Three-dimensional counter-examples to the Nash problem

TL;DR

This paper presents three-dimensional counter-examples to the Nash problem, demonstrating its negative resolution in dimension 3 and highlighting differences between algebraic and analytic contexts.

Contribution

It provides the first known three-dimensional counter-examples to the Nash problem, expanding understanding of its limitations beyond dimensions 2 and 4.

Findings

Counter-examples in dimension 3 show the Nash problem has a negative answer.

The nature of the problem differs between algebraic and analytic settings.

The results extend the known dimensions where the Nash problem fails.

Abstract

The Nash problem asks about the existence of a correspondence between families of arcs through singularities of complex varieties and certain types of divisorial valuations. It has been positively settled in dimension 2 by Fern\'andez de Bobadilla and Pe Pereira, and it was shown to have a negative answer in all dimensions $\ge 4$ by Ishii and Koll\'ar. In this note we discuss examples which show that the problem has a negative answer in dimension 3 as well. These examples bring also to light the different nature of the problem depending on whether it is formulated the algebraic setting or in the analytic setting.