On the topology of the complements of quartic and line configurations

Abstract

For a reduced plane curve $C$ and a line $L$ in ${\mathbb{P}}^2$, we put ${ℂ}_L^2:={\mathbb{P}}^2-L$, and $C_L:=C-(C\cap L)$. If $C$ and $L$ intersect transversaly and ${\pi }_1({\mathbb{P}}^2-C,b_0)$ is abelian, it is known that ${\pi }_1({ℂ}_L^2-C_L)$ is also abelian. In this article, we study ${\pi }_1({ℂ}_L^2-C_L)$ and the Alexander polynomial for the case when a quartic curve $C$ and a line $L$ do not intersect transversaly.