# Sylvester Gallai for complex projective plane

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!