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This paper of Zariski contains this statement: If $C$ is a curve in $\mathbb{CP}^2$, and $L$ is a generic line, then the injection $L\setminus C \hookrightarrow \mathbb{CP}^2\setminus C$ induces an isomorphism a surjection of fundamental groups.

  1. Edited version: Let $C$ be a fixed curve (not necessarily smooth) in $\mathbb{CP}^2$. Does there exist a characterization of curves $D$ such that the injection $D\setminus C \hookrightarrow \mathbb{CP}^2\setminus C$ induces a surjection of fundamental groups? In other words, how to check if every loop in $\mathbb{CP}^2\setminus C$ can be deformed into a loop in $D\setminus C$?

  2. More precisely, if $V$ is a base point free linear family of curves, then is it true that the answer to question 1 will be true for generic elements in $V$? This question has a positive answer if $V$ is a pencil of lines (by the statement written in the first line) - so the question is what happens in other cases.

  3. How about Questions 1 and 2 with $\mathbb{CP}^2$ replaced by $\mathbb{C}^2$?

I just got interested into this sort of questions, and am a bit overwhelmed with the vast literature on topology of complements of hypersurfaces. Any introductory reference would be greatly appreciated as well.

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    $\begingroup$ Perhaps I am missing something. $L-C$ has the homotopy type of a wedge of circles, so its fundamental group is a free group. On the other hand, for sufficiently nice curves $C$ the fundamental group of $\mathbb{P}^2 -C$ is a cyclic group. I believe is true that every loop in the later space can be deformed to a loop in $L -C$, so that $\pi_1(L-C)\to \pi_1(\mathbb{P}^2-C)$ is automatically surjective, but there are typically many more relations in the latter group, arising from the fact that we can vary the line in a pencil and obtain a nontrivial monodromy action. $\endgroup$ – Jack Huizenga Nov 28 '13 at 6:59
  • $\begingroup$ @Jack: You are right. But I want to know how, for a fixed (not necessarily nice) $C$, to choose $D$ so that every loop in $\mathbb{P}^2-C$ can be deformed into a loop in $D-C$. Even in this case $D$ can be chosen to be a generic line. But can we take it to be a generic element in any other linear system (not necessarily containing lines)? $\endgroup$ – auniket Nov 28 '13 at 10:58
  • $\begingroup$ @auniket: Since Jack is correct, and since you wrote something wrong, you should either edit your question or, perhaps, delete it entirely. $\endgroup$ – Jason Starr Nov 28 '13 at 13:17
  • $\begingroup$ The book by Orlik-Terao seems nice: springer.com/mathematics/geometry/book/… $\endgroup$ – Ian Agol Nov 28 '13 at 16:35
  • $\begingroup$ @Jason: I don't think I have written anything wrong. In any event, I edited the question to make it (hopefully) clearer. Thanks. $\endgroup$ – auniket Nov 28 '13 at 17:50
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I think the answer is no for generic curves: the fundamental group of the complement of a generic curve is Abelian. There is a nice survey of the fundamental groups of complements of plane curves:

MR2342909 Libgober, A. Lectures on topology of complements and fundamental groups. Singularity theory, 71–137, World Sci. Publ., Hackensack, NJ, 2007.

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