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arXiv · 1710.04146

Classifying Fano Complexity-One $T$-Varieties via Divisorial Polytopes

Nathan Ilten·Marni Mishna·Charlotte Trainor
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Abstract

The correspondence between Gorenstein Fano toric varieties and reflexive polytopes has been generalized by Ilten and S\"u{\ss} 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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Nathan Ilten, Marni Mishna, Charlotte Trainor. 2017-10-11. Classifying Fano Complexity-One $T$-Varieties via Divisorial Polytopes. https://doi.org/10.1007/s00229-018-1036-x

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