20 Chapter 1. Affine Toric Varieties (0,0,1) (0,1,0) (1,0,0) (1,1,−1) Figure 1. Cone containing the lattice points corresponding to V = V(xy − zw) polyhedral cone. In Exercise 1.1.5 you will show that S consists of all lattice points lying in the cone in Figure 1. We will use this in §1.3 to prove that V is normal. ♦ Exercises for §1.1. 1.1.1. As in Example 1.1.6, let I = xix j+1 − xi+1x j | 0 ≤ i j ≤ d − 1⊆ C[x0,...,xd] and let Cd be the surface parametrized by Φ(s,t) = (sd,sd−1t,...,st d−1 ,t d ) ∈ Cd+1. (a) Prove that V(I) = Φ(C2) ⊆ Cd+1. Thus Cd = V(I). (b) Prove that I(Cd) is homogeneous. (c) Consider lex monomial order with x0 x1 ··· xd. Let f ∈ I(Cd) be homogeneous of degree and let r be the remainder of f on division by the generators of I. Prove that r can be written r = h0(x0,x1)+ h1(x1,x2)+ ··· + hd−1(xd−1,xd) where hi is homogeneous of degree . Also explain why we may assume that the coefficient of xi in hi is zero for 1 ≤ i ≤ d − 1. (d) Use part (c) and r(sd,sd−1t,...,st d−1 ,t d ) = 0 to show that r = 0. (e) Use parts (b), (c) and (d) to prove that I = I(Cd). Also explain why the generators of I are a Grobner ¨ basis for the above lex order. 1.1.2. Let L ⊆ Zs be a sublattice. Prove that x + − x − | ∈ L = xα − xβ | α,β ∈ Ns, α − β ∈ L . Note that when ∈ L, the vectors + ,− ∈ Ns have disjoint support (i.e., no coordinate is positive in both), while this may fail for arbitrary α,β ∈ Ns with α − β ∈ L.
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