Three-dimensional counter-examples to the Nash problem

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.