Connectedness and combinatorial interplay in the moduli space of line arrangements

TL;DR

This paper explores the structure of the moduli space of line arrangements, establishing conditions for connectedness based on combinatorial properties and providing bounds on the number of connected components.

Contribution

It introduces the concept of arrangements with a rigid pencil form, unifies existing classes, and provides bounds and examples for the connectedness of the moduli space.

Findings

Connectedness characterized by rigid pencil arrangements.

Unified classes of arrangements with connected moduli space.

Provided sharp upper bounds and examples with many components.

Abstract

This paper aims to undertake an exploration of the behavior of the moduli space of line arrangements while establishing its combinatorial interplay with the incidence structure of the arrangement. In the first part, we investigate combinatorial classes of arrangements whose moduli space is connected. We unify the classes of simple and inductively connected arrangements appearing in the literature. Then, we introduce the notion of arrangements with a rigid pencil form. It ensures the connectedness of the moduli space and is less restrictive that the class of $C_3$ arrangements of simple type. In the last part, we obtain a combinatorial upper bound on the number of connected components of the moduli space. Then, we exhibit examples with an arbitrarily large number of connected components for which this upper bound is sharp.