The Local Nash problem on arc families of singularities
Annales de l'Institut Fourier, Volume 56 (2006) no. 4, pp. 1207-1223.

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.

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.

Received:
Accepted:
DOI: 10.5802/aif.2210
Classification: 14J17,  14M25
Keywords: Arc space, Nash problem, singularities
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Ishii, Shihoko. The Local Nash problem on arc families of singularities. Annales de l'Institut Fourier, Volume 56 (2006) no. 4, pp. 1207-1223. doi : 10.5802/aif.2210. https://aif.centre-mersenne.org/articles/10.5802/aif.2210/

[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)

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