Laplace Equations and the Weak Lefschetz Property

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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The weak Lefschetz property for quotients by quadratic monomials

MATHEMATICA SCANDINAVICA ◽

10.7146/math.scand.a-116681 ◽

2020 ◽

Vol 126 (1) ◽

pp. 41-60

Author(s):

Juan Migliore ◽

Uwe Nagel ◽

Hal Schenck

Keyword(s):

Linear Form ◽

Monomial Ideals ◽

Geometric Characterization ◽

General Linear ◽

Laplace Equations ◽

Weak Lefschetz Property ◽

Lefschetz Property

Michałek and Miró-Roig, in J. Combin. Theory Ser. A 143 (2016), 66–87, give a beautiful geometric characterization of Artinian quotients by ideals generated by quadratic or cubic monomials, such that the multiplication map by a general linear form fails to be injective in the first nontrivial degree. Their work was motivated by conjectures of Ilardi and Mezzetti, Miró-Roig and Ottaviani, connecting the failure to Laplace equations and classical results of Togliatti on osculating planes. We study quotients by quadratic monomial ideals, explaining failure of the Weak Lefschetz Property for some cases not covered by Michałek and Miró-Roig.

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On the weak Lefschetz property for almost complete intersections generated by uniform powers of general linear forms

Journal of Algebra ◽

10.1016/j.jalgebra.2019.12.029 ◽

2020 ◽

Vol 551 ◽

pp. 209-231 ◽

Cited By ~ 1

Author(s):

Rosa M. Miró-Roig ◽

Quang Hoa Tran

Keyword(s):

General Linear ◽

Complete Intersections ◽

Linear Forms ◽

Weak Lefschetz Property ◽

Lefschetz Property

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On the weak Lefschetz property for vector bundles on P2

Journal of Algebra ◽

10.1016/j.jalgebra.2020.10.005 ◽

2021 ◽

Vol 568 ◽

pp. 22-34

Author(s):

Gioia Failla ◽

Zachary Flores ◽

Chris Peterson

Keyword(s):

Vector Bundles ◽

Weak Lefschetz Property ◽

Lefschetz Property

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The Geometry of the Weak Lefschetz Property and Level Sets of Points

Canadian Journal of Mathematics ◽

10.4153/cjm-2008-019-2 ◽

2008 ◽

Vol 60 (2) ◽

pp. 391-411 ◽

Cited By ~ 6

Author(s):

Juan C. Migliore

Keyword(s):

Level Set ◽

Level Sets ◽

Hilbert Function ◽

Worst Case ◽

Weak Lefschetz Property ◽

Lefschetz Property ◽

Strong Lefschetz Property ◽

Set Of Points

AbstractIn a recent paper, F. Zanello showed that level Artinian algebras in 3 variables can fail to have the Weak Lefschetz Property (WLP), and can even fail to have unimodal Hilbert function. We show that the same is true for the Artinian reduction of reduced, level sets of points in projective 3-space. Our main goal is to begin an understanding of how the geometry of a set of points can prevent its Artinian reduction from having WLP, which in itself is a very algebraic notion. More precisely, we produce level sets of points whose Artinian reductions have socle types 3 and 4 and arbitrary socle degree ≥ 12 (in the worst case), but fail to have WLP. We also produce a level set of points whose Artinian reduction fails to have unimodal Hilbert function; our example is based on Zanello's example. Finally, we show that a level set of points can have Artinian reduction that has WLP but fails to have the Strong Lefschetz Property. While our constructions are all based on basic double G-linkage, the implementations use very different methods.

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