nonsingular variety

A variety over an algebraically closed field $k$ is nonsingular at a point $x$ if the local ring ${𝒪}_x$ 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 $F_1,\mathrm{\dots },F_n$ of $m$ variables $x_1,\mathrm{\dots },x_m$, then it is nonsingular at $x$ if the Jacobian 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 numbers, nonsingularity corresponds to “smoothness”: at nonsingular points, varieties are locally real or complex manifolds (this is simply the implicit function theorem). Singular points generally have “corners” or self intersections. Typical examples are the curves $x^2=y^3$, which has a cusp at $(0,0)$ and is nonsingular everywhere else, and $x^2(x+1)=y^2$, 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