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

By Sam Payne. 2006

55 citations recorded by the source corpus.

We associate to each toric vector bundle on a toric variety X(∆) a “branched cover” of the fan ∆ 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.

Subfields: Geometry and Topology, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/565d0f615fe6cc3c6c1c442914e40cbea96da1c3

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Geometry and Topology

1. Philipp Gross
2. David Rydh
3. David Anderson
4. Sam Payne
5. Eric Katz
6. Jack Hall
7. David Jensen
8. Kiumars Kaveh

In Statistical and Nonlinear Physics

1. David Rydh
2. Philipp Gross
3. David Anderson
4. Sam Payne
5. David Jensen
6. Dhruv Ranganathan
7. Matthew Baker

Citations in the corpus

• Noetherian approximation of algebraic spaces and stacks (2009, background): “ere is not a single example of a non-basic separated scheme or Deligne– Mumford stack [Tot04, Question 1] although S. Payne has given some evidence that a certain proper toric three-fold is not basic [Pay09]. The following example shows that many global quotient stacks are of global type. Example (2”
• Operational K-Theory (2013, background): “In the terminology of [Pa2], this piecewise exponential function is the trace of the exponential of the piecewise linear function on a branched cover of ∆ associated to E .”
• Tensor generators on schemes and stacks (2013, background): “if every algebraic stack with quasifinite and affine diagonal has the resolution property, even in case of normal, separated algebraic schemes of an algebraically closed field (like toric threefolds, see [Pay09]). Étale locally, everyalgebraicstackwith quasifinite and locallyseparateddiagonal has the res”