Finite automorphism groups of hypersurfaces via Bott vanishing

Abstract

We give a cohomological criterion for the automorphism group scheme of a hypersurface in a smooth Deligne-Mumford stack to be discrete in tame characteristics. The criterion converts Bott-type vanishing on the ambient stack into vanishing of infinitesimal automorphisms of every quasismooth hypersurface in a linear system. We apply it to toric orbifolds, obtain uniform finiteness results for quasismooth weighted-projective hypersurfaces in tame characteristic and hypersurfaces in products of projective spaces, and identify the precise exceptional windows in which the method fails.