Nash Blowing- Up
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Please note that the content of this book primarily consists of articles available from Wikipedia or other free sources online. 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_text{reg}:=Xsetminus text{Sing}(X) it is possible to define a map tau:X_text{reg}rightarrow Xtimes G_r^n, wher...
Please note that the content of this book primarily consists of articles available from Wikipedia or other free sources online. 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_text{reg}:=Xsetminus text{Sing}(X) it is possible to define a map tau:X_text{reg}rightarrow Xtimes G_r^n, where G_{r}^{n} is the Grassmanian of r-planes in n-space, by (a): = (a,TX,a), where TX,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.
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