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

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.