Vol. 288, No. 2, 2017

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ISSN: 0030-8730
Asymptotic order-of-vanishing functions on the pseudoeffective cone

Shin-Yao Jow

Vol. 288 (2017), No. 2, 377–380
Abstract

Let v be a discrete valuation on the function field of a normal projective variety X. Ein, Lazarsfeld, Mustaţă, Nakamaye, and Popa showed that v induces a nonnegative real-valued continuous function on the big cone of X, which they called the asymptotic order of vanishing along v. The case where v is given by the order of vanishing along a prime divisor was studied earlier by Nakayama, who extended the domain of the function to the pseudoeffective cone and investigated the continuity of the extended function.

Here we generalize Nakayama’s results to any discrete valuation v, using an approach inspired by Lazarsfeld and Mustaţă’s construction of the global Okounkov body, which has a quite different flavor from the arguments employed by Nakayama.

A corollary is that the asymptotic order-of-vanishing function can be extended continuously to the pseudoeffective cone PE(X) of X if PE(X) is polyhedral (note that we do not require PE(X) to be rational polyhedral).

Keywords
asymptotic order of vanishing, pseudoeffective cone, global Okounkov body, Nakayama's $\sigma$-decomposition
Mathematical Subject Classification 2010
Primary: 14C20
Milestones
Received: 13 May 2016
Accepted: 7 September 2016
Published: 28 April 2017
Authors
Shin-Yao Jow
Department of Mathematics
National Tsing Hua University
No. 101, Sec. 2, Guangfu Rd
Hsinchu 300
Taiwan