Divisorial Valuations via Arcs

  • Lawrence Ein

    University of Illinois at Chicago, United States
  • Tommaso de Fernex

    University of Utah, Salt Lake City, United States
  • Shihoko Ishii

    University of Tokyo, Japan


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

DOI 10.2977/PRIMS/1210167333