Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

The general ternary form can be recovered by its Hessian

Ciro Ciliberto, Giorgio Ottaviani, Jerson Caro, Juanita Duque-Rosero

TL;DR

The paper proves that the general Hessian map $h_{d,2}$ is birational on its image for all $d\ge 4$, $d\neq 5$, by leveraging an ${\rm SO}(3)$-equivariant approach and a harmonic decomposition of ternary forms to control indeterminacy. It develops a criterion for birationality based on closed orbits of the ${\rm SO}(3)$-action and analyzes the differential via explicit ${\rm SO}(3)$-equivariant maps between harmonic components, with separate treatment of even and odd degrees. A key technical ingredient is an indeterminacy analysis (Theorem 'divides') showing divisibility phenomena near cone forms, supported by a representation-theoretic framework and arithmetic input. The Appendix establishes integral-point results on two elliptic curves to bound possible degeneracies, providing the needed maximal-rank checks at critical points. Overall, the work extends the binary-form result to ternary forms and clarifies how the Hessian determines a general ternary form up to birational equivalence, aside from a known exception at $d=5$.

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.

The general ternary form can be recovered by its Hessian

TL;DR

The paper proves that the general Hessian map is birational on its image for all , , by leveraging an -equivariant approach and a harmonic decomposition of ternary forms to control indeterminacy. It develops a criterion for birationality based on closed orbits of the -action and analyzes the differential via explicit -equivariant maps between harmonic components, with separate treatment of even and odd degrees. A key technical ingredient is an indeterminacy analysis (Theorem 'divides') showing divisibility phenomena near cone forms, supported by a representation-theoretic framework and arithmetic input. The Appendix establishes integral-point results on two elliptic curves to bound possible degeneracies, providing the needed maximal-rank checks at critical points. Overall, the work extends the binary-form result to ternary forms and clarifies how the Hessian determines a general ternary form up to birational equivalence, aside from a known exception at .

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 , , by considering the action of the orthogonal group. In a previous paper we proved the analogous result for binary forms, with more geometric techniques.
Paper Structure (6 sections, 31 theorems, 105 equations)

This paper contains 6 sections, 31 theorems, 105 equations.

Table of Contents

  1. Introduction
  2. The Hessian map and the ${\rm SO}(r+1)$-action on ${\mathbb P}({\mathrm{Sym}}^d({\mathbb C}^{r+1}))$
  3. The indeterminacy values assumed by the Hessian map for ternary forms
  4. Appendix: Integral points on two elliptic curves
  5. The curve $y^2(2x + 1) + y(x + 2) - 3x(x + 1) = 0$
  6. The curve $2xy^2 + y(-3x + 3) - x(3x - 1) = 0$

Key Result

Theorem 1.1

The Hessian map $h_{d,2}$ is birational on its image for any $d\geqslant 4$, $d\neq 5$.

Theorems & Definitions (70)

  • Theorem 1.1
  • Proposition 2.1
  • proof
  • Remark 2.2
  • Lemma 2.3
  • Proposition 2.4
  • proof
  • Remark 2.5
  • Proposition 2.6
  • proof
  • ...and 60 more