Triangular curves and cyclotomic Zariski tuples

• Published in: Collectanea mathematica (Barcelona) Vol. 71; no. 3; pp. 427 - 441
• Main Authors: Artal Bartolo, Enrique, Cogolludo-Agustín, José Ignacio, Martín-Morales, Jorge
• Format: Journal Article
• Language: English
• Published: Milan Springer Milan 01-09-2020; Springer Nature B.V
• Subjects: Algebra; Analysis; Applications of Mathematics; Conjugation; Geometry; Mathematics; Mathematics and Statistics; Number theory; 14N20; 32S22; Linking numbers; Arithmetic Zariski pairs; 14G32; 14H50; 14F35; Zariski pairs; 14F45
• ISSN: 0010-0757; 2038-4815
• DOI: 10.1007/s13348-019-00269-y

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
≥
4
we find Zariski
d
2
+
1
-tuples parametrized by the
d
-roots of unity up to complex conjugation. As a consequence, for any divisor
m
of
d
,
m
≠
1
,
2
,
3
,
4
,
6
, we find arithmetic Zariski
ϕ
(
m
)
2
-tuples with coefficients in the corresponding cyclotomic field. These curves have abelian fundamental group and they are distinguished using a linking invariant.