## Advanced Studies in Pure Mathematics

### Topology of abelian pencils of curves

Mutsuo Oka

#### 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 $C(\vec{\tau}) = C(\tau_1) \cup \dots \cup C(\tau_r)$.

#### Article information

Dates
Revised: 15 July 2008
First available in Project Euclid: 28 November 2018

https://projecteuclid.org/ euclid.aspm/1543448020

Digital Object Identifier
doi:10.2969/aspm/05610225

Mathematical Reviews number (MathSciNet)
MR2604085

Zentralblatt MATH identifier
1201.14022

#### Citation

Oka, Mutsuo. Topology of abelian pencils of curves. Singularities — Niigata–Toyama 2007, 225--248, Mathematical Society of Japan, Tokyo, Japan, 2009. doi:10.2969/aspm/05610225. https://projecteuclid.org/euclid.aspm/1543448020