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Smoothness and Tangent Spaces

  • Joe Harris
Part of the Graduate Texts in Mathematics book series (GTM, volume 133)

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 nk at every point of X, and we say a point pX is a smooth point of X if the rank of the matrix M, evaluated at the point p, exactly nk. 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.)

Keywords

Tangent Space Projective Variety Smooth Point Nonempty Open Subset Affine Variety 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer Science+Business Media New York 1992

Authors and Affiliations

  • Joe Harris
    • 1
  1. 1.Department of MathematicsHarvard UniversityCambridgeUSA

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