The Nash problem of arcs and the rational double point E6

Abstract

This paper deals with the Nash problem, which consists in proving that the number of families of arcs on a germ of a normal isolated singularity coincides with the number of essential components of the exceptional set in any resolution of this singularity. We propose a program for an affirmative solution of the Nash problem for special types of normal isolated hypersurface singularities. We illustrate this program by giving an affirmative solution of the Nash problem for the rational double point E₆. 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 E₆.