§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 = | 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 = s i=1 xai. i Rescaling if necessary, 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 = s i=1 xbi i 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 IL by the second description of IL. It follows that f + 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 = | α,β Ns and α β L is called a lattice ideal. (b) A prime lattice ideal is called a toric ideal.
Previous Page Next Page