On the maximum dual volume of a canonical Fano polytope

Balletti, Gabriele; Kasprzyk, Alexander M.; Nill, Benjamin

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.

• Journal Article Type: Article
• Acceptance Date: Jun 26, 2022
• Online Publication Date: Dec 13, 2022
• Publication Date: Dec 13, 2022
• Deposit Date: Sep 11, 2022
• Publicly Available Date: Dec 13, 2022
• Journal: Forum of Mathematics, Sigma
• Print ISSN: 2050-5094
• Electronic ISSN: 2050-5094
• Publisher: Cambridge University Press (CUP)
• Peer Reviewed: Peer Reviewed
• Volume: 10
• Article Number: e109
• DOI: https://doi.org/10.1017/fms.2022.93
• Keywords: Computational Mathematics; Discrete Mathematics and Combinatorics; Geometry and Topology; Mathematical Physics; Statistics and Probability; Algebra and Number Theory; Theoretical Computer Science; Analysis
• Public URL: https://nottingham-repository.worktribe.com/output/2461955
• Publisher URL: https://www.cambridge.org/core/journals/forum-of-mathematics-sigma/article/on-the-maximum-dual-volume-of-a-canonical-fano-polytope/36121F80310E40113B38A45B164C4967
• Additional Information: Copyright: © The Author(s), 2022. Published by Cambridge University Press; License: This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (https://creativecommons.org/licenses/by/4.0/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.; Free to read: This content has been made available to all.