Smoothness and Tangent Spaces

Abstract

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.)