The general ternary form can be recovered by its Hessian (with an appendix by Jerson Caro and Juanita Duque-Rosero)

Abstract

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 $d\ge 4$, $d\ne 5$, by considering the action of the orthogonal group. In a previous paper, we proved the analogous result for binary forms, with more geometric techniques.