Cox rings and combinatorics II

Details Cox rings and combinatorics II Cox rings and combinatorics II

2008-01-25

arxiv.org/abs/0801.3995v2

Moscow Math J. Vol. 8, no. 4, 711-757 (2008)

Mathematics

Algebraic Geometry

41 pages, minor changes, to appear in Moscow Math. J

Scientific paper

We study varieties with a finitely generated Cox ring. In a first part, we generalize a combinatorial approach developed in earlier work for varieties with a torsion free divisor class group to the case of torsion. Then we turn to modifications, e.g., blow ups, and the question how the Cox ring changes under such maps. We answer this question for a certain class of modifications induced from modifications of ambient toric varieties. Moreover, we show that every variety with finitely generated Cox ring can be explicitly constructed in a finite series of toric ambient modifications from a combinatorially minimal one.

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