coordinatering

F%>;( ; -

. For more on tensor products, see [2, pp. 2427] or [25, A2.2].Example 1.0.10. Let ; be an affine variety. Since

+

R

BC

D.47A%

(

+ - 8

, theproduct ;

+

has coordinate ringA%>;S-

&%

(

+

-

R

A%>;S- %

(

+

-;- %

1

(

(

1

(

+

-


Example 1.0.11. Consider

R

. By definition, a basis for the productof the Zariski topologies consists of sets

4 ( 4

where4

' are Zariski open in

. Such a set is the complement of a union of collections of horizontal andvertical lines in

. This makes it easy to see that Zariski closed sets in

suchas354

R '

8

cannot be closed in the product topology.

Exercises for 1.0.1.0.1. Prove Lemma 1.0.1. Hint: You will need the Nullstellensatz.

1.0.2. Let be a commutative -algebra. A subset is a multipliciative subsetprovided , , and is closed under multiplication. The localization consistsof all formal expressions , , , modulo the equivalence relation

!"$#%&')(* for some +(a) Show that the usual formulas for adding and multiplying fractions induce well-defined

binary operations that make into -algebra.(b) If has no nonzero nilpotents, then prove that the same is true for .For more on localization, see [2, Ch. 3] or [25, Ch. 2].1.0.3. Let be a finitely generated -algebra without nilpotents as in Lemma 1.0.1 andlet ,-. be nonzero. Then (0/1123,23,54&26+6+7+98 is a multiplicative set. The localization: is denoted :; and is called the localization of at , .(a) Show that ; is a finitely generated -algebra without nilpotents.(b) Show that ; satisfies =?7@1A ; 'B(C=D?E@1F' ; .(c) Show that ; is given by (1.0.2) when is an integral domain.1.0.4. Let G be an affine variety with coordinate ring IH G:J . Given a point KLMG , let%(N/7H GJPO&QRKS'UT(VW8 .

(a) Show that is a multiplicative set. The localization IH G:J is denoted XZYW[ \ and iscalled the local ring of G at K .

(b) Show that every ]%X^YW[ \ has a well-defined value ]PRK' and that_

YW[ \:(N/]%XZY>[ \O]PRKS'B(V`8

is the unique maximal ideal of XaYW[ \ .(c) When G is irreducible, show that XaY>[ \ agrees with the definition given in the text.1.0.5. Prove that a UFD is normal.

1.0.6. In the setting of Example 1.0.5, show that HAbc bd JefFg' is the integral closure ofH gUJ and that the normalization -hig is defined by kjhlA 4m2npo7' .

1.0.7. In this exercise, you will prove some properties of normal domains needed for 1.3.(a) Let be a normal domain with field of fractions q and let .f be a multiplicative

subset. Prove that the localization is normal.(b) Let :r , stu , be normal domains with the same field of fractions q . Prove that the

intersection vr`wmx

yr is normal.

1.0.8. Prove that zW{}|f~ \ A')(V for all K .


1.0.9. Use Lemma 1.0.6 to prove the claim made in the text that smoothness is determinedby the rank of the Jacobian matrix (1.0.3).1.0.10. Let G be irreducible and suppose that K%G is smooth. The goal of this exerciseis to prove that X^Y>[ \ is normal using standard results from commutative algebra. Set (zW{}|VG and consider the ring of formal power series H H d 26+7+6+92 d

J}J . This is a local ringwith maximal ideal _ (

d

26+7+6+92

d

. We will use three facts:

IH}H

d

27+6+7+92

d

J J is a UFD by [102, p. 148] and hence normal by Exercise 1.0.5. Since K G is smooth, [70, 1C] proves the existence of a -algebra homomorphismXZYW[ \h IH}H

d

26+6+7+62

d

J}J that induces isomorphisms

X YW[ \

_

Y>[ \

IH H

d

26+7+6+72

d

J}JA

_

for all f . This implies that the completion [2, Ch. 10]

X

YW[ \

( {}|

X

Y>[ \

_

YW[ \

is isomorphic to a formal power series ring, i.e.,

X

Y>[ \

H H

d

26+7+6+62

d

J}J . Such anisomorphism captures the intuitive idea that at a smooth point, functions should havepower series expansions in local coordinates d 26+7+6+92 d

.

If NX YW[ \ is an ideal, then

(v

_

YW[ \

'9+

This theorem of Krull holds for any ideal in a Noetherian local ring u and followsfrom [2, Cor. 10.19] with (Vu: .

Now assume that KG is smooth.(a) Use the third bullet to show that XaY>[ \h IH}H d 26+7+6+72 d

J}J is injective.(b) Suppose that 2 X YW[ \ satisfy O in IH}H d 26+7+6+72 d

J}J . Prove that O in X YW[ \ . Hint:Use the second bullet to show XaY>[ \ _

Y>[ \

and then use the third bullet.(c) Prove that X^Y>[ \ is normal. Hint: Use part (b) and the first bullet.This argument can be continued to show that XaYW[ \ is a UFD. See [70, (1.28)]1.0.11. Let G and be affine varieties and let VCG be a subset. Prove that ! " (# .

1.0.12. Let G and be irreducible affine varieties. Prove that G$ % is irreducible. Hint:Suppose G& % ((' *) '

4

, where ' 2'4

are closed. Let G,+ ( /.- G OW/.-D8/ % '0+ 8 .Prove that GM(CG 1) G

4

and that G,+ is closed in G . Exercise 1.0.11 will be useful.

1.1. Introduction to Affine Toric VarietiesWe first discuss what we mean by torus and then explore various constructionsof affine toric varieties.

The Torus. The affine variety 47 M 8+

is a group under component-wise multipli-cation. A torus is an affine variety isomorphic to

47

M

8

+

, where inherits agroup structure from the isomorphism. Associated to are its characters and one-parameter subgroups. We discuss each of these briefly.

1.1. Introduction to Affine Toric Varieties 11

A character of a torus is a morphism '

M that is a group homomor-phism. For example, , R

4

*

(

*

+ 8

.

+

gives a character'

P

47

M

8

+

M

defined by(1.1.1) '

P

4

(

+ 8

R

(

+

One can show that all characters of 47 M 8+

arise this way (see [53, 16]). Thus thecharacters of

47

M

8

+

form a group isomorphic to

+

.

For an arbitrary torus , its characters form a free abelian group ; of rankequal to the dimension of . It is customary to say that ,. ; gives the character'

P

M

.

We will need the following results concerning tori (see [53, 16] for proofs).Proposition 1.1.1.(a) Let ( and be tori and let ( be a morphism that is a group

homomorphism. Then the image of is a torus and is closed in .(b) Let be a torus and let O ! be an irreducible subvariety of that is a

subgroup. Then O is a torus.

Now assume that a torus acts linearly on a finite dimensional vector space over

, where the action of . on

.

is denoted . Given ,. ; ,we get the eigenspace

P

R

.

R

'

P

4

8

for all . .

If P

P

R

, then every

.

P

S=

is a simultaneous eigenvector for all

.

, with eigenvalue given by 'P

4

8

.

Proposition 1.1.2. In the above situation, we have R P P .

This proposition is a sophisticated way of saying that a family of commutingdiagonalizable linear maps can be simultaneously diagonalized.

A one-parameter subgroup of a torus is a morphism )

M

that is agroup homomorphism. For example, R

4

(

+98

.

+

gives a one-parametersubgroup )

M

47

M

8

+

defined by(1.1.2) )

4

8

R

4

8

All one-parameter subgroups of 47 M 8+

arise this way (see [53, 16]). It followsthat the group of one-parameter subgroups of 47 M 8

+

is naturally isomorphic to

+

.

For an arbitrary torus , the one-parameter subgroups form a free abelian group? of rank equal to the dimension of . As with the character group, an element

.

? gives the one-parameter subgroup )

M

.

There is a natural bilinear pairing K L

;

?

defined as follows. (Intrinsic) Given a character '

P

and a one-parameter subgroup )

, the com-position

'

P

)

M

M is character of

M

, which is given by

forsome

.

. Then K ,

L

R

.


(Concrete) If R 47 M 8+

with , R4

*

(

*

+ 8

.

+

,

R

4

(

+ 8

.

+

, then one computes that

(1.1.3) K , L R+

')(

(

*

'

'

i.e., the pairing is the usual dot product.It follows that the characters and one-parameter subgroups of a torus form

free abelian groups ; and ? of finite rank with a pairing K L

;

?

that identifies ? with , I.* 4

;

JX8

and ; with , I.* 4

?

JX8

. In terms of tensorproducts, one obtains a canonical isomorphism ? 2*S M via )

4

8

.

Hence it is customary to write a torus as 5

.

From this point of view, picking an isomorphism 5

47

M

8

+

induces dualbases of ; and ? , i.e., isomorphisms ;

+

and ? +

that turn charactersinto Laurent monomials (1.1.1), one-parameter subgroups into monomial curves(1.1.2), and the pairing into dot product (1.1.3).

The Definition of Affine Toric Variety. We now define the main object of study ofthis chapter.

Definition 1.1.3. An affine toric variety is an irreducible affine variety ; contain-ing a torus

5

47

M

8

+

as a Zariski open subset such that the action of 5

on

itself extends to an action of 5

on;

.

Obvious examples of affine toric varieties are 47 M 8+

and +

. Here are someless trivial examples.

Example 1.1.4. The plane curve

R

354 ' #R

8

!

has a cusp at the origin.This is an affine toric variety with torus

R

R

9

47

M

8

R

4

8

.

M

M

where the isomorphism is4

8

. Example 1.0.4 shows that

is a non-normal toric variety.

Example 1.1.5. The variety;

R

354 '

R 8

!

is a toric variety with torus;

9

47

M

8

R

4

(

(

"

(

8

' .

M

47

M

8

where the isomorphism is4

(

(

"

(

894

(

8

. We will see later that;

is normal.

Example 1.1.6. Consider the surface in ?

U

(

parametrized by the map

R+

?

U

(

defined by 4 8 4?

?

"

(

?

"

(

?

8

. Thus is defined using all degree@ monomials in

.


Let the coordinates of ?

U

(

be ' *'2? and let 6 ! &% ' *',?- be theideal generated by the minors of the matrix

' ' ( '2?

"

',?

"

(

')( ' '2?

"

( ',?

so6

R

K

'

'

'

U

( R '

'

U

(J'

@

R

L

. In Exercise 1.1.1 you will verifythat

47

8

R

354768

, so that

?

R

47

8

is an affine variety. You will also provethat

:.4

?8

R

6

, so that6

is the ideal of all polynomials vanishing on

?

. It followsthat

6

is prime since3547698

is irreducible by Proposition 1.1.8 below. The affinesurface

?

is called the rational normal cone of degree @ and is an example of adeterminantal variety. We will see below that 6 is a toric ideal.

It is straightforward to show that

?

is a toric variety with torus

4*47

M

8

8

R

?

9

47

M

8

?

U

(

47

M

8

We will study this example from the projective point of view in Chapter 2. We next explore three equivalent ways of constructing affine toric varieties.

Lattice Points. In this book, a lattice is a free abelian group of finite rank. Thus alattice of rank / is isomorphic to

+

. For example, a torus 5

has lattices ; (ofcharacters) and ? (of one-parameter subgroups).

Given a torus 5

with character lattice ; , a set$

R

,

(

,

"

! ;

gives characters'

P

5

M

. Then consider the map

(1.1.4) 3 5 R "

defined by

34

8

R

'

P

4

8 ('

P

4

8

.

"

Definition 1.1.7. Given a finite set $ ! ; , the affine toric variety 3 is definedto be the Zariski closure of the image of the map 3 from (1.1.4).

This definition is justified by the following proposition.Proposition 1.1.8. Given $ ! ; as above, let %$ ! ; be the sublatticegenerated by $ . Then "3 is an affine toric variety whose torus has characterlattice %$ . In particular, the dimension of 3 is the rank of %$ .Proof. The map (1.1.4) can be regarded as a map

3

5

R 47

M

8

"

of tori. By Proposition 1.1.1, the image R 34

5

8

is a torus that is closed in47

M

8F"

. The latter implies that 3

9

47

M

8F"

R

since3

is the Zariski closureof the image. It follows that the image is Zariski open in

3

. Furthermore, isirreducible (it is a torus), so the same is true for its Zariski closure 3 .


We next consider the action of . Since ! 47 M 8F" , an element . acts on

"

and takes varieties to varieties. Then

R

!

3

shows that

3 is a variety containing . Hence

3

!

3 by the definitionof Zariski closure. Replacing with "

(

leads to 3 R 3 , so that the action of induces an action on "3 . We conclude that 3 is an affine toric variety.

It remains to compute the character lattice ;4

8

of . Since R 34 5

8

,

the map 3

gives the commutative diagram

5

//

"" ""FFF

FFFF

FF

47

M

8 "

?

OO

where denotes a surjective map and an injective map. This diagram of toriinduces a commutative diagram of character lattices

;

"

oo

;

4

8

2 R

ddHHHHHHHHH

Since

3

"

; takes the standard basis ( " to , ( , " , the imageof

3

is%$

. By the diagram, we obtain ; 4 8 %$ . Then we are done sincethe dimension of a torus equals the rank of its character lattice.

In concrete terms, fix a basis of ; , so that we may assume ; R +

. Thenthe

vectors in$

!

+

can be regarded as the columns of an / matrix

with integer entries. In this case, the dimension of 3 is simply the rank of thematrix

.

We will see below that every affine toric variety is isomorphic to 3 for somefinite subset $ of a lattice.

Toric Ideals. Let 3 ! "

R

B9C

D47&% ')(*'

"

- 8 be the affine toric varietycoming from a finite set

$

R

,

(

,

"

! ; . We can describe the ideal:.4 3=8

!

A% ')( *'

"

-

as follows. As in the proof of Proposition 1.1.8, (1.1.4)induces a map of character lattices

3

"

R

;

that sends the standard basis ( " to , ( , " . Let be the kernel of thismap, so that we have an exact sequence

R+

R+

"

R+

;


In down to earth terms, elements R 4 ( " 8 of satisfy &"

'(

(

',-'

R

andhence record the linear relations among the , ' .

Given R 4 ( " 8 . , set

U

R

'

' and " R R

'

'

Note that

R

U

R

" and that U " . "

. It follows easily that the binomial'

R '

R

'

'

R

'

"

'

vanishes on the image of (1.1.4) and hence on 3 since 3 is the Zariski closureof the image.

Proposition 1.1.9. The ideal of the affine toric variety 3 ! " is:.4 3=8

R

'

R '

.

R

'

R '

"H

.

"

and " R .

Proof. We leave it to the reader to prove equality of the two ideals on the right(Exercise 1.1.2). Let 6 denote this ideal and note that 6 ! :.4 3 8 . We provethe opposite inclusion following [97, Lem. 4.1]. Pick a monomial order onF% ' (*'

"

-

and an isomorphism 5

47

M

8

+

. Thus we may assume ; R

+

and the map

47

M

8

+

"

is given by Laurent monomials P

in variables

(

+

. If6

P

R

:.4

3

8

, then we can pick O .:.4

3

8

S

6

with minimal leadingmonomial

'

R

"

'(

(

'

'

. Rescaling if necessary,'

becomes the leading termof O .

Since O 4 P

P

8

is identically zero as a polynomial in ( + , theremust be cancellation involving the term coming from ' . In other words, O mustcontain a monomial

'

R

"

'(

(

'

'

'

such that"

'(

(

4

P

8

R

"

')(

(

4

P

8

This implies that"

'(

(

*

',-'

R

"

'(

(

' ,-'

so that" R

R

&

"

'(

(

4

*

'

R

'

8

' .

. Then'

R'

.

6 by the second descriptionof6

. It follows that OR '

'

also lies in:.4 3 8

S

6

and has strictly smallerleading term. This contradiction completes the proof.

Given a finite set $ ! ; , there are several methods to compute the ideal:.4 3=8

R

6

of Proposition 1.1.9. For simple examples, the rational implicitizationalgorithm of [17, Ch. 3,3] can be used. It is also possible to compute 6> using abasis of and ideal quotients (Exercise 1.1.3). Further comments on computing6

can be found in [97, Ch. 12].Inspired by Proposition 1.1.9, we make the following definition.


Definition 1.1.10. Let ! " be a sublattice.(a) The ideal 6 R ' R '

"H

.

"

and "-R . is called a lattice ideal.(b) A prime lattice ideal is called a toric ideal.

Since toric varieties are irreducible, the ideals appearing in Proposition 1.1.9are toric ideals. Examples of toric ideals include:

Example 1.1.4

K

'

R

L

!

&% '

-

Example 1.1.5 K ' GR L ! A% ' X-

Example 1.1.6

K

'

'

'

U

( R '

'

U

(*'

@

R

L

!

A% ' *',? -


Given an affine semigroup2

! ; , the semigroup algebra F%2

-

is the vectorspace over

with2

as basis and multiplication induced by the semigroup structureof2

. To make this precise, we think of ; as the character lattice of a torus 5

, so

that ,. ; gives the character'

P

. Then&%

2

-

R

P

P '

P

P

.

and

P

R

for all but finitely many ,

with multiplication induced by'

P

'

P

R

'

P

U

P

If2

R

$

for$

R

,

(

,

"

, thenF%

2

-

R

A% '

P

('

P

-

.

Here are two basic examples.Example 1.1.12. The affine semigroup

+

!

+

gives the polynomial ringA%

+

-

R

F% '

(

*'

+

-


This proves thatBC

D.47A%

2

- 8

R

"3

. Since2

R

#$

implies

2

R

%$

, the torusof 3

R

BC

D47A%

2

- 8

has the desired character lattice by Proposition 1.1.8.

Here is an example of this proposition.

Example 1.1.15. Consider the affine semigroup2

!

generated by and , sothat

2

R

. To study the semigroup algebra F%2

-

, we use (1.1.5). If$

R

, then 34

8

R

4

8

and the toric ideal is :.4 3 8 R K ' R

L byExample 1.1.4. Hence

F%

2

-

R

&%

-

A% '

-

K

'

R

L

and the affine toric variety 3 is the curve '

R

.

Equivalence of Constructions. We can now state the main result of this section,which asserts that our various approaches to affine toric varieties all give the sameclass of objects.Theorem 1.1.16. Let ; be an affine variety. The following are equivalent:(a) ; is an affine toric variety according to Definition 1.1.3.(b) ; R 3 for a finite set $ in a lattice.(c) ; is an affine variety defined by a toric ideal.(d) ; R BC D47A% 2 - 8 for an affine semigroup 2 .Proof. The implications (b) (c) (d) (a) follow from Propositions 1.1.8,1.1.9 and 1.1.14. For (a) (d), let ; be an affine toric variety containing the torus

5

with character lattice ; . Since the coordinate ring of 5

is the semigroupalgebra A% ; - , the inclusion

5

!

;

induces a map of coordinate rings&%>;-8R+ A%

;

-;S-as a subalgebra of A% ; - . Let

2

R

,.

;

'

P

.

A%>;S-

and note that2

is a semigroup. We will show that2

is finitely generated withsemigroup algebra equal to F%>;- . This will complete the proof of the theorem.

The inclusionF%

2

-

!

A%>;S-

is obvious. For the opposite inclusion, pick OQPR inA%>;S-

. Using&%>;-

!

F%

;

-

, we can write

O

R

P

P '

P

where

!; is finite and

P

P

R

for all ,.

. Let

R

BC+B

)4'

P

,.

8

!

F%

;

-

be the subspace spanned by the 'P

. Thus O . 9F%>; -

. If . 5

, then the actionof on

;

induces an action on the coordinate ring F%>; - , so that we get O . A%>;S- .


By Definition 1.1.3, this extends the usual action of on 5

, which in terms of thecoordinate ring F% ; - is given by '

P

R

'

P

4

8 '

P

. It follows that and hence 9

F%>; -

are stable under the action of 5

. Since 9 A%>;S- is finite-dimensional,Proposition 1.1.2 implies that 9

A%>;S-

is spanned by simultaneous eigenvectorsof

5

. But this is taking place inF%

;

-

, where the simultaneous eigenvectors arethe characters! It follows that 9

A%>;S-

is spanned by characters. Then the aboveexpression for O . 9

A%>;S-

implies that'

P

.

&%>;-

for , .

. It follows thatO

.

F%

2

-

, proving thatA%>;S-

R

A%

2

-

.

It remains to show that2

is finitely generated. Since A%>;S- is finitely generated,we can find O ( O " . &%>;- with &%>;- R &% O ( O " - . Expressing each O ' interms of characters as above gives the desired finite generating set of

2

. Hence2

isan affine semigroup.

Here is one way to think about the above proof. When an irreducible affinevariety

;

contains a torus 5

as a Zariski open subset, we have the inclusionA%>;S-

!

F%

;

-; -

consists of those functions on the torus 5

that extend to polynomialfunctions on

;

. Then the key insight is that;

is a toric variety precisely when thefunctions that extend are determined by the characters that extend.Example 1.1.17. Weve seen that

;

R

354 '

R 8

!

is a toric varietywith toric ideal K ' R L ! A% ' X- . Also, the torus is 47 M 8

via the map4

(

8

4

(

(

"

(

8

. The lattice points used in this map can be repre-sented as the columns of the matrix

(1.1.6)

R

The corresponding semigroup2

!

consists of the

-linear combinations ofthe column vectors. Hence the elements of

2

are lattice points lying in the poly-hedral region in

pictured in Figure 1 on the next page. In this figure, the fourvectors generating

2

are shown in bold, and the boundary of the polyhedral regionis partially shaded. In the terminology of 1.2, this polyhedral region is a rationalpolyhedral cone. In Exercise 1.1.5 you will show that

2

consists of all lattice pointslying in the cone in Figure 1. We will use this in 1.3 to prove that ; is normal.

Exercises for 1.1.1.1.1. As in Example 1.1.6, let

$(

d

+

d

#

d

+

d

Om#f

-IH

d

27+6+7+62

d

J

and let

g

be the surface parametrized by

F12nn'B(MF

23

26+7+6+92 6

2

'a

+


(0,0,1)

(0,1,0)

(1,0,0)

(1,1,1)

Figure 1. Cone containing the lattice points corresponding to

(a) Prove that 'Z( F 4 'a

. Thus

g

( ' .

(b) Prove that

g

' is homogeneous.(c) Consider lex monomial order with d *d *d . Let ,%

g

' be homoge-neous of degree and let be the remainder of , on division by the generators of .Prove that can be written

(

d

2

d

' .

d

2

d

4

'

.

d

2

d

'

where + is homogeneous of degree . Also explain why we may assume that thecoefficient of d

+

in ,+ is zero for # .(d) Use part (c) and WF

23

27+6+7+6237

2n

')(C to show that I(V .(e) Use parts (b), (c) and (d) to prove that $(

g

' . Also explain why the generators of are a Gr obner basis for the above lex order.

1.1.2. Let - "! be a sublattice. Prove that

d

$#

#

d

&%

O U'

(

d

r

#

d)(

OsB2+* ,

!

2 s#-* .

+

Note that when y. , the vectors 2 ., ! have disjoint support (i.e., no coordinate ispositive in both), while this may fail for arbitrary sZ2/* ', ! with s#-*%' .1.1.3. Let 0 be a toric ideal and let

26+7+6+92 1 be a basis of the sublattice "! . Define

32 (

d

54

#

#

d

$4

%

O (L126+6+7+92/

+

Prove that 0 (3276

d

8pd

!

. Hint: Given sZ2/* ., ! with s#-*%. , write s#*(9

1

+

+

+

, + . This implies

d

r

(

#f(;:

d

$4

#

d

4

%?

2 2

'Z'

o

O 0

:|>z (W8:

o

+

Also compute primary decomposition of to show that is not prime.

1.1.11. Let ~ be a torus with character lattice . Then every point *~ gives anevaluation map ] 6 h

defined by ] '( !n' . Prove that ] is a grouphomomorphism and that the map )jh ] induces a group isomorphism

~

1|S 23

' +

1.1.12. Consider tori ~ and ~4

with character lattices and 4

. By Example 1.1.13,the coordinate rings of ~ and ~

4

are IH J and IH 4

J . Let

6m~

h ~

4

be a morphismthat is a group homomorphism. Then

induces maps

6

4

#h

and 6mIH 4

Je#Sh IH

J

by composition. Prove that

is the map of semigroup algebras induced by the map

ofaffine semigroups.

1.1.13. A commutative semigroup is cancellative if #- (N $ implies - ( $ forall P2 -2 $ and torsion-free if Q%( implies %( for all - ,:/EW8 and t .Prove that is affine if and only if it is finitely generated, cancellative, and torsion-free.

1.1.14. The requirement that an affine semigroup be finitely generated is important sincelattices contain semigroups that are not finitely generated. For example, let beirrational and consider the semigroup

(N/> 2 9'Z',

4

O *8$

4

+

Prove that is not finitely generated. (When satisfies a quadratic equation with integercoefficients, the generators of are related to continued fractions. For example, when(Mn ' ( is the golden ratio, the minimal generators of are nm231' and

4

2

4

'

for (Mm2 (`26+7+6+ , where

is the th Fibonacci number. See [95] for further details.)1.1.15. Suppose that ] 6, h is a group isomorphism. Fix a finite set andlet ! ( ]P ' . Prove that the toric varieties

and#"

are equivariantly isomorphic(meaning that the isomorphism respects the torus action).

1.2. Cones and Affine Toric VarietiesWe begin with a brief discussion of rational polyhedral cones and then explain howthey relate to affine toric varieties.

1.2. Cones and Affine Toric Varieties 23

Convex Polyhedral Cones. Fix a pair of dual vector spaces ; @ and ? @ . Ourdiscussion of cones will omit most proofswe refer the reader to [30] for moredetails and [76, App. A.1] for careful statements. See also [10, 38, 87].Definition 1.2.1. A convex polyhedral cone in ? @ is a set of the form

6

R

=

I> 4 4.8 R .

One easily checks that a convex polyhedral cone 6 is in fact convex, meaningthat

'

.

6

) '

4 1R)28

.

6 for all )

, and is a cone, meaning'

.

6

) '

.

6 for all)

W

. Since we will only consider convex cones, thecones satisfying Definition 1.2.1 will be called simply polyhedral cones.

Examples of polyhedral cones include the first quadrant in

or first octant in

. For another example, the cone=

I> 4

(

(

8

!

is picturedin Figure 2 below. It is also possible to have cones that contain entire lines. Forexample,

=

I> 4

(4R

( 8

!

is the'

-axis, while=

I> 4

(4R

(

8

is the closedupper half-plane

4 '

8

.

W

. As we will see below, these last twoexamples are not strongly convex.

We can also create cones using polytopes, which are defined as follows.

Definition 1.2.2. A polytope in ?@ is a set of the form

R

=

I>)4


Polytopes include all polygons in

and bounded polyhedra in . As we willsee in later chapters, polytopes play a prominent role in the theory of toric varieties.Here, however, we simply observe that a polytope !? @ gives a polyhedral conein ?@

by taking the cone

6

R

) 4

8

.

?@

.

)

W

.

If

R

=

I>)4 4


if and only if , . 6N7QS . Furthermore, if , ( , " generate 6N7 , then it isstraightforward to check that(1.2.1) 6 R O UP 9 9 O UP

Thus every polyhedral cone is an intersection of finitely many closed half-spaces.We can use supporting hyperplanes and half-spaces to define faces of a cone.

Definition 1.2.5. A face of a cone of the polyhedral cone 6 is H R OP 9 6 forsome , . 6 7 . Using , R shows that 6 is a face of itself. Faces H PR 6 arecalled proper faces.

The faces of a polyhedral cone have the following obvious properties.

Lemma 1.2.6. Let 6 R=

I> 4


Proposition 1.2.8. Let 6 ! ?:@ +

be a polyhedral cone. Then:(a) If ! 6 R / and the facets of 6 are H ' R O P 96 for , ' . 6 7 , ,

then

6

R

O

U

P

9

9

O

U

P

and 6 7 R=

I> 4

,

(

,

"

8

(b) Every proper face H of 6 is the intersection of the facets of 6 containing H .

Note how part (a) of the proposition refines (1.2.1) when ! 6 R ! ? @ .When working in

+

, dot product allows us to identify the dual with +

. Fromthis point of view, the vectors ,

(

,

" in part (a) of the proposition are facetnormals, i.e., perpendicular to the facets. This makes it easy to compute examples.

Example 1.2.9. It easy to see that the facet normals to the cone 6 ! in Figure 2are ,

(

R

(

,

R

,

R

,

R

(

R

. Hence

6

7

R

=

I> 4

(

(

%R

8

!

This is the cone of Figure 1 at the end of 1.1 whose lattice points describe thesemigroup of the affine toric variety

354 '

R S8 (see Example 1.1.17). As wewill see, this is part of how cones describe normal affine toric varieties.

Now consider 6N7 , which is the cone in Figure 1. The reader can check that thefacet normals of this cone are

(

(

. Using duality and part (b) ofProposition 1.2.8, we obtain

6

R

4

6

7

8

7

R

=

I> 4

(

(

8

Hence we recover our original description of 6 .

In this example, facets of the cone correspond to edges of its dual. More gen-erally, given a face H of a polyhedral cone 6J! ? @ , we define

H

I

R

, .

;V@

K

,

L

R

for all . H

H

M

R

, .

6

7

K

,

L

R

for all . H

R

6

7

9

H

I

We call H M the dual face of H because of the following proposition.Proposition 1.2.10. If H is a face of a polyhedral cone 6 and H M R 6 7 9 H0I , then:(a) H0M is a face of 6N7 .(b) The map H H M is a bijective inclusion-reversing correspondence between

the faces of 6 and the faces of 687 .(c) ! H ! H M R / .

Here is an example of Proposition 1.2.10 when

!

6

!

? @ .


x

y

z

x

y

z

Figure 5. A -dimensional cone

and its dual

Example 1.2.11. Let 6 R=

I> 4

(

8

!

. Figure 5 shows 6 and 687 . Youshould check that the maximal face of 6 , namely 6 itself, gives the minimal face6

M of 67 , namely the

-axis. Note also that

!

6

!

6

M

R

even though 6 has dimension .

Relative Interiors. As already noted, the span of a cone 6 ! ? @ is the smallestsubspace of ?G@ containing 6 . Then the relative interior of 6 , denoted C

D4

6

8

, isthe interior of 6 in its span. Exercise 1.2.2 will characterize

C

D4

6

8

as follows:

.

C

D4

6

8

K

,

L

for all ,. 6 7 S 6 I

When the span equals ?G@ , the relative interior is just the interior, denoted EF D4 6 8 .For an example of how relative interiors arise naturally, let H be a face of a

cone 6 . This gives the dual face H M R 67Q9 H I of 67 . Furthermore, if , . 687 ,then one easily sees that

,.

H

M

H

!

O P

9 6

In Exercise 1.2.2, you will show that if ,. 6 7 , then,.

CX

D4

H

M

8

H

R

O P

9 6

Thus the relative interiorC

D4

H M

8

tells us exactly which supporting hyperplanesof 6 cut out the face H .

Strong Convexity. Of the cones shown in Figures 15, all but 6 7 in Figure 5 havethe nice property that the origin is a face. Such cones are called strongly convex.This condition can be stated several ways.


Proposition 1.2.12. Let 6J! ?:@ +

be a polyhedral cone. Then:6 is strongly convex is a face of 6

6 contains no positive-dimensional subspace of ? @

6T9

4FR

6

8

R

6

7

R

/

You will prove Proposition 1.2.12 in Exercise 1.2.3. One corollary is that if apolyhedral cone 6 is strongly convex of maximal dimension, then so is 6 7 . Thecones pictured in Figures 14 satisfy this condition.

In general, a polyhedral cone 6 always has a minimal face that is the largestsubspace contained in 6 . Furthermore:

R

6T9

4FR

6

8

.

R

O P

9-6 whenever ,.C

D4

6

7

8

.

6

R

6 ! ? @ is a strongly convex polyhedral cone.See Exercise 1.2.4.

Separation. When two cones intersect in a face of each, we can separate the coneswith the following result, often called the Separation Lemma.

Lemma 1.2.13 (Separation Lemma). Let 6 ( 6 be polyhedral cones in ? @ thatmeet along a common face H R 6 ( 9 6 . Then

H

R

O P

9 6

(

R

O P

9 6

for any ,. CX D4 6 7( 9 4FR 6 8 7 8 .Proof. Given

! ?G@ , we set

R

R

*

R

*

.

.

. A standardresult from cone theory tells us that

6

7

(

9

4FR

6

8

7

R

4

6

( R

6

8

7

Now fix ,.C

D4

6

7

(

9

4FR

6

8

7

8

. The above equation and Exercise 1.2.4 implythat

O P

cuts out the minimal face of 6( R

6

, i.e.,O

P

9

4

6

(

R

6

8

R

4

6

(

R

6

8

9

4

6

R

6

(

8

However, we also have4

6

( R

6

8

9

4

6

%R

6

( 8

R

H

R

H

One inclusion is obvious since H R 6 ( 9V6 . For the other inclusion, write .4

6

( R

6

8

9

4

6

%R

6

( 8

as

R

*

( R

*

R

XR

(

*

(

(

.

6

(

*

.

6

Then *(

(

R

*

implies that this element lies in H R 6(

9 6

. Since*

(

(

.

6

(

, we have *(

(

.

H by Lemma 1.2.7, and *

.

H follows similarly.Hence R *

( R

*

.

H

R

H

, as desired.


We conclude that OTP 9 4 6 ( R 6 8 R H R H . Intersecting with 6 ( , we obtainO P

9 6

(

R

4

H

R

H

8

9-6

(

R

H

where the last equality again uses Lemma 1.2.7 (Exercise 1.2.5). If instead weintersect with

R

6

, we obtainO P

9

4FR

6

8

R

4

H

R

H

8

9

4FR

6

8

R

R

H

and O P 9 4FR 6 8

R

H follows.

In the situation of Lemma 1.2.13 we call O P a separating hyperplane.

Rational Polyhedral Cones. Let ? and ; be dual lattices with associated vectorspaces ? @ R ?

*

and ;J@ R ; +* . For +

we usually use the lattice

+

.

Definition 1.2.14. A polyhedral cone 6 ! ?@ is rational if 6 R = I> 4 4


In a similar way, a rational polyhedral cone 6 of maximal dimension has uniquefacet normals, which are the ray generators of the dual 6 7 , which is strongly con-vex by Proposition 1.2.12.

Here are some especially important strongly convex cones.

Definition 1.2.16. Let 6J! ? @ be a strongly convex rational polyhedral cone.(a) 6 is smooth or regular if its minimal generators form part of a -basis of ? ,(b) 6 is simplicial if its minimal generators are linearly independent over .

The cones pictured in Figure 5 are smooth, while those in Figures 1 and 2 arenot even simplicial. Note also that the dual of a smooth (resp. simplicial) coneis again smooth (resp. simplicial). Later in the section we will give examples ofsimplicial cones that are not smooth.

Semigroup Algebras and Affine Toric Varieties. Given a rational polyhedral cone6J! ? @ , the lattice points

23

R

6

7

9-; !;

form a semigroup. A key fact is that this semigroup is finitely generated.

Proposition 1.2.17 (Gordans Lemma). 2 3 R 67

9J; is finitely generated andhence is an affine semigroup.

Proof. Since 6 7 is rational polyhedral, 6 7 R=

I> 4

8

for a finite set ! ; .Then R & P *

P

,

P

is a bounded region of ; @ +

, so that

9 ; is finite since ;

+

. Note that 4

9-;

8

!

2

3

.

We claim 4

9 ;

8

generates2+3

as a semigroup. To prove this, take

.

23

and write R & P * )P , where )0P W . Then )+P R )0P

P

with

)0P

.

and P , so that

R

P

*

)0P

,

P

*

P

,

The second sum is in 9X; (remember . ; ). It follows that is a nonnegativeinteger combination of elements of 4 9 ; 8 .

Since affine semigroups give affine toric varieties, we get the following.

Theorem 1.2.18. Let 6 ! ?:@ +

be a rational polyhedral cone with semigroup23

R

67:9-; . Then4

3

R

BC

D47A%

2

3

- 8

R

B9C

D47&%

6

7

9 ;

- 8

is an affine toric variety. Furthermore,

4

3

R

/

the torus of 4 3 is 5 R ? *S M 6 is strongly convex


Proof. By Gordans Lemma and Proposition 1.1.14, 43

is an affine toric varietywhose torus has character lattice

2 3

!; . To study 2 3

, note that

203

R

203

R

23

R

,

( R

,

,

(

,

.

23

.

Now suppose that

, .

2+3

for some

and ,. ; . Then

,

R

,

( R

,

for ,(

,

.

2 3

R

6 7 9 ; . Since ,(

and , lie in the convex set 6 7 , we have,

,

R

(

,

(

"

(

,

.

6

7

It follows that , R4

,

,

8 R

,

.

23

, so that ;-

203

is torsion-free. Hence

(1.2.3) the torus of 4 3 is 5 203 R ; B 23 R /

Since 6 is strongly convex if and only if

!

6

7

R

/ (Proposition 1.2.12), itremains to show that

! 4

3

R

/

B

23

R

/

6

7

R

/

The first equivalence follows since the dimension of an affine toric variety is thedimension of its torus, which is the rank of its character lattice. We leave the proofof the second equivalence to the reader (Exercise 1.2.6).

Since we want our affine toric varieties to contain the torus 5

, we consideronly those affine toric varieties

4

3

for which 6 !? @ is strongly convex.Our first example of Theorem 1.2.18 is an affine toric variety we know well.

Example 1.2.19. Let 6 R=

I> 4

(

(

8

! ?@

R

with ? R

.

This is the cone pictured in Figure 2. By Example 1.2.9, 6 7 is the cone picturedin Figure 1, and by Example 1.1.17, the lattice points in this cone are generatedby columns of matrix (1.1.6). It follows from Example 1.1.17 that 4 3 is the affinetoric variety

354 '

R 8

.

Here are two further examples of Theorem 1.2.18.

Example 1.2.20. Fix / and set 6 R=

I> 4

(

8

!

+

. Then

6

7

R

=

I> 4

(

U

(

+ 8

and the corresponding affine toric variety is4

3

R

BC

D.47A% ')( *'

*'

1

(

U

(

*'

1

(

+

- 8

R

547

M

8

+

"

(Exercise 1.2.7). This implies the general fact that if 6 ! ? @ +

is a smoothcone of dimension , then 4

3

547

M

8

+

"

.

Figure 5 illustrates the cones in Example 1.2.20 when R and / R .

Example 1.2.21. Fix a positive integer @ and let 6 R=

I> 4

@

(%R

8

!

.

This has dual cone 6 7 R=

I> 4

(

(

@

8

. Figure 7 on the next page shows 6 7when @ R . The affine semigroup

L

3

R

6

7

9

is generated by the lattice points


Figure 7. The cone when "

4

8

for

@ . When @ R , these are the white dots in Figure 7. (You willprove these assertions in Exercise 1.2.8.)

By 1.1, the affine toric variety 4 3 is the Zariski closure of the image of themap

47

M

8

?

U

(

defined by

4

8

R

4.

?

8

This map has the same image as the map4

8

4

?

?

"

(

?

"

(

?

8

usedin Example 1.1.6. Thus 4

3

is isomorphic to the rational normal cone

?

!

?

U

(

whose ideal is generated by the minors of the matrix

' ')( '2?

"

',?

"

(

')( ' '2?

"

(

',?

Note that the cones 6 and 6 7 are simplicial but not smooth.

We will return to this example often. One thing evident in Example 1.1.6 is thedifference between cone generators and semigroup generators: the cone 6 7 hastwo generators but the semigroup L

3

R

6

7

9

has @

.

When 6 ! ? @ has maximal dimension, the semigroup L3

R

67Q9 ; has aunique minimal generating set constructed as follows. Define an element , PR of23

to be irreducible if , R , Z

,

Z Z

for ,Z7

,

Z Z

.

23

implies ,Z

R

or ,Z Z

R

.

Proposition 1.2.22. Let 6 ! ?:@ be strongly convex of maximal dimension and let23

R

6

7

9-; . Then 4

(

.

(

8

pictured in Figure 1. ThenTheorem 1.3.5 implies that ; is normal, as claimed in Example 1.1.5. Example 1.3.7. By Example 1.2.21, the rational normal cone

?

!

?

U

(

is theaffine toric variety of a strongly convex rational polyhedral cone and hence is nor-mal by Theorem 1.3.5.

It is instructive to view this example using the parametrization

34.

8

R

4

?

?

"

(

?

"

(

?

8

from Example 1.1.6. Plotting the lattice points in $ for @ R gives the whitesquares in Figure 9 (a) below. These generate the semigroup 2 R #$ , and theproof of Theorem 1.3.5 gives the cone 6 7 R

=

I> 4

(

8

, which is the first quad-rant in the figure. At first glance, something seems wrong. The affine variety

is normal, yet in Figure 9 (a) the semigroup generated by the white squares missessome lattice points in 6 7 . This semigroup does not look saturated. How can theaffine toric variety be normal?

(a)

(b)

Figure 9. Lattice points for the rational normal cone

The problem is that we are using the wrong lattice! Proposition 1.1.8 tells usto use the lattice

%$

, which gives the white dots and squares in Figure 9 (b). Thisfigure shows that the white squares generate the semigroup of lattice points in 6 7 .Hence

2

is saturated and everything is fine.

1.3. Properties of Affine Toric Varieties 39

This example points out the importance of working with the correct lattice.

The Normalization of an Affine Toric Variety. The normalization of an affine toricvariety is easy to describe. Let ; R

BC

D47&%

2

- 8

for an affine semigroup2

, so thatthe torus of

;

has character lattice ; R

2

. Let=

I> 4

2

8 denote the cone of anyfinite generating set of

2

and set 6 R=

I> 4

2

8

7 !? @ . In Exercise 1.3.6 you willprove the following.

Proposition 1.3.8. The above cone 6 is a strongly convex rational polyhedral conein ? @ and the inclusion F%

2

-

!

A%

687A9 ;

-

induces a morphism 43

;

that isthe normalization map of ; .

The normalization of an affine toric variety of the form 3

is constructed byapplying Proposition 1.3.8 to the affine semigroup

1$

and the lattice %$ .

Example 1.3.9. Let $ R 4 8 48 4 8 4 8 !

. Then

3 4

8

R

4

8

parametrizes the surface 3

!

considered in Exercise 1.1.7. This is almostthe rational normal cone

, except that we have omitted

. Using4 .8

R

(

4

8

4

8

, we see that1$

is not saturated, so that 3 is not normal.Applying Proposition 1.3.8, one sees that the normalization of 3 is

. Thisis an affine variety in

, and the normalization map is induced by the obviousprojection .

In Chapter 3 we will see that the normalization map 43

;

constructed inProposition 1.3.8 is onto but not necessarily one-to-one.

Smooth Affine Toric Varieties. Our next goal is to characterize when an affine toricvariety is smooth. Since smooth affine varieties are normal (Proposition 1.0.9),we need only consider toric varieties 4

3

coming from strongly convex rationalpolyhedral cones 6J! ?G@ .

We first study 43

when 6 has maximal dimension. Then 6 7 is strongly convex,so that

203

R

6

7

9 ; has a Hilbert basisL3

-

, we obtain

R

P

(

'

P

R

P

irreducible

%'

P

P

reducible

%'

P

R

P

'

P


It follows that

!

R

z 8 .

1.3.12. Let ' and 4

0

4

'

be strongly convex rational polyhedral cones.This gives the cone

4

4

' . Prove that

. Also explainhow this result applies to (1.3.2).1.3.13. By Proposition 1.3.1, a point K of an affine toric variety G (C=?7@ AIH J!' is repre-sented by a semigroup homomorphism 6 h . Prove that K lies in the torus of G ifand only if never vanishes, i.e., k 'UT(V for all .


Appendix: Tensor Products of Coordinate RingsIn this appendix, we will prove the following result used in 1.0 in our discussion of prod-ucts of affine varieties.

Proposition 1.A.1. If and are finitely generated -algebras without nilpotents, thenthe same is true for .Proof. Since the tensor product is obviously a finitely generated -algebra, we need onlyprove that has no nilpotents. If we write

H

d

27+6+7+62

d

J! , then is radicaland hence has a primary decomposition ( v !

+

+ , where each + is prime ([17, Ch. 4,7]). This gives

IH

d

26+7+6+62

d

JA #Dh

!

+

IH

d

27+6+6+72

d

J!

+

where the second map is injective. Each quotient IH d 26+7+6+92 d

JA + is an integral domainand hence injects into its field of fractions q1+ . This yields an injection

M#Dh

!

+

q

+

2

and since tensoring over a field preserves exactness, we get an injection

h

!

+

q

+

+

Hence it suffices to prove that q has no nilpotents when q is a finitely generatedfield extension of . A similar argument using then reduces us to showing that q has no nilpotents when q and are finitely generated field extensions of .

Since has characteristic , the extension i has a separating transcendencebasis ([56, p. 519]). This means that we can find c 26+7+6+92 c such that c 26+7+6+72 c are algebraically independent over and ( c 26+6+7+62 c ' is a finite separableextension. Then

q

q

Z'

Aq

I'

+

But gN(Cq N(Cq I c 26+7+6+62 c p')(*q c 26+7+6+92 c p' is a field, so that we are reducedto considering

g

where g and are extensions of and is finite and separable. The latter and thetheorem of the primitive element imply that

H JA

, '

, where , ' has distinctroots in some extension of . Then

g

g

H JA

, '

gH J!

, '

+

Since , ' has distinct roots, this quotient ring has no nilpotents. Our result follows.

A final remark is that we can replace with any perfect field since finitely generatedextensions of perfect fields have separating transcendence bases ([56, p. 519]).

Chapter 2

Projective Toric Varieties

2.0. Background: Projective VarietiesOur discussion assumes that the reader is familiar with the elementary theory ofprojective varieties, at the level of [17, Ch. 8].

In Chapter 1, we introduced affine toric varieties. In general, a toric varietyis an irreducible variety over containing a torus

5

47

M

8

+

as a Zariskiopen set such that the action of

47

M

8

+

on itself extends to an action on . Wewill learn in Chapter 3 that the concept of variety is somewhat subtle. Hence wewill defer the formal definition of toric variety until then and instead concentrateon toric varieties that live in projective space

+

, defined by

(2.0.1) +

R

47

+

U

(

S

8

M

where

M acts via homotheties, i.e.,).4

*

*

+98

R

4 )

*

)

*

+ 8

for)

.

M

and 4 * * +98 . +

U

(

. Thus4

*

*

+ 8

are homogeneous coordinates of apoint in

+

and are well-defined up to homothety.The goal of this chapter is to use lattice points and polytopes to create toric

varieties that lie in

+

. We will use the affine semigroups and polyhedral conesintroduced in Chapter 1 to describe the local structure of these varieties.

Homogeneous Coordinate Rings. A projective variety ; ! +

is defined bythe vanishing of finitely many homogeneous polynomials in the polynomial ringL

R

F% ' *' +.-

. The homogeneous coordinate ring of ; is the quotient ringA%>;S-

R

L

:.4

50 Chapter 2. Projective Toric Varieties

The polynomial ringL

is graded by setting

,4 '

'

8

R

. This gives the decom-position

L

R

?

(

LV?

, whereL?

is the vector space of homogeneous polynomialsof degree @ . Homogeneous ideals decompose similarly, and the above coordinatering

F%>; -

inherits a grading whereF%>; -?

R

LV?

:.4

2.0. Background: Projective Varieties 51

Rational Functions on Irreducible Projective Varieties. A homogeneous polyno-mial O .

L

of degree @ does not give a function on +

sinceO

4 ) ' ) ' + 8

R

)

?

O

4 ' *' + 8

However, the quotient of two such polynomials O

.

L ?

gives the well-definedfunction

O

+

S

35498 S

provided PR . We write this as O

+

and say that O is a rationalfunction on

+

.

More generally, suppose that;

!

+

is irreducible, and let O . F%>; - RF%

; - be homogeneous of the same degree with PR . Then O and give functionson the affine cone

;

and hence an element O . &4

;A8

. By (2.0.2), this induces arational function O

;

. Thus&4;S-

homogeneous of the same degree, PR

is the field of rational functions on ; . It is customary to write the set on the left asF4

;=8

since it consists of the degree elements of A4

;&8

.

Affine Pieces of Projective Varieties. A projective variety ; ! +

is a union ofZariski open sets that are affine. To see why, let

4

'

R

+

S

354 '

'

8

. Then4

'

+

via the map(2.0.3) 4 * * +98 R

so that in the notation of Chapter 1, we have4

'

R

BC

D

Then;

9

4

' is a Zariski open subset of ; that maps via (2.0.3) to the affine varietyin

+

defined by the equations(2.0.4) O

R

as O varies over all homogeneous polynomials in:.4;- , so that we getthe localization(2.0.5) F%>; - R O ' ' O . A%>;S-;S-

has a well-defined -gradinggiven by

V4

O

'

'

8

R

V4

O

8/R

when O is homogeneous. Then

(2.0.6) 47A%>;S- 8 R O ' ' . &%>;- O is homogeneous of degree

is the subring of F%>; -

consisting of all elements of degree . This gives an affinepiece of

;

as follows.


Lemma 2.0.3. The affine piece ; 9 4 ' of ; has coordinate ringF%>;

9

4

'

-

47A%>; - 8

Proof. We have an exact sequence

R :.4; -NR

If we localize at'

'

, we get the exact sequence

(2.0.7) R :.4;S- R since localization preserves exactness (Exercises 2.0.1 and 2.0.2). These sequencespreserve degrees, so that taking elements of degree 0 gives the exact sequence

R+ 47:.4;-

8 R+

Note that47A% ' *' + -

8

R

. If O .:.4; 9 4 ' 9 4 - 47&%>;- 8

We will apply this to projective toric varieties later in the chapter.We will use these results in 2.2 when we explore the structure of projective

toric varieties. We will also see later in the book that Lemma 2.0.3 is related to theProj construction, where Proj of a graded ring gives a projective variety, just asSpec of an ordinary ring gives an affine variety.

Products of Projective Spaces. One can study the product +

P

of projectivespaces using the bigraded ring F% '!*',+ P - , where ' ' has bidegree4

8

and ' has bidegree 4 8 . Then a bihomogeneous polynomial O of bidegree4

*

8

gives a well-defined equation O R in +

P

. This allows us to definevarieties in

+

P

using bihomogeneous ideals. In particular, the ideal :.4

2.0. Background: Projective Varieties 53

defined by mapping 4 * * + P8 to the point4

*

*

(

*

P

*

(

*

(

P

*

+

*

+

P 8

This map is studied in [17, Ex. 14 of Ch. 8, 4]. If + P

U

+

U

P

has homogeneouscoordinates ' ' for / , , then

+

P

!

+ P

U

+

U

P

is definedby the vanishing of the minors of the 4 /

8%54

,

8

matrix

' ' FP

.

.

.

.

.

.

' +/ ' + P

This follows since an4

/

81 4

,

8

matrix has rank

if and only if it is aproduct

, where

and

are nonzero row matrices of lengths /

and ,

.

These approaches give the same notion of what it means to be a subvariety of

+

P

. A homogeneous polynomial 4 '

'

8

of degree @ gives the bihomogeneouspolynomial

4 '

'

8

of bidegree 4 @ @ 8 . Hence any subvariety of + P

U

+

U

P

lyingin

+

P

can be defined by a bihomogeneous ideal in F% ' *' +, P- .Going the other way takes more thought and is discussed in Exercise 2.0.5.

We also have the following useful result proved in Exercise 2.0.6.

Proposition 2.0.4. Given subvarieties ; ! +

and ! P

, the product ; is a subvariety of

+

P

.

Weighted Projective Space. The graded ring associated to +

is&% ' *',+.-

,

where each variable ' ' has degree . More generally, let -+ be positiveintegers with

.D

4

+

8

R

and define 4+ 8

R

47

+

U

(

SX

8

%

where%

is the equivalence relation4

*

*

+ 8% 4

+ 8

*

'

R

)

'

R

/ for some)

.

M

We call 4 +98

a weighted projective space. Note that +

R

4 8

.

The ring corresponding to 4 + 8 is the graded ring F% ' *' + - where'

' now has degree ' . A polynomial O is weighted homogeneous of degree @ if ev-ery monomial

'

appearing in O satisfies" 4 +98

R

@ . The O R is well-defined on 4+ 8 when O is weighted homogeneous, so that one can definevarieties in

4 + 8

using weighted homogeneous ideals of F% ' *' +.- .

Example 2.0.5. We can embed the weighted projective plane 4 .8 in usingthe monomials

'

*'

'

(

*'

(

*'

of weighted degree . In other words, the map 4 .8#R

given by4

*

*

(

*

8

R 4

*

*

*

(

*

(

*

8


is well-defined and injective. One can check that this map induces 4 .8

354

R

(

8

!

where

(

are homogeneous coordinates on

.

Later in the book we will use toric methods to construct projective embeddingsof arbitrary weighted projective spaces.

Exercises for 2.0.2.0.1. Let be a commutative -algebra. Given , t:/&W8 and an exact sequence of -modules h h

4

h

o

h , prove that

#Dh

y; #h

4

y; #h

o

y; #Dh

is also exact, where ; is the localization of at , defined in Exercises 1.0.2 and 1.0.3.

2.0.2. A projective variety G

has coordinate ring H GJ( FG$' , where (H

d

26+7+6+62

d

J . Let bd + be the image of d + in IH G:J .(a) Note the IH G:J is an -module. Prove that H GJ

4

IH GJ

I

4

.

(b) Use part (a) and the previous exercise to prove that (2.0.7) is exact.2.0.3. Prove the claim made in (2.0.8).

2.0.4. Let G

be a projective variety. Take , 26+7+6+723,

such that the intersectionG%F,

26+6+7+62 ,

' is empty. Prove that the map

26+7+6+62

')j #hlF,

26+7+6+92

'926+7+6+62 ,

27+6+6+72

'n'

induces a well-defined map function 6 GMh .

2.0.5. Let G

be defined by ,

d

+n2

c

'B(V , where ,

d

+32

c

' is bihomogenousof bidegree

2

' , t( m26+7+6+723 . The goal of this exercise is to show that when weembed

in

via the Segre embedding described in the text, G becomesa subvariety of

.

(a) For each , pick an integer

*| S/

2

8 and consider the polynomials ,

[

r

[

(

(

d

r

c

(

,+

d

+ 2

c

' where (*`27+6+6+623 and O s

Cox D., Little J., Schenck H. Toric Varieties (Draft, 2009)(530s)_MAg

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Transcript of Cox D., Little J., Schenck H. Toric Varieties (Draft, 2009)(530s)_MAg