The Hessian map

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 $d\ge 5$; 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 $r\ge 2$ and $d\ge 3$, 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 $r\ge 2$ and $d\ge 3$.