Nash blowing-up

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 .

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