Here is an interesting problem, which I could not solve and would appreciate any comment in solving it;


Assume that we have omitted infinitely many lines from $\mathbb{C}P^2$ to obtain $\mathbb{C}^2$ and now, consider the following lines in which $(z,w)$ is the coordinate of $\mathbb{C}^2$: $z=0, z=1, w=0, w=1, z=w.$ 


What is the fundamental group of the complement of these lines in $\mathbb{C}^2$? 



fundamental group of the complement of lines in C^2
What you have is an arrangement of hyperplanes in $\mathbb C^2$ obtained by complexifying a real arrangement. There is a simple way to describe the fundamental group of its complement, due to Randell. You can find the details and references in the beautiful book *Arrangement of hyperplanes* by Peter Orlik and Hiroaki Terao, in section 5.3.


