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

<p>

   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.