Cox rings of rational complexity-one \(T\)-varieties
type: Item
instance of: scholarly article
title: Cox rings of rational complexity-one \(T\)-varieties
zbMATH Open document ID: 1251.14032
author: Klaus Altmann
author: Lars Petersen
published in: Journal of Pure and Applied Algebra
publication date: 2012-07-30T00:00:00Z
full work available at URL: https://arxiv.org/abs/1009.0478
reviewed by: Alessandro Ruzzi
Mathematics Subject Classification ID: 14L30
Mathematics Subject Classification ID: 14H30
Mathematics Subject Classification ID: 14M99
zbMATH DE Number: 6060935
zbMATH Keywords: cox ring
zbMATH Keywords: Mori dream space
zbMATH Keywords: toric varieties
MaRDI profile type: Publication
OpenAlex ID: W2963339108
arXiv ID: 1009.0478
cites work: Polyhedral divisors and algebraic torus actions
cites work: Gluing affine torus actions via divisorial fans
cites work: A fansy divisor on \(\overline {M}_{0,n}\)
cites work: Polyhedral divisors of Cox rings
cites work: Cox rings and combinatorics
cites work: Q4324883
cites work: The Cox ring of an algebraic variety with torus action
cites work: Mori dream spaces and GIT.
cites work: Polarized complexity-1 T-varieties
cites work: Deformations of rational \(T\)-varieties
cites work: Upgrading and downgrading torus actions
cites work: Q4821063
cites work: Normal singularities with torus actions
cites work: Torus invariant divisors
cites work: Q5485133
DOI: 10.1016/J.JPAA.2011.12.018
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review text: Let \(X\) be a normal and complete variety with a finitely generated divisor class group \(Cl(X)\). Then one can defines the Cox ring of \(X\): \(\mathrm{Cox}(X)=\) \(\bigoplus_{D\in \mathrm{Cl}(X)} \Gamma(X,O_X(D))\). The authors say that \(X\) is a Mori dream space if \(\mathrm{Cox}(X)\) is a finitely generated \(\mathbb{C}\)-algebra. By [\textit{Y. Hu} and \textit{S. Keel}, Mich. Math. J. 48, Spec. Vol., 331--348 (2000; Zbl 1077.14554)], when \(X\) is \(\mathbb{Q}\)-factorial and projective, this definition coincides with the usual ones in the Mori theory.NEWLINENEWLINEOne of the most important class of Mori dream space is the one of toric varieties. Indeed, a normal complete variety with a finitely generated free divisor class group is a toric variety if and only if its Cox ring is a polynomial ring (see [\textit{F. Berchtold} and \textit{J. Hausen}, Trans. Am. Math. Soc. 359, No. 3, 1205--1252 (2007; Zbl 1117.14009)]). In particular the spectrum of the Cox ring of a toric variety is again an (affine) toric variety.NEWLINENEWLINEThe normal affine varieties \(X\) with an effective action of a torus \(T\) can be combinatorial described in terms of characters of \(T\) and of the so called polyhedral divisors (or p-divisors) on \(Y=X//^{\mathrm{ch}}T\) (see [\textit{K. Altmann} and \textit{J. Hausen}, Math. Ann. 334, No. 3, 557--607 (2006; Zbl 1193.14060)]). This description can be seriously simplified if \(Z\) is a curve (see [\textit{N. Ilten} and \textit{H. Süss}, Mich. Math. J. 60, No. 3, 561--578 (2011; Zbl 1230.14075)]). Moreover it generalizes the one of affine toric varieties in terms of cones.NEWLINENEWLINEIf \(X\) is a Mori dream space with \(\mathrm{Cl}(X)\) torsion free, then the spectrum of \(\mathrm{Cox}(X)\) is a normal affine toric variety and the \(\mathrm{Cl}(X)\)-grading gives an action of the Picard torus \(S=\Hom(\mathrm{Cl}(X),C^*)\). In such case \(\mathrm{Spec}(\mathrm{Cox}(X)\) has been described as \(S\)-variety in [\textit{K. Altmann} and …
description: scientific article; zbMATH DE number 6060935