Gorenstein spherical Fano varieties

TL;DR

This paper provides a combinatorial framework for Gorenstein spherical Fano varieties using polytopes, extending known descriptions from toric and horospherical cases, and establishes bounds on their Picard group rank.

Contribution

It introduces a new combinatorial description of Gorenstein spherical Fano varieties and proves a bound on the Picard group rank for these varieties.

Findings

Describes Gorenstein spherical Fano varieties via specific polytopes.

Establishes that the Picard group rank is at most twice the dimension.

Provides an overview of the anticanonical divisor class in spherical varieties.

Abstract

We obtain a combinatorial description of Gorenstein spherical Fano varieties in terms of certain polytopes, generalizing the combinatorial description of Gorenstein toric Fano varieties by reflexive polytopes and its extension to Gorenstein horospherical Fano varieties due to Pasquier. Using this description, we show that the rank of the Picard group of an arbitrary $d$-dimensional $\mathbb{Q}$-factorial Gorenstein spherical Fano variety is bounded by $2d$. This paper also contains an overview of the description of the natural representative of the anticanonical divisor class of a spherical variety due to Brion.