§1.1. Introduction to Affine Toric Varieties 13 Let the coordinates of Cd+1 be x0,...,xd and let I ⊆ C[x0,...,xd] be the ideal generated by the 2 × 2 minors of the matrix x0 x1 ··· xd−2 xd−1 x1 x2 ··· xd−1 xd , so I = xix j+1 − xi+1x j | 0 ≤ i j ≤ d − 1 . In Exercise 1.1.1 you will verify that Φ(C2) = V(I), so that Cd = Φ(C2) is an affine variety. You will also prove that I(Cd) = I, so that I is the ideal of all polynomials vanishing on Cd. It follows that I is prime since V(I) is irreducible by Proposition 1.1.8 below. The affine surface Cd is called the rational normal cone of degree d and is an example of a determinantal variety. We will see below that I is a toric ideal. It is straightforward to show that Cd is a toric variety with torus Φ((C∗)2) = Cd ∩ (C∗)d+1 (C∗)2. We will study this example from the projective point of view in Chapter 2. ♦ We next explore three equivalent ways of constructing affine toric varieties. Lattice Points. In this book, a lattice is a free abelian group of finite rank. Thus a lattice of rank n is isomorphic to Zn. For example, a torus TN has lattices M (of characters) and N (of one-parameter subgroups). Given a torus TN with character lattice M, a set A = {m1,...,ms} ⊆ M gives characters χmi : TN → C∗. Then consider the map (1.1.4) ΦA : TN −→ Cs defined by ΦA (t) = ( χm1(t),...,χms(t) ) ∈ Cs. Definition 1.1.7. Given a finite set A ⊆ M, the affine toric variety YA is defined to be the Zariski closure of the image of the map ΦA from (1.1.4). This definition is justified by the following proposition. Proposition 1.1.8. Given A ⊆ M as above, let ZA ⊆ M be the sublattice gener- ated by A . Then YA is an affine toric variety whose torus has character lattice ZA . In particular, the dimension of YA is the rank of ZA . Proof. The map (1.1.4) can be regarded as a map ΦA : TN −→ (C∗)s of tori. By Proposition 1.1.1, the image T = ΦA (TN) is a torus that is closed in (C∗)s. The latter implies that YA ∩ (C∗)s = T since YA is the Zariski closure of the image. It follows that the image is Zariski open in YA . Furthermore, T is irreducible (it is a torus), so the same is true for its Zariski closure YA .

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