Cohomology of toric line bundles via simplicial Alexander duality

TL;DR

This paper provides a rigorous proof for a toric sheaf cohomology algorithm, including a faster version, and addresses related conjectures on Serre duality for Betti numbers, advancing computational methods in algebraic geometry.

Contribution

It offers a new, independent, and simplified proof of the toric sheaf cohomology algorithm and proves a conjecture on Serre duality for Betti numbers.

Findings

Validated the original toric sheaf cohomology algorithm

Developed a faster version of the algorithm

Proved the conjecture on Serre duality for Betti numbers

Abstract

We give a rigorous mathematical proof for the validity of the toric sheaf cohomology algorithm conjectured in the recent paper by R. Blumenhagen, B. Jurke, T. Rahn, and H. Roschy (arXiv:1003.5217). We actually prove not only the original algorithm but also a speed-up version of it. Our proof is independent from (in fact appeared earlier on the arXiv than) the proof by H. Roschy and T. Rahn (arXiv:1006.2392), and has several advantages such as being shorter and cleaner and can also settle the additional conjecture on "Serre duality for Betti numbers" which was raised but unresolved in arXiv:1006.2392.