Asymptotic orders of vanishing along base loci separate Mori chambers

Chih-Wei Chang and Shin-Yao Jow

Vol. 304 (2020), No. 1, 55–64

DOI: 10.2140/pjm.2020.304.55

Abstract

It is known that there exists a Mori dream space such that the Mori chamber decomposition of its effective cone is strictly finer than the stable base locus decomposition. In other words, stable base loci of line bundles do not contain enough information to separate different Mori chambers in general. Here we show, however, that different Mori chambers can be separated if the scheme structures of the base loci are taken into account. More precisely, we show that for any two distinct Mori chambers $\Gamma$ and ${\Gamma }^{\prime }$, the asymptotic order of vanishing along some divisorial valuation is linear on $\Gamma$ and on ${\Gamma }^{\prime }$ respectively, but not simultaneously on their union $\Gamma \cup {\Gamma }^{\prime }$. Two toric examples are given to illustrate our result: the first one exhibits two adjacent Mori chambers where the base schemes have the same underlying set but different embedded components; the second one shows that it is not always possible to separate two adjacent Mori chambers by the asymptotic order of vanishing along an associated component of the base schemes.

Keywords

Mori dream space, Mori chamber, stable base locus, asymptotic order of vanishing

Mathematical Subject Classification 2010

Primary: 14C20

Milestones

Received: 30 January 2019

Revised: 12 August 2019

Accepted: 20 August 2019

Published: 18 January 2020