Fundamental groups of highly symmetrical curves and Fermat line arrangments

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., ${\pi }_1({\mathbb{CP}}^2-\mathcal{C},\overline{\epsilon })=G⋊F_n$, where $G$ and $F_n$ is defined in 4.3, and 1.2 respectively.

21 pages, 10 figures