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In [[Algebraic geometry|algebraic geometry]], '''Nash blowing-up''' is a process in which, roughly speaking, each [[Singular point of an algebraic variety|singular point]] is replaced by all limiting positions of the tangent spaces at the non-singular points. More formally, let <math>X</math> be an [[Algebraic variety|algebraic variety]] of pure [[Physics:Dimension|dimension]] ''r'' [[Embedding|embedded]] in a smooth variety <math>Y</math> of dimension ''n'', and let <math>X_\text{reg}</math> be the complement of the singular locus of <math>X</math>. Define a map <math>\tau:X_\text{reg}\rightarrow X\times G_{r}(TY)</math>, where <math>G_{r}(TY)</math> is the [[Grassmannian]] of ''r''-planes in the tangent bundle of <math>Y</math>, by <math>\tau(a):=(a,T_{X,a})</math>, where <math>T_{X,a}</math> is the tangent space of <math>X</math> at <math>a</math>. The closure of the image of this map together with the projection to <math>X</math> is called the Nash blow-up of <math>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 [[Blowing up|monoidal transformation]].
* If ''X'' is a [[Complete intersection|complete intersection]] defined by the vanishing of <math>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''&nbsp;&minus;&nbsp;''r'')-minors of the matrix with entries <math>\partial f_i/\partial x_j</math>.
* For a variety over a field of characteristic zero, the Nash blow-up is an [[Isomorphism|isomorphism]] if and only if ''X'' is non-singular. 
* For an algebraic curve over an [[Algebraically closed field|algebraically closed field]] of characteristic zero, repeated Nash blowing-up leads to [[Resolution of singularities|desingularization]] after a finite number of steps. 
* Both of the prior properties may fail in positive characteristic. For example, in characteristic ''q''&nbsp;>&nbsp;0, the curve <math>y^2-x^q=0</math> has a Nash blow-up which is the monoidal transformation with center given by the ideal <math>(x^{q})</math>, for ''q''&nbsp;=&nbsp;2, or <math>(y^2)</math>, for <math>q>2</math>. Since the center is a hypersurface the blow-up is an isomorphism.

==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
 | doi = 10.2140/pjm.1975.60.297| doi-access = free
 }}

[[Category:Algebraic geometry]]



{{Sourceattribution|Nash blowing-up}}