A variety over an algebraically closed field is nonsingular at a point if the local ring 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 of variables , then it is nonsingular at if the Jacobian has maximal rank at that point. Otherwise, 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 , which has a cusp at and is nonsingular everywhere else, and , which has a self-intersection at and is nonsingular everywhere else.