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Mathematics of Computation

ISSN 1088-6842(online) ISSN 0025-5718(print)



Computing Cox rings

Authors: Jürgen Hausen, Simon Keicher and Antonio Laface
Journal: Math. Comp. 85 (2016), 467-502
MSC (2010): Primary 14L24, 14L30, 14C20, 14Q10, 14Q15, 13A30, 52B55
Published electronically: June 10, 2015
MathSciNet review: 3404458
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Abstract: We consider modifications, for example blow ups, of Mori dream spaces and provide algorithms for investigating the effect on the Cox ring, for example verifying finite generation or computing an explicit presentation in terms of generators and relations. As a first application, we compute the Cox rings of all Gorenstein log del Pezzo surfaces of Picard number one. Moreover, we show computationally that all smooth rational surfaces of Picard numbers at most six are Mori dream surfaces and we provide explicit presentations of the Cox ring for those not admitting a torus action. Finally, we provide the Cox rings of projective spaces blown up at certain special point configurations.

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  • [1] Valery Alexeev and Viacheslav V. Nikulin, Del Pezzo and $ K3$ Surfaces, MSJ Memoirs, vol. 15, Mathematical Society of Japan, Tokyo, 2006. MR 2227002 (2007e:14059)
  • [2] M. Artebani, A. Garbagnati, A. Laface: Cox rings of extremal rational elliptic surfaces. To appear in Transactions of the AMS. Preprint, arXiv:1302.4361.
  • [3] I. V. Arzhantsev, On the factoriality of Cox rings, Mat. Zametki 85 (2009), no. 5, 643-651 (Russian, with Russian summary); English transl., Math. Notes 85 (2009), no. 5-6, 623-629. MR 2572855 (2010k:14009),
  • [4] Ivan Arzhantsev, Ulrich Derenthal, Jürgen Hausen, and Antonio Laface, Cox rings, Cambridge Studies in Advanced Mathematics, vol. 144, Cambridge University Press, Cambridge, 2015. MR 3307753
  • [5] H. Bäker, J. Hausen, S. Keicher: Chow quotients of torus actions. Preprint, arXiv:1203.3759.
  • [6] Lynn Margaret Batten and Albrecht Beutelspacher, The Theory of Finite Linear Spaces, Combinatorics of Points and Lines, Cambridge University Press, Cambridge, 1993. MR 1253067 (94m:51019)
  • [7] Victor V. Batyrev and Oleg N. Popov, The Cox ring of a del Pezzo surface, Arithmetic of higher-dimensional algebraic varieties (Palo Alto, CA, 2002), Progr. Math., vol. 226, Birkhäuser Boston, Boston, MA, 2004, pp. 85-103. MR 2029863 (2005h:14091)
  • [8] Benjamin Bechtold, Factorially graded rings and Cox rings, J. Algebra 369 (2012), 351-359. MR 2959797,
  • [9] Florian Berchtold and Jürgen Hausen, Homogeneous coordinates for algebraic varieties, J. Algebra 266 (2003), no. 2, 636-670. MR 1995130 (2004e:14004),
  • [10] C. Bertone, Modular absolute decomposition of equidimensional polynomial ideals. Preprint, arXiv:1012.5210.
  • [11] Wieb Bosma, John Cannon, and Catherine Playoust, The Magma algebra system. I. The user language, J. Symbolic Comput. 24 (1997), no. 3-4, 235-265. Computational algebra and number theory (London, 1993). MR 1484478,
  • [12] Régis de la Bretèche, Tim D. Browning, and Ulrich Derenthal, On Manin's conjecture for a certain singular cubic surface, Ann. Sci. École Norm. Sup. (4) 40 (2007), no. 1, 1-50 (English, with English and French summaries). MR 2332351 (2008e:11038),
  • [13] Ana-Maria Castravet and Jenia Tevelev, Hilbert's 14th problem and Cox rings, Compos. Math. 142 (2006), no. 6, 1479-1498. MR 2278756 (2007i:14044),
  • [14] Guillaume Chèze and Grégoire Lecerf, Lifting and recombination techniques for absolute factorization, J. Complexity 23 (2007), no. 3, 380-420. MR 2330992 (2008f:68174),
  • [15] Jean-Louis Colliot-Thélène and Jean-Jacques Sansuc, Torseurs sous des groupes de type multiplicatif; applications à l'étude des points rationnels de certaines variétés algébriques, C. R. Acad. Sci. Paris Sér. A-B 282 (1976), no. 18, Aii, A1113-A1116 (French, with English summary). MR 0414556 (54 #2657)
  • [16] Jean-Louis Colliot-Thélène and Jean-Jacques Sansuc, La descente sur une variété rationnelle définie sur un corps de nombres, C. R. Acad. Sci. Paris Sér. A-B 284 (1977), no. 19, A1215-A1218 (French, with English summary). MR 0447250 (56 #5565)
  • [17] David A. Cox, The homogeneous coordinate ring of a toric variety, J. Algebraic Geom. 4 (1995), no. 1, 17-50. MR 1299003 (95i:14046)
  • [18] W. Decker, G.-M. Greuel, G. Pfister, H. Schönemann, Singular 3-1-6 -- A computer algebra system for polynomial computations. 2012. Available at
  • [19] Ulrich Derenthal, Singular del Pezzo surfaces whose universal torsors are hypersurfaces, Proc. Lond. Math. Soc. (3) 108 (2014), no. 3, 638-681. MR 3180592,
  • [20] U. Derenthal: Geometry of universal torsors. PhD thesis. Universität Göttingen, 2006.
  • [21] Angela Gibney and Diane Maclagan, Equations for Chow and Hilbert quotients, Algebra Number Theory 4 (2010), no. 7, 855-885. MR 2776876 (2012c:14093),
  • [22] D. Grayson, M. Stillman: Macaulay2, a software system for research in algebraic geometry. Available at
  • [23] Gert-Martin Greuel and Gerhard Pfister, A Singular Introduction to Commutative Algebra, Second, extended edition, Springer, Berlin, 2008. With contributions by Olaf Bachmann, Christoph Lossen and Hans Schönemann; With 1 CD-ROM (Windows, Macintosh and UNIX). MR 2363237 (2008j:13001)
  • [24] A. Grothendieck, Éléments de géométrie algébrique. II. Étude globale élémentaire de quelques classes de morphismes, Inst. Hautes Études Sci. Publ. Math. 8 (1961), 222. MR 0217084 (36 #177b)
  • [25] Brendan Hassett and Yuri Tschinkel, Universal torsors and Cox rings, Arithmetic of higher-dimensional algebraic varieties (Palo Alto, CA, 2002), Progr. Math., vol. 226, Birkhäuser Boston, Boston, MA, 2004, pp. 149-173. MR 2029868 (2005a:14049)
  • [26] Jürgen Hausen, Cox rings and combinatorics. II, Mosc. Math. J. 8 (2008), no. 4, 711-757, 847 (English, with English and Russian summaries). MR 2499353 (2010b:14011)
  • [27] Jürgen Hausen, Three lectures on Cox rings, Torsors, étale homotopy and applications to rational points, London Math. Soc. Lecture Note Ser., vol. 405, Cambridge Univ. Press, Cambridge, 2013, pp. 3-60. MR 3077165
  • [28] Jürgen Hausen and Hendrik Süß, The Cox ring of an algebraic variety with torus action, Adv. Math. 225 (2010), no. 2, 977-1012. MR 2671185 (2011h:14007),
  • [29] E. Huggenberger: Fano Varieties with Torus Action of Complexity One. PhD thesis. Universität Tübingen, 2013,
  • [30] Yi Hu and Sean Keel, Mori dream spaces and GIT, Michigan Math. J. 48 (2000), 331-348. Dedicated to William Fulton on the occasion of his 60th birthday. MR 1786494 (2001i:14059),
  • [31] A. Jensen, Algorithmic aspects of Gröbner fans and tropical varieties, PhD thesis, University of Aarhus, 2007.
  • [32] Hanspeter Kraft, Geometrische Methoden in der Invariantentheorie, Aspects of Mathematics, D1, Friedr. Vieweg & Sohn, Braunschweig, 1984 (German). MR 768181 (86j:14006)
  • [33] S. Keicher, Algorithms for Mori Dream Spaces, PhD thesis, Universität Tübingen, 2014,
  • [34] R. Koelman, The number of moduli of families of curves on toric surfaces, Thesis, Univ. Nijmegen, 1991.
  • [35] Robert Lazarsfeld, Positivity in Algebraic Geometry. I, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics], vol. 48, Springer-Verlag, Berlin, 2004. Classical setting: line bundles and linear series. MR 2095471 (2005k:14001a)
  • [36] Maple 10. Maplesoft, a division of Waterloo Maple Inc., Waterloo, Ontario. See
  • [37] Kenji Matsuki, Introduction to the Mori Program, Universitext, Springer-Verlag, New York, 2002. MR 1875410 (2002m:14011)
  • [38] James McKernan, Mori dream spaces, Jpn. J. Math. 5 (2010), no. 1, 127-151. MR 2609325 (2011c:14045),
  • [39] Ezra Miller and Bernd Sturmfels, Combinatorial Commutative Algebra, Graduate Texts in Mathematics, vol. 227, Springer-Verlag, New York, 2005. MR 2110098 (2006d:13001)
  • [40] S. Okawa, On images of Mori dream spaces. Preprint, arXiv:1104.1326.
  • [41] P. Orlik and P. Wagreich, Algebraic surfaces with $ k^*$-action, Acta Math. 138 (1977), no. 1-2, 43-81. MR 0460342 (57 #336)
  • [42] Andrew J. Sommese, Jan Verschelde, and Charles W. Wampler, Numerical decomposition of the solution sets of polynomial systems into irreducible components, SIAM J. Numer. Anal. 38 (2001), no. 6, 2022-2046. MR 1856241 (2002g:65064),
  • [43] Mike Stillman, Damiano Testa, and Mauricio Velasco, Gröbner bases, monomial group actions, and the Cox rings of del Pezzo surfaces, J. Algebra 316 (2007), no. 2, 777-801. MR 2358614 (2008i:14054),
  • [44] Bernd Sturmfels, Gröbner Bases and Convex Polytopes, University Lecture Series, vol. 8, American Mathematical Society, Providence, RI, 1996. MR 1363949 (97b:13034)
  • [45] Bernd Sturmfels and Mauricio Velasco, Blow-ups of $ \mathbb{P}^{n-3}$ at $ n$ points and spinor varieties, J. Commut. Algebra 2 (2010), no. 2, 223-244. MR 2647477 (2011g:20072),
  • [46] Bernd Sturmfels and Zhiqiang Xu, Sagbi bases of Cox-Nagata rings, J. Eur. Math. Soc. (JEMS) 12 (2010), no. 2, 429-459. MR 2608947 (2011f:13009),

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Additional Information

Jürgen Hausen
Affiliation: Mathematisches Institut, Universität Tübingen, Auf der Morgenstelle 10, 72076 Tübingen, Germany

Simon Keicher
Affiliation: Mathematisches Institut, Universität Tübingen, Auf der Morgenstelle 10, 72076 Tübingen, Germany

Antonio Laface
Affiliation: Departamento de Matemática, Universidad de Concepción, Casilla 160-C, Concepción, Chile

Received by editor(s): October 20, 3013
Received by editor(s) in revised form: April 24, 2014, July 21, 2014, and August 10, 2014
Published electronically: June 10, 2015
Additional Notes: The second author was partially supported by the DFG Priority Program SPP 1489
The third author was partially supported by Proyecto FONDECYT Regular N. 1110096
Article copyright: © Copyright 2015 American Mathematical Society

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