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, $\text{Sing}(X)$ denotes the set of its singular points and $X_\text{reg}:=X\setminus \text{Sing}(X)$ it is possible to define a map $\tau:X_\text{reg}\rightarrow X\times G_r^n$ , where $G_{r}^{n}$ is the Grassmannian of r-planes in n-space, by $\tau(a):=(a,T_{X,a})$ , where $T_{X,a}$ 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.

Properties

The 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 blowing-up is the blowing-up with center given by the ideal generated by the (n − r)-minors of the matrix with entries $\partial f_i/\partial x_j$ .
For a variety over a field of characteristic zero, the Nash blowing-up is an isomorphism if and only if X is non-singular.
For an algebraic curve over an algebraically closed field of characteristic zero the application of Nash blowings-up leads to desingularization after a finite number of steps.
In characteristic q > 0, for the curve $y^2-x^q=0$ the Nash blowing-up 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 blowing-up is an isomorphism. Then the two previous points are not true in positive characteristic.