Geodesic
Laplace Equations and the Weak Lefschetz Property
Canadian journal of mathematics, Tome 65 (2013) no. 3, pp. 634-654

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DOI

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 the Weak 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.
DOI : 10.4153/CJM-2012-033-x
Mots-clés : 13E10, 14M25, 14N05, 14N15, 53A20, osculating space, weak Lefschetz property, Laplace equations, toric threefold 
Mezzettiaaa, Emilia; Miré-Roig, Rosa M.; Ottaviani, Giorgio. Laplace Equations and the Weak Lefschetz Property. Canadian journal of mathematics, Tome 65 (2013) no. 3, pp. 634-654. doi: 10.4153/CJM-2012-033-x
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     title = {Laplace {Equations} and the {Weak} {Lefschetz} {Property}},
     journal = {Canadian journal of mathematics},
     pages = {634--654},
     year = {2013},
     volume = {65},
     number = {3},
     doi = {10.4153/CJM-2012-033-x},
     url = {http://geodesic.mathdoc.fr/articles/10.4153/CJM-2012-033-x/}
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