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 ℂ3 must determine at least one ordinary complex line (a line incident to 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 n / 2 points are contained in any plane, we prove stronger bounds that take advantage of the existence of lines with four or more points (in the spirit of Melchior’s and Hirzebruch’s inequalities). Furthermore, when the points span four or more dimensions, with at most n / 2 points contained in any three-dimensional affine subspace, we show that there must be a quadratic number of ordinary lines.