Algebraic Geometry pp 174-185 | Cite as

# 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 A^{n} = ℂ^{n} in a neighborhood of *p*, or that *X* is a real submanifold of ℂ*n* 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## Preview

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