# Computing fundamental groups of the complement of plane curves

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.

• 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. – Jack Huizenga Nov 28 '13 at 6:59
• @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)? – auniket Nov 28 '13 at 10:58
• @auniket: Since Jack is correct, and since you wrote something wrong, you should either edit your question or, perhaps, delete it entirely. – Jason Starr Nov 28 '13 at 13:17
• The book by Orlik-Terao seems nice: springer.com/mathematics/geometry/book/… – Ian Agol Nov 28 '13 at 16:35
• @Jason: I don't think I have written anything wrong. In any event, I edited the question to make it (hopefully) clearer. Thanks. – auniket Nov 28 '13 at 17:50