Hyperpolygonal arrangements

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.

15 pages; v2 updated acknowledgments; v3 updated bibliographic info; to appear in Glasgow Mathematical Journal