Let {C \subset \mathbb{P}^2} be a smooth plane quartic, so that {C} is a nonhyperelliptic genus 3 curve imbedded canonically. In the previous post, we saw that bitangent lines to {C} were in natural bijection with effective theta characteristics on {C}, or equivalently spin structures (or framings) of the underlying smooth manifold.

It is a classical fact that there are {28} bitangents on a smooth plane quartic. In other words, of the {64} theta characteristics, exactly {28} of them are effective. A bitangent here will mean a line {L \subset \mathbb{P}^2} such that the intersection {L \cap C} is a divisor of the form {2(p + q)} for {p, q \in C} points, not necessarily distinct. So a line intersecting {C} 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 {24} flex lines. This is a special case of the Plücker formulas, and this post will describe a couple of the relevant ideas.  (more…)