A Sylvester-Gallai theorem for cubic curves

TL;DR

This paper proves a new geometric theorem for cubic curves, showing that a large enough point set not on a single cubic must have a cubic passing through exactly nine points, advancing the understanding of algebraic curve configurations.

Contribution

It establishes the first known case of a Sylvester-Gallai type theorem for cubic curves, confirming a long-standing conjecture for degree three.

Findings

Proves a Sylvester-Gallai variant for cubic curves in the plane.

Shows existence of a cubic passing through exactly nine points in certain configurations.

Resolves the first open case of a conjecture by Wiseman and Wilson from 1988.

Abstract

We prove a variant of the Sylvester-Gallai theorem for cubics (algebraic curves of degree three): If a finite set of sufficiently many points in $\mathbb{R}^2$ is not contained in a cubic, then there is a cubic that contains exactly nine of the points. This resolves the first unknown case of a conjecture of Wiseman and Wilson from 1988, who proved a variant of Sylvester-Gallai for conics and conjectured that similar statements hold for curves of any degree.