This paper shows a ﬁniteness property of a divisorial valuation in terms of arcs. First we show that every divisorial valuation over an algebraic variety corresponds to an irreducible closed subset of the arc space. Then we deﬁne the codimension for this subset and give a formula of the codimension in terms of “relative Mather canonical class”. By using this subset, we prove that a divisorial valuation is determined by assigning the values of ﬁnite functions. We also have a criterion for a divisorial valuation to be a monomial valuation by assigning the values of ﬁnite functions.
Cite this article
Lawrence Ein, Tommaso de Fernex, Shihoko Ishii, Divisorial Valuations via Arcs. Publ. Res. Inst. Math. Sci. 44 (2008), no. 2, pp. 425–448