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

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.