Hello,
Suppose that $C_1,C_2\subset\mathbb P^2$ are projective curves (over $\mathbb C$); $C_1$ and $C_2$ may be reducible but they must not have a common component. Let $L\subset \mathbb P^2$ be a line in general position, and denote by $H$ the kernel of the natural epimorphism $\pi_1(\mathbb P^2\setminus(C_1\cup C_2))\to \pi_1(\mathbb P^2\setminus C_1)$.
Is it true that $H$ is the smallest normal subgroup in $\pi_1(\mathbb P^2\setminus(C_1\cup C_2))$ containg small loops in $L$ around the points of $L\cap C_2$?
Thank you in advance,
Serge
$\pi_1$is commutative and generated by loops about components:) If I am not mistaken, the answer is also positive if $C_2$ has only normal crossings and each component of $C_2$ is transversal to $C_1$. I wonder what happens in general... $\endgroup$