Laplace Equations and the Weak Lefschetz Property

Canadian Journal of Mathematics - Tập 65 Số 3 - Trang 634-654 - 2013
Emilia Mezzetti1, Rosa M. Miró‐Roig, Giorgio Ottaviani
1Dipartimento di Matematica e Geoscienze

Tóm tắt

AbstractWe 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.

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