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…)