Abstract. -Let (S, 0) be a germ of complex analytic normal surface. On its minimal resolution, we consider the reduced exceptional divisor E and its irreducible components E i , i ∈ I. The Nash map associates to each irreducible component C k of the space of arcs through 0 on S the unique component of E cut by the strict transform of the generic arc in C k . Nash proved its injectivity and asked if it was bijective. As a particular case of our main theorem, we prove that this is the case if E · E i < 0 for any i ∈ I.Résumé (Une classe de singularités non-rationnelles de surfaces ayant une application de Nash bijective) Soit (S, 0) un germe de surface analytique complexe normale. Nous considérons le diviseur exceptionnel réduit E et ses composantes irréductibles E i , i ∈ I sur sa réso-lution minimale. L'application de Nash associeà chaque composante irréductible C k de l'espace des arcs passant par 0 sur S, l'unique composante de E rencontrée par la transformée stricte de l'arc générique dans C k . Nash a prouvé son injectivité et a demandé si elleétait bijective. Nous prouvons que c'est le cas si E · E i < 0 pour tout i ∈ I comme cas particulier de notre théorème principal.
Let ðX ; 0Þ be a germ of complex analytic normal variety, non-singular outside 0. An essential divisor over ðX ; 0Þ is a divisorial valuation of the field of meromorphic functions on ðX ; 0Þ, whose center on any resolution of the germ is an irreducible component of the exceptional locus. The Nash map associates to each irreducible component of the space of arcs through 0 on X the unique essential divisor intersected by the strict transform of the generic arc in the component. Nash proved its injectivity and asked if it was bijective. We prove that this is the case if there exists a divisorial resolution p of ðX ; 0Þ such that its reduced exceptional divisor carries su‰ciently many p-ample divisors (in a sense we define). Then we apply this criterion to construct an infinite number of families of 3-dimensional examples, which are not analytically isomorphic to germs of toric 3-folds (the only class of normal 3-fold germs with bijective Nash map known before).
A propos du problème des arcs de Nash Tome 55, n o 3 (2005), p. 805-823.
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.
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