# 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.

• There might be useful information in Hirzebruch's "Algebraische Flachen und Geradenkonfigurationen". – inkspot Feb 20 '17 at 8:07

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.

• Is this 12-point configuration just the dual of the Hesse configuration? – potentially dense Feb 20 '17 at 13:54
• No. The dual of the Hesse configuration has $9$ lines, whereas the Sylvester-Gallai configuration of cardinality $12$ (which over $\mathbb{C}$ is the dual of the so-called Ceva configuration) has $16$ lines. See this paper by Dolgachev, Section 5: arxiv.org/abs/math/0304258 – Francesco Polizzi Feb 20 '17 at 14:57
• OK, thanks for the extra infomation! – potentially dense Feb 20 '17 at 15:24
• I'm confused about something. Are you sure that the Sylvester-Gallai configuration of cardinality 12 has 16 lines and not 19? Unless I'm misreading the Kelly-Nwankpa paper it seems that their two configurations of cardinality 12 both have 19 lines. – Timothy Chow Feb 22 '17 at 5:02
• @Timoty Chow. You are right. We must also add to the $4^2=16$ points of the Ceva configuration the three coordinate vertices, obtaining a configuration with $19$ points and $12$ lines. The dual of this configuration is a Sylvester-Gallai configuration with $12$ points and $19$ lines. Thanks for the remark. – Francesco Polizzi Feb 22 '17 at 9:17

$$(x^n - y^n)(y^n - z^n)(z^n - x^n) = 0, \qquad n \ge 3.$$
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.