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# Laplace Equations and the Weak Lefschetz Property

Published:2012-09-21
Printed: Jun 2013
• Emilia Mezzetti,
Dipartimento di Matematica e Geoscienze, Università di Trieste, Via Valerio 12/1, 34127 Trieste, Italy
• Rosa M. Miró-Roig,
Facultat de Matemàtiques, Department d'Algebra i Geometria, Gran Via des les Corts Catalanes 585, 08007 Barcelona, Spain
• Giorgio Ottaviani,
Dipartimento di Matematica, Università di Firenze, Viale Morgagni 67/A I-50134 Firenze, Italy
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## 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.
 Keywords: osculating space, weak Lefschetz property, Laplace equations, toric threefold
 MSC Classifications: 13E10 - Artinian rings and modules, finite-dimensional algebras 14M25 - Toric varieties, Newton polyhedra [See also 52B20] 14N05 - Projective techniques [See also 51N35] 14N15 - Classical problems, Schubert calculus 53A20 - Projective differential geometry