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§1.1. Introduction to Affine Toric Varieties 15 Given = ( 1 ,...,s) ∈ L, set + = i 0 i ei and − = − i 0 i ei. Note that = + − − and that + ,− ∈ Ns. It follows easily that the binomial x + − x − = i 0 x i i − i 0 x− i i vanishes on the image of ΦA and hence on YA since YA is the Zariski closure of the image. Proposition 1.1.9. The ideal of the affine toric variety YA ⊆ Cs is I(YA ) = x + − x − | ∈ L = xα − xβ | α,β ∈ Ns and α − β ∈ L . Proof. We leave it to the reader to prove equality of the two ideals on the right (Exercise 1.1.2). Let IL denote this ideal and note that IL ⊆ I(YA ). We prove the opposite inclusion following [265, Lem. 4.1]. Pick a monomial order on C[x1,...,xs] and an isomorphism TN (C∗)n. Thus we may assume M = Zn and the map Φ : (C∗)n → Cs is given by Laurent monomials t mi in variables t1,...,tn. If IL = I(YA ), then we can pick f ∈ I(YA ) \ IL with minimal leading monomial xα = s i=1 xai. i Rescaling if necessary, xα becomes the leading term of f . Since f (t m 1 ,...,t ms ) is identically zero as a polynomial in t1,...,tn, there must be cancellation involving the term coming from xα. In other words, f must contain a monomial xβ = s i=1 xbi i xα such that s i=1 (t mi )ai = s i=1 (t mi )bi. This implies that s i=1 aimi = s i=1 bimi, so that α − β = ∑ s i=1 (ai − bi)ei ∈ L. Then xα − xβ ∈ IL by the second description of IL. It follows that f − xα + xβ also lies in I(YA ) \ IL and has strictly smaller leading term. This contradiction completes the proof. Given A ⊆ M, there are several ways to compute the ideal I(YA ) = IL of Proposition 1.1.9. In simple cases, the rational implicitization algorithm of [69, Ch. 3, §3] can be used. One can also compute IL using a basis of L and ideal quotients (Exercise 1.1.3). For more on computing IL, see [265, Ch. 12]. Inspired by Proposition 1.1.9, we make the following definition. Definition 1.1.10. Let L ⊆ Zs be a sublattice. (a) The ideal IL = xα − xβ | α,β ∈ Ns and α − β ∈ L is called a lattice ideal. (b) A prime lattice ideal is called a toric ideal.