Picard-graded Betti numbers and the defining ideals of Cox rings

Details Picard-graded Betti numbers and the defining ideals of Cox rings Picard-graded Betti numbers and the defining ideals of Cox rings

2007-07-22

arxiv.org/abs/0707.3251v1

Mathematics

Algebraic Geometry

21 pages, 3 figures

Scientific paper

Let X be a smooth projective variety with torsion-free Picard group. We introduce complexes of vector spaces whose homology determines the structure of the minimal free resolution of the Cox ring of X over the polynomial ring and show how the homology of these complexes can be studied by purely geometric methods. As an application of these techniques we give a simple new proof of a characterization of the Cox rings of Del Pezzo surfaces (of degree >1) conjectured by Batyrev and Popov.

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