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 .

  • $\begingroup$ 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 $\pi_{1}(\mathbb{CP}^{2} \setminus C) \cong \mathbb{Z}_{d}$ $\endgroup$ – Nick L Mar 23 at 5:14
  • $\begingroup$ @NickL $f_3$ has degree $3$, and $f_3^2$ has degree $6$ $\endgroup$ – 6666 Mar 23 at 18:32
  • $\begingroup$ I see, I will delete my first comment. $\endgroup$ – Nick L Mar 24 at 0:57
  • $\begingroup$ @NickL can I ask why if $C$ is a degree $d$ smooth curve, then $\pi_1(Cp^2\setminus C)=Z/d$? $\endgroup$ – 6666 Apr 30 at 4:02
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    $\begingroup$ 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. $\endgroup$ – Nick L Apr 30 at 15:12

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