Arcs and wedges on rational surface singularities

Abstract

Given a rational surface singularity (S,P0) over an algebraically closed field of characteristic 0, we prove that its minimal desingularization satisfies the property of lifting wedges centered at those stable points Pα of the space of arcs S∞ which correspond to the essential divisorial valuations. This proves the Nash problem for rational surface singularities and, more generally, reduces the Nash problem for surfaces to quasirational normal singularities which are not rational. In positive characteristic, we give counterexamples to the k-wedge lifting problem for the surface singularity x3+y5+z2=0.