An explicit wall crossing for the moduli space of hyperplane arrangements

TL;DR

This paper investigates a specific wall crossing in the moduli space of hyperplane arrangements, revealing how changing weights affects the compactification and its algebraic properties, with detailed results for planar lines.

Contribution

It provides the first explicit analysis of a wall crossing transforming a toric into a non-toric compactification in hyperplane arrangement moduli spaces.

Findings

The wall crossing alters the structure of the moduli space.

Certain blow-ups are not Mori dream spaces for large hyperplane counts.

A detailed description of the wall crossing for lines in the plane.

Abstract

The moduli space of hyperplanes in projective space has a family of geometric and modular compactifications that parametrize stable hyperplane arrangements with respect to a weight vector. Among these, there is a toric compactification that generalizes the Losev-Manin moduli space of points on the line. We study the first natural wall crossing that modifies this compactification into a non-toric one by varying the weights. As an application of our work, we show that any $\mathbb{Q}$-factorialization of the blow up at the identity of the torus of the generalized Losev-Manin space is not a Mori dream space for a sufficiently high number of hyperplanes. Additionally, for lines in the plane, we provide a precise description of the wall crossing.