14 Chapter 1. Affine Toric Varieties We next consider the action of T . Since T (C∗)s, an element t T acts on Cs and takes varieties to varieties. Then T = t · T t ·YA shows that t ·YA is a variety containing T. Hence YA t ·YA by the definition of Zariski closure. Replacing t with t−1 leads to YA = t ·YA , so that the action of T induces an action on YA . We conclude that YA is an affine toric variety. It remains to compute the character lattice of T, which we will temporarily denote by M . Since T = ΦA (TN), the map ΦA gives the commutative diagram TN ΦA ❊❊ ❊❊❊ ❊❊❊ (C∗)s T where denotes a surjective map and an injective map. This diagram of tori induces a commutative diagram of character lattices M Zs Φ A M . ❇❇❇ ❇❇❇ ❇❇ Since ΦA : Zs M takes the standard basis e1,...,es to m1,...,ms, the image of ΦA is ZA . By the diagram, we obtain M ZA . Then we are done since the dimension of a torus equals the rank of its character lattice. In concrete terms, fix a basis of M, so that we may assume M = Zn. Then the s vectors in A Zn can be regarded as the columns of an n×s matrix A with integer entries. In this case, the dimension of YA is simply the rank of the matrix A. We will see below that every affine toric variety is isomorphic to YA for some finite subset A of a lattice. Toric Ideals. Let YA Cs = Spec(C[x1,...,xs]) be the affine toric variety com- ing from a finite set A = {m1,...,ms} M. We can describe the ideal I(YA ) C[x1,...,xs] as follows. As in the proof of Proposition 1.1.8, ΦA induces a map of character lattices ΦA : Zs −→ M that sends the standard basis e1,...,es to m1,...,ms. Let L be the kernel of this map, so that we have an exact sequence 0 −→ L −→ Zs −→ M. In down to earth terms, elements = ( 1 ,...,s) of L satisfy s i=1 i mi = 0 and hence record the linear relations among the mi.
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