What are Sylvester-Gallai configurations in the complex projective plane? 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.

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?

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.
 A: 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 for references and more details. 
A: I asked around about this question a while ago and the best answer I got was from Konrad Swanepoel.  There are the well-known "Fermat" examples
$$(x^n - y^n)(y^n - z^n)(z^n - x^n) = 0, \qquad n \ge 3.$$
The $3n$ lines here, together with the $n^2$ points of intersection and the 3 coordinate vertices, form a dual Sylvester–Gallai configuration.  So these give rise to a Sylvester–Gallai configuration with $3n$ points and $n^2+3$ lines.  If you want an even number of points, just take $n$ to be even.
There are only two other ("sporadic") examples known, one due to Klein and one due to Wiman, which are described for example in this paper.  They have 21 and 
45 points respectively so they don't give you what you want.
According to Swanepoel the complete classification is still an open problem, but he believes that there are at most finitely many more sporadic examples. 
In 1973, Kelly and Nwankpa classified all Sylvester–Gallai designs (a more general concept than configurations) with at most 14 points.  Swanepoel warned me that there are some errors in this paper, but if we take its results at face value then it shows that there are no other complex Sylvester–Gallai configurations this small. It should be possible to extend the computational search beyond 14 points but nobody seems have done so (or at least has not published the results).  By considering complex reflection groups, a colleague of mine has performed an unpublished computation that, if correct, shows that there are no other examples with reflective symmetry in every line.
