Ehrhart Theory for Lawrence Polytopes and Orbifold Cohomology of Hypertoric Varieties

TL;DR

This paper links the orbifold cohomology of hypertoric varieties with Ehrhart theory of Lawrence polytopes, providing new formulas and inequalities for Ehrhart polynomials through geometric and combinatorial methods.

Contribution

It establishes a novel connection between orbifold cohomology and Ehrhart polynomials of Lawrence polytopes, including a new formula and inequalities for the $ ext{δ}$-polynomial.

Findings

Dimensions of orbifold cohomology equal Ehrhart $ ext{δ}$-polynomial coefficients.

Derived a formula for the Ehrhart $ ext{δ}$-polynomial of Lawrence polytopes.

Established inequalities between $ ext{δ}$-polynomial coefficients using Hard Lefschetz theorem.

Abstract

We establish a connection between the orbifold cohomology of hypertoric varieties and the Ehrhart theory of Lawrence polytopes. More specifically, we show that the dimensions of the orbifold cohomology groups of a hypertoric variety are equal to the coefficients of the Ehrhart $\delta$-polynomial of the associated Lawrence polytope. As a consequence, we deduce a formula for the Ehrhart $\delta$-polynomial of a Lawrence polytope and use the injective part of the Hard Lefschetz Theorem for hypertoric varieties to deduce some inequalities between the coefficients of the $\delta$-polynomial.