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).

arXiv comment26 pages

References in corpus (5)

• On the fundamental group of the complement of a complex hyperplane arrangement
• The Braid Monodromy of Plane Algebraic Curves and Hyperplane Arrangements
• Topology and combinatorics of real line arrangements
• An arithmetic Zariski 4-tuple of twelve lines
• On complex line arrangements and their boundary manifolds

Cited by in corpus (1)

• Fundamental groups of real arrangements and torsion in the lower central series quotients