[Arcs et coins sur une singularité rationnelle de surface]
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.
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.
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@article{CRMATH_2011__349_19-20_1083_0,
author = {Reguera, Ana J.},
title = {Arcs and wedges on rational surface singularities},
journal = {Comptes Rendus. Math\'ematique},
pages = {1083--1087},
publisher = {Elsevier},
volume = {349},
number = {19-20},
year = {2011},
doi = {10.1016/j.crma.2011.08.022},
language = {en},
url = {http://www.numdam.org/articles/10.1016/j.crma.2011.08.022/}
}
TY - JOUR AU - Reguera, Ana J. TI - Arcs and wedges on rational surface singularities JO - Comptes Rendus. Mathématique PY - 2011 SP - 1083 EP - 1087 VL - 349 IS - 19-20 PB - Elsevier UR - http://www.numdam.org/articles/10.1016/j.crma.2011.08.022/ DO - 10.1016/j.crma.2011.08.022 LA - en ID - CRMATH_2011__349_19-20_1083_0 ER -
%0 Journal Article %A Reguera, Ana J. %T Arcs and wedges on rational surface singularities %J Comptes Rendus. Mathématique %D 2011 %P 1083-1087 %V 349 %N 19-20 %I Elsevier %U http://www.numdam.org/articles/10.1016/j.crma.2011.08.022/ %R 10.1016/j.crma.2011.08.022 %G en %F CRMATH_2011__349_19-20_1083_0
Reguera, Ana J. Arcs and wedges on rational surface singularities. Comptes Rendus. Mathématique, Tome 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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