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

Hesse claimed that an irreducible projective hypersurface in $\P^n$ defined by an equation with vanishing hessian determinant is necessarily a cone. Gordan and Noether proved that this is true for n=3 and constructed counterexamples for every $n\geq 4$. Gordan and Noether and Franchetta gave classification of hypersurfaces in $\P^4$ with vanishing hessian and which are not cones. Here we translate in geometric terms Gordan and Noether approach, providing direct geometrical proofs of these results.