Irreducible components of contact loci in arc spaces -- Shihoko Ishii

We obtain a map from the set of a certain kind of irreducible subsets of the arc space of a variety to the set of discrete valuations over the variety. The Nash problem is generalized into the problem to determine the valuations corresponding to the irreducible components of the contact locus on the arc space. For this generalized Nash problem, it is essential to translate the inclusion relation between two closed subsets in the arc space to a relation between the corresponding divisorial valuations. The most natural candidate for the translated relation is the value-inequality relation. We determine the necessary and sufficient condition for the value-inequality relation in terms of arc spaces. As a result we show that this most natural relation is not the translation of the inclusion relation of the closed subsets in the arc space.