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 ${\mathbb{K}}^{\ast }$-surfaces. Furthermore, we show that any three-dimensional Fano divisorial polytope is equivalent to one involving only eight functions.

18 pages, 8 figures; minor revisions

References in corpus (4)

• K-Stability for Fano Manifolds with Torus Action of Complexity One
• On terminal Fano 3-folds with 2-torus action
• On the classification of Kahler-Ricci solitons on Gorenstein del Pezzo surfaces
• Smooth projective varieties with a torus action of complexity 1 and Picard number 2

Cited by in corpus (1)

• Classifying log del Pezzo surfaces with torus action