# An arithmetic Zariski pair of line arrangements with non-isomorphic fundamental group

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## 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).

### Keywords

Line arrangements Zariski pairs Number fields Fundamental group### Mathematics Subject Classification

14N20 32S22 14F35 14H50 14F45 14G32## Notes

### Acknowledgements

The authors want to thank the anonymous referees for their suggestions that have helped in the exposition of this paper.

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