Let $k$ be an algebraically closed field of characteristic zero, let $f(X,Y,Z) \in k[X,Y,Z]$ be a homogeneous polynomial of degree $d \ge 3$ such that $\Big(\dfrac {\partial f}{\partial X} (p) ,\dfrac {\partial f}{\partial Y} (p) , \dfrac {\partial f}{\partial Z} (p) \Big)\ne (0,0,0), \forall 0\ne p \in k^3$ i.e. the projective curve given by $f$ is smooth. Consider the determinant of the Hessian matrix (https://en.wikipedia.org/wiki/Hessian_matrix) of $f$, denote it by $H(f)$ . Then is it true that $H(f)$ is a homogeneous polynomial of degree $3(d-2)$ ?
Due to my conditions, it s easy to see that $f$ is irreducible (by Bezout's Theorem) and that if we have a projective change of coordinates given by $A \in Gl(3,k)$ , then $H(f)(X,Y,Z)=\det(A)^2 H(g)((X, Y,Z)A^t)$, where $g(X,Y,Z)=f((X,Y,Z))(A^{-1})^t)$; so the degree of the Hessian doesn't change with a projective change of co-ordinates. I'm unable to deduce anything else.
Please help.