Spanning Lattice Polytopes and the Uniform Position Principle

TL;DR

This paper extends Ehrhart theory inequalities from IDP lattice polytopes to spanning polytopes, broadening the class of polytopes for which these inequalities hold, and generalizes key algebraic geometry principles to weighted projective spaces.

Contribution

It demonstrates that Ehrhart inequalities hold for spanning polytopes, a wider class than IDP, and generalizes Bertini's theorem and Harris' Uniform Position Principle to new geometric contexts.

Findings

Ehrhart inequalities apply to spanning polytopes, not just IDP.

Generalization of Hibi's Lower Bound Theorem.

Extension of Bertini's theorem and Harris' principle to weighted projective spaces.

Abstract

A lattice polytope $P$ is called IDP if any lattice point in its $k$th dilate is a sum of $k$ lattice points in $P$. In 1991 Stanley proved a strong inequality in Ehrhart theory for IDP lattice polytopes. We show that his conclusion holds under much milder assumptions, namely if the lattice polytope $P$ is spanning, i.e., any lattice point of the ambient lattice is an integer affine combination of lattice points in $P$. As an application, we get a generalization of Hibi's Lower Bound Theorem. Our proof relies on generalizing Bertini's theorem to the semistandard situation and Harris' Uniform Position Principle to certain curves in weighted projective space.