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 be an algebraic variety of pure dimension r embedded in a smooth variety of dimension n, and let be the complement of the singular locus of. Define a map, where is the Grassmannian of r-planes in the tangent bundle of, by, where is the tangent space of at. The closure of the image of this map together with the projection to is called the Nash blow-up of.
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 then the Nash blow-up is the blow-up with center given by the ideal generated by the -minors of the matrix with entries.
- 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 has a Nash blow-up which is the monoidal transformation with center given by the ideal, for q = 2, or, for. Since the center is a hypersurface the blow-up is an isomorphism.