Algebraic Geometry
Arcs and wedges on rational surface singularities
Comptes Rendus. Mathématique, Volume 349 (2011) no. 19-20, pp. 1083-1087.

Let (S,P0) be a rational surface singularity over an algebraically closed field k of characteristic 0, let να be an essential divisorial valuation over (S,P0), and Pα the stable point of the space of arcs S corresponding to να. We prove that any wedge centered at Pα lifts to the minimal desingularization. This proves the Nash problem for rational surface singularities, and reduces the Nash problem for surfaces to quasirational normal singularities which are not rational. In positive characteristic, we give a counterexample to the k-wedge lifting problem for a surface for which the Nash map is bijective.

Soit (S,P0) une singularité rationnelle de surface sur un corps algébriquement clos k de caractéristique 0, soit να une valuation divisorielle essentielle sur (S,P0), et Pα le point stable de lʼespace des arcs S qui correspond à να. On démontre que tout coin centré en Pα se relève à la désingularisation minimale. Cela démontre le problème de Nash pour les singularités rationnelles de surface, et réduit le problème de Nash pour les surfaces aux singularités quasi-rationnelles qui ne sont pas rationnelles. En caractéristique positive, on donne un contre-exemple au problème de relèvement de k-coins pour une surface dont lʼapplication de Nash est bijective.

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DOI: 10.1016/j.crma.2011.08.022
Reguera, Ana J. 1

1 Universidad de Valladolid, Dep. de Álgebra, Geometría y Topología, Prado de la Magdalena, 47005 Valladolid, Spain
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Reguera, Ana J. Arcs and wedges on rational surface singularities. Comptes Rendus. Mathématique, Volume 349 (2011) no. 19-20, pp. 1083-1087. doi : 10.1016/j.crma.2011.08.022. http://www.numdam.org/articles/10.1016/j.crma.2011.08.022/

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