# nonsingular variety

A variety over an algebraically closed field $k$ is nonsingular at a point $x$ if the local ring $\mathcal{O}_{x}$ 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 $F_{1},\ldots,F_{n}$ of $m$ variables $x_{1},\ldots,x_{m}$, 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