×

Some numerical criteria for the Nash problem on arcs for surfaces. (English) Zbl 1178.14004

The Nash problem asks the bijectiveness of the Nash map from the set of families of arcs with the center in the singular locus of a variety, to the set of essential divisors over the singular locus of the variety. This problem was posed by Nash in 1968 and negatively answered in 2003 by J. Kollár and the reviewer [Duke Math. J. 120, No. 3, 601–620 (2003; Zbl 1052.14011)] for dimension higher than 3. The problem is still open for 2 and 3 dimensional cases. This paper shows some 2 dimensional singularities for which the Nash problem is affirmatively answered. The author generalized the result of [C. Plenat and P. Popescu-Pampur, Bull. Soc. Math. France, Bull. Soc. Math. Fr. 134, No. 3, 383–394 (2006; Zbl 1119.14007)] which showed a sufficient condition of the intersection matrix of the minimal resolution of a surface singularity for the affirmative answer to the Nash problem. The condition of this paper for the intersection matrix is not so simple as their condition, but the author obtains many affirmative examples which are not obtained by Plénat and Popescu-Pampu’s result.

MSC:

14B05 Singularities in algebraic geometry
PDF BibTeX XML Cite
Full Text: DOI Euclid

References:

[1] J. Denef and F. Loeser, Germs of arcs on singular varieties and motivic integration , Inv. Math., 135 (1999), 201–232. · Zbl 0928.14004
[2] J. Fernandez-Sanchez, Equivalence of the Nash conjecture for primitive and sandwiched singularities , Proc. Amer. Math. Soc., 133 (2005), 677–679. · Zbl 1056.14004
[3] H. Grauert, Uber modifikationen und exceptionnelle analytische Mengen , Math. Annalen, 146 (1962), 331–368. · Zbl 0178.42702
[4] G. Gonzalez-Sprinberg and M. Lejeune-Jalabert, Families of smooth curves on surface singularities and wedges , Annales Polonici Mathematici, LXVII .2 (1997), 179–190. · Zbl 0894.14017
[5] S. Ishii and J. Kollar, The Nash problem on arc families of singularities , Duke Math. J., 120 (2003), no. 3, 601–620. · Zbl 1052.14011
[6] M. Lejeune-Jalabert, Courbes tracées sur un germe d’hypersurface , Amer. J. of Math., 112 (1990), 525–568. JSTOR: · Zbl 0743.14002
[7] M. Lejeune-Jalabert and A. Reguera, Arcs and wedges on sandwiched surfaces singularities , Amer. J. of Math., 121 (1999), 1191–1213. · Zbl 0960.14015
[8] M. Morales, Clôture intégrale d’idéaux et anneaux gradués Cohen-Macaulay , Géométrie algébrique et applications, La Rabida 1984 (J-M. Aroca, et als, eds.), Hermann, pp. 15–172.
[9] J. F. Nash Jr., Arcs structure of singularities , Duke Math. J., 81 (1995), no. 1, 31–38. · Zbl 0880.14010
[10] C. Plénat, A Propos du problème des arcs de Nash , Ann. Inst. Fourier., 55 (2005), no. 3, 805–823. · Zbl 1080.14021
[11] C. Plénat, Résolution du problème des arcs de Nash pour les points doubles rationnels \(D_n\), Thèse Univ. Paul Sabatier. Toulouse, 2004. · Zbl 1072.14004
[12] C. Plénat and P. Popescu-Pampu, A class of non-rational surfaces singularities for which the Nash map is bijective , Bulletin Soc. Math. France, to be published.
[13] A. Reguera, Families of Arcs on rational surface singularities , Manuscripta Math., 88 (1995), 321–333. · Zbl 0867.14012
[14] A. Reguera, A curve selection lemma in spaces of arcs and the image of the Nash map , Compos. Math., 142 (2006), no. 1, 119–130. · Zbl 1118.14004
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.