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Homenonsingular variety

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# 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.

## Mathematics Subject Classification

14-00*no label found*

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