On the maximum dual volume of a canonical Fano polytope

TL;DR

This paper establishes a sharp upper bound on the volume of the dual of a canonical Fano polytope, linking geometric properties of polytopes to bounds on toric Fano varieties' anti-canonical degrees.

Contribution

It provides the first sharp upper bound on the dual volume of canonical Fano polytopes, expressed via the Sylvester sequence, with implications for toric geometry.

Findings

Bound is sharp and achieved by a specific reflexive simplex

Imposes a sharp upper limit on the volume of reflexive polytopes

Translates to a bound on the anti-canonical degree of toric Fano varieties

Abstract

We give an upper bound on the volume vol(P*) of a polytope P* dual to a d-dimensional lattice polytope P with exactly one interior lattice point, in each dimension d. This bound, expressed in terms of the Sylvester sequence, is sharp, and is achieved by the dual to a particular reflexive simplex. Our result implies a sharp upper bound on the volume of a d-dimensional reflexive polytope. Translated into toric geometry, this gives a sharp upper bound on the anti-canonical degree $(-K_X)^d$ of a d-dimensional toric Fano variety X with at worst canonical singularities.