Title: Classifying Fano complexity-one T-varieties via divisorial polytopes
Abstract: The correspondence between Gorenstein Fano toric varieties and reflexive polytopes has been generalized by Ilten and Süß to a correspondence between Gorenstein Fano complexity-one T-varieties and Fano divisorial polytopes. Motivated by the finiteness of reflexive polytopes in fixed dimension, we show that over a fixed base polytope, there are only finitely many Fano divisorial polytopes, up to equivalence. We classify two-dimensional Fano divisorial polytopes, recovering Huggenberger’s classification of Gorenstein del Pezzo $$\mathbb {K}^*$$ -surfaces. Furthermore, we show that any three-dimensional Fano divisorial polytope is equivalent to one involving only eight functions.