R\'esolution du probl\`eme des arcs de Nash pour une famille d'hypersurfaces quasi-rationnelles

TL;DR

This paper proves the Nash problem for a family of non-rational quasi-rational hypersurfaces, extending known results to specific singularity types and providing new proofs for others.

Contribution

It offers a positive solution to the Nash problem for certain hypersurfaces and singularity types, using a unified method applicable to multiple cases.

Findings

Confirmed the Nash problem for a family of non-rational quasi-rational hypersurfaces.

Extended the solution to E_6 and E_7 singularities.

Provided new proofs for D_n (n ≥ 4) singularities.

Abstract

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> =4, type singularities.