A theorem of Gordan and Noether via Gorenstein rings

Abstract

Gordan and Noether proved in their fundamental theorem that an hypersurface $X=V(F)\subseteq {\mathbb{P}}^n$ with $n\le 3$ 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 $n\ge 4$, 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.

21 pages

References in corpus (2)

• Infinitesimal variation functions for families of smooth varieties
• Lefschetz properties for jacobian rings of cubic fourfolds and other Artinian algebras

Cited by in corpus (4)

• Lefschetz properties for jacobian rings of cubic fourfolds and other Artinian algebras
• On the Hessian of cubic hypersurfaces
• The weak Lefschetz property for Standard graded, Artinian Gorenstein algebras of embedding dimension four and socle degree three
• On the irreducibility of Hessian loci of cubic hypersurfaces