Toric vector bundles, branched covers of fans, and the resolution property

Details Toric vector bundles, branched covers of fans, and the resolution property Toric vector bundles, branched covers of fans, and the resolution property

2006-05-18

arxiv.org/abs/math/0605537v3

J. Algebraic Geom. 18 (2009), 1--36.

Mathematics

Algebraic Geometry

29 pages, 4 figures. v3: corrected inconsistencies and improved explanation for sign conventions. To appear in J. Alg. Geom

Scientific paper

We associate to each toric vector bundle on a toric variety X(Delta) a "branched cover" of the fan Delta together with a piecewise-linear function on the branched cover. This construction generalizes the usual correspondence between toric Cartier divisors and piecewise-linear functions. We apply this combinatorial geometric technique to investigate the existence of resolutions of coherent sheaves by vector bundles, using singular nonquasiprojective toric threefolds as a testing ground. Our main new result is the construction of complete toric threefolds that have no nontrivial toric vector bundles of rank less than or equal to three. The combinatorial geometric sections of the paper, which develop a theory of cone complexes and their branched covers, can be read independently.

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