On the generic injectivity of Hessian maps of ternary forms

TL;DR

This paper investigates the conditions under which the Hessian map of ternary forms is generically injective, confirming a conjecture and exploring geometric properties related to the Hessian curve and polar map.

Contribution

It proves the conjecture on the generic injectivity of the Hessian map for ternary forms and provides a geometric analysis of the associated Hessian curve and polar map.

Findings

Confirmed the conjecture on the generic injectivity of the Hessian map.

Provided a geometric description of the Hessian curve and its relation to the polar map.

Enhanced understanding of the ramification divisor in the context of ternary forms.

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.