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A solution to the Nash problem of arcs for rational double points \(D_n\). (Résolution du probléme des arcs de Nash pour les points doubles rationnels \(D_{n}\).) (French) Zbl 1072.14004

Summary: This note deals with the Nash problem, which claims that there are as many families of arcs on a singular germ of surface \(U\) as there are essential components of the exceptional divisor in the desingularisation of this singularity. We prove that this claim holds for the rational double points \(D_n (n \geqslant 4)\).

MSC:

14B05 Singularities in algebraic geometry
14J17 Singularities of surfaces or higher-dimensional varieties
32S25 Complex surface and hypersurface singularities
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[1] Fernandez-Sanchez, J., Equivalence of the Nash conjecture for primitive and sandwiched singularities, Proc. amer. math. soc., 133, 677-679, (2005) · Zbl 1056.14004
[2] Ishii, S.; Kollár, J., The Nash problem on arc families of singularities · Zbl 1052.14011
[3] Lejeune-Jalabert, M., Courbes tracées sur un germe d’hypersurface, Amer. J. math., 112, 525-568, (1990) · Zbl 0743.14002
[4] Lejeune-Jalabert, M.; Reguera, A., Arcs and wedges on sandwiched surface singularities, Amer. J. math., 121, 1191-1213, (1999) · Zbl 0960.14015
[5] Nash, J.F., Arc structure of singularities. A celebration of John F. Nash, jr., Duke math. J., 81, 1, 31-38, (1995)
[6] C. Plénat, A propos du problème des arcs de Nash, a paraître dans l’I.F.A., vol. 55
[7] C. Plénat, Résolution du problème des arcs de Nash pour les points doubles rationnels, Thèse
[8] Plénat, C.; Popescu-Pampu, P., A class of non-rational surface singularities for which the Nash map is bijective · Zbl 1119.14007
[9] Reguera, A., Families of arcs on rational surface singularities, Manuscripta math., 88, 3, 321-333, (1995) · Zbl 0867.14012
[10] Reguera, A., Image of the Nash map in terms of wedges, C. R. acad. sci. Paris, ser. I, 338, 385-390, (2004) · Zbl 1044.14032
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