logo ANNALES DE L'INSTITUT FOURIER

With cedram.org
Table of contents for this issue | Next article
Camille Plénat
The Nash problem of arcs and the rational double points $D_n$
(Résolution du problème de Nash pour les singularités $D_n$)
Annales de l'institut Fourier, 58 no. 7 (2008), p. 2249-2278, doi: 10.5802/aif.2413
Article PDF | Reviews MR 2498350 | Zbl 1168.14004 | 1 citation in Cedram
Class. Math.: 14B05, 14J17
Keywords: Space of arcs, Nash map, Nash problem, rational double points

Résumé - Abstract

This paper deals with the Nash problem, which consists in comparing the number of families of arcs on a singular germ of surface $U$ with the number of essential components of the exceptional divisor in the minimal resolution of this singularity. We prove their equality in the case of the rational double points $D_n$ ($n \ge 4$).

Bibliography

[1] M. Artin, “On isolated rational singularities of surfaces”, Amer. J. Math. 88 (1966), p. 129-136 Article |  MR 199191 |  Zbl 0142.18602
[2] C. Bouvier, “Diviseurs essentiels, composantes essentielles des variétés toriques singulières”, Duke Math. J. 91 (1998), p. 609-620 Article |  MR 1604179 |  Zbl 0966.14038
[3] C. Bouvier & G. Gonzales-Sprinberg, “Système générateur minimal, diviseurs essentiels et G-désingularisations de varitétés toriques”, Tohoku Math. J. 47 (1995), p. 125-149 Article |  MR 1311446 |  Zbl 0823.14006
[4] D. Eisenbud, Commutative Algebra with a view toward Algebraic Geometry, Graduate Texts in Mathematics 150, Springer-Verlag, New York, 1995  MR 1322960 |  Zbl 0819.13001
[5] J. Fernandez-Sanchez, “Equivalence of the Nash conjecture for primitive and sandwiched singularities”, Proc. Amer. Math. Soc. 133 (2005), p. 677-679 Article |  MR 2113914 |  Zbl 1056.14004
[6] S. Ishii, “Arcs, valuations and the Nash map”, arXiv: math.AG/0410526 arXiv |  Zbl 1082.14007
[7] S. Ishii, “The local Nash problem on arc families of singularities”, arXiv: math.AG/0507530 Cedram |  Zbl 1116.14030
[8] S. Ishii & J. Kollár, “The Nash problem on arc families of singularities”, Duke Math. J. 120, 3 (2003), p. 601-620 Article |  MR 2030097 |  Zbl 1052.14011
[9] M. Lejeune–Jalabert, Arcs analytiques et résolution minimale des singularités des surfaces quasi-homogènes, in Séminaire sur les Singularités des Surfaces, Lecture Notes in Math., Springer-Verlag, 1980, p. 303-336 Numdam |  Zbl 0432.14020
[10] M. Lejeune–Jalabert, “Désingularisation explicite des surfaces quasi-homogènes dans $\mathbb{C}^3$”, Nova Acta Leopoldina NF 52, Nr 240 (1981), p. 139-160  MR 642702 |  Zbl 0474.14021
[11] M. Lejeune–Jalabert, “Courbes tracées sur un germe d’hypersurface”, Amer. J. Math. 112 (1990), p. 525-568 Article |  Zbl 0743.14002
[12] M. Lejeune–Jalabert & A. Reguera, “Arcs and wedges on sandwiched surface singularities”, Amer. J. Math. 121 (1999), p. 1191-1213 Article |  MR 1719822 |  Zbl 0960.14015
[13] H. Matsumura, Commutative ring theory. Translated from the Japanese by M. Reid, Cambridge Studies in Advanced Mathematics 8, Cambridge University Press, Cambridge, 1986  MR 879273 |  Zbl 0603.13001
[14] J. F. Jr. Nash, “Arc structure of singularities”, A celebration of John F. Nash, Jr. Duke Math. J. 81, 1 (1995), p. 31-38 Article |  MR 1381967 |  Zbl 0880.14010
[15] C. Plénat, “A propos du problème des arcs de Nash”, Annales de l’Institut Fourier 55 (2005) no. 3, p. 805-823 Cedram |  Zbl 1080.14021
[16] C. Plénat, “Résolution du problème des arcs de Nash pour les points doubles rationnels $D_n \: (n \ge 4)$.”, Note C.R.A.S, Série I 340 (2005), p. 747-750  MR 2141063 |  Zbl 1072.14004
[17] C. Plénat & P. Popescu-Pampu, “A class of non-rational surface singularities for which the Nash map is bijective”, Bulletin de la SMF 134 (2006) no. 3, p. 383-394 Numdam |  MR 2245998 |  Zbl 1119.14007
[18] A. Reguera, “Families of arcs on rational surface singularities”, Manuscripta Math 88, 3 (1995), p. 321-333 Article |  MR 1359701 |  Zbl 0867.14012
[19] A. Reguera, “Image of the Nash map in terms of wedges”, C. R. Acad. Sci. Paris, Ser. I 338 (2004), p. 385-390  MR 2057169 |  Zbl 1044.14032
top