The Nash problem on arcs for surface singularities

Abstract

Let $(X,O)$ be a germ of a normal surface singularity, $\pi : \tilde X\longrightarrow X$ be the minimal resolution of singularities and let $A=(a_{i,j})$ be the $n\times n$ symmetrical intersection matrix of the exceptional set of $\tilde X$. In an old preprint Nash proves that the set of arcs on a surface singularity is a scheme ${\cal H}$, and defines a map ${\cal N}$ from the set of irreducible components of ${\cal H}$ to the set of exceptional components of the minimal resolution of singularities of $(X,O)$. He proved that this map is injective and ask if it is surjective. In this paper we consider the canonical decomposition ${\cal H}=\cup_{i=1}^n \bar{\cal N}_i$: o For any couple $(E_i,E_j)$ of distinct exceptional components, we define Numerical Nash condition $(NN_{(i,j)})$. We have that $(NN_{(i,j)})$ implies $\bar{\cal N}_{i}\not\subset \bar{\cal N}_{j} $. In this paper we prove that $(NN_{(i,j)})$ is always true for at least the half of couples $(i,j)$. o The condition $(NN_{(i,j)})$ is true for all couples $(i,j)$ with $i\not=j$, characterizes a certain class of negative definite matrices, that we call Nash matrices. If $A$ is a Nash matrix then the Nash map ${\cal N}$ is bijective. In particular our results depends only on $A$ and not on the topological type of the exceptional set. o We recover and improve considerably almost all results known on this topic and our proofs are new and elementary. o We give infinitely many other classes of singularities where Nash Conjecture is true. The proofs are based on my old work \cite{M} and in Plenat \cite{P}.