On the non-connectivity of moduli spaces of arrangements: the splitting-polygon structure

TL;DR

This paper introduces a method to construct complex projective line arrangements that are lattice-equivalent but belong to different connected components of their moduli space, aiding in understanding their topological properties.

Contribution

It provides a novel construction technique for arrangements with disconnected moduli spaces and applies it to both classical and new examples.

Findings

Reconstructed classical arrangements with disconnected moduli spaces.

Produced new arrangements with four connected components in their moduli space.

Demonstrated the method's efficiency in generating lattice-equivalent arrangements.

Abstract

Questions that seek to determine whether a hyperplane arrangement property, be it geometric, arithmetic or topological, is of a combinatorial nature (that is determined by the intersection lattice) are abundant in the literature. To tackle such questions and provide a negative answer, one of the most effective methods is to produce a counterexample. To this end, it is essential to know how to construct arrangements that are lattice-equivalent. The more different they are, the more efficient it will be. In this paper, we present a method to construct arrangements of complex projective lines that are lattice-equivalent but lie in distinct connected components of their moduli space. To illustrate the efficiency of the method, we apply it to reconstruct all the classical examples of arrangements with disconnected moduli spaces: MacLane, Falk-Sturmfels, Nasir-Yoshinaga and Rybnikov. Moreover, we employ this method to produce novel examples of arrangements of eleven lines whose moduli spaces are formed by four connected components.