Terminal valuations and the Nash problem

Abstract

Let X be an algebraic variety of characteristic zero. Terminal valuations are defined in the sense of the minimal model program, as those valuations given by the exceptional divisors on a minimal model over X. We prove that every terminal valuation over X is in the image of the Nash map, and thus it corresponds to a maximal family of arcs through the singular locus of X. In dimension two, this result gives a new proof of the theorem of Fern\'andez de Bobadilla and Pe Pereira stating that, for surfaces, the Nash map is a bijection.