Hostname: page-component-6b989bf9dc-476zt Total loading time: 0 Render date: 2024-04-15T00:47:24.151Z Has data issue: false hasContentIssue false


Published online by Cambridge University Press:  21 July 2015

Instytut Matematyki UW, Banacha 2, PL-02097 Warsaw, Poland e-mail:
Rights & Permissions [Opens in a new window]


Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We investigate Cox rings of minimal resolutions of surface quotient singularities and provide two descriptions of these rings. The first one is the equation for the spectrum of a Cox ring, which is a hypersurface in an affine space. The second is the set of generators of the Cox ring viewed as a subring of the coordinate ring of a product of a torus and another surface quotient singularity. In addition, we obtain an explicit description of the minimal resolution as a divisor in a toric variety.

Research Article
Copyright © Glasgow Mathematical Journal Trust 2015 



1.Andreatta, M. and Wiśniewski, J. A., On the Kummer construction, Rev. Mat. Complut. 23 (1) (2010), 191251.Google Scholar
2.Arzhantsev, I., Derenthal, U., Hausen, J. and Laface, A., Cox rings (Cambridge University Press, New York, 2014).Google Scholar
3.Arzhantsev, I. V. and Gaifullin, S. A., Cox rings, semigroups and automorphisms of affine algebraic varieties, Sb. Math. 201 (1) (2010), 324.CrossRefGoogle Scholar
4.Benson, D. J., Polynomial invariants of finite groups, LMS Lecture Notes Series, vol. 190 (Cambridge University Press, Cambridge, 1993).Google Scholar
5.Berchtold, F. and Hausen, J., Cox rings and combinatorics, Trans. Amer. Math. Soc. 359 (3) (2007), 12051252.Google Scholar
6.Białynicki-Birula, A. and Święcicka, J., Open subset of projective spaces with a good quotient by an action of a reductive group, Transform. Groups 1 (3) (1996), 153185.CrossRefGoogle Scholar
7.Brieskorn, E., Rationale Singularitäten komplexer Flächen, Invent. Math. 4 (1968), 336358.Google Scholar
8.Cox, D., Little, J. and Schenck, H., Toric varieties, Grad. Stud. Math., vol. 124 (American Mathematical Society, 2011).CrossRefGoogle Scholar
9.Decker, W., Greuel, G.-M., Pfister, G. and Schönemann, H., Singular 3-1-5 – A computer algebra system for polynomial computations, (2012). Available at: Scholar
10.Deturck, D. and Ziller, W., Spherical minimal immersions of spherical space forms. Proc. Sympos. Pure Math. 54, Part I (1993), 111120.Google Scholar
11.Donten, M., On Kummer 3-folds, Rev. Mat. Complut. 24 (2) (2011), 465492.Google Scholar
12.Donten-Bury, M. and Wiśniewski, J. A., On 81 symplectic resolutions of a 4-dimensional quotient by a group of order 32. arXiv:1409.4204 [math.AG], 2014.Google Scholar
13.Facchini, L., González-Alonso, V. and Lasoń, M., Cox rings of Du Val singularities, Matematiche (Catania) 66 (2) (2011), 115136.Google Scholar
14.The GAP Group. GAP – Groups, Algorithms, and Programming, Version 4.5.6, 2012. Available at: Scholar
15.Hausen, J., Herppich, E. and Süß, H., Multigraded factorial rings and Fano varieties with torus action, Documenta Math. 16 (2011), 71109.Google Scholar
16.Hausen, J. and Süß, H., The Cox ring of an algebraic variety with torus action, Adv. Math. 225 (2) (2010), 9771012.CrossRefGoogle Scholar
17.Kollár, J., Shafarevich maps and plurigenera of algebraic varieties, Invent. Math. 113 (1993), 177215.CrossRefGoogle Scholar
18.Laface, A. and Velasco, M., A survey on Cox rings, Geom. Dedicata 139 (1) (2009), 269287.Google Scholar
19.Luna, D., Slices étales, Bull. Soc. Math. de France 33 (1973), 81105.Google Scholar
20.Reid, M., The Du Val singularities An, Dn, E6, E7, E8, Available at: Scholar
21.Reid, M., Surface cyclic quotient singularities and Hirzebruch-Jung resolutions, Available at:, (1997).Google Scholar
22.Riemenschneider, O., Die Invarianten der endlichen Untergruppen von GL(2,C), Math. Z. 153 (1) (1977), 3750.Google Scholar
23.Stanley, R. P., Invariants of finite groups and their applications to combinatorics, Bull. Amer. Math. Soc. 1 (3) (1979), 475511.Google Scholar
24.Sturmfels, B., Algorithms in Invariant Theory (Springer-Verlag, Vienna and New York, 1993).Google Scholar
25.Sumihiro, H., Equivariant completion, J. Math. Kyoto Univ. 14 (1974), 128.Google Scholar
26.Sumihiro, H., Equivariant completion II, J. Math. Kyoto Univ. 15 (1975), 573605.Google Scholar