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

TL;DR

This paper extends the classification of Fano varieties to complexity-one $T$-varieties using divisorial polytopes, showing finiteness results and providing explicit classifications in low dimensions.

Contribution

It proves finiteness of Fano divisorial polytopes over fixed base polytopes and classifies all two-dimensional cases, also reducing three-dimensional cases to a finite set of functions.

Findings

Finiteness of Fano divisorial polytopes over fixed base polytope.

Complete classification of two-dimensional Fano divisorial polytopes.

Any three-dimensional Fano divisorial polytope is equivalent to one with only eight functions.

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.