6 Chapter 1. Affine Toric Varieties At first glance, the definition of normal does not seem very intuitive. Once we enter the world of toric varieties, however, we will see that normality has a very nice combinatorial interpretation and that the nicest toric varieties are the normal ones. We will also see that normality leads to a nice theory of divisors. In Exercise 1.0.7 you will prove some properties of normal domains that will be used in §1.3 when we study normal affine toric varieties. Smooth Points of Affine Varieties. In order to define a smooth point of an affine variety V, we first need to define local rings and Zariski tangent spaces. When V is irreducible, the local ring of V at p is OV,p = { f /g C(V) | f ,g C[V] and g(p) = 0}. Thus OV ,p consists of all rational functions on V that are defined at p. Inside of OV,p we have the maximal ideal mV ,p = OV,p | φ(p) = 0}. In fact, mV,p is the unique maximal ideal of OV,p, so that OV,p is a local ring. Exercises 1.0.2 and 1.0.4 explain how to define OV,p when V is not irreducible. The Zariski tangent space of V at p is defined to be Tp(V) = HomC(mV,p/mV 2 ,p ,C). In Exercise 1.0.8 you will verify that dim Tp(Cn) = n for every p Cn. According to [131, p. 32], we can compute the Zariski tangent space of a point in an affine variety as follows. Lemma 1.0.6. Let V Cn be an affine variety and let p V. Also assume that I(V) = f1,..., fs⊆ C[x1,...,xn]. For each i, let dp( fi) = fi ∂x1 (p)x1 + ··· + fi ∂xn (p)xn. Then the Zariski tangent space Tp(V) is isomorphic to the subspace of Cn defined by the equations dp( f1) = ··· = dp( fs) = 0. In particular, dim Tp(V) n. Definition 1.0.7. A point p of an affine variety V is smooth or nonsingular if dim Tp(V) = dimpV, where dimpV is the maximum of the dimensions of the irre- ducible components of V containing p. The point p is singular if it is not smooth. Finally, V is smooth if every point of V is smooth. Points lying in the intersection of two or more irreducible components of V are always singular (see [69, Thm. 8 of Ch. 9, §6]). Since dim Tp(Cn) = n for every p Cn, we see that Cn is smooth. For an irreducible affine variety V Cn of dimension d, fix p V and write I(V) =
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