12 Chapter 1. Affine Toric Varieties • (Concrete) If T = (C∗)n with m = (a1,...,an) ∈ Zn, u = (b1,...,bn) ∈ Zn, then one computes that (1.1.3) m,u = n i=1 aibi, i.e., the pairing is the usual dot product. It follows that the characters and one-parameter subgroups of a torus T form free abelian groups M and N of finite rank with a pairing , : M × N → Z that identifies N with HomZ(M,Z) and M with HomZ(N,Z). In terms of tensor prod- ucts, one obtains a canonical isomorphism N ⊗Z C∗ T via u ⊗t → λu(t). Hence it is customary to write a torus as TN. From this point of view, picking an isomorphism TN (C∗)n induces dual bases of M and N, i.e., isomorphisms M Zn and N Zn that turn characters into Laurent monomials (1.1.1), one-parameter subgroups into monomial curves (1.1.2), and the pairing into dot product (1.1.3). The Definition of Affine Toric Variety. We now define the main object of study of this chapter. Definition 1.1.3. An affine toric variety is an irreducible affine variety V contain- ing a torus TN (C∗)n as a Zariski open subset such that the action of TN on itself extends to an algebraic action of TN on V. (By algebraic action, we mean an action TN ×V → V given by a morphism.) Obvious examples of affine toric varieties are (C∗)n and Cn. Here are some less trivial examples. Example 1.1.4. The plane curve C = V(x3 − y2) ⊆ C2 has a cusp at the origin. This is an affine toric variety with torus C \{0} = C ∩ (C∗)2 = {(t2,t3) | t ∈ C∗} C∗, where the isomorphism is t → (t2,t3). Example 1.0.4 shows that C is a nonnormal toric variety. ♦ Example 1.1.5. The variety V = V(xy − zw) ⊆ C4 is a toric variety with torus V ∩ (C∗)4 = {(t1,t2,t3,t1t2t3 −1 ) | ti ∈ C∗} (C∗)3, where the isomorphism is (t1,t2,t3) → (t1,t2,t3,t1t2t3 −1 ). We will see later that V is normal. ♦ Example 1.1.6. Consider the surface in Cd+1 parametrized by the map Φ : C2 −→ Cd+1 defined by (s,t) → (sd,sd−1t,...,st d−1 ,t d ). Thus Φ is defined using all degree d monomials in s,t.

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