×

A solution to the Nash problem on arcs for a family of quasi-rational hypersurfaces. (Résolution du problème des arcs de Nash pour une famille d’hypersurfaces quasi-rationnelles.) (French. English summary) Zbl 1244.14004

For an algebraic variety \(X\), the Nash problem is asking if the set of Nash components over \(X_{\mathrm{sing}}\) in its arc space \(X_\infty\), and the set of essential divisors over \(X\) are in bijection by the natural Nash map between them, which is always injective [S. Ishii and J. Kollár, Duke Math. J. 120, No. 3, 601–620 (2003; Zbl 1052.14011)]. The answer is positive for \(A_n\) [J. F. Nash, jun., Duke Math. J. 81, No.1, 31–38 (1995; Zbl 0880.14010)], for \(D_n\) [C. Plénat, C. R., Math., Acad. Sci. Paris 340, No. 10, 747–750 (2005; Zbl 1072.14004)], for sandwiched surface singularities [M. Lejeune-Jalabert and A. J. Reguera-López, Am. J. Math. 121, No. 6, 1191–1213 (1999; Zbl 0960.14015)], for toric varieties [Zbl 1052.14011], and for some other classes. From [J. F. de Bobadilla and M. Pe Pereira, “Nash problem for surfaces”, arXiv:1102.2212], it is positive for normal surfaces, in particular for \(\mathbb E_6, \mathbb E_7, \mathbb E_8\) singularities (see also C. Plénat and M. Spivakovsky, “The Nash problem of arcs and the rational double point \(\mathbb E_6\)”, arXiv:1011.2426]). There are counter examples in dimension 4 [Zbl 1052.14011], and recently, in dimension 3 [T. de Fernex, “Three-dimensional counter-examples to the Nash problem”, arXiv:1205.0603].
It is known that Nash problem for normal surface singularities could be reduced to the case of quasi-rational singularities, i.e. such that the divisors appearing in the minimal resolution are all rational. In the article under review is developed an approach to solve it for a family of quasi-rational surfaces \(S(p, h_q ) : z^p + h_q (x, y) = 0\), where hq is a degree q reduced homogeneous polynomial, and \(p, q \geq 2\) are coprime. Then using the same method is obtained positive answer for \(\mathbb E_6, \mathbb E_7\) and \(D_n\) surface singularities. It works in higher dimensions as well, as demonstrated in the author’s thesis, providing new examples of dimension 3 hypersurface singularities having bijective Nash map (http://tel.archives-ouvertes.fr/tel-00630100).
The main theorem is claiming that for any \(p, q\) as above, the Nash problem for \(S(p, h_q )\) has a positive answer. To prove it, an embedded resolution of \(S(p, h_q )\) is constructed first. The cases to be checked could be reduced now to \(p > q \geq 3\), and coprime. The idea is to apply the theorem of Lejeune- Jalabert and Reguera [A. J. Reguera, Compos. Math. 142, No. 1, 119–130 (2006; Zbl 1118.14004)], that for a given resolution\( f :\tilde X \to X\) an essential component \(E \subset \tilde X\) belongs to the image of the Nash map iff any wedge over \(X\) whose special arc is the generic point of the family \(N_E \subset X_{\infty}\), corresponding to E, and whose generic arc is over \(X_{\mathrm{sing}}\), could be lifted to \(\tilde X\). The key proposition needed is that when \(\tilde X\) is the minimal resolution for \(S(p, h_q )\), for a given wedge this holds if the three associated with the wedge power series in \(K[[s, t]]\) are invertible. Finally it is shown their invertibility for any wedge as above.
With the same method the Nash problem is then solved positively for \(\mathbb E_6\) and \(\mathbb E_7\) surface singularities. The approach here is different from the topological one accepted by de Bobadilla and Pereira, whose paper appeared after this article was submitted. At the end with this method is obtained also a new and simpler proof of the problem for the case of \(D_n\) surface singularities, \(n \geq 4\).
In this article the ground field is always assumed to be of characteristic \(0\).

MSC:

14B05 Singularities in algebraic geometry
14J70 Hypersurfaces and algebraic geometry
PDF BibTeX XML Cite
Full Text: DOI arXiv EuDML

References:

[1] 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). · Zbl 0823.14006
[2] Campillo (A.), Gonzalez-Sprinberg (G.), and Lejeune-Jalabert (M.).— Clusters of infinitely near points. Math. Ann., 306(1) p. 169-194 (1996). · Zbl 0853.14002
[3] 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). · Zbl 1181.14019
[4] Flenner (H.) and Zaidenberg (M.).— Rational curves and rational singularities. Math. Z., 244(3) p. 549-575 (2003). · Zbl 1043.14008
[5] 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). · Zbl 1129.14004
[6] Gonzalez-Sprinberg (G.) and Lejeune-Jalabert (M.).— Modèles canoniques plongés. I. Kodai Math. J., 14(2) p. 194-209 (1991). · Zbl 0772.14008
[7] 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. · Zbl 0894.14017
[8] Ishii (S.) and Kollár (J.).— The Nash problem on arc families of singularities. Duke Math. J., 120(3) p. 601-620 (2003). · Zbl 1052.14011
[9] Ishii (S.).— Arcs, valuations and the Nash map. J. Reine Angew. Math., 588 p. 71-92 (2005). · Zbl 1082.14007
[10] Ishii (S.).— The local Nash problem on arc families of singularities. Ann. Inst. Fourier (Grenoble), 56(4) p. 1207-1224 (2006). · Zbl 1116.14030
[11] Ishii (S.).— Jet schemes, arc spaces and the Nash problem. C. R. Math. Acad. Sci. Soc. R. Can., 29(1) p. 1-21 (2007). · Zbl 1162.14006
[12] Kempf (G.), Knudsen (F.), Mumford (D.), and Saint-Donat (B.).— Toroidal embeddings. I. Lecture Notes in Mathematics, Vol. 339. Springer-Verlag, Berlin (1973). · Zbl 0271.14017
[13] 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). · Zbl 1221.14017
[14] Lipman (J.).— On complete ideals in regular local rings. In Algebraic geometry and commutative algebra, Vol. I, p. 203-231. Kinokuniya, Tokyo (1988). · Zbl 0693.13011
[15] 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). · Zbl 0432.14020
[16] Lejeune-Jalabert (M.) and Reguera-López (A.J.).— Arcs and wedges on sandwiched surface singularities. Amer. J. Math., 121(6) p. 1191-1213 (1999). · Zbl 0960.14015
[17] 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). · Zbl 0456.14007
[18] Morales (M.).— Some numerical criteria for the Nash problem on arcs for surfaces. Nagoya Math. J., 191 p. 1-19 (2008). · Zbl 1178.14004
[19] 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. · Zbl 0880.14010
[20] 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). · Zbl 0622.14012
[21] Petrov (P.).— Nash problem for stable toric varieties. Math. Nachr., 282(11) p. 1575-1583 (2009). · Zbl 1182.14008
[22] Plénat (C.).— À propos du problème des arcs de Nash. Ann. Inst. Fourier (Grenoble), 55(3) p. 805-823 (2005). · Zbl 1080.14021
[23] Plénat (C.).— The Nash problem of arcs and the rational double points Dn. Ann. Inst. Fourier (Grenoble), 58(7) p. 2249-2278 (2008). · Zbl 1168.14004
[24] Pe-Pereira (M.).— Nash problem for quotient surface singularities. Preprint, arXiv :1011.3792v1 [math.AG] (2010).
[25] 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). · Zbl 1119.14007
[26] Plénat (C.) and Popescu-Pampu (P.).— Families of higher dimensional germs with bijective Nash map. Kodai Math. J., 31(2) p. 199-218 (2008). · Zbl 1210.14008
[27] Plénat (C.) and Spivakovsky (M.).— The nash problem of arcs and the rational double point E6. Preprint, arXiv :1011.2426v1 [math.AG] (2010).
[28] Reguera (A.-J.).— Families of arcs on rational surface singularities. Manuscripta Math., 88(3) p. 321-333, 1995. · Zbl 0867.14012
[29] 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). · Zbl 1118.14004
[30] Sumihiro (H.).— Equivariant completion. J. Math. Kyoto Univ., 14 p. 1-28 (1974). · Zbl 0277.14008
[31] Varchenko (A. N.).— Zeta-function of monodromy and Newton’s diagram. Invent. Math., 37(3) p. 253-262 (1976). · Zbl 0333.14007
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.