Mobile Membranes: Difference between revisions

Revision as of 14:43, 6 June 2013

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, $Sing (X)$ denotes the set of its singular points and $X_{reg} : = X ∖ Sing (X)$ it is possible to define a map $τ : X_{reg} \to X \times G_{r}^{n}$ , where $G_{r}^{n}$ is the Grassmannian of r-planes in n-space, by $τ (a) : = (a, T_{X, a})$ , where $T_{X, a}$ 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 $f_{1}, f_{2}, \dots, f_{n - r}$ 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 $\partial f_{i} / \partial x_{j}$ .
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 $y^{2} - x^{q} = 0$ the Nash blowing-up 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 blowing-up is an isomorphism. Then the two previous points are not true in positive characteristic.

References

Nobile, A. Some properties of the Nash blowing-up PACIFIC JOURNAL OF MATHEMATICS, Vol. 60, No. I, 1975

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+In [[algebraic geometry]], a '''Nash blowing-up''' is a process in which, roughly speaking, each [[Singular point of an algebraic variety|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'' [[embedding|embedded]] in a [[smooth variety]] of dimension ''n'', <math>\text{Sing}(X)</math> denotes the set of its singular points and <math>X_\text{reg}:=X\setminus \text{Sing}(X)</math> it is possible to define a map <math>\tau:X_\text{reg}\rightarrow X\times G_r^n</math>, where <math>G_{r}^{n}</math> is the [[Grassmannian]] of ''r''-planes in ''n''-space, by <math>\tau(a):=(a,T_{X,a})</math>, where <math>T_{X,a}</math> 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 [[blowing up|monoidal transformation]].
+* If ''X'' is a [[complete intersection]] defined by the vanishing of <math>f_1,f_2,\ldots,f_{n-r}</math> then the Nash blowing-up is the blowing-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 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 [[resolution of singularities|desingularization]] after a finite number of steps.
+* In characteristic ''q''&nbsp;>&nbsp;0, for the curve <math>y^2-x^q=0</math> the Nash blowing-up 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 blowing-up is an isomorphism. Then the two previous points are not true in positive characteristic.
+==See also==
+* [[Blowing up]]
+* [[Resolution of singularities]]
+==References==
+* Nobile, A. ''Some properties of the Nash blowing-up'' PACIFIC JOURNAL OF MATHEMATICS, Vol. 60, No. I, 1975
+[[Category:Algebraic geometry]]
+{{math-stub}}

Mobile Membranes: Difference between revisions

Revision as of 14:43, 6 June 2013

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