Let be a smooth plane quartic, so that is a nonhyperelliptic genus 3 curve imbedded canonically. In the previous post, we saw that bitangent lines to were in natural bijection with effective theta characteristics on , or equivalently spin structures (or framings) of the underlying smooth manifold.
It is a classical fact that there are bitangents on a smooth plane quartic. In other words, of the theta characteristics, exactly of them are effective. A bitangent here will mean a line such that the intersection is a divisor of the form for points, not necessarily distinct. So a line intersecting in a single point (with contact necessarily to order four) is counted as a bitangent line. In this post, I’d like to discuss a proof of a closely related claim, that there are flex lines. This is a special case of the Plücker formulas, and this post will describe a couple of the relevant ideas. (more…)