# Sylvester--Gallai configurations on algebraic curves in C^2

## TL;DR

This paper investigates complex Sylvester-Gallai configurations, proving a conjecture that the Fermat configurations are the only infinite class when most points lie on a low-degree algebraic curve.

## Contribution

It proves the conjecture that Fermat configurations are the only infinite class under the '99% structure' assumption involving low-degree algebraic curves.

## Key findings

- Fermat configurations are unique among infinite classes under the given conditions.
- Most points lying on a low-degree algebraic curve implies the configuration is Fermat.
- The conjecture is confirmed in the specified structured case.

## Abstract

The Sylvester-Gallai theorem says that for any finite set of non-collinear points in $\R^2$, there is some line passing through exactly two points of the set. Over the complex numbers, this theorem fails: there are finite configurations with the property that any line through two points also passes through a third. Only one infinite class of examples (the Fermat configurations) is known, and it is a folklore conjecture that this is the only infinite class of examples. We prove this conjecture in the ``99\% structure'' case where we assume most of the points lie on a low degree algebraic curve.