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Togliatti systems and Artinian ideals failing weak Lefschetz Mezzetti, Emilia


In a joint work with Rosa Maria Mir`o-Roig and Giorgio Ottaviani (Canad. J. Math. 65, 2013), we established a closed relationship, due to apolarity, between homogeneous Artinian ideals I of the polynomial ring which fail the Weak Lefschetz Property - WLP - and projective varieties X satisfying at least one Laplace equationof order s2, i.e. such that all the s-osculating spaces have dimension strictly less than expected. Thanks to this connection, we were able to classify the smooth (and quasi smooth) toric rational threefolds parametrized by cubics, satisfying a Laplace equation of order 2, extending a classical theorem of E. Togliatti for surfaces. Equivalently, we classified the monomial Artinian ideals of cubics in 4 variables. We also formulated a conjecture to extend this result to ideals generated by cubic monomials in any dimension. The conjecture has been successively proved by Rosa Maria Mir-Roig and Mateusz Michalek (arXiv 1310.2529). The assumption that the variety is toric allows to exploit the combinatorial methods, studying the associated polytope. More recently in a joint work with Rosa Maria Mir`o-Roig (arXiv 1506.05914), we have started to investigate the same problems for Artinian ideals of the polynomial ring generated by monomials of any degree d in any number of variables and failing the WLP. These are also called Togliatti systems. Since the picture becomes soon much more involved than in the case of cubics, we have restricted our attention mainly to Togliatti systems that are minimal and smooth, adressing the question of their minimal and maximal number of generators. We have solved this question, and also classified the systems with minimal number of generators, or number of generators close to the minimal, and found new classes of examples. After shortly recalling the relationship mentioned above, I would like to speak of the more recent results on Togliatti systems of any degree d

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