A propos du problème des arcs de Nash  [ On Nash problem of arc ]
Annales de l'Institut Fourier, Volume 55 (2005) no. 3, p. 805-823

Let ℋ=⋃N i the canonical decomposition of the space of arcs passing through a normal surface singularity. In this paper, we give two new conditions to ensure that N i is not included in N j . These conditions allow us to give two different direct demonstration of Nah problem for minimal sandwiched singularities.

Soit ℋ=⋃N i la décomposition canonique de l’espace des arcs ℋ passant par une singularité normale de surface. Dans cet article, on propose deux nouvelles conditions qui si elles sont vérifiées permettent de montrer que N i n’est pas inclus dans N j . On applique ces conditions pour donner deux nouvelles preuves du problème de Nash pour les singularités sandwich minimales.

DOI : https://doi.org/10.5802/aif.2115
Classification:  14J17
Keywords: geometry algebraic, space of arcs, minimal sandwiched singularities, problem of Nash 
@article{AIF_2005__55_3_805_0,
     author = {Pl\'enat, Camille},
     title = {A propos du probl\`eme des arcs de Nash},
     journal = {Annales de l'Institut Fourier},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {55},
     number = {3},
     year = {2005},
     pages = {805-823},
     doi = {10.5802/aif.2115},
     zbl = {1080.14021},
     mrnumber = {2149404},
     language = {fr},
     url = {http://www.numdam.org/item/AIF_2005__55_3_805_0}
}

Plénat, Camille. A propos du problème des arcs de Nash. Annales de l'Institut Fourier, Volume 55 (2005) no. 3, pp. 805-823. doi : 10.5802/aif.2115. http://www.numdam.org/item/AIF_2005__55_3_805_0/

[Ar] M. Artin On isolated rational singularities of surfaces, Amer. J. Math., Tome 88 (1966), pp. 129-136 | Article | MR 199191 | Zbl 0142.18602

[Fe] J. Fernandez-Sanchez Equivalence of the Nash conjecture for primitive and sandwiched singularities, Proc. Amer. Math. Soc., Tome 133 (2005), pp. 677-679 | Article | MR 2113914 | Zbl 1056.14004

[IK] S. Ishii; J. Kollár The Nash problem on arc families of singularities, Duke Math. J., Tome 120 (2003), pp. 601-620 | Article | MR 2030097 | Zbl 1052.14011

[La] H.B. Laufer Taut two-dimensional singularities, Math. Ann., Tome 205 (1973), pp. 131-164 | Article | MR 333238 | Zbl 0281.32010

[Le] M. Lejeune-Jalabert Courbes tracées sur un germe d'hypersurface, Amer. J. Math., Tome 112 (1990), pp. 525-568 | Article | MR 1064990 | Zbl 0743.14002

[Na] J.F. Nash Jr. Arc structure of singularities, A celebration of John F. Nash, Jr., Duke Math. J., Tome 81 (1995) no. 1, pp. 31-38 | MR 1381967 | Zbl 0880.14010

[Re] A. Reguera Families of arcs on rational surface singularities, Manuscripta Math., Tome 88 (1995) no. 3, pp. 321-333 | MR 1359701 | Zbl 0867.14012

[Sp1] Mark Spivakovsky Sandwiched singularities and desingularization of surfaces by normalized Nash transformations, Annals of Math., Tome 131 (1990), pp. 411-491 | Article | MR 1053487 | Zbl 0719.14005

[Sp2] Mark Spivakovsky Valuation in function fields of surfaces, Amer. J. Math., Tome 112 (1990), pp. 107-156 | Article | MR 1037606 | Zbl 0716.13003