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

## TL;DR

This paper explicitly computes the fundamental group of partial compactifications of the complement of a real line arrangement in projective space, providing counterexamples to a conjecture relating finiteness of the group to its abelianization.

## Contribution

It offers an explicit algebraic description of the fundamental group for these compactifications and challenges a previous conjecture about the finiteness of such groups.

## Key findings

- Explicit fundamental group expressions in terms of Randell's generators
- Counterexample to the conjecture linking group finiteness to abelianization
- Insight into the topology of partial compactifications of line arrangement complements

## 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.