Laplace Equations and the Weak Lefschetz Property

Details Laplace Equations and the Weak Lefschetz Property Laplace Equations and the Weak Lefschetz Property

2011-10-24

arxiv.org/abs/1110.5239v1

Mathematics

Algebraic Geometry

21 pages, 4 figures

Scientific paper

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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