(Dual curve)
Let \(C \subseteq \PP^2\) be an irreducible plane curve over an algebraically closed field of characteristic \(0\) that is not a line. The dual curve \(C^\vee \subseteq (\PP^2)^\vee\) is the closure of the set of tangent lines \(T_p C\) at the smooth points \(p\) of \(C\). Its degree is the class of \(C\): the number of tangent lines to \(C\) through a general point of \(\PP^2\).
(Biduality) \((C^\vee)^\vee = C\).
(Plücker) If \(C\) has degree \(d\) and its only singularities are \(\delta\) nodes and \(\kappa\) ordinary cusps, then \[\deg C^\vee = d(d-1) - 2\delta - 3\kappa.\] In particular a smooth plane curve of degree \(d \geq 2\) has class \(d(d-1)\).
The dual of a smooth conic is a smooth conic. The dual of a smooth plane cubic has degree \(6\); its \(9\) cusps correspond to the \(9\) flexes of the cubic, and it has no nodes because a smooth cubic has no bitangents. Its geometric genus is \(\binom{5}{2} - 9 = 1\), equal to the genus of the cubic, since the Gauss map \(C\to C^\vee\) is birational.
Dependencies
D-CRVPLSINGNode, cusp, and tacnode; the delta invariant and the genus formulauses