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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Cox rings and combinatorics


Authors: Florian Berchtold and Jürgen Hausen
Journal: Trans. Amer. Math. Soc. 359 (2007), 1205-1252
MSC (2000): Primary 14C20, 14J45, 14J70, 14M20, 14M25, 14Q15
Published electronically: October 16, 2006
MathSciNet review: 2262848
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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.


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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: 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
Email: hausen@mail.mathematik.uni-tuebingen.de

DOI: http://dx.doi.org/10.1090/S0002-9947-06-03904-3
PII: S 0002-9947(06)03904-3
Received by editor(s): August 23, 2004
Received by editor(s) in revised form: December 10, 2004
Published electronically: October 16, 2006
Article copyright: © Copyright 2006 American Mathematical Society