The Nash problem of arcs and the rational double point E6

Abstract

This paper deals with the Nash problem, which consists in proving that the number of families of arcs on a singular germ of a surface S coincides with the number of irreducible components of the exceptional divisor in the minimal resolution of this singularity. We propose a program for an affirmative solution of the Nash problem in the case of normal 2-dimensional hypersurface singularities. We illustrate this program by giving an affirmative solution of the Nash problem for the rational double point E 6 . We also prove some results on the algebraic structure of the space of k-jets of an arbitrary hypersurface singularity and apply them to the specific case of E 6 . Problem 1.2 Is it true that N i ⊂ N j for all i = j?This question has been answered affirmatively in the following special cases: for A n singularities by Nash, for minimal surface singularities by A. Reguera [23] (with other proofs in J. Fernandez-Sanchez [6] and C. Plénat [18]), for sandwiched singularities by M. Lejeune-Jalabert and A. Reguera (cf. [15] and [24]), for toric vareties by S. Ishii and J. Kollar ([11] using earlier work of C. Bouvier and G. Gonzalez-Sprinberg [2] and [3]), for rational double points D n by Plénat [20], for a family of non-rational surface singularities, as well as for a family of singularities in dimension higher than 2 by P. Popescu-Pampu and C.Plénat ([21], [22]).In [11], S. Ishii and J. Kollar gave a counter-example to the Nash problem in dimension greater than or equal to 4.In this paper we prove the following theorem:Theorem 1.3 The Nash problem has an affirmative answer for rational double points E 6 .But the principal aim of this paper is to present a general strategy for attacking normal 2-dimensional hypersurface singularities which has so far been successful in the case of D n ([19]) and E 6 (the present paper).Once this theorem is proved, we have the following corollary (cf.[18] for a proof):Corollary 1.4 Let (S, 0) be a normal surface singularity whose dual graph is obtained from E 6 by increasing the weights (that is, allowing the exceptional curves to have self-intersection numbers of the form −n for n 2). Then the problem also has an affirmative answer for (S, 0).Our program for solving the Nash problem for a normal 2-dimensional hypersurface singularity with equation F = c αβγ x α y β z γ = 0 is divided into two main steps. For the first step we use the following valuative criterion: Proposition 1.5 Let (S, 0) be a normal surface singularity. If there exists an element f in O S,0 such that ord E i f < ord E j f then N i ⊂ N j .This result is stated and proved in ([20], Proposition 1.1) for arbitrary singularities in any dimension. It was first proved by A. Reguera [23] in a different, but equivalent formulation for rational surface singularities.

The Nash problem for surfaces

“…On the other hand, bijectivity of the Nash mapping has been shown for many classes of surfaces (see [6], [10], [8], [9], [12], [13], [17], [19], [20], [21], [22], [24], [25], [26]). The techniques leading to the proof of each of these cases are different in nature, and the proofs are often complicated.…”

“…The techniques leading to the proof of each of these cases are different in nature, and the proofs are often complicated. It is worth noting that even for the case of the rational double points not solved by Nash, a complete proof had to be awaited until 2010; see [19], where the problem is solved for any quotient surface singularity, and also [21] and [24] for the cases of D n and E 6 . In [4] it is shown that the Nash problem for surfaces only depends on the topological type of the singularity.…”

Familles d'espaces de m -jets et d'espaces d'arcs

“…Étant donné explicitement une variété V et une désingularisation π, on remarque que déterminer si une composante irréductible de la fibre exceptionnelle de π est ou non un diviseur essentiel reste un problème difficile. L'approche de Nash est donc très De nombreux mathématiciens, qu'on ne peut pas tous citer ici, ont apporté de nouvelles contributions originales à l'étude du problème de Nash, surtout dans le cas de variétés de dimension 2 et 3 (voir par exemple [LJ80], [Reg95], [GSLJ97], [LJRL99], [IK03], [Ish05], [Ish06], [PPP06], [GP07], [Mor08], [Plé08], [PPP08], [Pet09], [LA11], [FdB12], [PS12], [dFD14]). …”

Nash problem for quotient surface singularities

“…Despite these advances, up to the moment of writing this article, the Nash problem was still open for the very basic cases of surface rational double points E 6 , E 7 and E 8 (an independent proof for E 6 was found by Plénat and Spivakovsky in [30]). Some time after this work, in [6], the author and Fernández de Bobadilla managed to prove the bijectivity of the Nash map in…”

“…For instance, in [18] it is proved that divisors that are not uniruled are in the image of the Nash map and that the so-called lifting wedge property for the case of quasi-rational surfaces would imply that the surjectivity of the Nash map for normal surface singularities. In [5], it is proved that the Nash problem for surfaces only depends on the topological type of the surface singularity and that a positive answer for the case of rational homology sphere links implies a positive answer for any normal surface; in [5,18,33] certain characterizations of the bijectivity of the Nash mapping are proved in terms of uniparametric families of arcs.Despite these advances, up to the moment of writing this article, the Nash problem was still open for the very basic cases of surface rational double points E 6 , E 7 and E 8 (an independent proof for E 6 was found by Plénat and Spivakovsky in [30]). Some time after this work, in [6], the author and Fernández de Bobadilla managed to prove the bijectivity of the Nash map in…”