nonsingular variety
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 neighborhoodwherein the variety is cut outby 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.
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 |