Laplace Equations and the Weak Lefschetz Property

2011/10/24 by Emilia Mezzetti, Mezzetti, Emilia, Rosa M. Miro'-Roig +3 · 1 citation
Mathematics · #13E10 #14M25 #14N05 #14N15 #53A20 #Algebraic Geometry (math.AG) #Commutative Algebra (math.AC) #FOS: Mathematics #math.AC #math.AG #msc:13E10 #msc:14M25 #msc:14N05 #msc:14N15 #msc:53A20

paper · pdf · doi:10.48550/arxiv.1110.5239

21 pages, 4 figures

arxiv created 2011/10/24 · arxiv updated 2011/10/25

Abstract

We prove that r independent homogeneous polynomials of the same degree d become dependent when restricted to any hyperplane if and only if their inverse system parameterizes a variety whose (d-1)-osculating spaces have dimension smaller than expected. This gives an equivalence between an algebraic notion (called Weak Lefschetz Property) and a differential geometric notion, concerning varieties which satisfy certain Laplace equations. In the toric case, some relevant examples are classified and as byproduct we provide counterexamples to Ilardi's conjecture.

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• The Weak Lefschetz Property for Tensor Products of Artinian Monomial Algebras and Its Applications to Lollipop Graphs

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