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

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 polytopes with rational vertices. We show that in dimension two, such polytopes are sporadic or are given by Farey sequences. We finally present the Cox ring iteration tree of the classified singularities.