# Pencil and fundamental group

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 .

• I should also say that it can be very complicated to describe the fundamental group of $\mathbb{CP}^{2} \setminus C$ where $C$ is an algebraic curve if $C$ is singular, the group will depend on the precise singularities that the curve (it is still not completely understood). However, if $C$ is a degree $d$ smooth curve in $\mathbb{CP}^{2}$ then $\pi_{1}(\mathbb{CP}^{2} \setminus C) \cong \mathbb{Z}_{d}$ – Nick L Mar 23 at 5:14
• @NickL $f_3$ has degree $3$, and $f_3^2$ has degree $6$ – 6666 Mar 23 at 18:32
• I see, I will delete my first comment. – Nick L Mar 24 at 0:57
• @NickL can I ask why if $C$ is a degree $d$ smooth curve, then $\pi_1(Cp^2\setminus C)=Z/d$? – 6666 Apr 30 at 4:02
• This was proved by Zariski in 1929, see page 23 here www.ims.nus.edu.sg/Programs/braids08/files/garber2_wkshp.pdf . You can probably find the original paper aswell. – Nick L Apr 30 at 15:12