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An ) ∈ k n is a tangent vector to Z at P if the line (a1 t + b1 , . . , an t + bn ) is tangent to Z at P . The advantage of considering tangent vectors is that it turns out they always form a subspace of k n . More precisely, we have the following. 5. Let X ⊆ Ank be an affine algebraic set, with f1 , . . , fm generating I(X). Then given a point P ∈ X, and a vector v ∈ k n , the following are equivalent: (a) v is a tangent vector to Z at P ; (b) we have (∂fi /∂x1 (P ), . . , ∂fi /∂xn (P )) · v = 0 for i = 1, .

Proof. If X is a variety, we can take the open subset to be all of X. Conversely, suppose the condition holds; we wish to show ∆(X) is closed. Thus, suppose (P, Q) is in the closure of ∆(X). By hypothesis, we can choose U containing P and Q and such that ∆(U ) is closed in U × U . Since U × U has the subset topology in X × X, and (P, Q) ∈ U × U , the hypothesis that (P, Q) is in the closure of ∆(X) implies it is in the closure of ∆(U ), thus in ∆(U ) ⊆ ∆(X), and since P and Q were arbitrary, we conclude ∆(X) is closed.

More generally, the foregoing algebra is saying that if we have P ∈ X an affine algebraic set of dimension n near P , and f1 , . . , fm ∈ mP span TP∗ (X), then in a neighborhood of P , we have Z(f1 ) ∩ · · · ∩ Z(fm ) = {P }, and since each fi can reduce the dimension by at most 1, we conclude that n m. Nonsingularity then corresponds to the dimension of mP /m2P being equal to n, the minimal possible value, or 31 equivalently, being able to find f1 , . . , fn ∈ A(X) whose zero sets cut out P in the strong sense that (f1 , .

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Algebraic Varieties by Brian Osserman


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