§1.1. Introduction to Affine Toric Varieties 21 1.1.3. Let IL be a toric ideal and let 1 ,...,r be a basis of the sublattice L Zs. Define ˜ L = x i + x i | i = 1,...,r . Prove that IL = ˜ L : x1 ···xs . Hint: Given α,β Ns with α β L, write α β = ∑r i=1 ai i , ai Z. This implies xα−β 1 = ai 0 x i + x i ai ai 0 x i x i + −ai 1. Show that multiplying this by (x1 ···xs)k gives an element of ˜ L for k 0. (By being more careful, one can show that this result holds for lattice ideals. See [205, Lem. 7.6].) 1.1.4. Fix an affine variety V . Then f1,..., fs C[V] give a polynomial map Φ : V Cs, which on coordinate rings is given by Φ∗ : C[x1,...,xs] −→ C[V], xi −→ fi. Let Y Cs be the Zariski closure of the image of Φ. (a) Prove that I(Y) = Ker(Φ∗). (b) Explain how this applies to the proof of Proposition 1.1.14. 1.1.5. Let m1 = (1,0,0),m2 = (0,1,0),m3 = (0,0,1),m4 = (1,1,−1) be the columns of the matrix in Example 1.1.18 and let C = 4 i=1 λimi | λi R≥0 R3 be the cone in Figure 1. Prove that C Z3 is a semigroup generated by m1,m2,m3,m4. 1.1.6. An interesting observation is that different sets of lattice points can parametrize the same affine toric variety, even though these parametrizations behave slightly differently. In this exercise you will consider the parametrizations Φ1(s,t) = (s2,st,st3) and Φ2(s,t) = (s3,st,t3). (a) Prove that Φ1 and Φ2 both give the affine toric variety Y = V(xz y3) C3. (b) We can regard Φ1 and Φ2 as maps Φ1 : C2 −→ Y and Φ2 : C2 −→ Y. Prove that Φ2 is surjective and that Φ1 is not. In general, a finite subset A Zn gives a rational map ΦA : Cn YA . The image of ΦA in Cs is called a toric set in the literature. Thus Φ1(C2) and Φ2(C2) are toric sets. The papers [169] and [240] study when a toric set equals the corresponding affine toric variety. 1.1.7. In Example 1.1.6 and Exercise 1.1.1 we constructed the rational normal cone Cd using all monomials of degree d in s,t. If we drop some of the monomials, things become more complicated. For example, consider the surface parametrized by Φ(s,t) = (s4,s3t,st3,t4) C4. This gives a toric variety Y C4. Show that the toric ideal of Y is given by I(Y) = xw yz,yw2 z3,xz2 y2w,x2z y3⊆ C[x,y,z,w].
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