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 Fernández de Bobadilla and Pe Pereira stating that, for surfaces, the Nash map is a bijection.