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

TL;DR

This paper provides a geometric reinterpretation and direct proofs of Gordan, Noether, and Franchetta's classification of hypersurfaces with vanishing hessian, clarifying their structure and counterexamples in projective geometry.

Contribution

It translates Gordan and Noether's algebraic approach into geometric terms, offering new geometric proofs for their classification results.

Findings

Classified hypersurfaces in $ ext{PP}^4$ with vanishing hessian that are not cones.

Provided geometric proofs of Gordan and Noether's results for $n \\leq 3$.

Constructed explicit counterexamples for $n \\geq 4$.

Abstract

Hesse claimed that an irreducible projective hypersurface in $\PP^n$ defined by an equation with vanishing hessian determinant is necessarily a cone. Gordan and Noether proved that this is true for $n\leq 3$ and constructed counterexamples for every $n\geq 4$. Gordan and Noether and Franchetta gave classification of hypersurfaces in $\PP^4$ 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.