The general ternary form can be recovered by its Hessian

TL;DR

This paper proves that the Hessian map is birational for ternary forms of degree at least 4 (excluding 5), revealing a deep connection between the form and its Hessian through group actions.

Contribution

It establishes the birationality of the Hessian map for ternary forms of degree d≥4, extending previous binary form results using group action techniques.

Findings

Hessian map is birational for ternary forms of degree d≥4, d≠5.

The proof involves the action of the orthogonal group.

Previous results for binary forms are extended to ternary forms.

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