Cox rings and combinatorics

Berchtold, Florian; Hausen, Jürgen

doi:10.1090/S0002-9947-06-03904-3

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.

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