A theorem of Gordan and Noether via Gorenstein rings. (English) Zbl 1540.14088
In the 1850’s [J. Reine Angew. Math. 42, 117–124 (1851; ERAM 042.1147cj); J. Reine Angew. Math. 56, 263–269 (1859; ERAM 056.1491cj)], O. Hesse claimed that any hypersurface \(X \subset \mathbb P^N\) with vanishing hessian is a cone. Subsequently in 1876 P. Gordan and M. Nöther [Math. Ann. 10, 547–568 (1876; JFM 08.0064.05)] showed that this is not true for \(N \geq 4\). In \(\mathbb P^4\) they gave a complete description of the hypersurfaces with vanishing hessian that are not cones. A nice treatment of this topic can be found in the book by F. Russo [On the geometry of some special projective varieties. Cham: Springer (2016; Zbl 1337.14001)]. In this context hessians were studied by
J. Watanabe in 2000 and by T. Maeno and J. Watanabe [Ill. J. Math. 53, No. 2, 591–603 (2009; Zbl 1200.13031)], connecting it to the Lefschetz properties for artinian Gorenstein algebras via Macaulay’s inverse systems. One class of such hypersurfaces that have been carefully studied (and is further studied in this paper) with an eye to the Lefschetz properties is that of Perazzo hypersurfaces. See for instance the paper of L. Fiorindo, E. Mezzetti and R. Miró-Roig [L. Fiorindo et al., J. Algebra 626, 56–81 (2023; Zbl 1516.14066)], where they study the connections between the Lefschetz properties and the Hilbert function for the associated artinian Gorenstein algebras. In the paper under review, the authors start by recalling the Gordan-Noether theorem and its connection to the Lefschetz properties. They take a different point of view for \(\mathbb P^4\), giving a direct proof of the fact that all artinian Gorenstein algebras \(R\) of codimension \(\leq 4\) have the property that there exists a linear form \(\ell\) such that \(\times \ell : [R]_1 \rightarrow [R]_{e-1}\) is an isomorphism (where \(e\) is the socle degree). From this they deduce the Gordan-Noether theorem. The authors use this point of view to prove that in \(S = k[x_0,\dots,x_4]\), if \(I\) is an ideal generated by a regular sequence of 5 quadrics then \(S/I\) satisfies the Strong Lefschetz Property (SLP), and in particular \(\times \ell^3 : [R]_1 \rightarrow [R]_4\) is an isomorphism. For example, this holds for Jacobian rings associated to smooth cubic threefolds. As a consequence, their work gives a new proof of a result of U. Nagel and the reviewer related to the Weak Lefschetz Property for complete intersections of quadrics in general.
Reviewer: Juan C. Migliore (Notre Dame)
MSC:
| 14J70 | Hypersurfaces and algebraic geometry |
| 14J30 | \(3\)-folds |
| 13E10 | Commutative Artinian rings and modules, finite-dimensional algebras |
| 14A25 | Elementary questions in algebraic geometry |
| 14A05 | Relevant commutative algebra |
Keywords:
Artinian Gorenstein algebras; cubic threefolds; Gordan-Noether; Jacobian rings; Lefschetz propertiesCitations:
ERAM 042.1147cj; ERAM 056.1491cj; JFM 08.0064.05; Zbl 1337.14001; Zbl 1200.13031; Zbl 1516.14066References:
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