Let $D$ be a divisor in $\mathbb P^2\_{\mathbb C}$ and let $X= \mathbb P^2\_{\mathbb C} - D$.


Under what condition on $D$ is the fundamental group of $X$ infinite? 



Fundamental groups of complements of divisors in $\mathbb P^2$
I'd leave this as a comment, but I don't have enough reputation. Consider the long exact sequence in homology of the pair $(\mathbb{P}^2, \mathbb{P}^2-D)$. Since $H\_1(\mathbb{P}^2,\mathbb{Z}) = 0$ and $H\_2(\mathbb{P}^2, \mathbb{Z}) = \mathbb{Z}$, this has a segment
$$\mathbb{Z}\rightarrow H\_2(\mathbb{P}^2, \mathbb{P}^2-D)\rightarrow H\_1(\mathbb{P}^2-D)\rightarrow 0.$$
The middle term is freely generated by the components of $D$. So if $D$ has 2 or more irreducible components, $H\_1(\mathbb{P}^2-D, \mathbb{Z})$ has positive rank and $\pi\_1(\mathbb{P}^2-D,\*)$ is infinite. 


