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).
Line arrangements Zariski pairs Number fields Fundamental group
Mathematics Subject Classification
14N20 32S22 14F35 14H50 14F45 14G32
This is a preview of subscription content, log in to check access.
The authors want to thank the anonymous referees for their suggestions that have helped in the exposition of this paper.
Artal, E., Carmona, J., Cogolludo-Agustín, J.I., Marco, M.Á.: Topology and combinatorics of real line arrangements. Compos. Math. 141(6), 1578–1588 (2005)MathSciNetCrossRefMATHGoogle Scholar
Artal, E., Carmona, J., Cogolludo-Agustín, J.I., Marco, M.Á.: Invariants of combinatorial line arrangements and Rybnikov’s example, Singularity theory and its applications. In: Izumiya, S., Ishikawa, G., Tokunaga, H., Shimada, I., Sano, T. (eds.) Advanced Studies in Pure Mathematics, vol. 43. Mathematical Society of Japan, Tokyo (2007)Google Scholar
Artal, E., Cogolludo-Agustín, J.I., Tokunaga, H., A survey on Zariski pairs, Algebraic geometry in East Asia-Hanoi, Advanced Studies in Pure Mathematics, vol. 50, Mathematical Society of Japan, Tokyo 2008, 1–100 (2005)Google Scholar
Artal, E., Florens, V., Guerville-Ballé, B.: A topological invariant of line arrangements. Preprint available at arXiv:1407.3387 [math.GT] (2014) (Submitted)
Charles, F.: Conjugate varieties with distinct real cohomology algebras. J. Reine Angew. Math. 630, 125–139 (2009)MathSciNetMATHGoogle Scholar
Shimada, I.: Non-homeomorphic conjugate complex varieties, Singularities–Niigata-Toyama, Advanced Studies in Pure Mathematics, vol. 56, Mathematical Society of Japan, Tokyo 2009, 285–301 (2007)Google Scholar
Stein, W.A., et al.: Sage Mathematics Software (Version 6.7), The Sage Development Team. http://www.sagemath.org (2015)