Let Z be a d-dimensional subscheme which can be embedded into a smooth variety A of dimension n. Let S(Z) be the singular points of Z, its complement in Z. Let be the Grassmannian bundle of d-dimensional subspaces of the tangent bundle TA over A.

Consider the morphism defined by

where is the tangent space of Z at z. Denote the closure of in . Then the morphism

induced by the first projection (locally ) is called the Nash blowing-up of . Note that the construction of is independent of the embedding up to unique -isomorphism.

The first basic result is