Asymptotic order-of-vanishing functions on the pseudoeffective cone

Shin-Yao Jow

Vol. 288 (2017), No. 2, 377–380

DOI: 10.2140/pjm.2017.288.377

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 $PE\left(X\right)$ of $X$ if $PE\left(X\right)$ is polyhedral (note that we do not require $PE\left(X\right)$ 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