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 $$X$$ be an algebraic variety of pure dimension r embedded in a smooth variety $$Y$$ of dimension n, and let $$X_\text{reg}$$ be the complement of the singular locus of $$X$$. Define a map $$\tau:X_\text{reg}\rightarrow X\times G_{r}(TY)$$, where $$G_{r}(TY)$$ is the Grassmannian of r-planes in the tangent bundle of $$Y$$, by $$\tau(a):=(a,T_{X,a})$$, where $$T_{X,a}$$ is the tangent space of $$X$$ at $$a$$. The closure of the image of this map together with the projection to $$X$$ is called the Nash blow-up of $$X$$.

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 $$f_1,f_2,\ldots,f_{n-r}$$ then the Nash blow-up is the blow-up with center given by the ideal generated by the (n &minus; r)-minors of the matrix with entries $$\partial f_i/\partial x_j$$.
 * 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 $$y^2-x^q=0$$ has a Nash blow-up which is the monoidal transformation with center given by the ideal $$(x^{q})$$, for q = 2, or $$(y^2)$$, for $$q>2$$. Since the center is a hypersurface the blow-up is an isomorphism.