# On $\mathrm{H}-$trivial line bundles on toric DM stacks of dim $\geq3$

## TL;DR

This paper investigates conditions under which infinitely many line bundles on smooth toric Deligne-Mumford stacks have trivial cohomology, providing both sufficient and necessary conditions in certain cases.

## Contribution

It establishes a sufficient condition for trivial cohomology of line bundles on toric DM stacks and proves its necessity in specific three-dimensional cases.

## Key findings

- Identifies a sufficient condition for infinitely many line bundles to have trivial cohomology.
- Shows that in dimension three, the sufficient condition is also necessary under certain geometric constraints.
- Advances understanding of line bundle cohomology on toric DM stacks of arbitrary dimension.

## Abstract

We study line bundles on smooth toric DM stacks $\mathbb{P}_{\mathbf{\Sigma}}$ of arbitrary dimension. A sufficient condition is given for when infinitely many line bundles on $\mathbb{P}_{\mathbf{\Sigma}}$ have trivial cohomology. In dimension three, the sufficient condition is also a necessary condition in the case that $\mathbf{\Sigma}$ has no more than one pair of collinear rays.