Cox rings of rational complexity-one \(T\)-varieties
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 \textit{J. Wiśniewski}, Mich. Math. J. 60, No. 2, 463--480 (2011; Zbl 1223.14006)].NEWLINENEWLINEThe authors consider \(T\)-varieties \(X\) of complexity one, i.e. \(\dim\,Y=1\). If \(X\) is a DMS then \(Y\) is \(\mathbb{P}^1\). In this case \(\mathrm{Cox}(X)\) has been described by generators and relations in [\textit{J. Hausen} and \textit{H. Süss}, Adv. Math. 225, No. 2, 977--1012 (2010; Zbl 1248.14008)]. In this case the action of \(T\) on \(X\) induces an action on \(\mathrm{Cox}(X)\), and by combining it with the action of the Picard torus, \(\mathrm{Spec}(\mathrm{Cox}(X))\) turns into a complexity-one variety. Unfortunately, the combination of the two torus actions might involve torsion and there is not a general description of complexity-one variety by the action of a diagonalizable group.NEWLINENEWLINERemoving the torsion gives rise to a finite covering \(C\rightarrow\mathbb{P}^1\), and the authors describe \(\mathrm{Cox}(X)\) in terms of a polyhedral divisor on \(C\). To do this, the authors present covering of \(\mathbb{P}^1\) together with an action of a finite abelian group \(A\) in terms of so-called \(A\)-divisors of degree zero on \(\mathbb{P}^1\). When \(\mathrm{Cl}(X)\) is free, then \(C=\mathbb{P}^1\) and the description is simpler.
scientific article; zbMATH DE number 6060935
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