{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,22]],"date-time":"2026-04-22T08:28:40Z","timestamp":1776846520504,"version":"3.51.2"},"reference-count":46,"publisher":"American Mathematical Society (AMS)","issue":"297","license":[{"start":{"date-parts":[[2016,6,10]],"date-time":"2016-06-10T00:00:00Z","timestamp":1465516800000},"content-version":"am","delay-in-days":366,"URL":"https:\/\/www.ams.org\/publications\/copyright-and-permissions"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Math. Comp."],"abstract":"<p>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.<\/p>","DOI":"10.1090\/mcom\/2989","type":"journal-article","created":{"date-parts":[[2015,6,10]],"date-time":"2015-06-10T15:11:27Z","timestamp":1433949087000},"page":"467-502","source":"Crossref","is-referenced-by-count":15,"title":["Computing Cox rings"],"prefix":"10.1090","volume":"85","author":[{"given":"J\u00fcrgen","family":"Hausen","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Simon","family":"Keicher","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Antonio","family":"Laface","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"14","published-online":{"date-parts":[[2015,6,10]]},"reference":[{"key":"1","series-title":"MSJ Memoirs","isbn-type":"print","doi-asserted-by":"publisher","DOI":"10.1142\/e002","volume-title":"Del Pezzo and $K3$ surfaces","volume":"15","author":"Alexeev, Valery","year":"2006","ISBN":"https:\/\/id.crossref.org\/isbn\/4931469345"},{"key":"2","unstructured":"M. Artebani, A. Garbagnati, A. Laface: Cox rings of extremal rational elliptic surfaces. To appear in Transactions of the AMS. Preprint, arXiv:1302.4361."},{"issue":"5","key":"3","doi-asserted-by":"publisher","first-page":"643","DOI":"10.1134\/S0001434609050022","article-title":"On the factoriality of Cox rings","volume":"85","author":"Arzhantsev, I. V.","year":"2009","journal-title":"Mat. Zametki","ISSN":"https:\/\/id.crossref.org\/issn\/0025-567X","issn-type":"print"},{"key":"4","series-title":"Cambridge Studies in Advanced Mathematics","isbn-type":"print","volume-title":"Cox rings","volume":"144","author":"Arzhantsev, Ivan","year":"2015","ISBN":"https:\/\/id.crossref.org\/isbn\/9781107024625"},{"key":"5","unstructured":"H. B\u00e4ker, J. Hausen, S. Keicher: Chow quotients of torus actions. Preprint, arXiv:1203.3759."},{"key":"6","isbn-type":"print","doi-asserted-by":"publisher","DOI":"10.1017\/CBO9780511666919","volume-title":"The theory of finite linear spaces","author":"Batten, Lynn Margaret","year":"1993","ISBN":"https:\/\/id.crossref.org\/isbn\/0521333172"},{"key":"7","isbn-type":"print","doi-asserted-by":"publisher","first-page":"85","DOI":"10.1007\/978-0-8176-8170-8_5","article-title":"The Cox ring of a del Pezzo surface","author":"Batyrev, Victor V.","year":"2004","ISBN":"https:\/\/id.crossref.org\/isbn\/081763259X"},{"key":"8","doi-asserted-by":"publisher","first-page":"351","DOI":"10.1016\/j.jalgebra.2012.07.030","article-title":"Factorially graded rings and Cox rings","volume":"369","author":"Bechtold, Benjamin","year":"2012","journal-title":"J. Algebra","ISSN":"https:\/\/id.crossref.org\/issn\/0021-8693","issn-type":"print"},{"issue":"2","key":"9","doi-asserted-by":"publisher","first-page":"636","DOI":"10.1016\/S0021-8693(03)00285-0","article-title":"Homogeneous coordinates for algebraic varieties","volume":"266","author":"Berchtold, Florian","year":"2003","journal-title":"J. Algebra","ISSN":"https:\/\/id.crossref.org\/issn\/0021-8693","issn-type":"print"},{"key":"10","unstructured":"C. Bertone, Modular absolute decomposition of equidimensional polynomial ideals. Preprint, arXiv:1012.5210."},{"issue":"3-4","key":"11","doi-asserted-by":"publisher","first-page":"235","DOI":"10.1006\/jsco.1996.0125","article-title":"The Magma algebra system. I. The user language","volume":"24","author":"Bosma, Wieb","year":"1997","journal-title":"J. Symbolic Comput.","ISSN":"https:\/\/id.crossref.org\/issn\/0747-7171","issn-type":"print"},{"issue":"1","key":"12","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1016\/j.ansens.2006.12.002","article-title":"On Manin\u2019s conjecture for a certain singular cubic surface","volume":"40","author":"de la Bret\u00e8che, R\u00e9gis","year":"2007","journal-title":"Ann. Sci. \\'{E}cole Norm. Sup. (4)","ISSN":"https:\/\/id.crossref.org\/issn\/0012-9593","issn-type":"print"},{"issue":"6","key":"13","doi-asserted-by":"publisher","first-page":"1479","DOI":"10.1112\/S0010437X06002284","article-title":"Hilbert\u2019s 14th problem and Cox rings","volume":"142","author":"Castravet, Ana-Maria","year":"2006","journal-title":"Compos. Math.","ISSN":"https:\/\/id.crossref.org\/issn\/0010-437X","issn-type":"print"},{"issue":"3","key":"14","doi-asserted-by":"publisher","first-page":"380","DOI":"10.1016\/j.jco.2007.01.008","article-title":"Lifting and recombination techniques for absolute factorization","volume":"23","author":"Ch\u00e8ze, Guillaume","year":"2007","journal-title":"J. Complexity","ISSN":"https:\/\/id.crossref.org\/issn\/0885-064X","issn-type":"print"},{"issue":"18","key":"15","first-page":"Aii, A1113--A1116","article-title":"Torseurs sous des groupes de type multiplicatif; applications \u00e0 l\u2019\u00e9tude des points rationnels de certaines vari\u00e9t\u00e9s alg\u00e9briques","volume":"282","author":"Colliot-Th\u00e9l\u00e8ne, Jean-Louis","year":"1976","journal-title":"C. R. Acad. Sci. Paris S\\'{e}r. A-B","ISSN":"https:\/\/id.crossref.org\/issn\/0151-0509","issn-type":"print"},{"issue":"19","key":"16","first-page":"A1215--A1218","article-title":"La descente sur une vari\u00e9t\u00e9 rationnelle d\u00e9finie sur un corps de nombres","volume":"284","author":"Colliot-Th\u00e9l\u00e8ne, Jean-Louis","year":"1977","journal-title":"C. R. Acad. Sci. Paris S\\'{e}r. A-B","ISSN":"https:\/\/id.crossref.org\/issn\/0151-0509","issn-type":"print"},{"issue":"1","key":"17","first-page":"17","article-title":"The homogeneous coordinate ring of a toric variety","volume":"4","author":"Cox, David A.","year":"1995","journal-title":"J. Algebraic Geom.","ISSN":"https:\/\/id.crossref.org\/issn\/1056-3911","issn-type":"print"},{"key":"18","unstructured":"W. Decker, G.-M. Greuel, G. Pfister, H. Sch\u00f6nemann, Singular 3-1-6 \u2014 A computer algebra system for polynomial computations. 2012. Available at \\url{http:\/\/www.singular.uni-kl.de}"},{"issue":"3","key":"19","doi-asserted-by":"publisher","first-page":"638","DOI":"10.1112\/plms\/pdt041","article-title":"Singular del Pezzo surfaces whose universal torsors are hypersurfaces","volume":"108","author":"Derenthal, Ulrich","year":"2014","journal-title":"Proc. Lond. Math. Soc. (3)","ISSN":"https:\/\/id.crossref.org\/issn\/0024-6115","issn-type":"print"},{"key":"20","unstructured":"U. Derenthal: Geometry of universal torsors. PhD thesis. Universit\u00e4t G\u00f6ttingen, 2006."},{"issue":"7","key":"21","doi-asserted-by":"publisher","first-page":"855","DOI":"10.2140\/ant.2010.4.855","article-title":"Equations for Chow and Hilbert quotients","volume":"4","author":"Gibney, Angela","year":"2010","journal-title":"Algebra Number Theory","ISSN":"https:\/\/id.crossref.org\/issn\/1937-0652","issn-type":"print"},{"key":"22","unstructured":"D. Grayson, M. Stillman: Macaulay2, a software system for research in algebraic geometry. Available at \\url{http:\/\/www.math.uiuc.edu\/Macaulay2\/}"},{"key":"23","isbn-type":"print","volume-title":"A {\\bf Singular} introduction to commutative algebra","author":"Greuel, Gert-Martin","year":"2008","ISBN":"https:\/\/id.crossref.org\/isbn\/9783540735410","edition":"2"},{"issue":"8","key":"24","first-page":"222","article-title":"\u00c9l\u00e9ments de g\u00e9om\u00e9trie alg\u00e9brique. II. \u00c9tude globale \u00e9l\u00e9mentaire de quelques classes de morphismes","author":"Grothendieck, A.","year":"1961","journal-title":"Inst. Hautes \\'{E}tudes Sci. Publ. Math.","ISSN":"https:\/\/id.crossref.org\/issn\/0073-8301","issn-type":"print"},{"key":"25","isbn-type":"print","doi-asserted-by":"publisher","first-page":"149","DOI":"10.1007\/978-0-8176-8170-8_10","article-title":"Universal torsors and Cox rings","author":"Hassett, Brendan","year":"2004","ISBN":"https:\/\/id.crossref.org\/isbn\/081763259X"},{"issue":"4","key":"26","doi-asserted-by":"publisher","first-page":"711","DOI":"10.17323\/1609-4514-2008-8-4-711-757","article-title":"Cox rings and combinatorics. II","volume":"8","author":"Hausen, J\u00fcrgen","year":"2008","journal-title":"Mosc. Math. J.","ISSN":"https:\/\/id.crossref.org\/issn\/1609-3321","issn-type":"print"},{"key":"27","isbn-type":"print","first-page":"3","article-title":"Three lectures on Cox rings","author":"Hausen, J\u00fcrgen","year":"2013","ISBN":"https:\/\/id.crossref.org\/isbn\/9781107616127"},{"issue":"2","key":"28","doi-asserted-by":"publisher","first-page":"977","DOI":"10.1016\/j.aim.2010.03.010","article-title":"The Cox ring of an algebraic variety with torus action","volume":"225","author":"Hausen, J\u00fcrgen","year":"2010","journal-title":"Adv. Math.","ISSN":"https:\/\/id.crossref.org\/issn\/0001-8708","issn-type":"print"},{"key":"29","unstructured":"E. Huggenberger: Fano Varieties with Torus Action of Complexity One. PhD thesis. Universit\u00e4t T\u00fcbingen, 2013, \\url{http:\/\/nbn-resolving.de\/urn:nbn:de:bsz:21-opus-69570}."},{"key":"30","doi-asserted-by":"publisher","first-page":"331","DOI":"10.1307\/mmj\/1030132722","article-title":"Mori dream spaces and GIT","volume":"48","author":"Hu, Yi","year":"2000","journal-title":"Michigan Math. J.","ISSN":"https:\/\/id.crossref.org\/issn\/0026-2285","issn-type":"print"},{"key":"31","doi-asserted-by":"crossref","unstructured":"A. Jensen, Algorithmic aspects of Gr\u00f6bner fans and tropical varieties, PhD thesis, University of Aarhus, 2007.","DOI":"10.1007\/978-0-387-78133-4_3"},{"key":"32","series-title":"Aspects of Mathematics, D1","isbn-type":"print","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-322-83813-1","volume-title":"Geometrische Methoden in der Invariantentheorie","author":"Kraft, Hanspeter","year":"1984","ISBN":"https:\/\/id.crossref.org\/isbn\/3528085258"},{"key":"33","unstructured":"S. Keicher, Algorithms for Mori Dream Spaces, PhD thesis, Universit\u00e4t T\u00fcbingen, 2014, \\url{http:\/\/nbn-resolving.de\/urn:nbn:de:bsz:21-dspace-540614}."},{"key":"34","unstructured":"R. Koelman, The number of moduli of families of curves on toric surfaces, Thesis, Univ. Nijmegen, 1991."},{"key":"35","series-title":"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]","isbn-type":"print","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-642-18808-4","volume-title":"Positivity in algebraic geometry. I","volume":"48","author":"Lazarsfeld, Robert","year":"2004","ISBN":"https:\/\/id.crossref.org\/isbn\/3540225331"},{"key":"36","unstructured":"Maple 10. Maplesoft, a division of Waterloo Maple Inc., Waterloo, Ontario. See \\url{http:\/\/www.maplesoft.com\/}"},{"key":"37","series-title":"Universitext","isbn-type":"print","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4757-5602-9","volume-title":"Introduction to the Mori program","author":"Matsuki, Kenji","year":"2002","ISBN":"https:\/\/id.crossref.org\/isbn\/0387984658"},{"issue":"1","key":"38","doi-asserted-by":"publisher","first-page":"127","DOI":"10.1007\/s11537-010-0944-7","article-title":"Mori dream spaces","volume":"5","author":"McKernan, James","year":"2010","journal-title":"Jpn. J. Math.","ISSN":"https:\/\/id.crossref.org\/issn\/0289-2316","issn-type":"print"},{"key":"39","series-title":"Graduate Texts in Mathematics","isbn-type":"print","volume-title":"Combinatorial commutative algebra","volume":"227","author":"Miller, Ezra","year":"2005","ISBN":"https:\/\/id.crossref.org\/isbn\/0387223568"},{"key":"40","unstructured":"S. Okawa, On images of Mori dream spaces. Preprint, arXiv:1104.1326."},{"issue":"1-2","key":"41","doi-asserted-by":"publisher","first-page":"43","DOI":"10.1007\/BF02392313","article-title":"Algebraic surfaces with \ud835\udc58*-action","volume":"138","author":"Orlik, P.","year":"1977","journal-title":"Acta Math.","ISSN":"https:\/\/id.crossref.org\/issn\/0001-5962","issn-type":"print"},{"issue":"6","key":"42","doi-asserted-by":"publisher","first-page":"2022","DOI":"10.1137\/S0036142900372549","article-title":"Numerical decomposition of the solution sets of polynomial systems into irreducible components","volume":"38","author":"Sommese, Andrew J.","year":"2001","journal-title":"SIAM J. Numer. Anal.","ISSN":"https:\/\/id.crossref.org\/issn\/0036-1429","issn-type":"print"},{"issue":"2","key":"43","doi-asserted-by":"publisher","first-page":"777","DOI":"10.1016\/j.jalgebra.2007.05.016","article-title":"Gr\u00f6bner bases, monomial group actions, and the Cox rings of del Pezzo surfaces","volume":"316","author":"Stillman, Mike","year":"2007","journal-title":"J. Algebra","ISSN":"https:\/\/id.crossref.org\/issn\/0021-8693","issn-type":"print"},{"key":"44","series-title":"University Lecture Series","isbn-type":"print","doi-asserted-by":"publisher","DOI":"10.1090\/ulect\/008","volume-title":"Gr\\\"{o}bner bases and convex polytopes","volume":"8","author":"Sturmfels, Bernd","year":"1996","ISBN":"https:\/\/id.crossref.org\/isbn\/0821804871"},{"issue":"2","key":"45","doi-asserted-by":"publisher","first-page":"223","DOI":"10.1216\/JCA-2010-2-2-223","article-title":"Blow-ups of \u2119\u207f\u207b\u00b3 at \ud835\udd5f points and spinor varieties","volume":"2","author":"Sturmfels, Bernd","year":"2010","journal-title":"J. Commut. Algebra","ISSN":"https:\/\/id.crossref.org\/issn\/1939-0807","issn-type":"print"},{"issue":"2","key":"46","doi-asserted-by":"publisher","first-page":"429","DOI":"10.4171\/JEMS\/204","article-title":"Sagbi bases of Cox-Nagata rings","volume":"12","author":"Sturmfels, Bernd","year":"2010","journal-title":"J. Eur. Math. Soc. (JEMS)","ISSN":"https:\/\/id.crossref.org\/issn\/1435-9855","issn-type":"print"}],"container-title":["Mathematics of Computation"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/www.ams.org\/mcom\/2016-85-297\/S0025-5718-2015-02989-5\/S0025-5718-2015-02989-5.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"},{"URL":"https:\/\/www.ams.org\/mcom\/2016-85-297\/S0025-5718-2015-02989-5\/S0025-5718-2015-02989-5.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,4,21]],"date-time":"2026-04-21T18:38:35Z","timestamp":1776796715000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.ams.org\/mcom\/2016-85-297\/S0025-5718-2015-02989-5\/"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2015,6,10]]},"references-count":46,"journal-issue":{"issue":"297","published-print":{"date-parts":[[2016,1]]}},"alternative-id":["S0025-5718-2015-02989-5"],"URL":"https:\/\/doi.org\/10.1090\/mcom\/2989","archive":["CLOCKSS","Portico"],"relation":{},"ISSN":["1088-6842","0025-5718"],"issn-type":[{"value":"1088-6842","type":"electronic"},{"value":"0025-5718","type":"print"}],"subject":[],"published":{"date-parts":[[2015,6,10]]}}}