Let ${C \subset \mathbb{P}^2}$ be a smooth degree ${d}$ curve. Then there is a dual curve

$\displaystyle C \rightarrow (\mathbb{P}^2)^*,$

which sends ${p \in C \mapsto \mathbb{T}_p C}$, to the (projectivized) tangent line at ${p \in C}$. Such lines live in the dual projective space ${(\mathbb{P}^2)^*}$ of lines in ${\mathbb{P}^2}$. We will denote the image by ${C^* \subset \mathbb{P}^2}$; it is another irreducible curve, birational to ${C}$.

This map is naturally of interest to us, because, for example, it lets us count bitangents. A bitangent to ${C}$ will correspond to a node of the image of the dual curve, or equivalently it will be a point in ${(\mathbb{P}^2)^*}$ where the dual map ${C \rightarrow (\mathbb{P}^2)^*}$ fails to be one-to-one. In fact, if ${C}$ is general, then ${C^*}$ will have only nodal and cuspidal singularities, and we we will be able to work out the degree of ${C^*}$. By the genus formula, this will determine the number of nodes in ${C^*}$ and let us count bitangents.

The purpose of this post is to describe this, and to discuss this map from the point of view of jet bundles, discussed in the previous post. (more…)

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…)