ARCS AND VALUATIONSSHIHOKO ISHIIIn [7], Nash posed a problem: if the set of the families of arcs throughthe singularities on a variety (these families are called the Nash components) corresponds bijectively to the set of essential divisors of resolutions of the singularities. This problem is affirmatively answered
for some 2-dimensional singularities by A. Reguera and M. LejeuneJalabert [8], [6], [9] and toric singularities of arbitrary dimension by S.
Ishii and J. Kollár [5]. On the other hand this problem is negatively
answered in general. The paper [5] gives a counter example of dimension greater than or equal to 4. Therefore the Nash problem should bechanged to a problem to determine the divisors corresponding to theNash components.
We can generalize this problem into the characterization problem
for valuations corresponding to the irreducible components of contactloci Cont≥m (a) which are introduced by L. Ein, R. Lazarsfeld and M.
Mustaţǎ ([1]). In this talk we introduce the maximal divisorial set
CX (v) in the arc space of a variety X corresponding to a divisorialvaluation v. An irreducible component of a contact locus is a maximal
divisorial set. In order to characterize the valuations corresponding to
the irreducible components of a contact locus, it is essential to translate
the inclusion relation between two maximal divisorial sets to a relation
between the corresponding divisorial valuations. The most natural candidate for the translated relation is the value-inequality relation, i.e.,v(f ) ≤ v 0 (f ) for every regular function f on the affine variety X. If
the variety X and the valuations v, v 0 are toric, we have the equivalence: v(f ) ≤ v 0 (f ) for every regular function f on X if and only if
CX (v) ⊃ CX (v 0 ). We study this value-inequality relation. We describe
a maximal divisorial set on the arc space of non-singular variety and determine the necessary and sufficient condition for the value-inequality
relation. As a result we show that this most natural relation is not thetranslation of the inclusion relation of the maximal divisorial sets.
References
1. L. Ein, R. Lazarsfeld and M. Mustaţǎ, Contact loci in arc spaces, Comp. Math.,
140, No 5 (2004) 1229–1244.
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2. W. Fulton, Introduction to Toric Varieties, Annals of Math. St. 131, (1993)
Princeton University Press.
3. S. Ishii, The arc space of a toric variety, J. Algebra 278 (2004) 666–683.
4. S. Ishii, Arcs, valuations and the Nash map, preprint math.AG/0410526, toappear in J. reine angew. Math.
5. S. Ishii and J. Kollár, The Nash problem on arc families of singularities, DukeMath. J. 120, No.3 (2003) 601-620.
6. M. Lejeune-Jalabert and A. J. Reguera-Lopez, Arcs and wedges on sandwichedsurface singularities, Amer. J. Math. 121, (1999) 1191–1213.
7. J. F. Nash, Arc structure of singularities, Duke Math. J. 81, (1995) 31–38.
8. A. J. Reguera-Lopez, Families of arcs on rational surface singularities,
Manuscr. Math. 88, (1995) 321–333.
9. A. J. Reguera, Image of Nash map in terms of wedges, C. R. Acad. Sci. Ser. I,338, (2004) 385–390.
Department of Mathematics, Tokyo Institute of Technology, OhOkayama, Meguro, Tokyo, Japane-mail : shihoko@math.titech.ac.jp