Let $X$ be a hypersurface in a $n$-dimensional projective space. The Hessian map is a rational map from $X$ to the projective space of symmetric matrices that sends a point $p\in X$ to the Hessian matrix of the defining polynomial of $X$ evaluated at $p$. The Hessian correspondence is the map that sends a hypersurface to its Hessian variety; i.e. the Zariski closure of its image via 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 $n\geq 2$, and for degree $4$, the Hessian correspondence is birational. In this study, we introduce the $k$-gradients varieties and analyze their main properties. We provide effective algorithms for recovering a hypersurface from its Hessian variety, for degree $3$ and $n\geq 1$, and for degree $4$ and $n$ even.PDF: The Hessian correspondence of hypersurfaces of degree 3 and 4.pdf
