An affine local criterion for toric projective space bundles

We study when an equidimensional toric morphism is forced to be a projective-space bundle. Our main result is an affine rigidity theorem: if the base space is affine, the toric relative canonical divisor is $\mathbb Q$-Cartier, and its negative has degree greater than the relative dimension on every complete curve, then the morphism is equivariantly a trivial projective-space bundle. As an application, we derive a projective-space-bundle theorem for equidimensional toric contractions associated to long extremal rays, without assuming $\mathbb Q$-factoriality.