# Triangular curves and cyclotomic Zariski tuples

## TL;DR

This paper constructs infinite families of conjugate projective curves over number fields with identical abelian fundamental groups but non-homeomorphic structures, using cyclotomic fields and linking invariants.

## Contribution

It introduces new Zariski tuples parametrized by roots of unity, demonstrating non-homeomorphic conjugate curves with the same fundamental group in cyclotomic fields.

## Key findings

- Existence of infinite conjugate curve families with identical abelian fundamental groups.
- Construction of Zariski tuples parametrized by roots of unity.
- Distinction of curves using linking invariants.

## 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 $d>3$ we find Zariski tuples parametrized by the $d$-roots of unity up to complex conjugation. As a consequence, for any divisor $m$ of $d$, $m\neq 1,2,3,4,6$, 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.