An arithmetic Zariski pair of line arrangements with non-isomorphic fundamental group

TL;DR

This paper presents the first example of Galois-conjugate plane curves with non-isomorphic fundamental groups of their complements, highlighting subtle topological differences not detectable by profinite completions.

Contribution

It constructs an explicit arithmetic Zariski pair of line arrangements with non-isomorphic fundamental groups, advancing understanding of topological invariants in algebraic geometry.

Findings

Fundamental groups of the complements are not isomorphic.

Arrangements are Galois-conjugate but topologically distinct.

First known example with non-isomorphic fundamental groups despite isomorphic profinite completions.

Abstract

In a previous work, the third named author found a combinatorics of line arrangements whose realizations live in the cyclotomic group of the fifth roots of unity and such that their non-complex-conjugate embedding are not topologically equivalent in the sense that they are not embedded in the same way in the complex projective plane. That work does not imply that the complements of the arrangements are not homeomorphic. In this work we prove that the fundamental groups of the complements are not isomorphic. It provides the first example of a pair of Galois-conjugate plane curves such that the fundamental groups of their complements are not isomorphic (despite the fact that they have isomorphic profinite completions).