2012
DOI: 10.1016/j.jpaa.2011.05.007
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Hyperplane sections and the subtlety of the Lefschetz properties

Abstract: The weak and strong Lefschetz properties are two basic properties that Artinian algebras may have. Both Lefschetz properties may vary under small perturbations or changes of the characteristic. We study these subtleties by proposing a systematic way of deforming a monomial ideal failing the weak Lefschetz property to an ideal with the same Hilbert function and the weak Lefschetz property. In particular, we lift a family of Artinian monomial ideals to finite level sets of points in projective space with the pro… Show more

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Cited by 14 publications
(9 citation statements)
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“…Perhaps a similar phenomenon will allow the WLP to continue to play a role in the study of pure O-sequences. A first approach using this philosophy was obtained by Cook and the second author [17], where they lifted a monomial ideal to an ideal of a reduced set of points in one more variable, showed that the general artinian reduction has the WLP, and concluded that the Hilbert function of the original monomial algebra is unimodal, regardless of whether it has the WLP or not.…”
Section: Monomial Level Algebrasmentioning
confidence: 99%
“…Perhaps a similar phenomenon will allow the WLP to continue to play a role in the study of pure O-sequences. A first approach using this philosophy was obtained by Cook and the second author [17], where they lifted a monomial ideal to an ideal of a reduced set of points in one more variable, showed that the general artinian reduction has the WLP, and concluded that the Hilbert function of the original monomial algebra is unimodal, regardless of whether it has the WLP or not.…”
Section: Monomial Level Algebrasmentioning
confidence: 99%
“…Consider the ideal I α+t,β+t,γ+t,α,β,γ in R where K has characteristic zero, 0 < α ≤ β ≤ γ ≤ 2(α + β), t ≥ 1 3 (α + β + γ), and α + β + γ is divisible by three. If (α, β, γ, t) is not (2,9,13,9) or (3,7,14,9), then R/I α+t,β+t,γ+t,α,β,γ fails to have the weak Lefschetz property if and only if t is even, α + β + γ is odd, and α = β or β = γ. Remark 7.6.…”
Section: Axis-centralmentioning
confidence: 99%
“…The weak Lefschetz property is known to be subtle to both deformations (see, e.g., [9], [24], and [25]) but also to field characteristic. The latter, considering the weak Lefschetz property in positive characteristic, is an exciting and active direction of research.…”
mentioning
confidence: 99%
“…The idea is to lift the given monomial ideal to the homogenous ideal of a set of points and then pass to a general hyperplane section of the latter. It is shown in [18] that this procedure does indeed produce ideals with the WLP for a certain class of monomial ideals without the WLP.…”
Section: Introductionmentioning
confidence: 98%
“…In [35], examples of monomial ideals were exhibited that did not have the WLP, but that could be deformed to ideals with the WLP. A systematic way for producing such deformations that preserve the Hilbert function has been proposed by Cook and the second author in [18]. The idea is to lift the given monomial ideal to the homogenous ideal of a set of points and then pass to a general hyperplane section of the latter.…”
Section: Introductionmentioning
confidence: 99%