Free arrangements with low exponents

TL;DR

This paper proves that free hyperplane arrangements with exponents only 1's and 2's are supersolvable and explores conjectures about arrangements with exponents including a single 3, providing proofs for certain ranks.

Contribution

It establishes that arrangements with exponents 1's and 2's are supersolvable and supports a conjecture about arrangements with a single 3 in their exponents.

Findings

Arrangements with exponents 1's and 2's are supersolvable.

Conjecture holds for ranks 4 and 5, and for inductively free arrangements.

Provides partial evidence for a broader conjecture about arrangements with a single 3.

Abstract

In this article we show that any free hyperplane arrangement with exponents 1's and 2's is a supersolvable arrangement. We conjecture that any free arrangement with exponents 1's, 2's and exactly one 3, is also supersolvable, and we show this conjecture for hyperplane arrangements of ranks 4 and 5, and for inductively free arrangements of any rank.