2018/07/20 by Lukas Braun, Braun, Lukas, Daniel Hättig +1
Mathematics · #11B57 #14B05 #14R05 #52B20 #Advanced Combinatorial Mathematics #Algebraic Geometry (math.AG) #Algebraic structures and combinatorial models #Commutative Algebra and Its Applications #FOS: Mathematics
paper · pdf · doi:10.48550/arxiv.1807.08022
openalex publication_date 2018/07/20 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We classify the canonical threefold singularities that allow an effective\ntwo-torus action. This extends classification results of Mori on terminal\nthreefold singularities and of Ishida and Iwashita on toric canonical threefold\nsingularities. Our classification relies on lattice point emptiness of certain\npolytopal complexes with rational vertices. Scaling the polytopes by the least\ncommon multiple k of the respective denominators, we investigate\nk-emptiness of polytopes with integer vertices. We show that two dimensional\nk-empty polytopes either are sporadic or come in series given by Farey\nsequences. We finally present the Cox ring iteration tree of the classified\nsingularities, where all roots, i.e. all spectra of factorial Cox rings, are\ngeneralized compound du Val singularities.\n