On the number of ordinary lines determined by sets in complex space

Abstract

Kelly's theorem states that a set of $n$ points affinely spanning $\mathbb{C}^3$ must determine at least one ordinary complex line (a line passing through exactly two of the points). Our main theorem shows that such sets determine at least $3n/2$ ordinary lines, unless the configuration has $n-1$ points in a plane and one point outside the plane (in which case there are at least $n-1$ ordinary lines). In addition, when at most $2n/3$ points are contained in any plane, we prove a theorem giving stronger bounds that take advantage of the existence of lines with 4 and more points (in the spirit of Melchior's and Hirzebruch's inequalities). Furthermore, when the points span 4 or more dimensions, with at most $2n/3$ points contained in any three dimensional affine subspace, we show that there must be a quadratic number of ordinary lines.