Moduli Spaces of One-Line Extensions of $(10_3)$ Configurations

TL;DR

This paper investigates the moduli spaces of line arrangements derived from $(10_3)$ configurations, identifying conditions under which these spaces are reducible, and provides a method to generate many such examples.

Contribution

It introduces a method to produce combinatorial line arrangements with reducible moduli spaces from known examples with irreducible spaces.

Findings

95 arrangements have reducible moduli spaces

76 arrangements remain reducible after complex conjugation quotient

Method to generate new arrangements with reducible moduli spaces

Abstract

Two line arrangements in $\mathbb{CP}^2$ can have different topological properties even if they are combinatorially isomorphic. Results by Dan Cohen and Suciu and by Randell show that a reducible moduli space under complex conjugation is a necessary condition. We present a method to produce many examples of combinatorial line arrangements with a reducible moduli space obtained from a set of examples with irreducible moduli spaces. In this paper, we determine the reducibility of the moduli spaces of a family of arrangements of 11 lines constructed by adding a line to one of the ten $(10_3)$ configurations. Out of the four hundred ninety-five combinatorial line arrangements in this family, ninety-five have a reducible moduli space, seventy-six of which are still reducible after the quotient by complex conjugation.