Arcs, valuations and the Nash map

Abstract

This paper gives a map from the set of families of arcs on a variety to the set of valuations on the rational function field of the variety We characterize a family of arcs which corresponds to a divisorial valuation by this map. We can see that both the Nash map and a certain McKay correspondence are the restrictions of this map. This paper also gives the affirmative answer to the Nash problem for a non-normal variety in a certain category. As a corollary, we get the affirmative answer for a non-normal toric variety.