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16 Chapter 1. Affine Toric Varieties Since toric varieties are irreducible, the ideals appearing in Proposition 1.1.9 are toric ideals. Examples of toric ideals include: Example 1.1.4 : x3 − y2⊆ C[x,y] Example 1.1.5 : xz − yw⊆ C[x,y,z,w] Example 1.1.6 : xix j+1 − xi+1x j | 0 ≤ i j ≤ d − 1⊆ C[x0,...,xd]. (The latter is the ideal of the rational normal cone Cd ⊆ Cd+1.) In each example, we have a prime ideal generated by binomials. As we now show, such ideals are automatically toric. Proposition 1.1.11. An ideal I ⊆ C[x1,...,xs] is toric if and only if it is prime and generated by binomials. Proof. One direction is obvious. So suppose that I is prime and generated by bino- mials xαi −xβi. Then observe that V(I)∩(C∗)s is nonempty (it contains (1,...,1)) and is a subgroup of (C∗)s (easy to check). Since V(I) ⊆ Cs is irreducible, it fol- lows that V(I) ∩(C∗)s is an irreducible subvariety of (C∗)s that is also a subgroup. By Proposition 1.1.1, we see that T = V(I) ∩ (C∗)s is a torus. Projecting on the ith coordinate of (C∗)s gives a character T → (C∗)s → C∗, which by our usual convention we write as χmi : T → C∗ for mi ∈ M. It follows easily that V(I) = YA for A = {m1,...,ms}, and since I is prime, we have I = I(YA ) by the Nullstellensatz. Then I is toric by Proposition 1.1.9. We will later see that all affine toric varieties arise from toric ideals. For more on toric ideals and lattice ideals, the reader should consult [205, Ch. 7]. Affine Semigroups. A semigroup is a set S with an associative binary operation and an identity element. To be an affine semigroup, we further require that: • The binary operation on S is commutative. We will write the operation as + and the identity element as 0. Thus a finite set A ⊆ S gives NA = ∑ m∈A amm | am ∈ N ⊆ S. • The semigroup is finitely generated, meaning that there is a finite set A ⊆ S such that NA = S. • The semigroup can be embedded in a lattice M. The simplest example of an affine semigroup is Nn ⊆ Zn. More generally, given a lattice M and a finite set A ⊆ M, we get the affine semigroup NA ⊆ M. Up to isomorphism, all affine semigroups are of this form. Given an affine semigroup S ⊆ M, the semigroup algebra C[S] is the vector space over C with S as basis and multiplication induced by the semigroup structure