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 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.
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 |