Proof of a conjecture of Batyrev and Juny on Gorenstein polytopes

TL;DR

This paper proves a conjecture relating the structure of Gorenstein polytopes to their dimension and degree, showing they are lattice pyramids under certain conditions, and refines this conjecture for IDP Gorenstein polytopes.

Contribution

It establishes a new dimension-degree relation for Gorenstein polytopes and proves the conjecture for IDP cases, advancing understanding of their geometric structure.

Findings

Gorenstein polytope is a lattice pyramid if dimension ≥ 3 × degree

Confirmed Batyrev and Juny's conjecture for a broad class of polytopes

Proposed and proved a refined conjecture for IDP Gorenstein polytopes

Abstract

A $d$-dimensional lattice polytope $P$ is Gorenstein if it has a multiple $r P$ that is a reflexive polytope up to translation by a lattice vector. The difference $d+1-r$ is called the degree of $P$. We show that a Gorenstein polytope is a lattice pyramid if its dimension is at least three times its degree. This was previously conjectured by Batyrev and Juny. We also present a refined conjecture and prove it for IDP Gorenstein polytopes.