I am trying to understand all details of the complex projective space CPn. Since surely CP1 must be the most simply to understand, I started out with it and even there I cannot gain full understanding. I would be eminently thankful for any help :D Nearly all texts trying to describe CPn get very general about it, using mostly language and not so much explicitness. Perhaps they assume some foreknowledge that I as a physicist lack :) I was wondering if someone could shed some light on this. 1) Firstly, neither on Wikipedia nor on Wolfram Mathworld can I find a clear, thorough and detailed definition of what a complex line is. Is it a collection of points of dimension 2? How does one parametrize it in the same way that a real line (x,y) in the plane has a parametrization (x(t),y(t)) for some interval of t. 2) Furthermore, since complex lines are fundamentally different to real lines, I have no idea what a complex line "through the origin" is supposed to mean. 3) Do we identify ANY two points in C^(n+1) that differ by ANY multiplicative factor lambda? Because from what I understand, any complex point z can be moved to any other complex point w if a suitable lambda is found, just use lambda = z / w This would mean that all points in C^(n+1) are identified. I hope that when I spell it out this way, you guys can see where I have gone wrong. 4) To specify a "complex line" in CPn, what is the minimum amount of information you will need to uniquely determine 1 exact "line" ? (points on the manifold need to be unique and fully defined).