Very ample and Koszul segmental fibrations

TL;DR

This paper constructs explicit examples of very ample 3D lattice polytopes with significant deviations from normality and describes a broad class of Koszul polytopes across dimensions, extending known classes and including many smooth cases.

Contribution

It introduces a simple construction method for lattice polytopes that yields very ample polytopes with large non-normality and characterizes a wide class of Koszul polytopes extending previous work.

Findings

Constructed a series of very ample 3D polytopes with arbitrarily large non-normality.

Described a large class of Koszul polytopes in arbitrary dimensions.

Extended the class of known Koszul polytopes to include many smooth examples.

Abstract

In the hierarchy of structural sophistication for lattice polytopes, normal polytopes mark a point of origin; very ample and Koszul polytopes occupy bottom and top spots in this hierarchy, respectively. In this paper we explore a simple construction for lattice polytopes with a twofold aim. On the one hand, we derive an explicit series of very ample 3-dimensional polytopes with arbitrarily large deviation from the normality property, measured via the highest discrepancy degree between the corresponding Hilbert functions and Hilbert polynomials. On the other hand, we describe a large class of Koszul polytopes of arbitrary dimensions, containing many smooth polytopes and extending the previously known class of Nakajima polytopes.