Freeness of Arrangements with Regular Underlying Matroids

Abstract

We classify freeness for finite central arrangements whose underlying matroids are regular. Let $\mathcal A$ be such an arrangement over an arbitrary field, and put $M=M(\mathcal A)$. Then $\mathcal A$ is free if and only if $M$ is supersolvable; equivalently, $M$ admits a nice partition; equivalently, $M$ is the cycle matroid of a chordal simple graph. Thus, for arrangements with regular underlying matroids, freeness has a complete combinatorial classification independent of the base field. We use Seymour's decomposition theorem for regular matroids to prove that freeness forces supersolvability. We also characterize nice partitions of finite simple binary matroids: a partition is nice if and only if it is independent and no line is contained in a single block. Consequently, a finite loopless binary matroid admits a nice partition if and only if it is simple and supersolvable.