Nash blowing up - Alchetron, The Free Social Encyclopedia

In algebraic geometry, a Nash blowing-up is a process in which, roughly speaking, each singular point is replaced by all the limiting positions of the tangent spaces at the non-singular points. Strictly speaking, if X is an algebraic variety of pure codimension r embedded in a smooth variety of dimension n, Sing ( X ) denotes the set of its singular points and X reg := X ∖ Sing ( X ) it is possible to define a map τ : X reg → X × G r n , where G r n is the Grassmannian of r-planes in n-space, by τ ( a ) := ( a , T X , a ) , where T X , a is the tangent space of X at a. Now, the closure of the image of this map together with the projection to X is called the Nash blowing-up of X.

Although (to emphasize its geometric interpretation) an embedding was used to define the Nash embedding it is possible to prove that it doesn't depend on it.

Properties

The Nash blowing-up is locally a monoidal transformation.

If X is a complete intersection defined by the vanishing of f 1 , f 2 , … , f n − r then the Nash blowing-up is the blowing-up with center given by the ideal generated by the (n − r)-minors of the matrix with entries ∂ f i / ∂ x j .

For a variety over a field of characteristic zero, the Nash blowing-up is an isomorphism if and only if X is non-singular.

For an algebraic curve over an algebraically closed field of characteristic zero the application of Nash blowings-up leads to desingularization after a finite number of steps.

In characteristic q > 0, for the curve y 2 − x q = 0 the Nash blowing-up is the monoidal transformation with center given by the ideal ( x q ) , for q = 2, or ( y 2 ) , for q > 2 . Since the center is a hypersurface the blowing-up is an isomorphism. Then the two previous points are not true in positive characteristic.

References

Nash blowing-up Wikipedia

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