Moduli Theory of the $r$-Braid Arrangement

TL;DR

This paper introduces a family of hyperplane arrangements called $r$-braid arrangements, generalizing the classical braid arrangement, and constructs associated moduli spaces of genus-zero curves with involutions, extending known compactification results.

Contribution

It generalizes the classical braid arrangement to $r$-braid arrangements and constructs new moduli spaces of genus-zero curves with involutions, linking combinatorics and algebraic geometry.

Findings

Construction of the $r$-braid arrangements as hyperplane arrangements.

Identification of the associated moduli space $ar{ ext{M}}^r_{n}$ with a wonderful compactification.

Relation of the new moduli space to previously studied spaces via weight changes.

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 $\overline{\mathcal{M}}^r_{n}$ 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 $\overline{\mathcal{L}}^r_n$ [arXiv:2104.06526], related via a change of weights on the markings.