Smoothness and Tangent Spaces
The basic definition of a smooth point of an algebraic variety is analogous to the corresponding one from differential geometry. We start with the affine case; suppose X ⊂A n is an affine variety of pure dimension k, with ideal I(X) =(f 1 ,..., f i ). Let M be the l × n matrix with entries ∂f i /∂x j . Then it’s not hard to see that the rank of M is at most n — k at every point of X, and we say a point p ∈ X is a smooth point of X if the rank of the matrix M, evaluated at the point p, exactly n — k. Note that in case the ground field K = ℂ, this is equivalent to saying that X is a complex submanifold of An = ℂn in a neighborhood of p, or that X is a real submanifold of ℂn near p. (It is not, however, equivalent, in the case of a variety X defined by polynomials f α with real coefficients, to saying that the locus of the f α in ℝ n is smooth; consider, for example, the origin p = (0, 0) on the plane curve x 3 + y 3 = 0.)
KeywordsTangent Space Projective Variety Smooth Point Nonempty Open Subset Affine Variety
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