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 n 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 n and the ideal B is the irrelevant ideal hx x i n 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 n On P there is a corresp ondence b etween sheaves and graded Cx x mo dules n 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 n of GLn C on C fg In x we will see that for any complete simplicial toric g 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 g 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 n 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 Z P 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 T m m Each m M gives a character T C and hence is a rational function on P m m X As is wellknown gives the Cartier divisor div hm n iD This gives a map P hm n iD M Z dened by m D m which is injective since spans N By Fulton x we have a commutative diagram R M Div X PicX T k M Z A X n 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 n For each introduce a variable x and consider the p olynomial ring S Cx Q a x determines a divisor We will usually write this as S Cx Note that a monomial P D 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 n 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 M D S C x D degx so that the ring S can b e written as the direct sum M S S A X n 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 n 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 n 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 n S H X O D D X where O D is the coherent sheaf on X determined by the Weil divisor D see Fulton X x 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 P Pro of Supp ose that D a D By Fulton x we know that H X O D X m C where P M D P fm M hm n i a for all g D R P Q hmn ia D D m Given m P M let D hm n iD so that x x Then D m D D m m x denes a map from P M to the monomials in S the monomial is D in S since m P and it has degree since m M This map is onetoone since D M Z is injective and it is easy to see that every monomial in S arises in this way D D m m Then x gives the desired isomorphism The pro of of the second part of D 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 P 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 n 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