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, and the general properties of this map and prove that $h_{d,1}$ is birational onto its image if $d\geqslant 5.$ We also study in detail the maps $h_{3,1}$, $h_{4,1}$, and $h_{3,2}$ and 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 $r\geqslant 2$ and $d\geqslant 3$, which implies that $h_{3,3}$ is birational onto its image. We also 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 $r\geqslant 2$ and $d\geqslant 3$.