Divisorial Valuations via Arcs

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.
Minor corrections, including in Remark 3.3 where it was incorrectly claimed that the codimension of a quasi-cylinder equals the Krull codimension; these corrections do not affect the rest of the paper