Some open questions on arithmetic Zariski pairs

E. Artal Bartolo; J. I. Cogolludo-Agustín

doi:10.1007/978-3-319-39339-1_3

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arXiv · 1506.05345

Some open questions on arithmetic Zariski pairs

E. Artal Bartolo·J. I. Cogolludo-Agustín

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Abstract

In this paper, complement-equivalent arithmetic Zariski pairs will be exhibited answering in the negative a question by Eyral-Oka 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 \'etale fundamental group. In the case of Galois-conjugate curves, their \'etale fundamental groups coincide. Braid monodromy factorization appears to be sensitive to the difference between \'etale fundamental groups and homeomorphism class of embeddings.

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E. Artal Bartolo, J. I. Cogolludo-Agustín. 2015-06-17. Some open questions on arithmetic Zariski pairs. https://doi.org/10.1007/978-3-319-39339-1_3

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