Let $f_i(x,y,z)$ be a homogeneous polynomial of degree $i$. Then for pencil $C_{a,b}=a(f_2)^3+b(f_3)^2$, we have a map $\phi:\mathbb{C}P^2\setminus B\to \mathbb{C}P^1$, where $B$ is the base locus of pencil $C_{a,b}$, and the map is given by $[x,y,z]\mapsto [a,b]=[(f_3)^2(x,y,z):(f_2)^3(x,y,z)]$.
Let $C$ be the union $\phi^{−1}(1,0)\cup \phi^{−1}(0,1)\cup\phi^{−1} (1,1)$ and let $L$ be a line containing only smooth points of $C$ and transversal to $C$.
Then how to see the map: $\pi_1(L\setminus L\cap C)\to\pi_1(\mathbb{C}P^2\setminus C)$ induced by inclusion is surjective? Moreover, how to compute these two fundamental groups?
I guess we can use Lefschetz hyperplane theorem to deduce the surjection. But can't go further .