A Sylvester-Gallai result for concurrent lines in the complex plane

TL;DR

This paper proves a new geometric property in the complex plane related to points on concurrent lines, confirming a conjecture and extending previous results with an optimal bound.

Contribution

It establishes an optimal bound for points on concurrent lines in c^2, resolving a conjecture and generalizing earlier work.

Findings

If a set of points on m concurrent lines has one line with more than m-2 points, a line with exactly two points exists.

The bound m-2 is proven to be optimal.

The result confirms a conjecture of Frank de Zeeuw and extends Kelly and Nwankpa's work.

Abstract

We show that if a set of points in $\mathbb{C}^2$ lies on a family of $m$ concurrent lines, and if one of those lines contains more than $m-2$ points, then there is a line passing through exactly two points of the set. The bound $m-2$ in our result is optimal. Our main theorem resolves a conjecture of Frank de Zeeuw, and generalizes a result of Kelly and Nwankpa.