Koszulness and supersolvability for Dirichlet arrangements

TL;DR

This paper establishes a precise equivalence between supersolvability and Koszulness for cones over Dirichlet arrangements, expanding known results to an infinite family of new, combinatorially distinct arrangements.

Contribution

It proves the equivalence for Dirichlet arrangements and introduces an infinite family of arrangements that are new and distinct from previously studied classes.

Findings

Supersolvability of cones over Dirichlet arrangements is equivalent to their Orlik-Solomon algebra being Koszul.

An infinite family of such arrangements is identified, expanding the known classes.

The results generalize previous findings for four other classes of arrangements.

Abstract

We prove that the cone over a Dirichlet arrangement is supersolvable if and only if its Orlik-Solomon algebra is Koszul. This was previously shown for four other classes of arrangements. We exhibit an infinite family of cones over Dirichlet arrangements that are combinatorially distinct from these other four classes.