The weak lefschetz property for quotients by quadratic monomials

Juan Migliore; Uwe Nagel; Hal Schenck

doi:10.7146/math.scand.a-116681

The weak lefschetz property for quotients by quadratic monomials

Juan Migliore, Uwe Nagel, Hal Schenck

Mathematics

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Abstract

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.

Original language	English
Pages (from-to)	41-60
Number of pages	20
Journal	Mathematica Scandinavica
Volume	126
Issue number	1
DOIs	https://doi.org/10.7146/math.scand.a-116681
State	Published - 2020

Bibliographical note

Funding Information:
by Grayson and Stillman, available at: http://www.math.uiuc.edu/Macaulay2/. Our collaboration began at the BIRS workshop “Artinian algebras and Lef-schetz properties”, organized by S. Faridi, A. Iarrobino, R. Miró-Roig, L. Smith and J. Watanabe, and we thank them and BIRS for a wonderful and stimulating environment. The first author was partially funded by the Simons Foundation under Grant #309556, the second author was partially supported by the Simons Foundation under grant #317096, and the third author was partially funded by NSF 1818646. We thank an anonymous referee for helpful comments.

Funding Information:
Computations were performed using Macaulay2, by Grayson and Stillman, available at: http://www.math.uiuc.edu/Macaulay2/. Our collaboration began at the BIRS workshop "Artinian algebras and Lefschetz properties", organized by S. Faridi,A. Iarrobino, R. Miró-Roig, L. Smith and J.Watanabe, and we thank them and BIRS for a wonderful and stimulating environment. The first author was partially funded by the Simons Foundation under Grant #309556, the second authorwas partially supported by the Simons Foundation under grant #317096, and the third author was partially funded by NSF 1818646. We thank an anonymous referee for helpful comments.

Publisher Copyright:
© 2020 Mathematica Scandinavica. All rights reserved.

ASJC Scopus subject areas

Mathematics (all)

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10.7146/math.scand.a-116681

Cite this

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abstract = "Micha{\l}ek and Mir{\'o}-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, Mir{\'o}-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 Micha{\l}ek and Mir{\'o}-Roig.",

author = "Juan Migliore and Uwe Nagel and Hal Schenck",

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The weak lefschetz property for quotients by quadratic monomials

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