The Local Nash problem on arc families of singularities  [ Le problème du Nash local sur les familles d’arc de singularités ]
Annales de l'Institut Fourier, Tome 56 (2006) no. 4, pp. 1207-1223.

Cet article présente la réponse positive au problème du Nash local pour une singularité torique ainsi que pour une singularité analytiquement prétorique. Il en résulte comme corollaire une réponse affirmative au problème du Nash local pour une singularité quasi ordinaire.

This paper shows the affirmative answer to the local Nash problem for a toric singularity and analytically pretoric singularity. As a corollary we obtain the affirmative answer to the local Nash problem for a quasi-ordinary singularity.

Reçu le : 2005-03-11
Accepté le : 2005-06-05
DOI : https://doi.org/10.5802/aif.2210
Classification : 14J17,  14M25
Mots clés: arc de singularité, problème de Nash, singularité
@article{AIF_2006__56_4_1207_0,
     author = {Ishii, Shihoko},
     title = {The Local Nash problem on arc families of singularities},
     journal = {Annales de l'Institut Fourier},
     pages = {1207--1223},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {56},
     number = {4},
     year = {2006},
     doi = {10.5802/aif.2210},
     zbl = {1116.14030},
     mrnumber = {2266888},
     language = {en},
     url = {aif.centre-mersenne.org/item/AIF_2006__56_4_1207_0/}
}
Ishii, Shihoko. The Local Nash problem on arc families of singularities. Annales de l'Institut Fourier, Tome 56 (2006) no. 4, pp. 1207-1223. doi : 10.5802/aif.2210. https://aif.centre-mersenne.org/item/AIF_2006__56_4_1207_0/

[1] Bouvier, C. Diviseurs essentiels, composantes essentielle des variétés toriques singulières, Duke Math. J., Tome 91 (1998), pp. 609-620 | Article | MR 1604179 | Zbl 0966.14038

[2] Bouvier, C.; Gonzalez-Sprinberg, G. Système générateur minimal, diviseurs essentiels et G-désingularisation, Tohoku Math. J., Tome 47 (1995), pp. 125-149 | Article | MR 1311446 | Zbl 0823.14006

[3] Fulton, W. Introduction to toric varieties, Annals of Mathematics Studies, Tome 131, Princeton University Press, Princeton, NJ, 1993 (The William H. Roever Lectures in Geometry) | MR 1234037 | Zbl 0813.14039

[4] Gau, Y.-N. Embedded topological classification of quasi-ordinary singularities, Mem. Amer. Math. Soc., Tome 74 (1988) no. 388, pp. 109-129 (With an appendix by Joseph Lipman) | MR 954948 | Zbl 0658.14004

[5] Hironaka, H. Resolution of singularities of an algebraic variety over a field of characteristic zero: I, II, Annals of Math., Tome 79 (1964), pp. 109-326 | Article | MR 199184 | Zbl 0122.38603

[6] Ishii, S. The arc space of a toric variety, J. Algebra, Tome 278 (2004), pp. 666-683 | Article | MR 2071659 | Zbl 1073.14066

[7] Ishii, S. Arcs, valuations and the Nash map, J. reine angew. Math, Tome 588 (2005), pp. 71-92 | Article | MR 2196729 | Zbl 02235895

[8] Ishii, S.; Kollár, J. The Nash problem on arc families of singularities, Duke Math. J., Tome 120 (2003) no. 3, pp. 601-620 | Article | MR 2030097 | Zbl 1052.14011

[9] Lejeune-Jalabert, M.; Reguera-Lopez, A. J. Arcs and wedges on sandwiched surface singularities, Amer. J. Math., Tome 121 (1999), pp. 1191-1213 | Article | MR 1719822 | Zbl 0960.14015

[10] Lipman, J. Quasi-ordinary singularities of embedded surfaces (1965) (Ph. D. Thesis)

[11] Lipman, J. Quasi-ordinary singularities of surfaces in C 3 , Singularities, Part 2 (Arcata, Calif., 1981) (Proc. Sympos. Pure Math.) Tome 40, Amer. Math. Soc., Providence, RI, 1983, pp. 161-172 | MR 713245 | Zbl 0521.14014

[12] Nash, J. F. Arc structure of singularities, Duke Math. J., Tome 81 (1995), pp. 31-38 | Article | MR 1381967 | Zbl 0880.14010

[13] Oh, K. Topological types of quasi-ordinary singularities, Proc. AMS, Tome 117 (1993), pp. 53-59 | Article | MR 1106181 | Zbl 0791.32018

[14] Pérez, P. D. Gonález Toric embedded resolutions of quasi-ordinary hypersurface singularities, Ann. Inst. Fourier (Grenoble), Tome 53 (2003), pp. 1819-1881 | Article | Numdam | MR 2038781 | Zbl 1052.32024

[15] Plenat, C.; Popescu-Pampu, P. A class of non-rational surface singularities for which the Nash map is bijective (ath.AG/0410145)

[16] Reguera, A. J. Image of Nash map in terms of wedges, C. R. Acad. Sci. Ser. I, Tome 338 (2004), pp. 385-390 | MR 2057169 | Zbl 1044.14032

[17] Reguera-Lopez, A. J. Families of arcs on rational surface singularities, Manuscr. Math., Tome 88 (1995), pp. 321-333 | Article | MR 1359701 | Zbl 0867.14012

[18] Vojta, P. Jets via Hasse-Schmidt derivations (preprint AG/0407113)