On the birationality of the Hessian maps of quartic curves and 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.

arXiv commentThe proofs of our results are incomplete. Indeed, the generic injectivity should be shown not only for the restriction of the Hessian map to the transversal, but for the restriction to the much bigger set, union of all orbits of points in the transversal