Nash problem for surface singularities is a topological problem

TL;DR

This paper advances the understanding of the Nash problem for surface singularities by refining wedge-based characterizations and linking the problem to topological and combinatorial properties of singularities.

Contribution

It provides a new wedge-based criterion for the bijectivity of the Nash map and relates the problem to the topology of the singularity's link, improving previous results.

Findings

Characterization of the Nash map's bijectivity using wedges over the base field.

Reformulation of Nash problem in terms of branched covers of surface singularities.

Reduction of the Nash problem to cases with rational homology sphere links.

Abstract

We address Nash problem for surface singularities using wedges. We give a refinement of the characterisation of A. Reguera of the image of the Nash map in terms of wedges. Our improvement consists in a characterisation of the bijectivity of the Nash mapping using wedges defined over the base field, which are convergent if the base field is the field of complex numbers, and whose generic arc has transverse lifting to the exceptional divisor of a resolution of singularities. This improves the recent results of M. Lejeune-Jalabert and A. Reguera for the surface case. In the way to do this we find a reformulation of Nash problem in terms of branched covers of normal surface singularities. As a corollary of this reformulation we prove that the image of the Nash mapping is characterised by the combinatorics of a resolution of the singularity, or, what is the same, by the topology of the abstract link of the singularity in the complex analytic case. Using these results we prove several reductions of the Nash problem, the most notable being that, if Nash problem is true for singularities having rational homology sphere links, then it is true in general.