On the maximum dual volume of a canonical Fano polytope

2016/11/08 by Gabriele Balletti, Balletti, Gabriele, Alexander Kasprzyk +3
Mathematics · #14M25 (Secondary) #52B20 (Primary) #Advanced Combinatorial Mathematics #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #Combinatorics (math.CO) #Commutative Algebra and Its Applications #FOS: Mathematics #Metric Geometry (math.MG)

paper · pdf · doi:10.48550/arxiv.1611.02455

openalex publication_date 2016/11/08 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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.

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