  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).
