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.
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Ilten, N., Mishna, M. & Trainor, C. Classifying Fano complexity-one T-varieties via divisorial polytopes. manuscripta math. 158, 463–486 (2019). https://doi.org/10.1007/s00229-018-1036-x
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DOI: https://doi.org/10.1007/s00229-018-1036-x