22 Chapter 1. Affine Toric Varieties The ideal C4 has quadratic generators by removing s2t2, we now get cubic generators. See Example B.1.1 for a computational approach to this exercise. See also Example 2.1.10, where we will study the paramaterization Φ from the projective point of view. 1.1.8. Instead of working over C, we will work over an algebraically closed field k of characteristic 2. Consider the affine toric variety V ⊆ k5 parametrized by Φ(s,t,u) = (s4,t4,u4,s8u,t12u3) ∈ k5. (a) Find generators for the toric ideal I = I(V) ⊆ k[x1,x2,x3,x4,x5]. (b) Show that dim V = 3. You may assume that Proposition 1.1.8 holds over k. (c) Show that I = x4 4 + x8x3,x4 1 5 + x12x3 2 3 . It follows that V ⊆ k5 has codimension two and can be defined by two equations, i.e., V is a set-theoretic complete intersection. The paper [12] shows that if we replace k with an alge- braically closed field of characteristic p 2, then the image of the above parametrization is never a set-theoretic complete intersection. 1.1.9. Prove that a lattice ideal IL for L ⊆ Zs is a toric ideal if and only if Zs/L is torsion- free. Hint: When Zs/L is torsion-free, it can be regarded as the character lattice of a torus. The other direction of the proof is more challenging. If you get stuck, see [205, Thm. 7.4]. 1.1.10. Prove that I = x2 − 1,xy − 1,yz − 1 is the lattice ideal for the lattice L = {(a,b,c) ∈ Z3 | a + b + c ≡ 0 mod 2} ⊆ Z3. Also compute the primary decomposition of I to show that I is not prime. 1.1.11. Let TN be a torus with character lattice M. Then every point t ∈ TN gives an evalua- tion map φt : M → C∗ defined by φt(m) = χm(t). Prove that φt is a group homomorphism and that the map t → φt induces a group isomorphism TN HomZ(M,C∗). 1.1.12. Consider tori T1 and T2 with character lattices M1 and M2. By Example 1.1.13, the coordinate rings of T1 and T2 are C[M1] and C[M2]. Let Φ : T1 → T2 be a morphism that is a group homomorphism. Then Φ induces maps Φ : M2 −→ M1 and Φ∗ : C[M2] −→ C[M1] by composition. Prove that Φ∗ is the map of semigroup algebras induced by the map Φ of affine semigroups. 1.1.13. A commutative semigroup S is cancellative if u + v = u + w implies v = w for all u,v,w ∈ S and torsion-free if nu = nv implies u = v for all n ∈ N \{0} and u,v ∈ S. Prove that S 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 since lattices contain semigroups that are not finitely generated. For example, let τ 0 be irra- tional and consider the semigroup S = {(a,b) ∈ N2 | b ≥ τa} ⊆ Z2. Prove that S is not finitely generated. (The generators of S are related to continued frac- tions. For example, when τ = (1 + √ 5)/2 is the golden ratio, the minimal generators of S are (0,1) and (F2n,F2n+1) for n = 1,2,..., where Fn is the nth Fibonacci number. See [232] and [260]. Continued fractions will play an important role in Chapter 10.

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