I would like to ask a question about plane projective curves. Let $C\subset{\mathbb P}_2={\mathbb P}(V)$ be a plane curve of degree $n\geq 3$. Then we have a non splitted exact sequence $$0\longrightarrow{\mathcal O}_C(-n)\longrightarrow\Omega_{{\mathbb P}_2|C}\longrightarrow\omega_C={\mathcal O}_C(n-3)\longrightarrow 0,$$ hence an element $\sigma$ of $H^1({\mathcal O}_C(3-2n))$. By using the exact sequence $$0\longrightarrow{\mathcal O}_{{\mathbb P}_2}(-n)\longrightarrow{\mathcal O}_{{\mathbb P}_2}\longrightarrow{\mathcal O}_C\longrightarrow 0$$ we get an inclusion $$H^1({\mathcal O}_C(3-2n))\ \subset\ H^2({\mathcal O}_{{\mathbb P}_2}(3-3n))\ =\ S^{3n-6}V \ . $$ Hence $\sigma$ corresponds to a curve of degree $3n-6$ in the dual projective plane ${\mathbb P}_2^*$.
My question is : what is this curve ? (or did I a mistake somewhere ?).
