The fundamental group of partial compactifications of the complement of a real line arrangement

Abstract

Let $\mathscr{A}$ be a real projective line arrangement and $M(\mathscr{A})$ its complement in $\mathbb{CP}^2$. We obtain an explicit expression in terms of Randell's generators of the meridians around the exceptional divisors in the blow-up $\bar{X}$ of $\mathbb{CP}^2$ in the singular points of $\mathscr{A}$. We use this to investigate the partial compactifications of $M(\mathscr{A})$ contained in $\bar{X}$ and give a counterexample to a statement suggested by A. Dimca and P. Eyssidieux to the effect that the fundamental group of such an algebraic variety is finite whenever its abelianization is.