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

Abstract

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