A canonicity criterion for toric varieties and the classification of canonical 4-simplices

Abstract

Based on the Reid-Shepherd-Barron-Tai criterion for canonical and terminal quotient singularities, we characterize canonicity and terminality of a toric variety in terms of its local class group actions. Specializing it to the Picard number one setting, we arrive at a classification algorithm for canonical and terminal fake weighted projective spaces in any dimension. In dimension four it gives, up to isomorphism, 710450 canonical fake weighted projective spaces. We take a look at the corresponding Calabi-Yau hypersurfaces, compute the Fine interior of the associated canonical simplices, and discuss the results.