Michałek and Miró-Roig, in J. Combin. Theory Ser. A 143 (2016), 66–87, give a beautiful geometric characterization of Artinian quotients by ideals generated by quadratic or cubic monomials, such that the multiplication map by a general linear form fails to be injective in the first nontrivial degree. Their work was motivated by conjectures of Ilardi and Mezzetti, Miró-Roig and Ottaviani, connecting the failure to Laplace equations and classical results of Togliatti on osculating planes. We study quotients by quadratic monomial ideals, explaining failure of the Weak Lefschetz Property for some cases not covered by Michałek and Miró-Roig.
On the weak Lefschetz property for almost complete intersections generated by uniform powers of general linear forms
