Laplace Equations and the Weak Lefschetz Property
Emilia Mezzetti, Rosa M. Miré-Roig, Giorgio Ottaviani · Canadian Journal of Mathematics · 2012
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 theWeak Lefschetz Property) and a differential geometric notion, concerning varieties that satisfy certain Laplace equations. In the toric case, some relevant examples are classified, and as a byproduct we provide counterexamples to Ilardi's conjecture.