Fundamental groups of highly symmetrical curves and Fermat line arrangments

TL;DR

This paper computes the fundamental group of the complement of Fermat line arrangements in the complex projective plane, revealing a semi-direct product structure influenced by the arrangement's symmetries.

Contribution

It provides the first explicit computation of the fundamental group for Fermat line arrangements, highlighting its semi-direct product structure and symmetry properties.

Findings

Fundamental group is a semi-direct product of G and F_n.

Explicit description of the fundamental group for Fermat arrangements.

Shows the influence of symmetries on the fundamental group's structure.

Abstract

We showcase a computation of the fundamental group of $\mathbb{CP}^2 - \mathcal{C}$ when $\mathcal{C}$ is a curve admitting a lot of symmetries. In particular, let $\mathcal{C}$ denote the Fermat line arrangement in $\mathbb{CP}^2$ defined by the vanishing locus of homogeneous polynomial $(x^n-y^n)(y^n-z^n)(z^n-x^n)$. In this article, we compute the fundamental group $\pi_1(\mathbb{CP}^2-\mathcal{C})$ of complement of this line arrangement in the complex projective plane. We show that this group is semi-direct product of $G$ and $F_n$, i.e., $\pi_1(\mathbb{CP}^2-\mathcal{C}, \overline{\epsilon}) = G \rtimes F_{n}$, where $G$ and $F_n$ is defined in 4.3, and 1.2 respectively.