On the dual of the mobile cone
type: Item
instance of: scholarly article
title: On the dual of the mobile cone
zbMATH Open document ID: 1252.14014
author: Sung Rak Choi
published in: Mathematische Zeitschrift
publication date: 2012-10-22T00:00:00Z
full work available at URL: https://arxiv.org/abs/1011.4646
review text: Let \(X\) be a normal projective complex variety. There are several cones one can define inside the vector spaces \(N^1(X)_{\mathbb R}\) and \(N_1(X)_{\mathbb R}\), and it is one of the basic questions which ones are dual to each other with respect to the intersection pairing. For instance, it is classical that the nef cone is dual to the cone of effective curves, and it is the main result of the paper [\textit{S. Boucksom, J.-P. Demailly, M. Păun} and \textit{T. Peternell}, ``The pseudo-effective cone of a compact Kähler manifold and varieties of negative Kodaira dimension'', \url{arXiv:math/0405285}] that the pseudoeffective cone is dual to the cone of movable curves. The paper under review considers the question which cone is dual to the cone of movable divisors; and it gives a partial answer in the spirit of [\textit{C. Araujo}, Math. Z. 264, No. 1, 179--193 (2010; Zbl 1189.14006)] and [\textit{B. Lehmann}, J. Algebr. Geom. 21, No. 3, 473--493 (2012; Zbl 1253.14014)]. The author defines the cone of curves which are birationally movable in codimension \(1\) (b-mov\(^1\) curves), and proves that it is dual (in a certain region) to the movable cone; the precise statement is Theorem 1.1 in the paper. In particular, if \(X\) is a Fano variety, it is shown that the movable cone is dual to the cone of b-mov\(^1\) curves.
Mathematics Subject Classification ID: 14E30
Mathematics Subject Classification ID: 14C20
zbMATH DE Number: 6097117
zbMATH Keywords: log minimal model program
zbMATH Keywords: mobile cone
zbMATH Keywords: movable curve
zbMATH Keywords: non-ample locus
zbMATH Keywords: non-nef locus
zbMATH Keywords: cone theorem
zbMATH Keywords: contraction theorem
reviewed by: Vladimir Lazic
MaRDI profile type: Publication
OpenAlex ID: W2170015103
arXiv ID: 1011.4646
cites work: The cone of pseudo-effective divisors of log varieties after Batyrev
cites work: The cone of moving curves of a smooth Fano three- or fourfold
cites work: Q4026912
cites work: Existence of minimal models for varieties of log general type
cites work: Uniruledness of stable base loci of adjoint linear systems via Mori theory
cites work: The pseudo-effective cone of a compact Kähler manifold and varieties of negative Kodaira dimension
cites work: Restricted volumes and base loci of linear series
cites work: Asymptotic invariants of base loci
cites work: Mori dream spaces and GIT.
cites work: Birational models and flips
cites work: Q4821063
cites work: Q4650504
cites work: Introduction to the Mori program
cites work: Stable base loci of linear series
cites work: Stable base loci, movable curves, and small modifications, for toric varieties
cites work: Towards the second main theorem on complements
cites work: 3-fold log models
cites work: Geography of log models: theory and applications
DOI: 10.1007/S00209-011-0922-7
description: scientific article; zbMATH DE number 6097117