The Hessian map

TL;DR

This paper investigates the properties of the Hessian map for hypersurfaces in projective space, establishing conditions for birationality, injectivity, and maximal rank of its differential in various cases.

Contribution

It provides new results on the birationality, injectivity, and differential properties of the Hessian map for hypersurfaces, including specific cases and rank conditions.

Findings

$h_{d,1}$ is birational onto its image for $d extgreater=5$

The restriction of the Hessian map to hypersurfaces with Waring rank $r+2$ is injective for $r extgreater=2$, $d extgreater=3$

The differential of the Hessian map has maximal rank on generic hypersurfaces with Waring rank $r+2$ in ${ m P}^r$ for $r extgreater=2$, $d extgreater=3$

Abstract

In this paper we study the Hessian map $h_{d,r}$ which associates to any hypersurface of degree $d$ in ${\mathbb P}^r$ its Hessian hypersurface. We study general properties of this map and we prove that: $h_{d,1}$ is birational onto its image if $d\geq 5$; we study in detail the maps $h_{3,1}$, $h_{4,1}$ and $h_{3,2}$; we study the restriction of the Hessian map to the locus of hypersurfaces of degree $d$ with Waring rank $r+2$ in ${\mathbb P}^r$, proving that this restriction is injective as soon as $r\geq 2$ and $d\geq 3$, which implies that $h_{3,3}$ is birational onto its image; we prove that the differential of the Hessian map is of maximal rank on the generic hypersurfaces of degree $d$ with Waring rank $r+2$ in ${\mathbb P}^r$, as soon as $r\geq 2$ and $d\geq 3$.