Let $C$ be a (singular, reducible) curve in the complex projective plane $\mathbb{P}^2$. An old problem is to study the topology of the complement $\mathbb{P}^2-C$. A famous result in this direction was originally stated by Zariski, but not proven until 1980 by Fulton. It states that if $C$ has only nodes for singularities (that is, double points with distinct tangents), then the fundamental group $\pi_1(\mathbb{P}^2-C)$ is abelian.

My problem is as follows: again, let $C$ be a (singular, reducible) curve in $\mathbb{P}^2$. Now, blow up the plane at one of the singular points of $C$. Denote the resulting surface $S_1$, the exceptional divisor by $E_1$, and the proper transform of $C$ by $\hat{C}_1$. Then blow up $S_1$ at a singular point of $\hat{C}_1$, and repeat this procedure to get a surface $S=S_n$, exceptional divisors $E_1,\ldots,E_n\subset S$, and the proper transform $\hat{C}\subset S$ of the original curve. I am interested in the topology of the complement $X=S-(\hat{C} \; \cup E_{i_1} \cup \ldots \cup E_{i_k})$. In other words, $X$ is obtained from $S$ by deleting $\hat{C}$ as well as some (but not all) of the exceptional divisors obtained from blowing up. In particular, I would like to prove that the fundamental group $\pi_1(X)$ is abelian.

If the $C$ has only nodes for singularities, then $\pi_1(\mathbb{P}^2-C)$ is already abelian, and it is not hard to see that $\pi_1(X)$ will also be abelian. However, I am more interested in some cases where $C$ has slightly more complicated singularities. For example, if $C$ has a triple point with distinct tangents, $\pi_1(\mathbb{P}^2-C)$ is non-abelian; but I suspect that if I blow up the plane at the triple point, then $\pi_1(S-\hat{C})$ will be abelian. Does anyone know of any techniques or references for dealing with this situation?

(Note: It is certainly not the case that $\pi_1(X)$ will always be abelian. I am interested in techniques that will allow me to prove it is abelian for certain specific curves and configurations of exceptional divisors.)