Lefschetz properties for Jacobian rings of cubic fourfolds and other Artinian algebras (Q6141129)

Let \(R=\oplus_{k=0}^N R^k\) be a standard graded Artinian Gorenstein algebra over a field of zero characteristic; this is called a SAGA by the authors of the article under review. It has been conjectured in [\textit{T. Harima} et al., The Lefschetz properties. Berlin: Springer (2013; Zbl 1284.13001)] that all complete intersection SAGA's satisfy the Strong Lefschetz Property -- SLP for short, which means that for any general homogeneous element \(x\) of degree \(1\) in \(R\) the multiplication map by \(x^s\) from \(R^k\) to \(R^{k+s}\) has maximal rank for any positive integers \(k,s\). The weaker Weak Lefschetz Property -- WLP -- means that the multiplication map by \(x\) from \(R^k\) to \(R^{k+1}\) has maximal rank for any \(k\). The conjecture has been proved only in codimension \(2\). On the other hand, the WLP holds true for general complete intersection SAGA's \(R\) with generators of fixed degrees, and for all \(R\) if the codimension is at most \(3\). The authors of the article under review are interested in the case of complete intersection SAGA's presented by quadrics, that include the particularly interesting case of jacobian rings of cubic hypersurfaces. Here the case of codimension \(6\) is considered and the main result (Theorem 3.1) is that the multiplication by \(x^4\) is a bijection. This extends recent results of \textit{A. Alzati} and \textit{R. Re} [Collect. Math. 70, No. 2, 283--294 (2019; Zbl 1432.13001)] and of \textit{D. Bricalli} et al. [Sel. Math., New Ser. 29, No. 5, Paper No. 74, 25 p. (2023; Zbl 07756886)] stating that the SLP holds for these algebras if the codimension is at most \(5\). The Authors are also interested in studying the non-general complete intersection SAGA's, for which some of the following loci are non empty: the Nihilpotent loci \(\mathcal N_i=\{[x]\in\mathbb P(R^1)\mid x^i=0\}\), and the non-Lefschetz locus \(\{[x]\in\mathbb P(R^1)\mid \cdot x\) is not injective\(\}\). Among other results, a careful analysis is performed of the geometrical behavior of the Nihilpotent locus \(\mathcal N_i\) when it contains a linear space. Moreover in Theorem 4.2 a sort of lifting theorem for the WLP in degree 2 is proved for non-general complete intersection SAGA's presented by quadrics of any codimension.