Nash blowing-up explained

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

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

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 \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

y ^{2-x} ^{q=0} has a Nash blow-up which is the monoidal transformation with center given by the ideal (x ^{q} ) , for q = 2, or (y ^{2)} , for q> 2 . Since the center is a hypersurface the blow-up is an isomorphism.

See also

• Blowing up
• Resolution of singularities