Moduli Theory of the $r$-Braid Arrangement

Abstract

We describe a family of hyperplane arrangements depending on a positive integer parameter $r$, which we refer to as the $r$-braid arrangements, and which can be viewed as a generalization of the classical braid arrangement. The wonderful compactification of the braid arrangement (with respect to its minimal building set) is well-known to yield the moduli space ${\overline{\mathcal{M}}}_{0,n}$, and, in this work, we generalize this result, constructing a moduli space ${\overline{\mathcal{M}}}_n^r$ of certain genus-zero curves with an order-$r$ involution that we identify with the corresponding wonderful compactification of the $r$-braid arrangement. The resulting space is a variant of the previously studied moduli space ${\overline{\mathcal{L}}}_n^r$ [arXiv:2104.06526], related via a change of weights on the markings.

22 pages