Cox rings of minimal resolutions of surface quotient singularities (Q2806138)

This article studies Cox rings of certain varieties. If \(X\) is an algebraic variety satisfying suitable conditions, there is an associated graded ring \({\text{Cox}}(X)\), essentially \(\bigoplus \Gamma(X, \mathcal O _X(D))\), for \(D\) a representative of an element of the class group \({\text{Cl}}(X)\). Thus, \({\text{Cox}}(X)\) is a multi-graded ring, in general not indexed by \(\mathbb Z\). More precisely, in the present article it is assumed that \(X\) is the minimal resolution of a surface quotient singularity \(Y={\mathbb C}^2/G\), where \(G\) is a subgroup of \(\mathrm{GL}(2,{\mathbb C})\). The main results are as follows, where \(f:X \to Y\) is the mentioned minimal resolution. {\parindent=6mm \begin{itemize}\item[(a)] Let \(n\) be the number of irreducible components of the exceptional divisor of the minimal resolution \(f\). Then there is a hypersurface \(S\) in \({\mathbb C}^{n+3}\) (whose defining equation is described) such that \(S \simeq {\text{Spec}}({\text{Cox}}(X))\). In the construction and proofs the author exploits an action of the ``Picard torus'' \(T={\text{Hom}} ({\text{Pic}}(X), {\mathbb C}^{\star}) \simeq {(\mathbb C ^{\star})}^n\) on \(\mathbb C ^{n+3}\) and uses, among other techniques, a good amount of toric geometry. \item[(b)] A description of the ring \({\text{Cox}} (X)\) is given as a subalgebra of the coordinate ring of the the product of the Picard torus \(T\) a certain singular surface (again with a quotient singularity). The generators of the subalgebra \({\text{Cox}} (X)\) are described. \item[(c)] An explicit description of the minimal resolution as a divisor in a toric variety. NEWLINENEWLINE\end{itemize}} These results are completely proved, although the author indicates that (a) could be shown, alternatively, using the methods of [\textit{J. Hausen} and \textit{H. Süß}, Adv. Math. 225, No. 2, 977--1012 (2010; Zbl 1248.14008)]. But she hopes that the techniques developed in the present paper could yield generalizations of her results to higher dimension, as well as a description of \(X\) as a quotient of an open set of \({\text{Spec}} ({\text{Cox}} (X)\), in case \({\text{Cox}}(X)\) is finitely generated.NEWLINENEWLINEThe article reviews much of the necessary background material and includes some interesting examples.