Canonical threefold singularities with a torus action of complexity one and $k$-empty polytopes

TL;DR

This paper classifies canonical threefold singularities with a two-torus action of complexity one, extending previous classifications by analyzing lattice point emptiness in polytopal complexes and exploring $k$-emptiness of scaled polytopes.

Contribution

It introduces a classification method based on $k$-emptiness of polytopes, extending Mori's and Ishida-Iwashita's work to a broader class of singularities.

Findings

Two-dimensional $k$-empty polytopes are either sporadic or form series related to Farey sequences.

The classification includes the Cox ring iteration tree with roots being generalized compound du Val singularities.

All spectra of factorial Cox rings are characterized as generalized compound du Val singularities.

Abstract

We classify the canonical threefold singularities that allow an effective two-torus action. This extends classification results of Mori on terminal threefold singularities and of Ishida and Iwashita on toric canonical threefold singularities. Our classification relies on lattice point emptiness of certain polytopal complexes with rational vertices. Scaling the polytopes by the least common multiple $k$ of the respective denominators, we investigate $k$-emptiness of polytopes with integer vertices. We show that two dimensional $k$-empty polytopes either are sporadic or come in series given by Farey sequences. We finally present the Cox ring iteration tree of the classified singularities, where all roots, i.e. all spectra of factorial Cox rings, are generalized compound du Val singularities.