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

TL;DR

This paper develops a criterion for canonicity in toric varieties, enabling classification of canonical and terminal fake weighted projective spaces, and applies it to classify 4-dimensional canonical simplices, with implications for Calabi-Yau hypersurfaces.

Contribution

It introduces a new canonicity criterion based on local class group actions and provides a comprehensive classification algorithm for canonical fake weighted projective spaces in any dimension.

Findings

Classified 710,450 canonical fake weighted projective spaces in dimension four.

Developed a classification algorithm for canonical and terminal fake weighted projective spaces.

Analyzed the associated Calabi-Yau hypersurfaces and computed their Fine interior.

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.