§1.1. Introduction to Affine Toric Varieties 19 (c) V is an affine variety defined by a toric ideal. (d) V = Spec(C[S]) for an affine semigroup S. Proof. The implications (b) ⇔ (c) ⇔ (d) ⇒ (a) follow from Propositions 1.1.8, 1.1.9 and 1.1.14. For (a) ⇒ (d), let V be an affine toric variety containing the torus TN with character lattice M. Since the coordinate ring of TN is the semigroup algebra C[M], the inclusion TN ⊆ V induces a map of coordinate rings C[V] −→ C[M]. This map is injective since TN is Zariski dense in V, so that we can regard C[V] as a subalgebra of C[M]. Since the action of TN on V is given by a morphism TN ×V → V, we see that if t ∈ TN and f ∈ C[V], then p → f (t−1 · p) is a morphism on V. It follows that C[V] ⊆ C[M] is stable under the action of TN. By Lemma 1.1.16, we obtain C[V] = χm∈C[V ] C · χm. Hence C[V] = C[S] for the semigroup S = {m ∈ M | χm ∈ C[V]}. Finally, since C[V] is finitely generated, we can find f1,..., fs ∈ C[V] with C[V] = C[ f1,..., fs]. Expressing each fi in terms of characters as above gives a finite generating set of S. It follows that S is an affine semigroup. Here is one way to think about the above proof. When an irreducible affine variety V contains a torus TN as a Zariski open subset, we have the inclusion C[V] ⊆ C[M]. Thus C[V] consists of those functions on the torus TN that extend to polynomial functions on V. Then the key insight is that V is a toric variety precisely when the functions that extend are determined by the characters that extend. Example 1.1.18. We have seen that V = V(xy − zw) ⊆ C4 is a toric variety with toric ideal xy−zw⊆ C[x,y,z,w]. Also, the torus is (C∗)3 via the map (t1,t2,t3) → (t1,t2,t3,t1t2t3 −1 ). The lattice points used in this map can be represented as the columns of the matrix (1.1.6) ⎛ ⎝0 1 0 0 1 1 0 1⎠. 0 0 1 −1 ⎞ The corresponding semigroup S ⊆ Z3 consists of the N-linear combinations of the column vectors. Hence the elements of S are lattice points lying in the polyhe- dral region in R3 pictured in Figure 1 on the next page. In this figure, the four vectors generating S are shown in bold, and the boundary of the polyhedral region is partially shaded. In the terminology of §1.2, this polyhedral region is a rational

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