On the birationality of the Hessian maps of quartic curves and cubic surfaces

TL;DR

This paper proves that the Hessian map for quartic plane curves is birational onto its image, providing new evidence for a conjecture and offering a simpler proof for cubic surfaces.

Contribution

It establishes the birationality of the Hessian map for quartic curves and simplifies the proof for cubic surfaces, advancing understanding of these geometric objects.

Findings

Hessian map of quartic plane curves is birational onto its image

Provides new evidence for a conjecture by Ciliberto and Ottaviani

Simplifies proof of birationality for cubic surfaces

Abstract

We show that the hessian map of quartic plane curves is a birational morphism onto its image, thus bringing new evidence for a very interesting conjecture of Ciro Ciliberto and Giorgio Ottaviani. Our new approach also yields a simpler proof of the similar property for cubic surfaces, which is already known by the work of these two authors.