Compactification of the moduli space of hyperplane arrangements

TL;DR

This paper explores the compactification of the moduli space of hyperplane arrangements in higher dimensions, identifying its boundary and additional components using stable pairs and Chow quotients.

Contribution

It provides an explicit description of the boundary of the moduli space of hyperplane arrangements and constructs new irreducible components within the stable pairs framework.

Findings

Identifies the closure of the moduli space as Kapranov's Chow quotient compactification.

Describes the boundary pairs explicitly.

Constructs additional irreducible components of the moduli space.

Abstract

Consider the moduli space M^0 of arrangements of n hyperplanes in general position in projective (r-1)-space. When r=2 the space has a compactification given by the moduli space of stable curves of genus 0 with n marked points. In higher dimensions, the analogue of the moduli space of stable curves is the moduli space of stable pairs: pairs (S,B) consisting of a variety S (possibly reducible) and a divisor B=B_1+..+B_n, satisfying various additional assumptions. We identify the closure of M^0 in the moduli space of stable pairs as Kapranov's Chow quotient compactification of M^0, and give an explicit description of the pairs at the boundary. We also construct additional irreducible components of the moduli space of stable pairs.