Nash blowing-up
Introduction 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 * 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 * Blowing up* Resolution of singularities
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