Cohomological aspects of power ideals

TL;DR

This paper explores the cohomological structure of line bundle sections on augmented wonderful varieties related to hyperplane arrangements, connecting them to power ideals and zonotopal algebras, and proving a conjectured Hilbert series formula.

Contribution

It establishes a cohomological framework linking hyperplane arrangement power ideals with sections of line bundles on augmented wonderful varieties, including new interpretations of superspace zonotopal algebras.

Findings

Cohomology vanishing results on augmented wonderful varieties.

Recovery of many results about zonotopal algebras.

Proved a conjectured formula for the Hilbert series of superspace zonotopal algebras.

Abstract

We show that the space of sections of any line bundle on the augmented wonderful variety of a hyperplane arrangement has the structure of a coalgebra. These coalgebras correspond to the hyperplane arrangement power ideals of Ardila and Postnikov, which include zonotopal algebras as a special case. By proving cohomology vanishing results on augmented wonderful varieties, we recover many results about zonotopal algebras. We also interpret the "superspace" zonotopal algebras of Rhoades, Tewari, and Wilson in terms of the sections of vector bundles on the augmented wonderful variety, and we use this interpretation to prove a formula that they conjectured for the Hilbert series of the superspace version of the central zonotopal algebra.