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

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.