Publications of the Research Institute for Mathematical Sciences

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Volume 44, Issue 2, 2008, pp. 425–448
DOI: 10.2977/prims/1210167333

Divisorial Valuations via Arcs

Tommaso de Fernex[1], Lawrence Ein[2] and Shihoko Ishii[3]

(1) Department of Mathematics, University of Utah, 155 South 1400 East, UT 84112-0090, SALT LAKE CITY, UNITED STATES
(2) Department of Mathematics, Statistics and Computer, University of Illinois at Chicago, 851 South Morgan Street, IL 60607-7045, CHICAGO, UNITED STATES
(3) Department of Mathematical Sciences, University of Tokyo, Komaba, Meguro, 153-8914, 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.

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de Fernex Tommaso, Ein Lawrence, Ishii Shihoko: Divisorial Valuations via Arcs. Publ. Res. Inst. Math. Sci. 44 (2008), 425-448. doi: 10.2977/prims/1210167333