Laplace Equations and the Weak Lefschetz Property

http://dx.doi.org/10.4153/CJM-2012-033-x
Canad. J. Math. 65(2013), 634-654

  Published:2012-09-21
 Printed: Jun 2013

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.