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…)