Hyperplane sections and the subtlety of the Lefschetz properties

Cook, David; Nagel, Uwe

doi:10.1016/j.jpaa.2011.05.007

Cited by 14 publications

(9 citation statements)

References 19 publications

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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%

Pure O-Sequences: Known Results, Applications, and Open Problems

Migliore

Nagel

Zanello

2012

Commutative Algebra

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Section: Monomial Level Algebrasmentioning

confidence: 99%

Pure O-Sequences: Known Results, Applications, and Open Problems

Migliore

Nagel

Zanello

2012

Commutative Algebra

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“…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.…”

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confidence: 99%

Enumerations deciding the weak Lefschetz property

Cook¹,

Nagel²

2011

Preprint

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We introduce a natural correspondence between artinian monomial almost complete intersections in three variables and punctured hexagonal regions. We use this correspondence to investigate the algebras for the presence of the weak Lefschetz property. In particular, we relate the field characteristics in which such an algebra fails to have the weak Lefschetz property to the prime divisors of the enumeration of signed lozenge tilings of the associated punctured hexagonal region. On the one side this allows us to establish the weak Lefschetz property in many new cases. On the other side we can determine some of the prime divisors of the enumerations by means of an algebraic argument.For numerous classes of punctured hexagonal regions we find closed formulae for the enumerations of signed lozenge tilings, and thus the field characteristics in which the associated algebras fail to the have the weak Lefschetz property. Further, we offer a conjecture for a closed formula for the enumerations of signed lozenge tilings of symmetric punctured hexagonal regions. These formulae are exploited to lend further evidence to a conjecture by Migliore, Miró-Roig, and the second author that classifies the level artinian monomial almost complete intersections in three variables that have the weak Lefschetz property in characteristic zero. Moreover, the formulae are used to generate families of algebras which never, or always, have the weak Lefschetz property, regardless of field characteristic. Finally, we determine (in one case, depending on the presence of the weak Lefschetz property) the splitting type of the syzygy bundle of an artinian monomial almost complete intersection in three variables, when the characteristic of the base field is zero.Our results convey an intriguing interplay between problems in algebra, combinatorics, and algebraic geometry, which raises new questions and deserves further investigation.

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“…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%

Survey Article: A tour of the weak and strong Lefschetz properties

Migliore¹,

Nagel²

2013

J. Commut. Algebra

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An artinian graded algebra, A, is said to have the Weak Lefschetz property (WLP) if multiplication by a general linear form has maximal rank in every degree. A vast quantity of work has been done studying and applying this property, touching on numerous and diverse areas of algebraic geometry, commutative algebra, and combinatorics. Amazingly, though, much of this work has a "common ancestor" in a theorem originally due to Stanley, although subsequently reproved by others. In this paper we describe the different directions in which research has moved starting with this theorem, and we discuss some of the open questions that continue to motivate current research.

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Hyperplane sections and the subtlety of the Lefschetz properties

Cited by 14 publications

References 19 publications

Pure O-Sequences: Known Results, Applications, and Open Problems

Pure O-Sequences: Known Results, Applications, and Open Problems

Enumerations deciding the weak Lefschetz property

Survey Article: A tour of the weak and strong Lefschetz properties

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