§1.0. Background: Affine Varieties 7 f1,..., fs . Using Lemma 1.0.6, it is straightforward to show that V is smooth at p if and only if the Jacobian matrix (1.0.3) Jp( f1,..., fs) = ∂ fi ∂x j (p) 1≤i≤s,1≤ j≤n has rank n − d (Exercise 1.0.9). Here is a simple example. Example 1.0.8. As noted in Example 1.0.4, the plane curve C defined by x3 = y2 has I(C) = x3 − y2⊆ C[x,y]. A point p = (a,b) ∈ C has Jacobian Jp = (3a2,−2b), so the origin is the only singular point of C. ♦ Since Tp(V) = HomC(mV ,p /m2 V,p ,C), we see that V is smooth at p when dim V equals the dimension of mV ,p /m2 V ,p as a vector space over OV ,p /mV ,p . In terms of commutative algebra, this means that p ∈ V is smooth if and only if OV,p is a regular local ring. See [10, p. 123] or [89, 10.3]. We can relate smoothness and normality as follows. Proposition 1.0.9. A smooth irreducible affine variety V is normal. Proof. Recall that C[X] ⊆ C(V) and OV,p ⊆ C(V) since V is irreducible. In §3.0 we will see that C[V] = p∈V OV,p. By Exercise 1.0.7, C[V] is normal once we prove that OV ,p is normal for all p ∈ V. Hence it suffices to show that OV,p is normal whenever p is smooth. This follows from some powerful results in commutative algebra: OV ,p is a regular local ring when p is a smooth point of V (see above), and every regular local ring is a UFD (see [89, Thm. 19.19]). Then we are done since every UFD is normal. A direct proof that OV,p is normal at a smooth point p ∈ V is sketched in Exercise 1.0.10. The converse of Propostion 1.0.9 can fail. We will see in §1.3 that the affine variety V(xy − zw) ⊆ C4 is normal, yet V(xy − zw) is singular at the origin. Products of Affine Varieties. Given affine varieties V1 and V2, there are several ways to show that the cartesian product V1 ×V2 is an affine variety. The most direct way is to proceed as follows. Let V1 ⊆ Cm = Spec(C[x1,...,xm]) and V2 ⊆ Cn = Spec(C[y1,...,yn]). Take I(V1) = f1,..., fs and I(V2) = g1,...,gt . Since the fi and g j depend on separate sets of variables, it follows that V1 ×V2 = V( f1,..., fs,g1,...,gt) ⊆ Cm+n is an affine variety. A fancier method is to use the mapping properties of the product. This will also give an intrinsic description of its coordinate ring. Given V1 and V2 as above,

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