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 {\displaystyle X} be an algebraic variety of pure dimension r embedded in a smooth variety Y {\displaystyle Y} of dimension n, and let X reg {\displaystyle X_{\text{reg}}} be the complement of the singular locus of X {\displaystyle X} . Define a map τ : X reg → X × G r ( T Y ) {\displaystyle \tau :X_{\text{reg}}\rightarrow X\times G_{r}(TY)} , where G r ( T Y ) {\displaystyle G_{r}(TY)} is the Grassmannian of r-planes in the tangent bundle of Y {\displaystyle Y} , by τ ( a ) := ( a , T X , a ) {\displaystyle \tau (a):=(a,T_{X,a})} , where T X , a {\displaystyle T_{X,a}} is the tangent space of X {\displaystyle X} at a {\displaystyle a} . The closure of the image of this map together with the projection to X {\displaystyle X} is called the Nash blow-up of X {\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 f 1 , f 2 , … , f n − r {\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 ∂ f i / ∂ x j {\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 y 2 − x q = 0 {\displaystyle y^{2}-x^{q}=0} has a Nash blow-up which is the monoidal transformation with center given by the ideal ( x q ) {\displaystyle (x^{q})} , for q = 2, or ( y 2 ) {\displaystyle (y^{2})} , for q > 2 {\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
