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

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 ${\displaystyle f_{1},f_{2},\ldots ,f_{n-r}}$ 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 ${\displaystyle \partial f_{i}/\partial x_{j}}$.
• 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 ${\displaystyle y^{2}-x^{q}=0}$ has a Nash blow-up which is the monoidal transformation with center given by the ideal ${\displaystyle (x^{q})}$, for q = 2, or ${\displaystyle (y^{2})}$, for ${\displaystyle q>2}$. Since the center is a hypersurface the blow-up is an isomorphism.

See also

• Blowing up
• Resolution of singularities

References

• Nobile, A. (1975), "Some properties of the Nash blowing-up", Pacific Journal of Mathematics, 60 (1): 297–305, doi:10.2140/pjm.1975.60.297, MR 0409462