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Nash blowing-up

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In algebraic geometry, Nash blowing-up is a process in which, roughly speaking, each singular point is replaced by all limiting positions of the tangent spaces at the non-singular points. More formally, let be an algebraic variety of pure dimension r embedded in a smooth variety of dimension n, and let be the complement of the singular locus of . Define a map , where is the Grassmannian of r-planes in the tangent bundle of , by , where is the tangent space of at . The closure of the image of this map together with the projection to is called the Nash blow-up of .

Although the above construction uses an embedding, the Nash blow-up itself is unique up to unique isomorphism.

Properties

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  • Nash blowing-up is locally a monoidal transformation.
  • If X is a complete intersection defined by the vanishing of then the Nash blow-up is the blow-up with center given by the ideal generated by the (n − r)-minors of the matrix with entries .
  • For a variety over a field of characteristic zero, the Nash blow-up is an isomorphism if and only if X is non-singular.
  • For an algebraic curve over an algebraically closed field of characteristic zero, repeated Nash blowing-up leads to desingularization after a finite number of steps.
  • Both of the prior properties may fail in positive characteristic. For example, in characteristic q > 0, the curve has a Nash blow-up which is the monoidal transformation with center given by the ideal , for q = 2, or , for . Since the center is a hypersurface the blow-up is an isomorphism.

See also

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References

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  • Nobile, A. (1975), "Some properties of the Nash blowing-up", Pacific Journal of Mathematics, 60 (1): 297–305, doi:10.2140/pjm.1975.60.297, MR 0409462