Cox rings and combinatorics

By F. Berchtold, Jürgen Hausen. 2003

88 citations recorded by the source corpus.

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.

Subfields: Algebra and Number Theory, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/fa8fd64daa6f4a39d292182301eefe23ff338467

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In Algebra and Number Theory

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Citations in the corpus

Cox Rings and Combinatorics II (2008, methodology): “In [8], D. Cox associated to any non-degenerate, e.g. complete, toric variety Z a multigraded homogeneous coordinate ring R(Z).”
Singular del Pezzo surfaces whose universal torsors are hypersurfaces (2006, unknown): “[HK00], [Muk01], [BH03a], [BH03b], [EKW04], [CT05].”
A geometric characterization of toric varieties (2016, methodology): “, is the union of the cones f∗ i Nef(Xi). The Cox ring of a variety with ﬁnitely generated class group was originally deﬁned in [10]; it is unique but ignores torsion in the class group. Subsequently [9] gave a reﬁned deﬁnition which takes into account torsion in the class group. As we would like to”
The Cox ring of an algebraic variety with torus action (2009, methodology): “the Cox ring. THE COX RING OF A T-VARIETY 3 In Section 5, we give some applications. The description of the Cox ring given in Theorem 1.3 allows us to apply the language of bunched rings presented in [6] and [11] in order to investigate complete normal rational varieties Xwith a complexity one torus”
On images of Mori dream spaces (2011, unknown): “cally well known as the Gelfand-Kapranov-Zelevinsky decomposition introduced by Oda and Park [OP91]. After the author ﬁnished the ﬁrst version of the draft, he found that Ju¨rgen Hausen introduced in [Hau08] the notion of GIT fan, starting from the VGIT of Cox rings. This 3 seems to coincide with ou”