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Désingularisation des surfaces par des modifications de Nash normalisées. (Dessingularisation of surfaces by normalized Nash modifications). [D’après M. Spivakovsky]. (French) Zbl 0614.14004
Sémin. Bourbaki, 38ème année, Vol. 1985/86, Exp. 661, Astérisque 145/146, 187-207 (1987).
[For the entire collection see Zbl 0601.00002.]
It is proved that a complex algebraic surface X can be desingularized by a finite succession of normalized Nash blowing-ups. Recall that if we let G be the Grassmannian of locally free quotients of $$\Omega^ 1_ X$$ of rank 2, $$\nu: G\to X$$ the canonical morphism and $$\tilde X$$ the closure of $$\nu^{-1}$$ (smooth points of X) in G then $$\nu: \tilde X\to X$$ is called the Nash blowing-up of X. Composing this with the normalization of $$\tilde X,$$ we get the normalized Nash blowing-up of X. A normal surface singularity X is called a ”sandwich singularity” if there exists a birational morphism $$X\to X_ 0$$ with $$X_ 0$$ a smooth surface. (If $$X'\to X$$ is a desingularization of X then we have $$X'\to X\to X_ 0$$ with X’, $$X_ 0$$ smooth, which explains the terminology.) The problem of desingularizing X by a finite succession of normalized Nash blowing-ups was reduced, by earlier results of the author and Hironaka, to proving such a desingularization for a sandwich singularity. This has been done by M. Spivakovsky in his thesis at Harvard in 1985. The article under review is an exposition of this result of Spivakovsky.
Reviewer: B.Singh
##### MSC:
 14E15 Global theory and resolution of singularities (algebro-geometric aspects) 14J17 Singularities of surfaces or higher-dimensional varieties 32S45 Modifications; resolution of singularities (complex-analytic aspects)
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