Let be a smooth degree curve. Then there is a dual curve
which sends , to the (projectivized) tangent line at . Such lines live in the dual projective space of lines in . We will denote the image by ; it is another irreducible curve, birational to .
This map is naturally of interest to us, because, for example, it lets us count bitangents. A bitangent to will correspond to a node of the image of the dual curve, or equivalently it will be a point in where the dual map fails to be one-to-one. In fact, if is general, then will have only nodal and cuspidal singularities, and we we will be able to work out the degree of . By the genus formula, this will determine the number of nodes in and let us count bitangents.