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Soit $\mathscr{H}=\bigcup N_i$ la décomposition canonique de l’espace des arcs $\mathscr{H}$ passant par une singularité normale de surface. Dans cet article, on propose deux nouvelles conditions qui si elles sont vérifiées permettent de montrer que $N_i$ n’est pas inclus dans $N_j$. On applique ces conditions pour donner deux nouvelles preuves du problème de Nash pour les singularités sandwich minimales.

This paper deals with the Nash problem, which consists in comparing the number of families of arcs on a singular germ of surface $U$ with the number of essential components of the exceptional divisor in the minimal resolution of this singularity. We prove their equality in the case of the rational double points $D_n$ ($n\ge 4$).

Let $(𝒮,0)$ be a germ of complex analytic normal surface. On its minimal resolution, we consider the reduced exceptional divisor $E$ and its irreducible components $E_i$, $i\in I$. The Nash map associates to each irreducible component $C_k$ of the space of arcs through $0$ on $𝒮$ the unique component of $E$ cut by the strict transform of the generic arc in $C_k$. Nash proved its injectivity and asked if it was bijective. As a particular case of our main theorem, we prove that this is the case if $E·E_i\<0$ for any $i\in I$.