Nash blowing-up

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", formula_1 denotes the set of its singular points and formula_2 it is possible to define a map formula_3, where formula_4 is the Grassmannian of "r"-planes in "n"-space, by formula_5, where formula_6 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. 