The classical Sylvester-Gallai theorem states the following: Given a finite set of points in the 2-dimensional Euclidean plane, not all collinear, there must exist a line containing exactly 2 points (referred to as an ordinary line). In a recent result, Green and Tao were able to give optimal lower bounds on the number of ordinary lines for large finite point sets. In this talk we will consider the situation over the complex numbers. While the Sylvester-Gallai theorem as stated is false in the complex plane, Kelly's theorem states that if a finite point set in 3-dimensional complex space is not contained in a plane, then there must exist an ordinary line. Using techniques developed for bounding ranks of design matrices, we will show that either such a point set must determine at least 3n/2 ordinary lines or at least n-1 of the points are contained in a plane.
Joint work with Abdul Basit, Zeev Dvir and Shubhangi Saraf.