Laplace Equations and the Weak Lefschetz Property
Algebraic Geometry
2011-10-25 v1 Commutative Algebra
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.
Cite
@article{arxiv.1110.5239,
title = {Laplace Equations and the Weak Lefschetz Property},
author = {Emilia Mezzetti and Rosa M. Miro'-Roig and Giorgio Ottaviani},
journal= {arXiv preprint arXiv:1110.5239},
year = {2011}
}
Comments
21 pages, 4 figures