Some Open Questions on Arithmetic Zariski Pairs

Abstract

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.

Dedicado con cariño a Pepe, singular matemático y amigo

Mathematics Subject Classification (2000). Primary 14N20, 32S22, 14F35; Secondary 14H50, 14F45, 14G32.