A theorem of Gordan and Noether via Gorenstein rings

Abstract

AbstractGordan and Noether proved in their fundamental theorem that an hypersurface $$X=V(F)\subseteq {{\mathbb {P}}}^n$$ X = V ( F ) ⊆ P n with $$n\le 3$$ 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 $$n\ge 4$$ 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 $${{\mathbb {K}}}$$ K -algebras and the Lefschetz properties. As a further application of our setting, we prove that a standard Artinian Gorenstein algebra $$R={{\mathbb {K}}}[x_0,\dots ,x_4]/J$$ R = K [ x 0 , ⋯ , 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.