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.

Laplace equations and the weak Lefschetz property / Emilia Mezzetti; Rosa Mirò Roig; Giorgio Ottaviani. - In: CANADIAN JOURNAL OF MATHEMATICS-JOURNAL CANADIEN DE MATHEMATIQUES. - ISSN 0008-414X. - STAMPA. - 65:(2013), pp. 634-654. [10.4153/CJM-2012-033-x]

Laplace equations and the weak Lefschetz property

OTTAVIANI, GIORGIO MARIA
2013

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.
2013
65
634
654
Emilia Mezzetti; Rosa Mirò Roig; Giorgio Ottaviani
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Utilizza questo identificatore per citare o creare un link a questa risorsa: https://hdl.handle.net/2158/811289
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