I understand that the Sylvester Gallai theorem doesn't hold for the projective complex plane. Can anyone explain why does Kelly's proof: Here doesn't hold for the complex projective plane?
A counter example for the theorem could be the following SG configuration from basic AG course: The 9 inflection points of a cubic in $\mathbb{P}^2$, it is a well known theorem that every line that passes through 2 of them must pass through a 3rd one, and a simple application of Bezout's theorem shows that each line intersecting a cubic curve will meet it in at most 3 points.
Thank you very much in advance!
I think the main problem is that lines in the complex projective plane aren't one dimensional (topologically), but instead they are topologically spheres, being isomorphic to complex projective one space.
Thus when the proof talks about points in a line being on one side or the other of each other, it doesn't make sense in the complex case. Since complex lines are topologically two dimensional, so they don't have a betweenness relation.