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Published in:
2017 | OriginalPaper | Chapter
Some Open Questions on Arithmetic Zariski Pairs
Authors : Enrique Artal Bartolo, José Ignacio Cogolludo-Agustín
Published in: Singularities in Geometry, Topology, Foliations and Dynamics
Publisher: Springer International Publishing
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In this paper, complement-equivalent arithmetic Zariski pairs will be exhibited answering in the negative a question by Eyral-Oka [14] on these curves and their groups. A complement-equivalent arithmetic Zariski pair is a pair of complex projective plane curves having Galois-conjugate equations in some number field whose complements are homeomorphic, but whose embeddings in $$ {\mathbb{P}}^2 $$ are not.Most of the known invariants used to detect Zariski pairs depend on the étale fundamental group. In the case of Galois-conjugate curves, their étale fundamental groups coincide. Braid monodromy factorization appears to be sensitive to the difference between étale fundamental groups and homeomorphism class of embeddings.