Abstract
The purpose of this paper is to exhibit infinite families of conjugate projective curves in a number field whose complement have the same abelian fundamental group, but are non-homeomorphic. In particular, for any \(d\ge 4\) we find Zariski \(\left( \left\lfloor \frac{d}{2}\right\rfloor +1\right) \)-tuples parametrized by the d-roots of unity up to complex conjugation. As a consequence, for any divisor m of d, \(m\ne 1,2,3,4,6\), we find arithmetic Zariski \(\frac{\phi (m)}{2}\)-tuples with coefficients in the corresponding cyclotomic field. These curves have abelian fundamental group and they are distinguished using a linking invariant.
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Acknowledgements
The second and third authors want to thank the Fulbright Program (within the José Castillejo and Salvador de Madariaga grants by Ministerio de Educación, Cultura y Deporte) for their financial support while writing this paper. They also want to thank the University of Illinois at Chicago, especially Anatoly Libgober and Lawrence Ein for their warmth welcome and support in hosting them.
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Partially supported by MTM2016-76868-C2-2-P and Gobierno de Aragón (Grupo de referencia “Álgebra y Geometría”) cofunded by Feder 2014–2020 “Construyendo Europa desde Aragón”. The third author is also partially supported by FQM-333 “Junta de Andalucía”
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Artal Bartolo, E., Cogolludo-Agustín, J.I. & Martín-Morales, J. Triangular curves and cyclotomic Zariski tuples. Collect. Math. 71, 427–441 (2020). https://doi.org/10.1007/s13348-019-00269-y
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DOI: https://doi.org/10.1007/s13348-019-00269-y