On \(\textrm{H}-\)trivial line bundles on toric DM stacks of dim \(\ge 3\)

We study line bundles on smooth toric Deligne-Mumford stacks \({\mathbb {P}}_{\mathbf {\Sigma }}\) ${\mathbb{P}}_{\mathrm{\Sigma }}$ of arbitrary dimension. We give a sufficient condition for when infinitely many line bundles on \({\mathbb {P}}_{\mathbf {\Sigma }}\) ${\mathbb{P}}_{\mathrm{\Sigma }}$ have trivial cohomology. In dimension three, this sufficient condition is also a necessary condition under the technical assumption that \(\mathbf {\Sigma }\) $\mathrm{\Sigma }$ has no more than one pair of collinear rays.