Homology inclusion of complex line arrangements

TL;DR

This paper introduces a new topological invariant for complex line arrangements in projective space, aiding in distinguishing Zariski pairs with identical combinatorics but different embeddings.

Contribution

It develops a novel homology-based invariant derived from boundary inclusion maps, extending previous ideas and providing computational tools to identify Zariski pairs.

Findings

Identified new Zariski pairs with the invariant

Developed a combinatorial presentation of the invariant group

Implemented computational methods in Sage for invariant calculation

Abstract

We introduce a new topological invariant of complex line arrangements in $\mathbb{CP}^2$, derived from the interaction between their complement and the boundary of a regular neighbourhood. The motivation is to identify Zariski pairs which have the same combinatorics but different embeddings. Building on ideas developed by B. Guerville-Ball\'e and W. Cadiegan-Schlieper, we consider the inclusion map of the boundary manifold to the exterior and its effect on homology classes. A careful study of the graph Waldhausen structure of the boundary manifold allows to identify specific generators of the homology. Their potential images are encoded in a group, the graph stabiliser, with a nice combinatorial presentation. The invariant related to the inclusion map is an element of this group. Using a computer implementation in Sage, we compute the invariant for some examples and exhibit new Zariski pairs.