Résolution du problème des arcs de Nash pour une famille d’hypersurfaces quasi-rationnelles

Leyton-Alvarez, Maximiliano

Annales de la Faculté des sciences de Toulouse : Mathématiques, Série 6, Tome 20 (2011) no. 3, pp. 613-667.

Le problème des arcs de Nash pour les singularités normales de surfaces affirme qu’il y aurait autant de familles d’arcs sur un germe de surface singulier $(S,O)$ que de diviseurs essentiels sur $(S,O)$. Il est connu que ce problème se réduit à étudier les singularités quasi-rationnelles. L’objet de cet article est de répondre positivement au problème de Nash pour une famille d’hypersurfaces quasi-rationnelles non rationnelles. On applique la même méthode pour répondre positivement à ce problème dans les cas de singularités de type ${𝔼}_6$ et ${𝔼}_7$ et pour fournir une nouvelle preuve dans le cas de singularités de type ${𝔻}_n$, $n\ge 4$.

A solution to the Nash problem on arcs for a family of quasi-rational hypersurfaces. The Nash problem on arcs for normal surface singularities states that there are as many arc families on a germ $(S,O)$ of a singular surface as there are essential divisors over $(S,O)$. It is known that this problem can be reduced to the study of quasi-rational singularities. In this paper we give a positive answer to the Nash problem for a family of non-rational quasi-rational hypersurfaces. The same method is applied to answer positively to this problem in the case of ${𝔼}_6$ and ${𝔼}_7$ type singularities, and to provide new proof in the case of ${𝔻}_n$, $n\ge 4$, type singularities.

[BGS95] Bouvier (C.) and Gonzalez-Sprinberg (G.).— Système générateur minimal, diviseurs essentiels et G-désingularisations de variétés toriques. Tohoku Math. J. (2), 47(1) p. 125-149 (1995). | MR | Zbl

[CGSLJ96] Campillo (A.), Gonzalez-Sprinberg (G.), and Lejeune-Jalabert (M.).— Clusters of infinitely near points. Math. Ann., 306(1) p. 169-194 (1996). | MR | Zbl

[EM09] Ein (L.) and Mustaţă (M.).— Jet schemes and singularities. In Algebraic geometry-Seattle 2005. Part 2, volume 80 of Proc. Sympos. Pure Math., pages 505-546. Amer. Math. Soc., Providence, RI (2009). | MR | Zbl

[FZ03] Flenner (H.) and Zaidenberg (M.).— Rational curves and rational singularities. Math. Z., 244(3) p. 549-575 (2003). | MR | Zbl

[GP07] González Pérez (P. D.).— Bijectiveness of the Nash map for quasi-ordinary hypersurface singularities. Int. Math. Res. Not. IMRN, (19) :Art. ID rnm076, 13 (2007). | MR | Zbl

[GSLJ91] Gonzalez-Sprinberg (G.) and Lejeune-Jalabert (M.).— Modèles canoniques plongés. I. Kodai Math. J., 14(2) p. 194-209 (1991). | MR | Zbl

[GSLJ97] Gonzalez-Sprinberg (G.) and Lejeune-Jalabert (M.).— Families of smooth curves on surface singularities and wedges. Ann. Polon. Math., 67(2) p. 179-190, 1997. | MR | Zbl

[IK03] Ishii (S.) and Kollár (J.).— The Nash problem on arc families of singularities. Duke Math. J., 120(3) p. 601-620 (2003). | MR | Zbl

[Ish05] Ishii (S.).— Arcs, valuations and the Nash map. J. Reine Angew. Math., 588 p. 71-92 (2005). | MR | Zbl

[Ish06] Ishii (S.).— The local Nash problem on arc families of singularities. Ann. Inst. Fourier (Grenoble), 56(4) p. 1207-1224 (2006). | Numdam | MR | Zbl

[Ish07] Ishii (S.).— Jet schemes, arc spaces and the Nash problem. C. R. Math. Acad. Sci. Soc. R. Can., 29(1) p. 1-21 (2007). | MR | Zbl

[KKMS73] Kempf (G.), Knudsen (F.), Mumford (D.), and Saint-Donat (B.).— Toroidal embeddings. I. Lecture Notes in Mathematics, Vol. 339. Springer-Verlag, Berlin (1973). | MR | Zbl

[LA11] Leyton-Alvarez (M.).— Une famille d’hypersurfaces quasi-rationnelles avec application de nash bijective. C. R. Acad. Sci. Paris Sér. I Math., 349(5-6) p. 323-326 (2011). | MR

[Lip88] Lipman (J.).— On complete ideals in regular local rings. In Algebraic geometry and commutative algebra, Vol. I, p. 203-231. Kinokuniya, Tokyo (1988). | MR | Zbl

[LJ80] Lejeune-Jalabert (M.).— Arcs analytiques et resolution minimale des singularites des surfaces quasi homogenes. In Séminaire sur les Singularités des Surfaces, volume 777 of Lecture Notes in Mathematics, p. 303-336. Springer Berlin / Heidelberg, Amsterdam (1980). | Numdam | Zbl

[LJRL99] Lejeune-Jalabert (M.) and Reguera-López (A.J.).— Arcs and wedges on sandwiched surface singularities. Amer. J. Math., 121(6) p. 1191-1213 (1999). | MR | Zbl

[Mer80] Merle (M.).— Polyèdre de newton, éventails et désingularisation, d’après a. n. varchenko. In Séminaire sur les Singularités des Surfaces, Palaiseau, France, 1976-1977, volume 777 of Lecture Notes in Mathematics, pages 289-294. Springer Berlin / Heidelberg, Amsterdam (1980). | Numdam | Zbl

[Mor08] Morales (M.).— Some numerical criteria for the Nash problem on arcs for surfaces. Nagoya Math. J., 191 p. 1-19 (2008). | MR | Zbl

[Nas95] Nash (J.F.) Jr.— Arc structure of singularities. Duke Math. J., 81(1) p. 31-38 (1996), 1995. A celebration of John F. Nash, Jr. | MR | Zbl

[Oka87] Oka (M.).— On the resolution of the hypersurface singularities. In Complex analytic singularities, volume 8 of Adv. Stud. Pure Math., pages 405-436. North-Holland, Amsterdam (1987). | MR | Zbl

[Pet09] Petrov (P.).— Nash problem for stable toric varieties. Math. Nachr., 282(11) p. 1575-1583 (2009). | MR | Zbl

[Plé05] Plénat (C.).— À propos du problème des arcs de Nash. Ann. Inst. Fourier (Grenoble), 55(3) p. 805-823 (2005). | Numdam | MR | Zbl

[Plé08] Plénat (C.).— The Nash problem of arcs and the rational double points Dn. Ann. Inst. Fourier (Grenoble), 58(7) p. 2249-2278 (2008). | Numdam | MR | Zbl

[PP10] Pe-Pereira (M.).— Nash problem for quotient surface singularities. Preprint, arXiv :1011.3792v1 [math.AG] (2010).

[PPP06] Plénat (C.) and Popescu-Pampu (P.).— A class of non-rational surface singularities with bijective Nash map. Bull. Soc. Math. France, 134(3) p. 383-394 (2006). | Numdam | MR | Zbl

[PPP08] Plénat (C.) and Popescu-Pampu (P.).— Families of higher dimensional germs with bijective Nash map. Kodai Math. J., 31(2) p. 199-218 (2008). | MR | Zbl

[PS10] Plénat (C.) and Spivakovsky (M.).— The nash problem of arcs and the rational double point E6. Preprint, arXiv :1011.2426v1 [math.AG] (2010).

[Reg95] Reguera (A.-J.).— Families of arcs on rational surface singularities. Manuscripta Math., 88(3) p. 321-333, 1995. | MR | Zbl

[Reg06] Reguera (A.-J.).— A curve selection lemma in spaces of arcs and the image of the Nash map. Compos. Math., 142(1) p. 119-130 (2006). | MR | Zbl

[Sum74] Sumihiro (H.).— Equivariant completion. J. Math. Kyoto Univ., 14 p. 1-28 (1974). | MR | Zbl

[Var76] Varchenko (A. N.).— Zeta-function of monodromy and Newton’s diagram. Invent. Math., 37(3) p. 253-262 (1976). | MR | Zbl