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,

  • $\begingroup$ If $C_1\cup C_2$ has only normal crossings, the answer is positive; I am not sure otherwise. $\endgroup$ – Misha Apr 19 '13 at 15:49
  • $\begingroup$ @Misha: you bet it's positive in this case, when $\pi_1$ 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... $\endgroup$ – Serge Lvovski Apr 20 '13 at 4:04

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