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

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).