Standard Artinian algebras and Lefschetz properties: a geometric approach (Q6630011)

In the first part of the article under review the authors survey some of their recent results, also obtained in collaboration with G.P. Pirola, about the following conjecture stated in 2013 [\textit{T. Harima} et al., The Lefschetz properties. Berlin: Springer (2013; Zbl 1284.13001)]: Every complete intersection SAGA (i.e. standard Artinian Gorenstein algebra) satisfies the Strong Lefschetz Property. The conjecture is still widely open. The main results obtained in [\textit{D. Bricalli} and \textit{F. F. Favale}, Collect. Math. 75, No. 1, 149--169 (2024; Zbl 1532.14074)] and [\textit{D. Bricalli} et al., Sel. Math., New Ser. 29, No. 5, Paper No. 74, 25 p. (2023; Zbl 1540.14088)] concern the Jacobian rings of smooth cubic threefolds and fourfolds. In particular in those articles they analyzed some geometric consequences of the existence of non-Lefschetz elements in degree 1. \N\NIn the second part of this paper the authors consider the existence of non-Lefschetz elements in degree 2, such that the kernels of the associated multiplication map have dimension 2, and in Corollary 2.5 they obtain some quite technical generalizations of their previous results. In the last Section they deal with the relation between linear factors of the hessian of a smooth cubic hypersurface and nilpotent elements of its Jacobian ring.\N\NFor the entire collection see [Zbl 1545.13003].