A Sylvester-Gallai configuration in the the complex projective plane is a finite number of $n\ge 2$ points in the complex projective plane such that there is no line through exactly two of them. Trivial examples are obtained by taking $n\ge 3$ points on the same line. There is also the classical Hessian configuration, obtained by taking the nine inflexion points of a smooth cubic.



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> **Question.** Is there any other example? Is there a way to classify these examples? I would be particularly interested in finding a non-collinear example with an even number of points. Does it exist?
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Remark: In higher dimension, it is known that the configuration of points has to be coplanar. If the points have coordinates defined over $\mathbb{R}$, the Sylvester-Gallai theorem shows that any configuration as above is in fact collinear. Over finite fields one can of course find plenty of configurations by taking all points.



What are Sylvester-Gallai configurations in the complex projective plane?
Yes, there are other Sylvester-Gallai configurations in $\mathbb{P}^2(\mathbb{C})$. Apart from the Hesse configuration (that contains $9$ points) the minimum number of points for a non-collinear configuration is $12$. 


A configuration with $12$ points actually exists over any field $\mathbb{K}$ of characteristic different from $2$ and containing a square root of $-1$, as proven by Kelly and Nwankpa.


See the answers to the MathOverflow question *[The Sylvester-Gallai theorem over $p$-adic fields](https://mathoverflow.net/questions/258459/the-sylvester-gallai-theorem-over-p-adic-fields)* for references and more details. 


