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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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