The Nash problem of arcs and the rational double point $\mathbf{E_6}$

This paper deals with the Nash problem, which consists in proving that the number of families of arcs on a singular germ of a surface $S$ coincides with the number of irreducible components of the exceptional divisor in the minimal resolution of this singularity. We propose a program for an affirmative solution of the Nash problem in the case of normal 2-dimensional hypersurface singularities. We illustrate this program by giving an affirmative solution of the Nash problem for the rational double point $\mathbf{E_6}$. 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 $\mathbf{E_6}$.