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, 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 Grassmanian of r-planes in n-space, by τ(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 isomophism 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 y² − x^q = 0 the Nash blowing-up is the monoidal transformation with center given by the ideal (x^q), for q = 2, or (y²), 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.

See also

• Blowing up
• Resolution of singularities

References

• Nobile, A. Some properties of the Nash blowing-up PACIFIC JOURNAL OF MATHEMATICS, Vol. 60, No. I, 1975