A propos de la conjecture de Nash

Abstract

This paper deals with the Nash problem, which claims that there are as many families of arcs on a singular germ of surface $U$ as there are essential components of the exceptional divisor in the desingularisation of this singularity. Let $\mathcal{H}=\bigcup \bar{N_\alpha}$ be a particular decomposition of the set of arcs on $U$, described later on. We give two new conditions to insure that $\bar{N_\alpha}\not \subset \bar{N_\beta}$, $\alpha \not = \beta$; more precisely,for the first one, we claim that if there exists $f \in {\mathcal{O}}_{U}$ such that $ord_{E_\alpha}(f)<ord_{E_\beta}(f)$, where $E_\alpha, E_\beta$ are exceptional divisors of the desingularisation, then $\bar{N_\alpha}\not \subset \bar{N_\beta}$. The second condition, used when the singularity is rational and of surface, is the following:let $(S,s)$ et $(S',s')$ be two rational surface singularities so that there exist a dominant and birational morphism $\pi$ from $(S,s)$ to $(S',s')$;then,let $E_\alpha, E_\beta$ be two essential components of the exceptional divisors in the minimal desingularisation of $(S,s)$, such that their image by $\pi$, $E'_\alpha$ and $E'_\beta$, are exceptional divisor for $(S',s')$; if $\bar{N'_\alpha}(S',s')\not \subset \bar{N'_\beta}(S',s')$ then $\bar{N_\alpha}(S,s)\not \subset \bar{N_\beta}(S,s)$. These two conditions are simple, but it allows us to prove quite directly the conjecture for the rational minimal surface singularities, using the decomposition of minimal suface singularities into cyclic quotient singularities of type $A_n$. A proof of the conjecture for these singularities has already been given by Ana Reguera.