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 [math]\displaystyle{ X }[/math] be an algebraic variety of pure dimension r embedded in a smooth variety [math]\displaystyle{ Y }[/math] of dimension n, and let [math]\displaystyle{ X_\text{reg} }[/math] be the complement of the singular locus of [math]\displaystyle{ X }[/math]. Define a map [math]\displaystyle{ \tau:X_\text{reg}\rightarrow X\times G_{r}(TY) }[/math], where [math]\displaystyle{ G_{r}(TY) }[/math] is the Grassmannian of r-planes in the tangent bundle of [math]\displaystyle{ Y }[/math], by [math]\displaystyle{ \tau(a):=(a,T_{X,a}) }[/math], where [math]\displaystyle{ T_{X,a} }[/math] is the tangent space of [math]\displaystyle{ X }[/math] at [math]\displaystyle{ a }[/math]. The closure of the image of this map together with the projection to [math]\displaystyle{ X }[/math] is called the Nash blow-up of [math]\displaystyle{ X }[/math]. 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 [math]\displaystyle{ f_1,f_2,\ldots,f_{n-r} }[/math] 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 [math]\displaystyle{ \partial f_i/\partial x_j }[/math].
- 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 [math]\displaystyle{ y^2-x^q=0 }[/math] has a Nash blow-up which is the monoidal transformation with center given by the ideal [math]\displaystyle{ (x^{q}) }[/math], for q = 2, or [math]\displaystyle{ (y^2) }[/math], for [math]\displaystyle{ q\gt 2 }[/math]. Since the center is a hypersurface the blow-up is an isomorphism.
See also
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, https://projecteuclid.org/euclid.pjm/1102868640
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