nonsingular variety

A variety over an algebraically closed field k is nonsingular at a point x if the local ring 𝒪x is a regular local ring. Equivalently, if around the point one has an open affine neighborhood wherein the variety is cut out by certain polynomials F1,,Fn of m variables x1,,xm, then it is nonsingular at x if the JacobianDlmfPlanetmath has maximal rank at that point. Otherwise, x is a singular point.

A variety is nonsingular if it is nonsingular at each point.

Over the real or complex numbersPlanetmathPlanetmath, nonsingularity corresponds to “smoothness”: at nonsingular points, varieties are locally real or complex manifolds (this is simply the implicit function theoremMathworldPlanetmath). Singular points generally have “corners” or self intersections. Typical examples are the curves x2=y3, which has a cusp at (0,0) and is nonsingular everywhere else, and x2(x+1)=y2, which has a self-intersection at (0,0) and is nonsingular everywhere else.

Title nonsingular variety
Canonical name NonsingularVariety
Date of creation 2013-03-22 12:03:47
Last modified on 2013-03-22 12:03:47
Owner CWoo (3771)
Last modified by CWoo (3771)
Numerical id 10
Author CWoo (3771)
Entry type Definition
Classification msc 14-00
Synonym non-singular variety
Defines nonsingular
Defines non-singular
Defines singular point
Defines nonsingular point
Defines non-singular point