Real line arrangements with Hirzebruch property

TL;DR

This paper proves that all real line arrangements in the complex projective plane satisfying Hirzebruch's property are related to finite complex reflection groups, confirming exactly four such arrangements exist.

Contribution

It provides a complete classification of real line arrangements with Hirzebruch property, showing they correspond to finite complex reflection groups and identifying the four unique arrangements.

Findings

Exactly four real arrangements satisfy Hirzebruch property

All such arrangements are related to finite complex reflection groups

Confirmed the conjecture for the real case

Abstract

A line arrangement of $3n$ lines in $\mathbb CP^2$ satisfies Hirzebruch property if each line intersect others in $n+1$ points. Hirzebruch asked if all such arrangements are related to finite complex reflection groups. We give a positive answer to this question in the case when the line arrangement in $\mathbb CP^2$ is real, confirming that there exist exactly four such arrangements.