Arc Valuations on Smooth Varieties.

Abstract

For a nonsingular $CC$-arc valuation $v$ on a nonsingular variety $X$ over a field $CC$, we describe the maximal irreducible subset $C(v)$ of the arc space of X such that $val_{C(v)}=v$. We describe $C(v)$ both algebraically, in terms of the sequence of valuation ideals of $v$, and geometrically, in terms of the sequence of infinitely near points associated to $v$. For a singular $CC$-arc valuation $v$, we show that after a finite number of blowups of centers, its becomes nonsingular. When $X$ is a surface, our construction also applies to any divisorial valuation $v$, and in this case $C(v)$ coincides with the construction of Ein, Lazarsfeld, and Mustata (cite[Example 2.5]{mustata}). We also investigate the situation for irrational valuations on surfaces. Our results suggest that a more natural place to look for these valuations are in spaces that generalize arc spaces. Also, we compute the motivic measure of $C(v)$ for some of the various types of valuations on surfaces.