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The Nash problem of arcs and the rational double points \(D_{n}\). (English) Zbl 1168.14004

The Nash problem is asking whether the number of irreducible families of arcs passing through the singular locus is the same as the number of the essential divisors of resolutions of singularities. This problem is negatively answered for varieties of dimension greater than or equal to 4. But for 2- and 3-dimensional cases, the problem is still open. This paper answers the problem affirmatively if the singularity is 2-dimensional rational and of type \(D_n\). The problem is reduced to show that there is no inclusion relation between the irreducible families of arcs corresponding to the essential divisors. The author first obtain an inclusion relation among the irreducible families using “usual order” of valuations and then check the other inclusion relations case by case.

MSC:

14B05 Singularities in algebraic geometry
14J17 Singularities of surfaces or higher-dimensional varieties
14J10 Families, moduli, classification: algebraic theory
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References:

[1] Artin, M., On isolated rational singularities of surfaces, Amer. J. Math., 88, 129-136, (1966) · Zbl 0142.18602
[2] Bouvier, C., Diviseurs essentiels, composantes essentielles des variétés toriques singulières, Duke Math. J., 91, 609-620, (1998) · Zbl 0966.14038
[3] Bouvier, C.; Gonzales-Sprinberg, G., Système générateur minimal, diviseurs essentiels et G-désingularisations de varitétés toriques, Tohoku Math. J., 47, 125-149, (1995) · Zbl 0823.14006
[4] Eisenbud, D., Commutative Algebra with a view toward Algebraic Geometry, 150, (1995), Springer-Verlag, New York · Zbl 0819.13001
[5] Fernandez-Sanchez, J., Equivalence of the Nash conjecture for primitive and sandwiched singularities, Proc. Amer. Math. Soc., 133, 677-679, (2005) · Zbl 1056.14004
[6] Ishii, S., Arcs, valuations and the Nash map, arXiv: math.AG/0410526 · Zbl 1082.14007
[7] Ishii, S., The local Nash problem on arc families of singularities, arXiv: math.AG/0507530 · Zbl 1116.14030
[8] Ishii, S.; Kollár, J., The Nash problem on arc families of singularities, Duke Math. J., 120, 3, 601-620, (2003) · Zbl 1052.14011
[9] Lejeune-Jalabert, M., Arcs analytiques et résolution minimale des singularités des surfaces quasi-homogènes, Séminaire sur les Singularités des Surfaces, Lecture Notes in Math., 777, 303-336, (1980), Springer-Verlag · Zbl 0432.14020
[10] Lejeune-Jalabert, M., Désingularisation explicite des surfaces quasi-homogènes dans \(\mathbb{C}^3,\) Nova Acta Leopoldina, NF 52, Nr 240, 139-160, (1981) · Zbl 0474.14021
[11] Lejeune-Jalabert, M., Courbes tracées sur un germe d’hypersurface, Amer. J. Math., 112, 525-568, (1990) · Zbl 0743.14002
[12] Lejeune-Jalabert, M.; Reguera, A., Arcs and wedges on sandwiched surface singularities, Amer. J. Math., 121, 1191-1213, (1999) · Zbl 0960.14015
[13] Matsumura, H., Commutative ring theory. Translated from the Japanese by M. Reid, 8, (1986), Cambridge University Press, Cambridge · Zbl 0603.13001
[14] Nash, J. F. Jr., Arc structure of singularities, A celebration of John F. Nash, Jr. Duke Math. J., 81, 1, 31-38, (1995) · Zbl 0880.14010
[15] Plénat, C., A propos du problème des arcs de Nash, Annales de l’Institut Fourier, 55, 3, 805-823, (2005) · Zbl 1080.14021
[16] Plénat, C., 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, 747-750, (2005) · Zbl 1072.14004
[17] Plénat, C.; Popescu-Pampu, P., A class of non-rational surface singularities for which the Nash map is bijective, Bulletin de la SMF, 134, 3, 383-394, (2006) · Zbl 1119.14007
[18] Reguera, A., Families of arcs on rational surface singularities, Manuscripta Math, 88, 3, 321-333, (1995) · Zbl 0867.14012
[19] Reguera, A., Image of the Nash map in terms of wedges, C. R. Acad. Sci. Paris, Ser. I , 338, 385-390, (2004) · Zbl 1044.14032
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