Hyperpolygonal arrangements

TL;DR

This paper systematically studies hyperpolygonal arrangements, a family of hyperplane arrangements linked to quiver varieties, revealing their unique local properties, projective uniqueness, and their role as a counterexample to a longstanding conjecture.

Contribution

It introduces and analyzes hyperpolygonal arrangements, showing they are projectively unique, combinatorially formal, and serve as a key counterexample in arrangement theory.

Findings

Hyperpolygonal arrangements discriminate all local properties.

They are projectively unique and combinatorially formal.

The arrangement H_5 counterexamples Edelman-Reiner's conjecture.

Abstract

In 2024, Bellamy, Craw, Rayan, Schedler, and Weiss introduced a particular family of real hyperplane arrangements stemming from hyperpolygonal spaces associated with certain quiver varieties which we thus call hyperpolygonal arrangements $\mathcal H_n$. In this note we study these arrangements and investigate their properties systematically. Remarkably the arrangements $\mathcal H_n$ discriminate between essentially all local properties of arrangements. In addition we show that hyperpolygonal arrangements are projectively unique and combinatorially formal. We note that the arrangement $\mathcal H_5$ is the famous counterexample of Edelman and Reiner from 1993 of Orlik's conjecture that the restriction of a free arrangement is again free.