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单词 NonsingularVariety
释义

nonsingular variety


A variety over an algebraically closed field kis nonsingular at a point xif the local ring 𝒪x is a regular local ring.Equivalently, if around the point one has an open affine neighborhoodwherein the variety is cut outby 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.

Titlenonsingular variety
Canonical nameNonsingularVariety
Date of creation2013-03-22 12:03:47
Last modified on2013-03-22 12:03:47
OwnerCWoo (3771)
Last modified byCWoo (3771)
Numerical id10
AuthorCWoo (3771)
Entry typeDefinition
Classificationmsc 14-00
Synonymnon-singular variety
Definesnonsingular
Definesnon-singular
Definessingular point
Definesnonsingular point
Definesnon-singular point
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