(1) Let $C\subseteq \mathbb{P}^2$ be a curve defined by a homogeneous equation $F(X,Y,Z)=0$. The dual curve of C, denoted by $C'$, is the image of the map $$[X,Y,Z]\to [\frac{\partial F}{\partial X},\frac{\partial F}{\partial Y},\frac{\partial F}{\partial Z}].$$
Ques1. How to write the homogeneous equation of $C'$?
(2) Let $F_1,F_2,F_3$ be homogeneous polynomials of degree $k$ on $\mathbb{P}^2$ and $J(X,Y,Z)$ be the determinant of the Jacobian matrix $$\frac{\partial(F_1,F_2,F_3)}{\partial(X,Y,Z)}.$$ Let $C$ be the curve defined by $J=0$.
Ques2. I wish find the precise equation of the dual curve of $C$ and give a direct proof that the dual curve has only nodes and ordinary cusps for sufficient genral $F_1,F_2,F_3 \in\mathcal{O}(k)$.
(3) Let $\pi:\mathbb{P}^2\to \mathbb{P}^2$ be a generic cover of projective plane, $R\subseteq \mathbb{P}^2$ be the ramified divisor and $B=\pi(R)$ be the determinant curve of $\pi$.
Ques3. Is $B$ the dual curve of $R$? I wish find an answer by a straightforwards computation of polynomial equations.
