(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.

  • 2
    $\begingroup$ Read the book of Gelfand-Kapranov-Zelevinsky. To make sense of question (3) you need first to identify one $P^2@ with the dual of the other. $\endgroup$ – Sasha Oct 27 '12 at 18:35

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