The Homogeneous Coordinate Ring of a Toric Variety

Home , Subvariety

The Homogeneous Co ordinate Ring of a Toric Variety

David A Cox

Department of Mathematics and Computer Science

Amherst College

Amherst MA

daccsamherstedu

This pap er will introduce the homogeneous coordinate ring S of a toric variety X

The ring S is a p olynomial ring with one variable for each onedimensional cone in the

fan determining X and S has a natural grading determined by the monoid of eective

divisor classes in the Chow group A X of X where n dim X Using this graded

ring we will show that X b ehaves like pro jective space in many ways

In the literature one often nds co ordinate rings attached to toric varieties This

is typically done for a pro jective toric variety where one uses the co ordinate ring of a

particular pro jective embedding Such an embedding is given by a convex p olytop e so

that the ring has a nice combinatorial description see for example x of Batyrev In

contrast the homogeneous co ordinate ring discussed here is intrinsic to the toric variety

and in fact the rings corresp onding to pro jective embeddings are subrings of our ring

The pap er is organized into four sections as follows

In x we dene the homogeneous co ordinate ring S of X and compute its graded

pieces in terms of global sections of certain coherent sheaves on X We also dene a

monomial ideal B S that describ es the combinatorial structure of the fan In the case

of pro jective space the ring S is just the usual homogeneous co ordinate ring Cx x

and the ideal B is the irrelevant ideal hx x i

n n

Pro jective space P can b e constructed as the quotient C fgC In x we will

see that there is a similar construction for any toric variety X In this case the algebraic

group G Hom A X C acts on an ane space C such that the categorical

Z n

quotient C Z G exists and is isomorphic to X The exceptional set Z is the zero

set of the ideal B dened in x If X is simplicial meaning that the fan is simplicial

then X C Z G is a geometric quotient so that elements of C Z can b e

regarded as homogeneous co ordinates for p oints of X

On P there is a corresp ondence b etween sheaves and graded Cx x mo dules

For any toric variety we will see in x that nitely generated graded S mo dules give rise

to a coherent sheaves on X and when X is simplicial every coherent sheaf arises in this

way In particular every closed subscheme of X is determined by a graded ideal of S We

will also study the extent to which this corresp ondence fails to b e onetoone

n n

Another feature of P is that the action of P GLn C on P lifts to an action

of GLn C on C fg In x we will see that for any complete simplicial toric

variety X the action of Aut X on X lifts to an action of an algebraic group AutX

on C Z This group is a extension of Aut X by the group G dened ab ove We

will give an explicit description of AutX and in particular we will see that the ro ots of

AutX as dened in Demazure have an esp ecially nice interpretation

For simplicity we will work over the complex numbers C Our notation will b e similar

to that used by Fulton and Oda though the reader should b e aware that Danilov is also

a basic reference for toric varieties I would like to thank Bernd Sturmfels for stimulating

my interest in toric varieties

x The Homogeneous Co ordinate Ring

Let X b e the toric variety determined by a fan in N Z As usual M will denote

the Zdual of N and cones in will b e denoted by The onedimensional cones of

form the set and given we let n denote the unique generator of N If

is any cone in then f g is the set of onedimensional faces of

We will assume that spans N N R

R Z

Each corresp onds to an irreducible T invariant Weil divisor D in X where

T N C is the torus acting on X The free ab elian group of T invariant Weil divisors

on X will b e denoted Z Thus an element D Z is a sum D a D The

T invariant Cartier divisors form a subgroup Div X Z

m m

Each m M gives a character T C and hence is a rational function on

m m

X As is wellknown gives the Cartier divisor div hm n iD This gives a

map

hm n iD M Z dened by m D

which is injective since spans N By Fulton x we have a commutative diagram

M Div X PicX

M Z A X

where the rows are exact and the vertical arrows are inclusions Thus a divisor D Z

determines an element D A X Note that n is the dimension of X

For each introduce a variable x and consider the p olynomial ring

S Cx

x determines a divisor We will usually write this as S Cx Note that a monomial

D We a D and to emphasize this relationship we will write the monomial as x

will grade S as follows

D D

the degree of a monomial x S is degx D A X

Q Q

a b

Using the exact sequence it follows that two monomials x and x in S have

the degree if and only if there is some m M such that a hm n i b for all Then

let

S C x

degx

so that the ring S can b e written as the direct sum

S S

A X

Note also that S S S We call S the homogeneous coordinate ring of the toric

variety X Of course homogeneous means with resp ect to the ab ove grading We

should mention that the ring S without the grading app ears in the DanilovJurkiewicz

description of the cohomology of a simplicial toric variety see Danilov x Fulton x

or Oda x

Here are some examples of what the ring S lo oks like

Pro jective Space When X P it is easy to check that S is usual homogeneous

co ordinate ring Cx x with the standard grading

Weighted Pro jective Space When X Pq q then S is the ring Cx x

n n

where the grading is determined by giving the variable x weight q

i i

n m

Pro duct of Pro jective Spaces When X P P then S Cx x y y

n m

Here the grading is the usual bigrading where a p olynomial has bidegree a b if it is

homogeneous of degree a resp b in the x resp y

i j

Our rst result ab out the ring S shows that the graded pieces of S are isomorphic to

the global sections of certain sheaves on X

Prop osition

i If D A X then there is an isomorphism

S H X O D

D X

where O D is the coherent sheaf on X determined by the Weil divisor D see Fulton

ii If D and E then there is a commutative diagram

S S S

H X O D H X O E H X O D E

X X X

where the top arrows is multiplication the b ottom arrow is tensor pro duct and the

vertical arrows are the isomorphisms and

D E D E

Pro of Supp ose that D a D By Fulton x we know that H X O D

C where

P M

P fm M hm n i a for all g

D R

P Q

hmn ia

D D

Given m P M let D hm n iD so that x x Then

D m

D D

m x denes a map from P M to the monomials in S the monomial is

in S since m P and it has degree since m M This map is onetoone since

M Z is injective and it is easy to see that every monomial in S arises in this way

D D m

Then x gives the desired isomorphism The pro of of the second part of

the prop osition is straightforward and is omitted

We get the following corollary see Fulton x

Corollary Let X b e a complete toric variety Then

i S is nite dimensional for every and in particular S C

ii If D for an eective divisor D a D then dim S jP M j

C D

In the complete case the graded ring S has some additional structure Namely there

D E

is a natural order relation on the monomials of S dened as follows if x x S then

set

D E

F D F D F F E

x x there is x S with x jx x x and deg x degx

This relation has the following prop erties

Lemma When X is a complete toric variety the order relation dened ab ove is

D E D F E F

transitive antisymmetric and multiplicative meaning that x x x x

on the monomials of S

Pro of The transitive and multiplicative prop erties are trivial to verify To prove antisym

D F F E

mety note that x jx and degx degx implies that E D F D F D

D E E D

is the class of an eective divisor Thus if x x and x x then E D and its

negative are eective classes Since X is complete irreducible and reduced we must have

F F D

D E Then for x as ab ove we would have x S C which would imply

F D D E

x x This contradicts the denition of x x and antisymmetry is proved

In the case of P this gives the usual ordering by total degree For a complete

simplicial toric variety X we will use the order relation of Lemma in x when we study

the automorphism group of X

An imp ortant observation is that the theory developed so far dep ends only on the

onedimensional cones of the fan More precisely if and are fans in N with

then A X A X and the corresp onding rings S and S are

n n

also equal as graded rings It follows that we cannot reconstruct the fan from S and its

gradingmore information is needed The ring tells us ab out but we need something

else to tell us which cones b elong in

The crucial ob ject is the following ideal of S For a cone let b e the divisor

P Q

D and let the corresp onding monomial in S b e x x Then dene

B B S to b e the ideal generated by the x ie

B hx i S

Note the B is in fact generated by the x as ranges over the maximal cones of Also

if and are fans in N with then if and only if the corresp onding

ideals B B S satisfy B B This follows b ecause fx is a maximal cone of g is

the unique minimal basis of the monomial ideal B

When X is a pro jective space or weighted pro jective space the ideal B is just the

irrelevant ideal hx x i Cx x A basic theme of this pap er is that the pair

n n

B S plays the role for an arbitrary toric variety that hx x i Cx x plays

n n

for pro jective space In particular we can think of the variety

Z V B fx C x for all g C

as the exceptional subset of C In x we will show how to construct the toric variety

X from C Z

Lemma Z C has co dimension at least two

Pro of Since B hx i B we have Z V B V B But V B is the

union of all co dimension two co ordinate subspaces and the lemma follows

x Homogeneous Co ordinates

Given a toric variety X the Chow group A X dened in x is a nitely generated

ab elian group of rank d n where d jj It follows that G Hom A X C is

Z n

isomorphic to a pro duct of the torus C and the nite group Hom A X QZ

Z n tor

Furthermore if we apply Hom C to the b ottom exact sequence of then we get

the exact sequence

G C T

where T N C Hom M C is the torus acting on X Note also that A X is

Z n

naturally isomorphic to the character group of G

Since C acts naturally on C the subgroup G C acts on C by

g t g D t

for g A X C in G and t t in C We get a corresp onding representation

of G on S Cx and given a character A X of G the graded piece S is the

eigenspace of this representation

Using the action of G on C we can describ e the toric variety X as follows

Theorem Let X b e the toric variety determined by the fan and let Z V B

C b e as in x Then

i The set C Z is invariant under the action of the group G

ii X is naturally isomorphic to the categorical quotient of C Z by G

iii X is the geometric quotient of C Z by G if and only if X is simplicial ie the

fan is simplicial

In the analytic category part iii of Theorem was known to M Audin see Chapter

VI of Audin which mentions earlier work by Delzant and Kirwan and versions of the

theorem were known to V Batyrev unpublished and J Fine just now app earing in

Fine As this pap er was b eing written I Musson discovered a result closely related to

part ii of the theorem see Musson and more recently he proved an analog of part iii

In the rst version of this pap er part iii only stated that the geometric quotient

exists when X is simplicial The referee asked if the converse were true and simultaneously

Musson proved this in a slightly dierent context His argument is used b elow

Pro of of Theorem Given a cone let

U C V x fx C x g

Note that U is a Ginvariant ane op en subset of C and that

C Z U

This proves part i of the theorem

For parts ii and iii of the theorem we rst need to study the quotient of G acting

on the ane variety U The co ordinate ring of U is the lo calization of S at x which

we will denote S Note that S has a natural grading by A X

The following lemma describ es the degree zero part of S

Lemma Let b e a cone and let M b e its dual cone Then there is a

natural isomorphism of rings C M S

hmn i

Pro of Given m M let x and note that x

x S hm n i for all m

Since a monomial has degree zero if and only if it is of the form x for some m M we

get a onetoone corresp ondence b etween M and monomials in S Then the map

sending m M to x S gives the desired ring isomorphism

The action of G on U induces an action on S and as we observed earlier ab out S

the graded pieces are the eigenspaces for the characters of G In particular the invariants

of G acting on S are precisely the p olynomials of degree zero Thus Lemma implies

S S C M

Since G is a reductive group acting on the ane scheme U standard results from invariant

theory see Fogarty and Mumford Theorem imply that the categorical quotient is

given by

U G Sp ecS Sp ecC M

Thus U G is the ane toric variety X determined by the cone Note that X is an

ane piece of the toric variety X

The next step is to see how the quotients U G t together as we vary So let b e

a face of a cone It is wellknown that fmg for some m M Then

we can relate S and S as follows

Lemma Let b e a face of and let m M b e as ab ove Under the isomorphism

hmn i

x S Then of Lemma m maps to the monomial x

i There is a natural identication

S S

ii There is a commutative diagram

S S S

C M C M C M

where the horizontal maps are lo calization maps and the vertical arrows are the iso

morphisms of Lemma

Pro of Since fmg x has p ositive exp onent for x when and

zero exp onent when It follows immediately that S S D Part i of the

lemma follows since lo calization commutes with taking elements of degree zero b ecause

x already has degree zero It is trivial to check that the diagram of part ii commutes

and the lemma is proved

We can now prove part ii of Theorem Namely Lemma gives us categorical

X and by the compatibility of Lemma these patch to give a quotients U G

X categorical quotient C Z G

For part iii of the theorem rst assume that X is simplicial To show that X is a

geometric quotient it suces to show that each X is the geometic quotient of G acting

on U By Fogarty and Mumford Amplication we need only show that the orbits

of G acting on U are closed

Fix a p oint t t of U In Lemma we saw that m M gives the monomial

hmn i

x S Then consider the following subvariety V U x

D D

m m

V fx x U x t for all m M g

is Ginvariant it We will prove that V G t First observe that t V and since x

has degree zero it follows that G t V

To prove the other inclusion let u u V We must nd g A X C such

that u g D t for all The rst step is to show that

for all u t

Note that is trivially true when So supp ose that Since X is

simplicial the set fn g is linearly indep endent Hence we can nd m M

such that

i for all fg hm n i and hm n

Setting a hm n i we have

hmn i

a D

u u u

hmn i

a D

t t t

are nonzero for it follows that u and t Since these are equal and since u

if and only if t This completes the pro of of

Now let A f t g f u g and note that A For A

set

g D

This denes a map g Z C To relate this to G Hom A X C let

A Z n

A A

i M Z b e the comp osition M Z Z where the last map is pro jection

Then we have the commutative diagram

M M

A A

Z Z Z

Z A X

Since the columns and two middle rows are exact the snake lemma gives us an exact

sequence

keri Z A X

A n

We next claim that g keri fg An easy diagram chase shows that for m

A A

keri we have

g m

D D

m m

so that g m whenever m keri t When m M we know that u

A A

If we can show that this semigroup generates the group keri then our claim will follow

But fn g is linearly indep endent since is simplicial Hence we can nd m M

such that

hm n i when A and hm n i when A

Note that m keri Now given any m keri we can pick an integer N such

A A

that m N m keri Then m m N m N m shows that keri generates

A A

keri as a group This proves that g keri fg

A A A

From the exact sequence it follows that g Z C induces a map

g A C where A A X is the image of Z A X Since C is

A n n

injective as an ab elian group g extends to g A X C Thus g G and it is

A n

straightforward to check that u g D t for all Then u G t and we conclude

that G t V Thus the orbits are closed and we have a geometric quotient

To prove the converse assume that X is not simplicial The following argument of

I Musson will show that G has a nonclosed orbit in C Z Let b e a non

simplicial cone so that we have a relation b n We can assume that b Z

and that at least one b Then set b for Applying Hom C to the

exact sequence at the b ottom of we see that tD t denes an element of G

for t C in fact t is a parameter subgroup of G Now consider u u C

where

if or b

otherwise

It is easy to check that u and lim tu are in C Z but lie in dierent Gorbits

Thus G u isnt closed in C Z and Theorem is proved

The description of X as a quotient of C Z gives a new way of lo oking at the

torus action on X Namely an element of the torus C gives an automorphism

of C which preserves Z It also commutes with the action of G so that we get an

automorphism of X by the universal prop erty of a categorical quotient This gives a map

C Aut X and the kernel is easily seen to b e G see x for a pro of Since we

have the exact sequence

G C T

it follows that the torus T is naturally a subgroup of Aut X In x we will discuss AutX

in more detail Note also that C C Z and thus taking the quotient with

resp ect to G we get the inclusion T X

We should also mention Odas toric interpretation of the quotient map C Z X

from Theorem Namely let N b e the lattice Hom Z Z Then the injection

where n M Z induces a surjection N N Further N has a natural basis n

maps to n N for all Now consider the fan in N determined as follows for each

for Then the toric variety generated by n cone let b e the cone in N

determined by is C Z with its natural action by the torus C and the map

C Z X is the equivariant map determined by the map of fans N N

When considering Theorem one should keep in mind our assumption that

spans N Although this is true in the most interesting cases eg X complete we should

discuss what happ ens when fails to span N In this case let N N Span

so that can b e regarded as a fan in N This gives a toric variety X canonically

embedded in X of dimension d rankN One can check that A X is naturally

isomorphic to A X and hence X and X have the same co ordinate ring Furthermore

Theorem shows that

X C Z G

To recover X note that N is a direct summand of N Then N N N and since

we can regard as the pro duct fan fg we get a noncanonical isomorphism X

nd nd N N

Since C Z C C W for some N where W C is a X C

union of co ordinate subspaces we get a quotient categorical or geometric as appropriate

X C W G

This construction is not canonical since the isomorphism dep ends on N Note also that

W has co dimension in C while Z has co dimension in C by Lemma

For the nal result of this section we will use Theorem to study closed subsets

reduced closed subschemes of a simplicial toric variety X Given a graded ideal I S

note that the set

V I Z ft C Z f t for all f I g C Z

is Ginvariant and closed in the Zariski top ology Since we have a geometric quotient the

quotient map C Z X is submersive see Fogarty and Mumford Denition

Thus there is a onetoone corresp ondence b etween Ginvariant closed sets in C Z

and closed sets in X Hence V I Z determines a closed subset of X which we will

denote V I By abuse of notation we can write

V I ft X f t for all f I g

provided we think of t as the homogeneous co ordinates of a p oint of X

The map sending I S to V I X has the following prop erties

Prop osition Let X b e a simplicial toric variety and let B hx i S b e

the ideal dened in x

i The Toric Nullstellensatz For any graded ideal I S we have V I if and

only if B I for some integer m

ii The Toric IdealVariety Corresp ondence The map I V I induces a onetoone

corresp ondence b etween radical graded ideals of S contained in B and closed subsets

of X

Pro of i Since V I in X if and only if V I Z V B in C the rst part

of the prop osition follows from the usual Nullstellensatz

ii Given a closed subset W C we get the ideal IW ff S f g It

is straightforward to show that W is Ginvariant if and only if IW is a graded ideal of S

Note also that B is radical since it is generated by squarefree monomials Thus B IZ

and we get onetoone corresp ondences

radical graded ideals of S contained in B

Ginvariant closed subsets of C containing Z

Ginvariant closed subsets of C Z

where the last corresp ondence is induced by Zariski closure since Z is Ginvariant The

prop osition now follows since C Z X is submersive

In x we will generalize this prop osition by showing that there is a corresp ondence

b etween graded S mo dules and quasicoherent sheaves on X

x Graded Mo dules and Quasicoherent Sheaves

In this section we will explore the relation b etween sheaves on a toric variety X and

graded mo dules over its homogeneous co ordinate ring S The theory we will sketch is

similar to what happ ens for sheaves on P as given in Hartshorne or Serre

An S mo dule F is graded if there is a direct sum decomp osition

F F

A X

such that S F F for all A X Given such a mo dule F we will construct

a quasicoherent sheaf F on X as follows First for let S b e the lo calization of S

x Then F F S is a graded S mo dule so that taking elements at x

of degree gives a S mo dule F From the pro of of Theorem we know that

X Sp ecS is an ane op en subset of X and then standard theory tells us that

F determines a quasicoherent sheaf F on X

To see how these sheaves patch together let b e a face of Then fm g for

some m M and as in Lemma one can prove

F F

hmn i

where x x It follows immediately that the sheaves F on X patch to

give a quasicoherent sheaf F on X

The map F F has the following basic prop erties of we omit the simple pro of

Prop osition The map sending F to F is an exact functor from graded S mo dules

to quasicoherent O mo dules

As an example of how this works consider the S mo dule S where S S

for all We will denote the sheaf S by O If D for some T invariant divisor

D then there is an isomorphism

O O D

X X

To prove this it suces to nd compatible isomorphisms

S H X O D

for each ane op en X of X If D a D then Fulton x shows that this is

equivalent to nding compatible isomorphisms

S C

hmn ia

for all

hmn ia m D D

x S As in Prop osition we can map to the monomial x

The compatibility of these maps is easily checked giving us the desired isomorphism

From we see that S O It follows easily that H X O is canonically

X X

isomorphic to S so that

H X O S

From we also conclude that O is a coherent sheaf see Fulton x However

it need not b e a line bundlein fact O is invertible if and only if is the class of a

Cartier divisor See Dolgachev x for an example of how O can fail to b e invertible

when X is a weighted pro jective space Also note that the natural map

O O O

X O X X

need not b e an isomorphismsee Dolgachev x for an example However when O

and O are invertible then is an isomorphism

Our next task is to see which sheaves on X come from graded S mo dules When X

is simplicial the answer is esp ecially nice

Theorem If X is a simplicial toric variety then every quasicoherent sheaf on X is

of the form F for some graded S mo dule F

Pro of The pro of is similar to the pro of of the corresp onding result in P as given in

Hartshorne or Serre in what follows we will give precise references to Hartshorne

though one could just as easily use the original pap er of Serre The argument requires

standard facts ab out the b ehavior of sections of line bundles see Hartshorne Lemma

I I As we observed ab ove O need not b e a line bundle However when X is

simplicial PicX has nite index in A X this follows from an exercise in Fulton

x Thus given any there is a p ositive integer k such that O k is a line bundle

Given a quasicoherent sheaf F on X let

F H O X F

X O

Using and the natural isomorphism S H X O we see that F has the natural

structure of a graded S mo dule Given let

F H X F O

O X

which is a graded mo dule over S There is an obvious map

F F H X F

which patch to give F F To show that is an isomorphism it suces to show that

is an isomorphism for all

D D

An element of F is written f x where f H X F O and degx

O X

D k D k D

Since f x x f x when k we can assume that O is a line bundle

D D

We can also assume that x divides x which implies that X fx X x g This

guarantees that Hartshorne Lemma I I applies and from here the pro of follows the

arguments in Hartshorne Prop osition I I without change

We next study coherent sheaves on X

Prop osition Let X b e a toric variety with co ordinate ring S

i If F is a nitely generated graded S mo dule then F is a coherent sheaf on X

ii If X is simplicial then every coherent sheaf on X is of the form F for some nitely

generated graded S mo dule F

Pro of i A set of homogeneous generators of F gives a surjective map S F

i i

e e

By Prop osition we get a surjection O F and it follows that F is coherent

i X i

ii Supp ose that F is coherent On the ane scheme X we can nd nitely many

sections f H X F which generate F over X Since X is simplicial arguing as

invertible in S such that in the pro of of Theorem shows that we can nd some x

D D

Now consider O where deg x f comes from a section g of F x

X i i O

O generated by the g Since the graded S mo dule F F H X F

X i O

e e

and F F F we see that F F is injective Over X we have f g x

i i

since these sections generate F over X it follows that F F

The map sending graded S mo dule F to the sheaf F is not injective and in fact there

are many nonzero mo dules which give the zero sheaf To analyze this phenomenon we

will use the subgroup PicX A X The following lemma will b e helpful

Lemma If PicX A X then for every we can write in the

form a D

Pro of It is wellknown see for example the exercises in Fulton x that if b D

is a Cartier divisor then for each there is m M such that hm n i b for all

It follows that D is linearly equivalent to the divisor b hm n iD

and the lemma is proved

We can now describ e which nitely generated graded S mo dules F corresp ond to the

zero sheaf on X In the pro jective case where S Cx x this happ ens if and

only if F fg for all k This can b e rephrased in terms of the irrelevant ideal

hx x i S namely F fg for k exactly when hx x i F fg for

n k n

some k In the case of a simplicial toric variety we can generalize this as follows

Prop osition Let X b e a simplicial toric variety and let B S b e the ideal dened

in x Then for a nitely generated graded S mo dule F we have F on X if and only

if there is some k such that B F fg for all PicX

Pro of First supp ose that B F fg for all PicX Then x and take

f x F Because X is simplicial we know that l D PicX for some l If

lD k lD

we let l D then x f F and hence x x f for some k This

easily implies f x and it follows that F fg Thus F is the zero sheaf

Conversely let F b e a nitely generated graded S mo dule such that F Fix

a D PicX and let f F By Lemma we can write D where D

D D

Then x is an invertible element in S and hence f x F fg This implies that

x f for some k To show that we can get a k that works for all f F and all

PicX note that SF F is a submo dule and hence is nitely generated

PicX

The prop osition now follows easily

When X is a smo oth toric variety we know that PicX A X Hence we get

the following corollary of Prop osition

Corollary If X is a smo oth toric variety and F is a nitely generated graded S

mo dule then F on X if and only if there is some k with B F fg

To see how Corollary can fail when X not smo oth consider X P Here

S Cx y z where x y have degree and z has degree and B hx y z i The mo dule

F S xS y S has only elements of o dd degree Then F fg since z has

degree and it is clear that F F fg It follows that F yet one easily

x y

k k

checks that B F z F fg for all k Thus Corollary do es not hold

We will end this section with a study of the idealvariety corresp ondence Our goal

is to rene what we did in Prop osition by taking the scheme structure into account

First note that if I S is a graded ideal then by exactness we get an ideal sheaf I O

Hence I S determines the closed subscheme Y X with O O I The following

Y X

theorem tells us which subschemes of X arise in this way

Theorem Let X b e a simplicial toric variety and let B S b e as usual

i Every closed subscheme of X comes from a graded ideal of S as describ ed ab ove

ii Two graded ideals I and J of S corresp ond to the same closed subscheme of X if and

only I B J B for all PicX

O Since X is Pro of i Given an ideal sheaf I O let I H X I

X X O

simplicial we know that I I The map I S factors I I S which gives maps

e e

I I O by exactness Since the comp osition is I O it follows that I I

X X

ii First note that if I is graded then so is I B Now given graded ideals I J S

e e

assume that I J Then for every we have I J in S Let PicX

and pick u I B For k suciently large we have x u I Increasing k if

necessary we can assume that k degx PicX Thus PicX so Lemma

k D

implies D where D a D Then x ux I J

k l

which implies that x u J for some l This is true for all so that u J B

The other inclusion J B I B follows by symmetry and equality is proved

e e

Conversely supp ose I B J B for all PicX To prove I J

it suces to show that I J for all But given u I we know that

u f x and we can assume that degf PicX Then f I I B

J B This easily implies u f x is in J Thus I J and the

opp osite inclusion follows by symmetry This completes the pro of of the theorem

As an example consider the ideal I hx i for some xed We will show

that the corresp onding subscheme of X is the divisor D Given we need to study

I If then it is easy to check that under the isomorphism S C M

we have

C I

hm i

hmn i

By an exercise in Fulton x this is exactly the ideal of D in the ane piece X X

On the other hand if then I hi which is consistent with D X

Thus D is dened schemetheoretically by the ideal hx i In particular D has the global

equation x even though it may not b e a Cartier divisor

In the smo oth case Theorem has some nice consequences We say that an ideal

I S is B saturated if I B I Then we have

Corollary Let X b e a smo oth toric variety

i Two graded ideals I and J of S give the same closed subscheme of X if and only if

I B J B

ii There is a onetoone corresp ondence b etween closed subschemes of X and graded

B saturated ideals of S

Pro of Part i follows from Theorem since PicX A X and part ii follows

immediately since an ideal is B saturated if and only if it is of the form J B

When X is smo oth Corollary gives a onetoone corresp ondence b etween radical

graded B saturated ideals of S and closed reduced subschemes of X This is slightly

dierent from the corresp ondence given in Prop osition where we used radical ideals

contained in B However the maps

I radical graded B saturated I B

J radical graded contained in B J B

give bijections b etween these two classes of radical graded ideals so the two descriptions

of reduced subschemes are compatible

Finally one could ask if there is an analog of Corollary in the simplicial case

There are two ways to approach this problem One way is to work with the following

sp ecial class of graded ideals We say that a graded ideal I S is Picgenerated if I is

generated by homogeneous elements whose degrees lie in PicX and I is Picsaturated if

I I B for all PicX For example if I is any graded ideal it is easy to check

that S I B is b oth Picgenerated and Picsaturated Then one can prove

PicX

the following result we will omit the pro of

Corollary If X is a simplicial toric variety then there is a onetoone corresp ondence

b etween closed subschemes of X and graded ideals of S which are Picgenerated and Pic

saturated

Another way to study sheaves on X is to replace the ring S with the subring

S R

PicX

When X is simplicial we know that PicX A X has nite index Thus

H Hom A X Pic X C G Hom A X C

Z n Z n

is a nite subgroup of G and R S under the action of H on S It follows that

Y Sp ecR is the quotient of C under the action of H Furthermore if we set

B B R and let Z V B Y then it is easy to see that Y Z C Z H

There is also a natural action of Hom Pic X C on Y Z and Theorem implies

the following result

Theorem If X is a simplicial toric variety and Y Sp ecR where R is the ring

then X is the geometric quotient of the action of Hom PicX C on Y Z

Although the ring R may not b e a p olynomial ring the relation b etween sheaves and

Rmo dules is nicer than the relation b etween sheaves and S mo dules It is easy to show

that a graded Rmo dule F where the grading is now given by PicX gives a sheaf F on

X Then one can prove the following result we omit the pro of since the arguments are

similar to what we did ab ove

Theorem If X is a simplicial toric variety and R is the ring dened in then

i The map sending a graded Rmo dule F to F is an exact functor

ii All quasicoherent sheaves on X arise in this way

iii F is coherent when F is nitely generated and this gives all coherent sheaves on X

iv If F is nitely generated then F if and only if B F fg for some k where

B B R

Finally a graded ideal I R gives an ideal sheaf I O so that I determines a

closed subscheme of X We will omit the pro of of the following result

Theorem If X is a simplicial toric variety and R is the ring dened in then

i Every closed subscheme of X comes from some graded ideal of R

ii Two graded ideals I J give the same closed subscheme if and only if I B J B

where B B R

iii There is a onetoone corresp ondence b etween graded B saturated ideals of R and

closed subschemes of X

x The Automorphism Group

This section will use the graded ring S to give an explicit description of the automor

phism group AutX of a complete simplicial toric variety X To see what sort of results

n n n

to exp ect consider P C fgC Notice that Aut P P GLn C do esnt

act on C fg rather we must use GLn C which is related to P GLn C

by the exact sequence

C GLn C P GLn C

For a complete simplicial toric variety X we know X C Z G by Theorem

Then the analog of GLn C will b e the following pair of groups

Denition

i AutX is the normalizer of G in the automorphism group of C Z

ii Aut X is the centralizer of G in the automorphism group of C Z

g g

It is clear that Aut X AutX is a normal subgroup Note also that every element

of Aut X preserves Gorbits and hence descends to an automorphism of X Thus we get

a natural homomorphism AutX AutX

We can now state the main result of this section

Theorem Let X b e a complete simplicial toric variety and let S Cx b e its

homogeneous co ordinate ring Then

g g

dim S and Aut X is i AutX is an ane algebraic group of dimension

degx

the connected comp onent of the identity

ii The natural map Aut X AutX induces an exact sequence

G AutX Aut X

iii Aut X is naturally isomorphic to the group Aut S of graded Calgebra automor

phisms of S

In the smo oth case this theorem follows from results of Demazure Before we can

b egin the pro of of Theorem we need to study the group Aut S Consider the set

End S of graded Calgebra homomorphisms S S which satisfy Since

X is complete we know that S C and it follows that S S is closed under

multiplication Thus there is a natural bijection End S End S Since S has no

g g

identity to worry ab out End S has a natural structure as a Calgebra and in this way

we can regard End S as a Calgebra

Note that an element End S is uniquely determined by x S for

degx

Furthermore each S is nite dimensional since X is complete so that

degx

End S is a nite dimensional Calgebra Since Aut S is the set of invertible elements

g g

of End S standard arguments show that Aut S is an ane algebraic group

g g

To describ e the structure of this group in more detail we will use the equivalence

relation on dened by

deg x deg x in A X

This partitions into disjoint subsets where each corresp onds to a

s i

set of variables of the same degree Then each S can b e written as

S S S

i i

is spanned by the remaining monomials is spanned by the x for and S where S

i i

all of which are the pro duct of at least two variables in S

Before we state our next result note that the torus C can b e regarded as a

subgroup of Aut S since t t C gives the graded automorphism S S

g t

dened by x t x Then the group structure of Aut S can b e describ ed as follows

t g

Prop osition Let X b e a complete toric variety and let S b e its homogeneous co or

dinate ring Then

and i Aut S is a connected ane algebraic group of dimension j j dim S

g i C

C Aut S is its maximal torus

ii The unip otent radical R of Aut S is isomorphic as a variety to an ane space of

u g

j jdim S j j dimension

i C i

of iii Aut S contains a subgroup G isomorphic to the reductive group GLS

g s

dimension j j

iv Aut S is isomorphic to the semidirect pro duct R o G

g u s

Pro of We rst study the structure of the Calgebra End S We have a natural vector

space isomorphism

End S Hom S S

g C

i i

since S is spanned by x for and a graded homomorphism S S is uniquely

determined by x S Then the direct sum decomp osition gives an exact sequence

s s

Hom S S End S End S

C g C

i i

i i i

Notice that is a ring homomorphism This follows immediately from the observation

that S S for any End S Thus we can regard N Hom S S as

g C

i i i i

an ideal of End S Note that N if and only if x S for

g i

We claim that every element of N is nilp otent To prove this we will use the order

relation on the monomials of S dened in Lemma For variables the relation is dened

D D D

jx and x if and only if there is a monomial x with deg x deg x x by x

D D D

x x If degx this is equivalent to the existence of x S with x jx

Given N let N the b e number of variables x for which x If

N jj then and we are done If N jj then lo ok at the variables

where do esnt vanish and pick one say x which is minimal with resp ect to We can do

this b ecause X is complete and hence is antisymmetric by Lemma If degx

By the then as we noted ab ove x x for any variable x dividing a monomial in S

This implies and consequently S minimality of x it follows that x

We conclude that N N since also vanishes that x since x S

on every variable killed by The same argument applies to N and it

follows that some p ower of must b e zero Thus is nilp otent as claimed

Once we know that N is a nilp otent ideal it follows that every element of N is

invertible so that N is a unip otent subgroup of Aut S From the exact sequence

we also see that a element End S is invertible if and only if its image in GLS

is invertible From this it follows that we get an exact sequence

N Aut S GLS

is a reductive group it follows that N is the unip otent radical of Since GLS

Aut S Then part ii of the prop osition follows b ecause as a variety we have N N

which is an ane space of the required dimension

To prove part iii observe that we have a map

End S s End S

g C

maps to the homomorphism dened by x x where End S

i i C

when It is easy to check that s is a ring homomorphism which is a section of the

map of It follows that s induces a map

s GLS Aut S

which is a section of the exact sequence The image G of s is a subgroup of Aut S

s g

isomorphic to GLS and thus Aut S is the semidirect pro duct of N o G

g s

This proves parts iii and iv and then part i now follows immediately

We will next study the roots of Aut X as dened by Demazure Recall that the ro ots

of the complete toric variety X are the set

RN fm M with hm n i and hm n i for g

see Demazure x or Oda x We divide the ro ots RN into two classes as

follows

R N RN RN R N RN R N

s u s

The ro ots in R N are called the semisimple ro ots

We can describ e RN and R N in terms of the graded ring S as follows

Lemma There is are onetoone corresp ondences

D D D

RN fx x x S is a monomial x x deg x deg x g

g degx deg x x x R N fx x

Pro of Given m RN we have with hm n i and hm n i for

Then consider the map

hmn i

m x x

Since m is a ro ot the second comp onent has nonnegative exp onents and hence is a mono

D D

mial x S It is also clear that x and x are distinct and have the same degree

This map is injective b ecause M Z is injective To see that it is surjective

D D

supp ose we have x x of the same degree in S A rst observation is that x x

D E E

for otherwise we could write x x x and it would follow that x S would have

degree zero Yet X complete implies S C so that x must b e constant This would

D D

force x x which is imp ossible Then writing x x the denition of

degx deg x implies there is m M with

hm n i hm n i a for

Thus m is a ro ot mapping to the pair x x

The corresp ondence for semisimple ro ots follows easily from the observation that if

D D D

m x x and m x x then x x

This lemma allows us to think of semisimple ro ots as pairs of distinct variables of the

same degree

Given a ro ot m RN we have the corresp onding pair x x Then for C

consider the map

y S S

for It is easy to see x dened by y x x x and y x

m m

that y gives a one parameter group of graded automorphisms of S so that y

m m

Aut S We can relate the y to Aut S as follows

g m g

Prop osition Let X b e a complete toric variety Then Aut S is generated by the

torus C and the one parameter subgroups y for m RN

Pro of In Prop osition we gave showed that Aut S was the semidirect pro duct

Aut S N o G where G GLS

g s s

In this decomp osition we know from Prop osition that C Aut S is the

maximal torus and hence lies in G

is in R N Then letting degx we have First supp ose that m x x

s i

x and y x in the direct sum Since y x x x S x x

m m

for it follows that y G and in fact the y corresp ond to the ro ots

m s m

in the usual sense of algebraic groups Since any reductive group of G GLS

is generated by its maximal torus and the one parameter subgroups given by its ro ots it

follows that G is generated by C and the y for m R N

s m s

Next supp ose that m x x is a ro ot in R N If we set deg x then

u i

in the direct sum It follows that y N We claim that N is x S

generated by the y for m R N To prove this supp ose that N If

m u

then dene the weight of to b e the sum x x u where u S

number of nonzero co ecients of u w

Note the is the identity if and only if w If is not the identity then we can use

the ordering of variables from the pro of of Prop osition to pick a x with u such

that x is maximal with resp ect to this prop erty Let x is a nonzero term of u so that

D D

u x u Then x x corresp onds to a ro ot m R N and we have

y x y x x u x u

m m

since none of the monomials in u x u involve x Next supp ose that Then

x do esnt app ear in any monomial in u for if it did we would have x x which

only involves variables u x would contradict the maximality of x Thus x

dierent from x and hence

for x y x

From and we conclude that w y w and from here it is easy to see

can b e written in terms of the y This shows that N is generated by the y

m m

for m R N Given the generators weve found for G and N the prop osition

u s

follows now immediately

Our next task is to relate Aut S to the group Aut X of Denition Every

Aut S gives an automorphism C C which sends t t C

to t x t the notation x t means the p olynomial x evaluated at the

p oint t It is easy to check that which commutes with the action of G Our next step

is to show that preserves C Z C

Prop osition Assume that X is a complete simplicial toric variety and let

C C is the automorphism determined by Aut S Then preserves

C Z commutes with G and hence lies in Aut X

Pro of Every element of C clearly preserves C Z so that by Prop osition

it suces to prove the prop osition when y where m is a ro ot in RN Also it

suces to show that y C Z C Z To prove this supp ose m x x

We will write a p oint of C as t t t where t is a vector indexed by fg

Then y is given by

y t t t t t

D D D

where t is the monomial x evaluated at t this makes sense since x x

As in the pro of of Theorem write C Z U where U ft C

t g Recall that x x Now let t t t U and set u y t

t t t We will show that u U for some by an argument similar to

Prop osition of Oda When we have u t so that u U So

supp ose that Since hm n i and hm n i for it follows that

fmg is a face of such that is on one side of fmg and all of the other one

dimensional cones of are on the other side But is complete so and fmg must

b e faces of some cone of which is on the same side of fmg as Since is simplicial it

follows that fmg must b e a cone of There are now two cases to consider

Case If t then since we have u t Thus u U

Case If t then t for some Since fmg this means

hmn i

which implies hm n i Then t

Y Y Y

hmn i

t t t u t t t

fg fg

It follows that u U

Thus y preserves C Z and the prop osition is proved

We can now b egin the pro of of Theorem

Pro of of Theorem We rst prove part iii of the theorem which asserts that

Aut S is naturally isomorphic to Aut X From Prop osition the map sending to

denes a group homomorphism

Aut S Aut X

This map is easily seen to b e injective To see that it is surjective supp ose that

C Z C Z is Gequivariant Then for any we get x C Z C

and since has co dimension by Lemma purity of the branch lo cus implies that

x extends to a p olynomial u dened on C Since commutes with the action

of G it follows that u is homogeneous and degu degx Then x u denes

a graded homomorphism S S with the prop erty that To see that is an

isomorphism apply the ab ove argument to This shows that is an isomorphism

and part iii of Theorem is done

We now turn to part ii of the theorem which asserts that we have an exact sequence

G AutX Aut X

It is obvious that every element of G gives the identity automorphism of X Conversely

supp ose that AutX induces the identity automorphism on X We need to show that

G Our assumption implies that Y Y for every Ginvariant Y C Z

so that in particular U U for every Since U Sp ecS we get an

induced map S S and hence S S It is easy to see that S S and

it follows that S S Thus for every we have x A S Note however

that A need not b e homogeneous of degree deg x

Since V x Z C Z is also Ginvariant it is stable under But the th

co ordinate of t is A t so that A vanishes on V x Z Since Z has co dimension

we conclude that A vanishes on V x The Nullstellensatz implies that x jA and

hence x x B for some B S The same argument applied to shows that

x x C for some C S Then

x x x C x B C

This implies that B C in the p olynomial ring S Cx so that B is a nonzero

constant Thus B C Since induces the identity on T X the exact

sequence G C T shows that G as claimed This proves that G

is the kernel of Aut X Aut X

We still need to prove that Aut X AutX is onto We will rst show that there

is an exact sequence

G Aut X AutX AutA X

The map AutX Aut A X is induced by direct image of divisors given

P P

AutX a divisor class a D maps to a D

i i i

i i

To show exactness let Aut X We need to show that the induced automorphism

of X which will also b e denoted is the identity on A X By part iii of the theorem

V x Z then we can write t u t where u S If D

degx

divu x D D

in C Z Since u and x have the same degree the quotient u x is Ginvariant

and descends to a rational function on X Then it follows immediately that

divu x D D

in X This shows that D D so that induces the identity on A X

Next assume that AutX induces the identity on A X We need to show

that comes from an element of Aut X Fix and let deg x D

Since D D in A X we can nd a rational function g on X such that

divg D D

Then divg D which implies g H X O D Using the isomorphism

H X O D S from Prop osition we can write g u x for some u S

i i

Note that u is only determined up to a constant We will choose the constant as follows

Q Q

hmn i hmn i

Given m M x and u are rational functions on X Then

Q Q Q

hmn i hmn i

hmn i

x u div u x div div

P P

hm n i D D hm n iD

P Q

hmn i

div hm n iD x

This implies that there is m C such that

Q Q

hmn i hmn i

m u x

It is easy to see that M C is a homomorphism which implies that T The map

hmn i

C T is surjective so that there is t t C such that m t

for all m M Thus if we replace u by t u then for m M we have

Q Q

hmn i hmn i

u x

Now dene S S by x u and consider the diagram

S S

i i

x x

u x

H X O D H X O D H X O D

X X X

Here the vertical arrows are the isomorphisms given by Prop osition Also maps

g H X O D to g H X O D and the nal map on the b ottom

X X

takes g H X O D to g u x H X O D To see that the diagram

X X

commutes let u S b e a monomial Using the left and b ottom maps we get

u u

H X O D

x x

hmn i

However since u and x have the same degree we can write ux x for some

m M Then easily implies that ux uu so that

u u u u u

x x u x x

This shows that the ab ove diagram commutes Since the maps on the b ottom are clearly

is an isomorphism This shows that Aut S and isomorphisms it follows that

then Prop osition gives Aut X

From we get at automorphism of X which we will also denote by Since

u x on C Z it follows that for any m M we have

Q Q

hmn i hmn i

x u

From we see that and agree as birational automorphisms of T which implies

as automorphisms of X This completes the pro of that the sequence is exact

The next step in showing that Aut X Aut X is onto is to study the role played

by the automorphisms of the fan Let AutN b e the group of lattice isomorphisms

of N which preserve the fan Such a p ermutes and this p ermutation determines

uniquely since spans N It follows that AutN is a nite group

We rst relate Aut N to AutX Given AutN the induced p ermutation

of can b e regarded as a p ermutation matrix P GLC We claim that induces

an automorphism b G G such that for any g G we have

g P P b g

To prove this rst note that induces a dual automorphism M M Consider the

following diagram

M Z A X

It is straightforward to check that the diagram commutes which shows that we get an

induced isomorphism b A X A X This in turn gives b G G since

n n

G Hom A X C and now follows easily

Z n

From we see that P normalizes G and notice also that P preserves Z b ecause

preserves Hence P Aut X for all Aut N Furthermore it is easy to

check that the induced automorphism of X is the usual way that AutN acts on

X By abuse of notation we will write P Aut X

Let H Aut X b e the image of Aut X AutX By and Prop osition

it follows that H Aut X G Aut S G is a connected ane algebraic group with

C G T as maximal torus Also implies that H is normal in Aut X

Now we can nally show that Aut X AutX is onto Supp ose that AutX

Then we have T H H so that T is a maximal torus in H Since

all maximal tori of H are conjugate there is some H such that T T

If we set it follows that T T in AutX This easily implies that

T T in X Thus is an automorphism of T But the automorphism group of a

torus is generated by translations and group automorphisms Thus g t where g is

a group automorphism and t T We claim that t X X is T equivariant This

is certainly true on T since t g and the claim follows since T is dense in X

By Theorem of Oda this implies t P for some AutN

It follows that P t in Aut X Thus AutX is generated by

H and the P for AutN Since every P comes from an element of AutX

AutX AutX is surjective This completes the pro of of part ii of Theorem

Finally consider part i of the theorem which asserts that AutX is an ane alge

braic group with Aut X as the connected comp onent of the identity To show this note

that since AutX is generated by the image of Aut X and the P it follows from part

g g

ii of the theorem that AutX is generated by Aut X and P for AutN Since

g g g

AutN is nite Aut X has nite index in Aut X We already know that Aut X

is a connected ane algebraic group and hence AutX is an ane algebraic group with

Aut X as the connected comp onent of the identity Theorem is now proved

From the ab ove pro of we can extract some additional information ab out AutX and

AutX For example AutX is generated by C the y for m RN and

the P for AutN To discuss AutX in more detail we need some notation for

a ro ot m RN we have y Aut X and we let x denote the resulting

m m

automorphism of X It is easy to check that this is the same automorphism x as

dened by Demazure x or Oda x Then we can describ e AutX as follows

Corollary Let X b e a complete simplicial toric variety Then

i AutX is an ane algebraic group with T as maximal torus

ii AutX is generated by T the x for m RN and the P for AutN

iii If Aut X is the connected comp onent of the identity of AutX then

Aut X Aut X G Aut S G

iv Aut X is the semidirect pro duct of its unip otent radical with a reductive group G

whose ro ot system consists of comp onents of type A

v From the partition where the x have the same degree for

s i

is the Weyl group of G denotes the symmetric group on Then let

s i

i i

Furthermore we can regard this group as a normal subgroup of AutN and there

are natural isomorphisms

g g

AutX Aut X AutX Aut X Aut N

Pro of Parts i iv of the corollary follow easily from the pro of of Theorem and

the other results of this section As for part v it is clear that we have an isomorphism

g g

AutX Aut X Aut X Aut X Furthermore we can regard AutN as a sub

group of AutX and then we have an isomorphism

g g g

Aut X Aut X AutN Aut N Aut X

Thus regarding elements of as p ermutation matrices it suces to prove that

Aut N Aut X

One direction is easy for if P AutN Aut X then by we know that P

induces the identity automorphism on A X This shows that as a p ermutation of the

s P preserves the degree of x and it follows immediately that P

Q Q Q

s s s

Going the other way supp ose that P Since GLS

i i i

Aut X P induces an automorphism of X On C Z P p ermutes the variables

so that on X p ermutes the D Since T X D it follows that T T As

we saw in the pro of of Theorem this implies that P t for some AutN

and t T Back in Aut X we then have P P u for some u C Since

P and P are p ermutation matrices we conclude that P P and it follows that P

AutN Aut X This completes the pro of of the corollary

When X is smo oth this corollary is due to Demazure see Demazure or Oda x

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