- Properties
- See also
- References
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, 
denotes the set of its singular points and 
it is possible to define a map 
, where 
is the Grassmannian of r-planes in n-space, by 
, where 
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
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 
. - 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
the Nash blowing-up is the monoidal transformation with center given by the ideal 
, for q = 2, or 
, for 
. 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
- {{citation
| last = Nobile | first = A.
| issue = 1
| journal = Pacific Journal of Mathematics
| mr = 0409462
| pages = 297–305
| title = Some properties of the Nash blowing-up
| url = https://projecteuclid.org/euclid.pjm/1102868640
| volume = 60
| year = 1975}}{{algebraic-geometry-stub}}