# On the Number of Ordinary Lines Determined by Sets in Complex Space

## Charles Wolf, Rutgers University

## December 6, 2016

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.