The Hessian correspondence of hypersurfaces of degree 3 and 4

Let $\mathcal{V}$ be a hypersurface in a $n$--dimensional projective space The Hessian map is a rational map from $\mathcal{V}$ to the projective space of symmetric matrices that sends $p\in \mathcal{V}$ to the Hessian matrix of the defining polynomial of $\mathcal{V}$ specialized at $p$. The Hessian correspondence is the map that sends a hypersurface to Zariski closure of its image through the Hessian map. In this paper, we study this correspondence for the cases of hypersurfaces of degree $3$ and $4$. We prove that, for degree $3$ and $n=1$, the map is two to one, and that, for degree $3$ and $n\ge 2$, and for degree $4$, the Hessian correspondence is birational. Moreover, we provide effective algorithms for recovering a hypersurface from its image through the Hessian map for degree $3$ and $n\ge 1$, and for degree $4$ and $n$ even.