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Comptes Rendus Mathématique
Volume 349, n° 19-20
pages 1083-1087 (novembre 2011)
Doi : 10.1016/j.crma.2011.08.022
Received : 17 May 2011 ;  accepted : 30 August 2011
Arcs and wedges on rational surface singularities
Arcs et coins sur une singularité rationnelle de surface
 

Ana J. Reguera
Universidad de Valladolid, Dep. de Álgebra, Geometría y Topología, Prado de la Magdalena, 47005 Valladolid, Spain 

Abstract

Let   be a rational surface singularity over an algebraically closed field k of characteristic 0, let   be an essential divisorial valuation over  , and   the stable point of the space of arcs   corresponding to  . We prove that any wedge centered at   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.

The full text of this article is available in PDF format.
Résumé

Soit   une singularité rationnelle de surface sur un corps algébriquement clos k de caractéristique 0, soit   une valuation divisorielle essentielle sur  , et   le point stable de lʼespace des arcs   qui correspond à  . On démontre que tout coin centré en   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.

The full text of this article is available in PDF format.


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