December 5, 2016
Consider a set of $n$ points in $\mathbb R^d$. The classical theorem of Sylvester-Gallai says that, if the points are not all collinear then there must be a line through exactly two of the points. Let us call such a line an "ordinary line". In a recent result, Green and Tao were able to give optimal linear lower bounds (roughly $n/2$) on the number of ordinary lines determined $n$ non-collinear points in $\mathbb R^d$. In this talk we will consider the analog over the complex numbers. While the Sylvester-Gallai theorem as stated above is known to be false over the field of complex numbers, it was shown by Kelly that for a set of $n$ points in $\mathbb C^d$, if the points don’t all lie on a $2$-dimensional plane then the points must determine an ordinary line. Using techniques developed for bounding the rank of design matrices, we will show that such a point set must determine at least $3n/2$ ordinary lines, except in the trivial case of $n - 1$ of the points being contained in a $2$ dimensional plane.