Arcs and wedges on rational surface singularities

Abstract

Let ( S , P 0 ) be a rational surface singularity over an algebraically closed field k of characteristic 0, let ν α be an essential divisorial valuation over ( S , P 0 ) , and P α the stable point of the space of arcs S ∞ corresponding to ν α . We prove that any wedge centered at P α lifts to the minimal desingularization. This proves the Nash problem for rational surface singularities, and reduces the Nash problem for surfaces to quasirational normal singularities which are not rational. In positive characteristic, we give a counterexample to the k-wedge lifting problem for a surface for which the Nash map is bijective.