Homogeneous Coordinate Ring

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1 Students: Kaiserslautern University Algebraic Group June 14, 2013

2 Outline Quotients in Algebraic Geometry 1 Quotients in Algebraic Geometry 2 3 4

3 Outline Quotients in Algebraic Geometry 1 Quotients in Algebraic Geometry 2 3 4

4 Definition Let G be a group acting on a variety X = Spec(R), R is a K-algebra. Then the following map G R R (g, f ) g.f defined by (g.f )(x) = f (g 1.x) for all x X is an action of G on R.

5 Remark: a) The group acting as above is induced by the group acting of G on X. b) The above acting gives two objects, namely the set G-orbits X /G = {G.x x X } and the ring of invariants R G = {f R g.f = f, for all g G}.

6 Definition Let G act on X and let π : X Y be morphism that is constant on G-orbits. Then π is called a good categorical quotient if: a) If U Y is open, then the natural map O Y (U) O X (π 1 (U)) induces an isomorphism O Y (U) O X (π 1 (U)) G. b) If W X is closed and G-invariant, then π(w ) Y is closed. c) If W 1, W 2 are closed, disjoint, and G-invariant in X, then π(w 1 ) and π(w 2 ) are disjoint in Y. We often write a good categorical quotient as π : X X //G.

7 Theorem: Let π : X X //G be a good categorical quotient. Then: a) Given any diagram where φ is a morphism of varieties such that φ(g.x) = φ(x) for g G and x X, there is unique morphism φ making the diagram commute, i.e., φ π = φ. b) π is surjective. c) A subset U X //G is open iff π 1 (U) X is open. d) x, y X, we have π(x) = π(y) G.x G.y.

8 Definition a) A subgroup G of GL n (C) is called an affine algebraic group if G is a subvariety of GL n (C). b) Let G be an affine algebraic group acting on a variety X. The G-action is called algebraic action if the action defines a morphism. G X X (g, x) g.x

9 Proposition Let an affine algebraic group G act algebraically on a variety X, and assume that a good categorical quotient π : X X //G. Then: a) If p X //G, then π 1 (p) contains a unique closed G-orbit. b) π induces a bijection {closed G-orbits in X } X //G.

10 Proposition Let π : X X //G be a good categorical quotient. Then the following are equivalent: a) All G-orbits are closed in X. b) Given x, y X, we have π(x) = π(y) x and y lie in the same G-orbit. c) π induces a bijection {G-orbits in X } X //G.

11 Definition A good categorical quotient is called a good geometric quotient if it satisfies the condition of the above proposition. We write a good geometric quotient as π : X X /G.

12 Definition An affine algebraic group G is called reductive if its maximal connected solvable subgroup is a torus. Proposition Let G be a reductive group acting algebraically on an affine variety X = Spec(R). Then a) R G is a finely generated C-algebra. b) The morphism π : X Spec(R G ) induced by R G R is a good categorical quotient.

13 Proposition Let G act on X and let π : X Y be a morphism of varieties that is constant on G-orbits. If Y has an open cover Y = α V α such that π π 1 (V α) : π 1 (V α ) V α is a good categorical quotient for every α, then π : X Y is a good categorical quotient.

14 Example: Let C act on C 2 \{0} by scalar multiplication, where C 2 = Spec(C[x 0, x 1 ]). Then C 2 \{0} = U 0 U 1, where U 0 = C 2 \V (x 0 ) = Spec(C[x ±1 0, x 1]) U 1 = C 2 \V (x 1 ) = Spec(C[x 0, x ±1 1 ]) U 0 U 1 = C 2 \V (x 0 x 1 ) = Spec(C[x ±1 0, x ±1 1 ]) The rings of invariants are C[x ±1 0, x 1] C = C[x 1 /x 0 ] C[x 0, x ±1 1 ]C = C[x 0 /x 1 ] C[x ±1 0, x ±1 1 ]C = C[(x 1 /x 0 ) ±1 ]

15 It follows that the V i = U i //C glue together in the usual way to create P 1. Since C -orbits are closed in C 2 \{0}, it follows that is a good geometric quotient. P 1 = (C 2 \{0})/C

16 Outline Quotients in Algebraic Geometry 1 Quotients in Algebraic Geometry 2 3 4

17 Let X Σ be the toric variety of a fan Σ in N R. The goal is to construct X Σ as a good categorical quotient X Σ (C r \Z)//G for an appropriate of affine space C r, exceptional set Z C r, and reductive group G.

18 Definition Let X Σ be the toric variety of fan Σ in N (R). Assume that X Σ has no torus factor. We define G = Hom Z (Cl(X Σ ), C ) where Cl (X Σ ) = Div (X Σ) /Div0 (X Σ ). Remark: By the above definition, we have the following short exact sequence of affine algebraic group 1 G (C ) Σ(1) T N 1.

19 Lemma Let G be as in the above definition. Then: a) Cl(X Σ ) is the character group of G. b) G is isomorphic to a product of a torus and a finite Abelian group. In particular, G is reductive. c) Give a basis e 1,..., e n of M. We have G = {(t ρ ) (C ) Σ(1) ρ = {(t ρ ) (C ) Σ(1) ρ t m,uρ ρ = 1 for all m M} t e i,u ρ ρ = 1 for 1 i n}.

20 Example: The ray generators of the fan for P n are u 0 = n e i, u 1 = e 1,..., u n = e n. i=1 By the above lemma, (t 0,..., t n ) (C ) n+1 lies in G if and only if t m, e 1... e n 0 t m,e t m,en n = 1 for all m M = Z n. Taking m equal to e 1,..., e n, we see that G is defined by t0 1 t 1 =... = t0 1 t n. Thus G = {(λ,..., λ) λ C } C, which is the action of C on C n+1 given by scalar multiplication.

21 Example: The fan for P 1 P 1 has ray generators u 1 = e 1, u 2 = e 1, u 3 = e 2, u 4 = e 2 in N = Z 2. By this lemma, (t 1, t 2, t 3, t 4 ) (C ) 4 lies in G if and only if t m,e 1 1 t m, e 1 2 t m,e 2 3 t m, e 2 4 = 1 for all m M = Z 2. Taking m equal to e 1, e 2, we obtain t 1 t 1 2 = t 3 t 1 4 = 1. Thus G = {(µ, µ, λ, λ) µ, λ C } (C ) 2.

22 Definition Let X Σ be the toric variety of fan Σ in N (R). S := C[x ρ ρ Σ(1)] is called the homogeneous coordinate ring of X Σ.

23 Definition Let X Σ be the toric variety of fan Σ in N (R). a) For each cone σ Σ, define the monomial x ˆσ = ρ/ σ(1) x ρ. b) B(Σ) := x ˆσ σ Σ S is called irrelevant ideal. Remark: a) Spec(S) = C Σ(1). b) x ˆτ is the multiple of x ˆσ whenever τ is a face of σ.

24 c) B(Σ) = x ˆσ σ Σ max. where Σ max is the set of maximal cones of Σ. Now define Z(Σ) = V (B(Σ)) C Σ(1). Example: The fan for P n consists of cones generated by proper subsets of {u 0,..., u n }, where u 0 = n i=1 e i, u 1 = e 1,..., u n = e n. Let u i generate ρ i for 0 i n and x i be the corresponding variable in the total coordinate ring. The maximal cones of the fan are σ i = Cone(u 0,..., û i,..., u n ). Then x ˆσ i = x i, so that B(Σ) = x 0,..., x n. Hence Z(Σ) = {0}.

25 Definition A subset P Σ(1) is a primitive collection if: a) P σ(1) for all σ Σ. b) For every proper subset Q P, there is a σ Σ with Q σ(1).

26 Proposition The Z(Σ) as a union of irreducible components is given by Z(Σ) = P V (x ρ ρ P), where the union is over all primitive collections P Σ(1). Example: The fan for P n consists of cones generated by proper subsets of {u 0,..., u n }, where u 0 = n i=1 e i, u 1 = e 1,..., u n = e n. The only primitive collection is {ρ 0,..., ρ n }, so Z(Σ) = V (x 0,..., x n ) = {0}.

27 Example: The fan for P 1 P 1 has ray generators u = e 1, u 2 = e 1, u 3 = e 2, u 4 = e 2. each u i gives a ray ρ i and a variable x i. We compute Z(Σ) in two ways: * The maximal cone Cone(u 1, u 3 ) gives the monomial x 2 x 4 and the others give x 1 x 4, x 1 x 3, x 2 x 3. Thus B(Σ) = x 2 x 4, x 1 x 4, x 1 x 3, x 2 x 3. We can check that Z(Σ) = {0} C 2 C 2 {0}. * The only primitive collections are {ρ 1, ρ 2 } and {ρ 3, ρ 4 }, so that Z(Σ) = V (x 1, x 2 ) V (x 3, x 4 ) = {0} C 2 C 2 {0} by the proposition, where B(Σ) = x 1, x 2 x 3, x 4.

28 Let {e ρ ρ Σ(1)} be the standard basis of the lattice Z Σ(1). For each σ Σ, define the cone σ = Cone(e ρ ρ σ(1)) R Σ(1). These cones and their faces form a fan Σ = { σ σ Σ} in (Z Σ(1) ) R = R Σ(1). This fan has the following properties.

29 Proposition Let Σ be the fan as above. a) C Σ(1) \Z(Σ) is the toric variety of the fan Σ. b) The map e ρ u ρ defines a map of lattices Z Σ(1) N that is compatible with the fans Σ and Σ in N R. c) The resulting toric morphism is constant on G-orbits. π : C Σ(1) \Z(Σ) X Σ

30 Theorem Let X Σ be the toric variety without torus factors and consider the toric morphism π : C Σ(1) \Z(Σ) X Σ from the above proposition. Then: a) π is a good categorical quotient for the action of G on C Σ(1) \Z(Σ), so that X Σ (C Σ(1) \Z(Σ))//G. b) π is a good geometric quotient if and only if Σ is simplicial.

31 We have a commutative diagram: X Σ (C Σ(1) \Z(Σ))//G T N (C ) Σ(1) /G

32 Example: From some examples above, the quotient representation of P n is P n = (C n+1 \{0})/C, where C acts by scalar multiplication. This is a good geometric quotient since Σ is a smooth and hence simplicial.

33 Example: Also from some example before, the quotient representation P 1 P 1 is C 1 C 1 = (C 4 \({0} C 2 C 2 {0}))/(C ) 2, where (C ) 2 acts via (µ, λ).(a, b, c, d) = (µa, µb, λa, λd). This is again a good geometric quotient.

34 Outline Quotients in Algebraic Geometry 1 Quotients in Algebraic Geometry 2 3 4

35 In this section we will explore how this ring relates to the algebra and geometry of X Σ.

36 Definition Let X Σ be a toric variety without torus factor. Its total coordinate ring is S = C[x ρ ρ Σ(1)]. We have the sequence: 0 M Z Σ(1) Cl(X Σ ) 0 where α = (a ρ ) Z Σ(1) maps to the class [Σ ρ a ρ D ρ ] Cl (X Σ ). Given x α = ρ x ρ aρ S, we define its degree: deg (x α ) = [Σ ρ a ρ D ρ ] Cl (X Σ ). For β Cl (X Σ ), S β denotes the corresponding grade piece of S.

37 Remark: The grading on S is closely related to G = Hom Z (Cl (X Σ ), C ). Cl (X Σ ) is the character group of G, where as usual β Cl (X Σ ) gives the character χ β : G C. The action of G on C Σ(1) induces an action on S with the following property: For given f S f S β g.f = χ β ( g 1) f for all g G f (g.x) = χ β (g) f (x) for all g G, x C Σ(1). We say that f S β is homogeneous of degree β.

38 Example: The total coordinate ring of P n is C[x 0,..., x n ]. The map Z n+1 Z is (a 0,..., a n ) a a n. This gives the grading on C[x 0,.., x n ] where each variable x i has degree 1, so that homogeneous polynomial has the usual meaning.

39 Example: The fan for P n P m is the product of the fans of P n and P m. The class group is Cl(P n P m ) Cl(P n ) Cl(P m ) Z 2. The total coordinate ring is C[x 0,..., x n, y 0,..., y m ], where deg(x i ) = (1, 0) deg(y i ) = (0, 1). For this ring, homogeneous polynomial means bihomogeneous polynomial.

40 Proposition Let S be the total coordinate ring of the simplicial toric variety X Σ. Then: a) If I S is a homogeneous ideal, then V (I ) = {π(x) X Σ f (x) = 0 for all f I } is a closed subvarieties of X Σ. b) All closed subvarieties of X Σ arise this way.

41 Proposition (The Toric Nullstellensazt) Let X Σ be the simplicial toric variety with total coordinate ring S and irrelevant ideal B(Σ) S. If I S is a homogeneous ideal, then V (I ) = in X Σ B(Σ) k I for some k 0.

42 Proposition (The Toric Ideal-Variety Correspondence) Let X Σ be a simplicial toric variety. Then there is a bijective correspondence { } radical homogeneous ideals {closed subvarieties of X Σ } I B(Σ) S

43 When X Σ is not simplicial, there is still a relation between ideals in the total coordinate ring and closed subvarieties of X Σ. Proposition Let S be the total coordinate ring of the toric variety X Σ. Then: a) If I S is a homogeneous ideal, then V (I ) = { p X Σ there is a x π 1 (p), f (x) = 0 f I } is a closed subvariety of X Σ. b) All closed subvarieties of X Σ arise this way.

44 Outline Quotients in Algebraic Geometry 1 Quotients in Algebraic Geometry 2 3 4

45 Definition a) A polytope M R is the convex hull of affine set of points. b) The dimension of is the dimension of the subspace spanned by the difference {m 1 m 2 m 1, m 2 }. c) is called integral if the vertice of lie in M. d) Let 1,..., k be polytopes. We define k = {m m k m i i ; i = 1,..., k}. We also denote k := for k N. We have k = {km m }.

46 Definition Let be a polytope. We define a) t k x m is called monomial, where m k. b) The monomials multiply by c) The degree of t k x m is d) The polytope ring of is t k x m.t l x n := t k+l x m+n. deg(t k x m ) := k. S := C[t k x m k N, m k ].

47 Remark: a) The definition of the monomials multiply is well-defined. Indeed, because m k, m l we get m + m (k + l). b) S = C[t k x m m k ] is a grade ring. Hence S = (S ) k. k=0

48 Definition Let S be a polytope ring of. a) S + := k 1 (S ) k is called irrelevant ideal. b) T := {P P is a homogeneous prime ideal of S and P S + }. P T is called relevant prime ideal of S. c) For any homogeneous ideal I of S, we define Z(I ) := {P P is a relevant prime of S and P I }.

49 Proposition a) If {I j } is a family of homogeneous ideals in S then j Z(I j ) = Z( j I j ). b) If I 1, I 2 are homogeneous ideals then Z(I 1 ) Z(I 2 ) = Z(I 1 I 2 ).

50 T is a topological space whose closed sets are Z(I ), I is a homogeneous ideal of S. Let f be any homogeneous element of S of degree 1. We set U f := T \Z( f ). We may identity U f with the topological space Spec(S [f 1 ] 0 ) and give it the corresponding structure of an affine scheme, where S [f 1 ] 0 = { g f s f, g homogeneous in S and deg(g) = deg(f s )}. We will write (Proj(S )) f for this open affine subscheme of Proj(S ). Let P = Proj(S ).

51 Definition Let be a polytope and F be a nonempty face of. We define a) σ v F := {λ(m m ) m, m F, λ 0} M R is a cone and its dual is a cone σ F N R. b) NF ( ) := {σ F F is nonempty face of } is normal face of.

52 Theorem P = X (NF ( )).

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