Reflexive polytopes and the Picard ranks of Gorenstein toric Fano varieties

TL;DR

This paper establishes an upper bound on the sum of Picard ranks for polar pairs of Gorenstein toric Fano varieties, linking it to the combinatorial properties of reflexive polytopes, and characterizes when this bound is tight.

Contribution

It generalizes Eikelberg's affine dependence theory to relate Picard ranks with reflexive polytope facets and vertices, providing a new bound and characterization for Gorenstein toric Fano varieties.

Findings

Upper bound on Picard rank sum related to polytope facets and vertices

Characterization of cases achieving the bound as simple-simplicial pairs

Extension of Eikelberg's affine dependence theory to Gorenstein toric Fano varieties

Abstract

We prove that the sum of the Picard ranks of a polar pair of Gorenstein toric Fano varieties of dimension $d\geq 3$ is at most the minimum of the number of facets and vertices of the corresponding pair of reflexive polytopes minus $(d-1)$. This is a generalization of Eikelberg's theory of affine dependences describing the Picard groups of toric varieties. The upper bound is achieved if and only if the polar pair is a simple-simplicial pair.