A Geometrical approach to Gordan–Noether’s and Franchetta’s contributions to a question posed by Hesse

Hesse claimed in [7] (and later also in [8]) that an irreducible projective hypersurface in $\mathbb{P}^n$ defined by an equation with vanishing hessian determinant is necessarily a cone. Gordan and Noether proved in [6] that this is true for $n\leq 3$ and constructed counterexamples for every $n\leq 4$. Gordan and Noether and Franchetta gave classification of hypersurfaces in $\mathbb {P}^4$ with vanishing hessian and which are not cones, see [6, 5]. Here we translate in geometric terms Gordan and Noether approach, providing direct geometrical proofs of these results.