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 − 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.