BIRS Workshop Lecture Videos
Large Lefschetz defects Cook II, David
Mezzetti, Mir´o-Roig, and Ottaviani showed that in some cases the failure of the weak Lefschetz property can be used to produce a variety satisfying a Laplace equation. We define a graded algebra to have a Lefschetz defect of ! in degree d if the rank of the multiplication map of a general linear form between the degree d−1 and degree d components has rank ! less than the expected rank. Mezzetti and Mir´o-Roig recently explored the minimal and maximal number of generators of ideals that fail to have the weak Lefschetz property, i.e., to have a positive Lefschetz defect. In contrast to this, we will discuss constructions of ideals that have large Lefschetz defects and thus can be used to produce toric varieties satisfying many Laplace equations.
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