Fundamental groups of complements of divisors in $\mathbb P^2$

Question

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?

Answer 1

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.