Cox rings and combinatorics
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- by Florian Berchtold and Jürgen Hausen PDF
- Trans. Amer. Math. Soc. 359 (2007), 1205-1252 Request permission
Abstract:
Given a variety $X$ with a finitely generated total coordinate ring, we describe basic geometric properties of $X$ in terms of certain combinatorial structures living in the divisor class group of $X$. For example, we describe the singularities, we calculate the ample cone, and we give simple Fano criteria. As we show by means of several examples, the results allow explicit computations. As immediate applications we obtain an effective version of the Kleiman-Chevalley quasiprojectivity criterion, and the following observation on surfaces: a normal complete surface with finitely generated total coordinate ring is projective if and only if any two of its non-factorial singularities admit a common affine neighbourhood.References
- 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, DOI 10.1007/978-0-8176-8170-8_{5}
- Florian Berchtold, Lifting of morphisms to quotient presentations, Manuscripta Math. 110 (2003), no. 1, 33–44. MR 1951798, DOI 10.1007/s00229-002-0297-5
- Florian Berchtold and Jürgen Hausen, Homogeneous coordinates for algebraic varieties, J. Algebra 266 (2003), no. 2, 636–670. MR 1995130, DOI 10.1016/S0021-8693(03)00285-0
- Florian Berchtold and Jürgen Hausen, Bunches of cones in the divisor class group—a new combinatorial language for toric varieties, Int. Math. Res. Not. 6 (2004), 261–302. MR 2041065, DOI 10.1155/S1073792804130973
- Andrzej Białynicki-Birula and Joanna Święcicka, Complete quotients by algebraic torus actions, Group actions and vector fields (Vancouver, B.C., 1981) Lecture Notes in Math., vol. 956, Springer, Berlin, 1982, pp. 10–22. MR 704983, DOI 10.1007/BFb0101505
- A. Białynicki-Birula and J. Święcicka, Three theorems on existence of good quotients, Math. Ann. 307 (1997), no. 1, 143–149. MR 1427680, DOI 10.1007/s002080050027
- Jean-François Boutot, Singularités rationnelles et quotients par les groupes réductifs, Invent. Math. 88 (1987), no. 1, 65–68 (French). MR 877006, DOI 10.1007/BF01405091
- David A. Cox, The homogeneous coordinate ring of a toric variety, J. Algebraic Geom. 4 (1995), no. 1, 17–50. MR 1299003
- E. Javier Elizondo, Kazuhiko Kurano, and Kei-ichi Watanabe, The total coordinate ring of a normal projective variety, J. Algebra 276 (2004), no. 2, 625–637. MR 2058459, DOI 10.1016/j.jalgebra.2003.07.007
- Günter Ewald, Polygons with hidden vertices, Beiträge Algebra Geom. 42 (2001), no. 2, 439–442. MR 1865531
- William Fulton, Introduction to toric varieties, Annals of Mathematics Studies, vol. 131, Princeton University Press, Princeton, NJ, 1993. The William H. Roever Lectures in Geometry. MR 1234037, DOI 10.1515/9781400882526
- Jacob Eli Goodman, Affine open subsets of algebraic varieties and ample divisors, Ann. of Math. (2) 89 (1969), 160–183. MR 242843, DOI 10.2307/1970814
- Robin Hartshorne, Algebraic geometry, Graduate Texts in Mathematics, No. 52, Springer-Verlag, New York-Heidelberg, 1977. MR 0463157, DOI 10.1007/978-1-4757-3849-0
- Jürgen Hausen, Producing good quotients by embedding into toric varieties, Geometry of toric varieties, Sémin. Congr., vol. 6, Soc. Math. France, Paris, 2002, pp. 193–212. MR 2075611
- Melvin Hochster and Joel L. Roberts, Rings of invariants of reductive groups acting on regular rings are Cohen-Macaulay, Advances in Math. 13 (1974), 115–175. MR 347810, DOI 10.1016/0001-8708(74)90067-X
- 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, DOI 10.1007/978-0-8176-8170-8_{1}0
- 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, DOI 10.1307/mmj/1030132722
- T. Kambayashi and P. Russell, On linearizing algebraic torus actions, J. Pure Appl. Algebra 23 (1982), no. 3, 243–250. MR 644276, DOI 10.1016/0022-4049(82)90100-1
- Steven L. Kleiman, Toward a numerical theory of ampleness, Ann. of Math. (2) 84 (1966), 293–344. MR 206009, DOI 10.2307/1970447
- Peter Kleinschmidt, A classification of toric varieties with few generators, Aequationes Math. 35 (1988), no. 2-3, 254–266. MR 954243, DOI 10.1007/BF01830946
- Friedrich Knop, Über Hilberts vierzehntes Problem für Varietäten mit Kompliziertheit eins, Math. Z. 213 (1993), no. 1, 33–36 (German). MR 1217668, DOI 10.1007/BF03025706
- Friedrich Knop, Hanspeter Kraft, and Thierry Vust, The Picard group of a $G$-variety, Algebraische Transformationsgruppen und Invariantentheorie, DMV Sem., vol. 13, Birkhäuser, Basel, 1989, pp. 77–87. MR 1044586
- János Kollár, Rational curves on algebraic varieties, 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. 32, Springer-Verlag, Berlin, 1996. MR 1440180, DOI 10.1007/978-3-662-03276-3
- Kurt Leichtweiss, Konvexe Mengen, Hochschultext [University Textbooks], Springer-Verlag, Berlin-New York, 1980 (German). MR 586235, DOI 10.1007/978-3-642-95335-4
- Kenji Matsuki, Introduction to the Mori program, Universitext, Springer-Verlag, New York, 2002. MR 1875410, DOI 10.1007/978-1-4757-5602-9
- D. Mumford, J. Fogarty, and F. Kirwan, Geometric invariant theory, 3rd ed., Ergebnisse der Mathematik und ihrer Grenzgebiete (2) [Results in Mathematics and Related Areas (2)], vol. 34, Springer-Verlag, Berlin, 1994. MR 1304906, DOI 10.1007/978-3-642-57916-5
- Tadao Oda and Hye Sook Park, Linear Gale transforms and Gel′fand-Kapranov-Zelevinskij decompositions, Tohoku Math. J. (2) 43 (1991), no. 3, 375–399. MR 1117211, DOI 10.2748/tmj/1178227461
- Joanna Święcicka, A combinatorial construction of sets with good quotients by an action of a reductive group, Colloq. Math. 87 (2001), no. 1, 85–102. MR 1812145, DOI 10.4064/cm87-1-5
- Balázs Szendrői, On a conjecture of Cox and Katz, Math. Z. 240 (2002), no. 2, 233–241. MR 1900310, DOI 10.1007/s002090100377
- Hassler Whitney, Complex analytic varieties, Addison-Wesley Publishing Co., Reading, Mass.-London-Don Mills, Ont., 1972. MR 0387634
- Jarosław Włodarczyk, Embeddings in toric varieties and prevarieties, J. Algebraic Geom. 2 (1993), no. 4, 705–726. MR 1227474
Additional Information
- Florian Berchtold
- Affiliation: Mathematisches Institut, Universität Heidelberg, 69221 Heidelberg, Germany
- Address at time of publication: Fachbereich Mathematik und Statistik, Universität Konstanz, D-78457 Konstanz, Germany
- Email: Florian.Berchtold@uni-konstanz.de
- Jürgen Hausen
- Affiliation: Mathematisches Forschungsinstitut Oberwolfach, Lorenzenhof, 77709 Oberwolfach–Walke, Germany
- Address at time of publication: Mathematisches Institut, Universität Tübingen, Auf der Morgenstelle 10, 72076 Tübingen, Germany
- MR Author ID: 361664
- Email: hausen@mail.mathematik.uni-tuebingen.de
- Received by editor(s): August 23, 2004
- Received by editor(s) in revised form: December 10, 2004
- Published electronically: October 16, 2006
- © Copyright 2006 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 359 (2007), 1205-1252
- MSC (2000): Primary 14C20, 14J45, 14J70, 14M20, 14M25, 14Q15
- DOI: https://doi.org/10.1090/S0002-9947-06-03904-3
- MathSciNet review: 2262848