A lattice polytope $P \subset {\bf R}^n$ is the convex hull of finitely many points in ${\bf Z}^n$. There is a natural hierarchy of structural sophistication for lattice polytopes, with various concepts motivated from toric geometry and commutative algebra. We will discuss three such concepts in this hierarchy, occupying a point of origin (normality), the bottom (very ampleness), and the top spot (Koszul property). More specifically, 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.

This joint work with Jessica Delgado, Joseph Gubeladze, and Mateusz Michalek. ​