The general ternary form can be recovered by its Hessian

The Hessian map is the rational map that sends a homogeneous polynomial to the determinant of its Hessian matrix. We prove that the Hessian map is birational on its image for ternary forms of degree d 4, d 5, by considering the action of the orthogonal group. In a previous paper we proved the analogous result for binary forms, with more geometric techniques.