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 |$d\geqslant 3$|⁠. [ABSTRACT FROM AUTHOR]