Quotients in Algebraic Geometry Quotient Construction of Toric Varieties The Total Coordinate Ring Toric Varieties via Polytopes Homogeneous Coordinate Ring Students: Tien Mai Nguyen, Bin Nguyen Kaiserslautern University Algebraic Group June 14, 2013 Tien Mai Nguyen, Bin Nguyen Homogeneous Coordinate Ring Quotients in Algebraic Geometry Quotient Construction of Toric Varieties The Total Coordinate Ring Toric Varieties via Polytopes Outline 1 Quotients in Algebraic Geometry 2 Quotient Construction of Toric Varieties 3 The Total Coordinate Ring 4 Toric Varieties via Polytopes Tien Mai Nguyen, Bin Nguyen Homogeneous Coordinate Ring Quotients in Algebraic Geometry Quotient Construction of Toric Varieties The Total Coordinate Ring Toric Varieties via Polytopes Outline 1 Quotients in Algebraic Geometry 2 Quotient Construction of Toric Varieties 3 The Total Coordinate Ring 4 Toric Varieties via Polytopes Tien Mai Nguyen, Bin Nguyen Homogeneous Coordinate Ring Quotients in Algebraic Geometry Quotient Construction of Toric Varieties The Total Coordinate Ring Toric Varieties via Polytopes 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 ) 7−! g:f defined by (g:f )(x) = f (g −1:x) for all x 2 X is an action of G on R. Tien Mai Nguyen, Bin Nguyen Homogeneous Coordinate Ring Quotients in Algebraic Geometry Quotient Construction of Toric Varieties The Total Coordinate Ring Toric Varieties via Polytopes 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 = fG:xjx 2 X g and the ring of invariants RG = ff 2 Rjg:f = f ; for all g 2 Gg. Tien Mai Nguyen, Bin Nguyen Homogeneous Coordinate Ring Quotients in Algebraic Geometry Quotient Construction of Toric Varieties The Total Coordinate Ring Toric Varieties via Polytopes 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 −1 OY (U) ! OX (π (U)) induces an isomorphism −1 G OY (U) ' OX (π (U)) : b) If W ⊆ X is closed and G-invariant, then π(W ) ⊆ Y is closed. c) If W1; W2 are closed, disjoint, and G-invariant in X , then π(W1) and π(W2) are disjoint in Y . We often write a good categorical quotient as π : X −! X ==G. Tien Mai Nguyen, Bin Nguyen Homogeneous Coordinate Ring Quotients in Algebraic Geometry Quotient Construction of Toric Varieties The Total Coordinate Ring Toric Varieties via Polytopes 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 2 G and x 2 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 2 X , we have π(x) = π(y) , G:x \ G:y 6= ;. Tien Mai Nguyen, Bin Nguyen Homogeneous Coordinate Ring Quotients in Algebraic Geometry Quotient Construction of Toric Varieties The Total Coordinate Ring Toric Varieties via Polytopes Definition a) A subgroup G of GLn(C) is called an affine algebraic group if G is a subvariety of GLn(C). b) Let G be an affine algebraic group acting on a variety X . The G-action is called algebraic action if the action G × X −! X (g; x) 7−! g:x defines a morphism. Tien Mai Nguyen, Bin Nguyen Homogeneous Coordinate Ring Quotients in Algebraic Geometry Quotient Construction of Toric Varieties The Total Coordinate Ring Toric Varieties via Polytopes 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 2 X ==G, then π−1(p) contains a unique closed G-orbit. b) π induces a bijection fclosed G-orbits in X g ' X ==G. Tien Mai Nguyen, Bin Nguyen Homogeneous Coordinate Ring Quotients in Algebraic Geometry Quotient Construction of Toric Varieties The Total Coordinate Ring Toric Varieties via Polytopes 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 2 X , we have π(x) = π(y) () x and y lie in the same G-orbit. c) π induces a bijection fG-orbits in X g ' X ==G. Tien Mai Nguyen, Bin Nguyen Homogeneous Coordinate Ring Quotients in Algebraic Geometry Quotient Construction of Toric Varieties The Total Coordinate Ring Toric Varieties via Polytopes 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. Tien Mai Nguyen, Bin Nguyen Homogeneous Coordinate Ring Quotients in Algebraic Geometry Quotient Construction of Toric Varieties The Total Coordinate Ring Toric Varieties via Polytopes 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 G a)R is a finely generated C-algebra. b) The morphism π : X −! Spec(RG ) induced by RG ⊆ R is a good categorical quotient. Tien Mai Nguyen, Bin Nguyen Homogeneous Coordinate Ring Quotients in Algebraic Geometry Quotient Construction of Toric Varieties The Total Coordinate Ring Toric Varieties via Polytopes 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 = [αVα such that −1 −1 πjπ (Vα) : π (Vα) −! Vα is a good categorical quotient for every α, then π : X −! Y is a good categorical quotient. Tien Mai Nguyen, Bin Nguyen Homogeneous Coordinate Ring Quotients in Algebraic Geometry Quotient Construction of Toric Varieties The Total Coordinate Ring Toric Varieties via Polytopes ∗ 2 Example: Let C act on C nf0g by scalar multiplication, where 2 2 C = Spec(C[x0; x1]). Then C nf0g = U0 [ U1, where 2 ±1 U0 = C nV (x0) = Spec(C[x0 ; x1]) 2 ±1 U1 = C nV (x1) = Spec(C[x0; x1 ]) 2 ±1 ±1 U0 \ U1 = C nV (x0x1) = Spec(C[x0 ; x1 ]) The rings of invariants are ∗ ±1 C C[x0 ; x1] = C[x1=x0] ∗ ±1 C C[x0; x1 ] = C[x0=x1] ∗ ±1 ±1 C ±1 C[x0 ; x1 ] = C[(x1=x0) ] Tien Mai Nguyen, Bin Nguyen Homogeneous Coordinate Ring Quotients in Algebraic Geometry Quotient Construction of Toric Varieties The Total Coordinate Ring Toric Varieties via Polytopes ∗ It follows that the Vi = Ui ==C glue together in the usual way to 1 ∗ 2 create P . Since C -orbits are closed in C nf0g, it follows that 1 2 ∗ P = (C nf0g)=C is a good geometric quotient. Tien Mai Nguyen, Bin Nguyen Homogeneous Coordinate Ring Quotients in Algebraic Geometry Quotient Construction of Toric Varieties The Total Coordinate Ring Toric Varieties via Polytopes Outline 1 Quotients in Algebraic Geometry 2 Quotient Construction of Toric Varieties 3 The Total Coordinate Ring 4 Toric Varieties via Polytopes Tien Mai Nguyen, Bin Nguyen Homogeneous Coordinate Ring Quotients in Algebraic Geometry Quotient Construction of Toric Varieties The Total Coordinate Ring Toric Varieties via Polytopes Let XΣ be the toric variety of a fan Σ in NR. The goal is to construct XΣ as a good categorical quotient r XΣ ' (C nZ)==G r r for an appropriate of affine space C , exceptional set Z ⊆ C , and reductive group G. Tien Mai Nguyen, Bin Nguyen Homogeneous Coordinate Ring Quotients in Algebraic Geometry Quotient Construction of Toric Varieties The Total Coordinate Ring Toric Varieties via Polytopes Definition Let XΣ be the toric variety of fan Σ in N(R). Assume that XΣ has no torus factor. We define ∗ G = HomZ(Cl(XΣ); C ) where Cl (X ) = Div (XΣ)/ . Σ Div0 (XΣ) Remark: By the above definition, we have the following short exact sequence of affine algebraic group ∗ Σ(1) 1 −! G −! (C ) −! TN −! 1: Tien Mai Nguyen, Bin Nguyen Homogeneous Coordinate Ring Quotients in Algebraic Geometry Quotient Construction of Toric Varieties The Total Coordinate Ring Toric Varieties via Polytopes Lemma Let G be as in the above definition. Then: a) Cl(XΣ) 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 e1; :::; en of M. We have ∗ Σ(1) Y hm;uρi G = f(tρ) 2 (C ) j tρ = 1 for all m 2 Mg ρ ∗ Σ(1) Y hei ;uρi = f(tρ) 2 (C ) j tρ = 1 for 1 ≤ i ≤ ng: ρ Tien Mai Nguyen, Bin Nguyen Homogeneous Coordinate Ring Quotients in Algebraic Geometry Quotient Construction of Toric Varieties The Total Coordinate Ring Toric Varieties via Polytopes n Example: The ray generators of the fan for P are n X u0 = − ei ; u1 = e1; :::; un = en: i=1 ∗ n+1 By the above lemma, (t0; :::; tn) 2 (C ) lies in G if and only if hm;−e1−:::−eni hm;e1i hm;eni t0 t1 :::tn = 1 n for all m 2 M = Z . Taking m equal to e1; :::; en, we see that G is defined by −1 −1 t0 t1 = ::: = t0 tn: Thus ∗ ∗ G = f(λ, :::; λ)jλ 2 C g ' C ; ∗ n+1 which is the action of C on C given by scalar multiplication. Tien Mai Nguyen, Bin Nguyen Homogeneous Coordinate Ring Quotients in Algebraic Geometry Quotient Construction of Toric Varieties The Total Coordinate Ring Toric Varieties via Polytopes 1 1 Example: The fan for P × P has ray generators 2 u1 = e1; u2 = −e1; u3 = e2; u4 = −e2 in N = Z .