An arithmetic Zariski 4-tuple of twelve lines

TL;DR

This paper constructs and distinguishes four line arrangements in the complex projective plane with identical combinatorial data but different topological embeddings, introducing new examples of arithmetic Zariski pairs.

Contribution

It develops an invariant to differentiate arrangements with the same combinatorics but different deformation classes, creating new arithmetic Zariski pairs.

Findings

Four arrangements with identical combinatorics but different deformation classes

Construction of arrangements with no orientation-preserving homeomorphism between them

Identification of new arithmetic Zariski pairs

Abstract

Using the invariant developed in [6], we differentiate four arrangements with the same combinatorial information but in different deformation classes. From these arrangements, we construct four other arrangements such that there is no orientation-preserving homeomorphism between them. Furthermore, some couples of arrangements among this 4-tuplet form new arithmetic Zariski pairs, i.e. a couple of arrangements with the same combinatorial information but with different embedding in $\mathbb{CP}^2$.