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Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.48.

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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


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.
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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:
  • 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:
  • 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:
  • MathSciNet review: 2262848