Computing the nonfree locus of the moduli space of arrangements and Terao's freeness conjecture

TL;DR

This paper develops a method using Fitting ideals to compute the nonfree locus in the moduli space of rank 3 arrangements, and verifies Terao's freeness conjecture for arrangements with up to 14 hyperplanes.

Contribution

It introduces a novel computational approach for the nonfree locus using Fitting ideals and confirms Terao's conjecture for specific rank 3 arrangements.

Findings

Computed nonfree locus for rank 3 arrangements using Fitting ideals.

Verified Terao's freeness conjecture for arrangements with up to 14 hyperplanes.

Established a link between Ziegler restriction, Yoshinaga's criterion, and the nonfree locus.

Abstract

In this paper, we show how to compute using Fitting ideals the nonfree locus of the moduli space of arrangements of a rank $3$ simple matroid, i.e., the subset of all points of the moduli space which parametrize nonfree arrangements. Our approach relies on the so-called Ziegler restriction and Yoshinaga's freeness criterion for multiarrangements. We use these computations to verify Terao's freeness conjecture for rank $3$ central arrangements with up to $14$ hyperplanes in any characteristic.