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Fundamental group of sextics of torus type

Mutsuo Oka·Duc Tai Pho
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Abstract

We show that the fundamental group of the complement of any irreducible tame torus sextics in $\bf P^2$ is isomorphic to $\bf Z_2*\bf Z_3$ except one class. The exceptional class has the configuration of the singularities $\{C_{3,9},3A_2\}$ and the fundamental group is bigger than $\bf Z_2*\bf Z_3$. In fact, the Alexander polynomial is given by $(t^2-t+1)^2$. For the proof, we first reduce the assertion to maximal curves and then we compute the fundamental groups for maximal tame torus curves.

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Mutsuo Oka, Duc Tai Pho. 2000-10-18. Fundamental group of sextics of torus type. https://arxiv.org/abs/math/0010182

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