A Geometrical approach to Gordan-Noether's and Franchetta's contributions to a question posed by Hesse

Abstract

Hesse claimed in [7] (and later also in [8]) that an irreducible projective hypersurface in ℙn defined by an equation with vanishing hessian determinant is necessarily a cone. Gordan and Noether proved in [6] that this is true forn≤3 and constructed counterexamples for everyn≥4. Gordan and Noether and Franchetta gave classification of hypersurfaces in ℙ4 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.