A theorem of Gordan and Noether via Gorenstein rings

Gordan and Noether proved in their fundamental theorem that an hypersurface \(X=V(F)\subseteq \mathbb{P}^n\) with \(n\leq 3\) is a cone if and only if \(F\) has vanishing hessian (i.e. the determinant of the Hessian matrix). They also showed that the statement is false if \(n\geq 4\), by giving some counterexamples. Since their proof, several others have been proposed in the literature. In this paper we give a new one by using a different perspective which involves the study of standard Artinian Gorenstein \(\mathbb{K}\)-algebras and the Lefschetz properties. As a further application of our setting, we prove that a standard Artinian Gorenstein algebra \(R=\mathbb{K}[x_0,\dots,x_4]/J\) with \(J\) generated by a regular sequence of quadrics has the strong Lefschetz property. In particular, this holds for Jacobian rings associated to smooth cubic threefolds.