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arXiv · 2201.07550

A theorem of Gordan and Noether via Gorenstein rings

Davide Bricalli·Filippo F. Favale·Gian Pietro Pirola
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Abstract

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.

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Davide Bricalli, Filippo F. Favale, Gian Pietro Pirola. 2022-01-19. A theorem of Gordan and Noether via Gorenstein rings. https://doi.org/10.1007/s00029-023-00882-7

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