Topology of abelian pencils of curves

Abstract

We study the geometry of a linear system of plane curves $C(\tau) (\tau \in \mathbb{C})$ spanned by two irreducible curves $C$, $C'$ of degree $d$ such that $\pi_1 (\mathbb{P}^2 - C \cup C')$ is abelian. We will show that the fundamental group $\pi_1 (\mathbb{C}^2 - C(\vec{\tau}))$ is isomorphic to $\mathbb{Z} \times F(r- 1)$ for a generic $\vec{\tau}$ where $\vec{\tau} = (\tau_1, \dots, \tau_r)$ and $C(\vec{\tau}) = C(\tau_1) \cup \dots \cup C(\tau_r)$.