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On the generic injectivity of Hessian maps of ternary forms

Valentina Beorchia

TL;DR

This work develops a geometric framework to investigate the generic injectivity of the Hessian map for ternary forms by examining the graph of the polar map and its ramification as encoded by the Hessian curve. It shows that when two forms share the same Hessian, equality of ramification forces proportional gradients (hence proportional polynomials), while noncoincident ramification leads to a condition under which the polar linear systems must coincide, provided a product-type ruled surface arises. The results connect ramification geometry, Hessian/Steinerian curves, and polar linear systems, and situate the problem within Mammana’s classification of forms with equal polar linear systems, highlighting cases where injectivity can be established and identifying open scenarios where the surface is not a product. Overall, the paper offers a concrete geometric strategy to tackle the generic injectivity question and clarifies how equal Hessians constrain the underlying gradients and linear systems.

Abstract

We study the problem of the generic injectivity of the Hessian map, associating with a proportionality class of a ternary form the class of its Hessian determinant, conjectured by C. Ciliberto and G. Ottaviani and recently proved by the same authors. Taking into account that the Hessian curve is the ramification divisor associated with the polar map, we perform a study of the problem using a geometric description of the graph of such a map.

On the generic injectivity of Hessian maps of ternary forms

TL;DR

This work develops a geometric framework to investigate the generic injectivity of the Hessian map for ternary forms by examining the graph of the polar map and its ramification as encoded by the Hessian curve. It shows that when two forms share the same Hessian, equality of ramification forces proportional gradients (hence proportional polynomials), while noncoincident ramification leads to a condition under which the polar linear systems must coincide, provided a product-type ruled surface arises. The results connect ramification geometry, Hessian/Steinerian curves, and polar linear systems, and situate the problem within Mammana’s classification of forms with equal polar linear systems, highlighting cases where injectivity can be established and identifying open scenarios where the surface is not a product. Overall, the paper offers a concrete geometric strategy to tackle the generic injectivity question and clarifies how equal Hessians constrain the underlying gradients and linear systems.

Abstract

We study the problem of the generic injectivity of the Hessian map, associating with a proportionality class of a ternary form the class of its Hessian determinant, conjectured by C. Ciliberto and G. Ottaviani and recently proved by the same authors. Taking into account that the Hessian curve is the ramification divisor associated with the polar map, we perform a study of the problem using a geometric description of the graph of such a map.
Paper Structure (5 sections, 5 theorems, 46 equations)

This paper contains 5 sections, 5 theorems, 46 equations.

Table of Contents

  1. Introduction
  2. Notation and Preliminaries
  3. Geometry of the polar map
  4. Hessians and polar linear systems
  5. Forms with equal polar linear systems

Key Result

Proposition 3.1

Let $V(f) \subset \mathbb{P}^2$ be a smooth curve. Then the graph $S_f \subset \mathbb{P}^2 \times \mathbb{P}^2$ of $\nabla f : \mathbb{P}^2 \to \mathbb{P}^2$ is a smooth surface with class Moreover, the ramification divisor $R_{f}$ satisfies

Theorems & Definitions (12)

  • Proposition 3.1
  • proof
  • Remark 3.2
  • Remark 3.3
  • Lemma 4.1
  • proof
  • Proposition 4.2
  • proof
  • Proposition 4.3
  • proof
  • ...and 2 more