# 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 ${F}_{1},\mathrm{\dots},{F}_{n}$ 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 |