Lines at infinity in the complex projective plane

Question

I have been learning about complex projective spaces and my professor was saying something about there being three complex projective lines at the points of infinity in a $\Bbb C \Bbb P _2$ (ie a complex projective plane). I don't really understand this. If anyone could explain this or perhaps point me towards some references for me to read, that would be greatly appreciated.

Answer 1

If we have CP2 and its coordinates are (x,y,z) then the three charts are (x,y,1) , (x,1,z) and (1,y,z) and the points at infinity are (0,0,1), (0,1,0), (1,0,0). Is there any way to relate CP1 (ie coordinates (x,y)) to any these points at infinity?

The corresponding lines at infinity, given these choices of affine planes within the complex plane, would be $\{((x,y,0)\mid x,y\in \mathbb C\}$, $\{((x,0,z)\mid x,z\in \mathbb C\}$ and $\{((0,y,z)\mid y,z\in \mathbb C\}$ respectively.

The relationship with the points you've given (respectively) is that they are orthogonal with respect to the inner product. Said another way, the $1$-d subspace represented by each of those points has a normal plane complement in $\mathbb C^3$, which when collapsed into $\mathbb{CP}^2$ is a line (the line of points at infinity.

To me, the points you gave are the origin of the chosen affine plane, not some point on the corresponding line at infinity.