Resumen de A theorem of Gordan and Noether via Gorenstein rings

Davide Bricalli, Filippo F. Favale, Gian Pietro Pirola

• Gordan and Noether proved in their fundamental theorem that an hypersurface X = V(F) ⊆ Pn with n ≤ 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 ≥ 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 K-algebras and the Lefschetz properties. As a further application of our setting, we prove that a standard Artinian Gorenstein algebra R = K[x0,..., x4]/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.