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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 osculating space, weak Lefschetz property, Laplace equations, toric threefold
MSC Classifications: 13E10, 14M25, 14N05, 14N15, 53A20 show english descriptions Artinian rings and modules, finite-dimensional algebras
Toric varieties, Newton polyhedra [See also 52B20]
Projective techniques [See also 51N35]
Classical problems, Schubert calculus
Projective differential geometry
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
 

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