The Lefschetz Properties

Harima, Tadahito; Maeno, Toshiaki; Morita, Hideaki; Numata, Yasuhide; Wachi, Akihito; Watanabe, Junzo

doi:10.1007/978-3-642-38206-2

Cited by 102 publications

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“…This also allows us to discuss Lefschetz properties. These important properties have been formalized in , and we refer to the monograph for further information.…”

Section: Introductionmentioning

confidence: 99%

Squeezed complexes

Juhnke-Kubitzke

Nagel

2019

J. London Math. Soc.

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Given a shifted order ideal U, we associate to it a family of simplicial complexes false(Δt(U)false)t⩾0 that we call squeezed complexes. In a special case, our construction gives squeezed balls that were defined and used by Kalai to show that there are many more simplicial spheres than boundaries of simplicial polytopes. We study combinatorial and algebraic properties of squeezed complexes. In particular, we show that they are vertex decomposable and characterize when they have the weak or the strong Lefschetz property. Moreover, we define a new combinatorial invariant of pure simplicial complexes, called the singularity index, that can be interpreted as a measure of how far a given simplicial complex is from being a manifold. In the case of squeezed complexes false(Δt(U)false)t⩾0, the singularity index turns out to be strictly decreasing until it reaches (and stays) zero if t grows.

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“…This also allows us to discuss Lefschetz properties. These important properties have been formalized in , and we refer to the monograph for further information.…”

Section: Introductionmentioning

confidence: 99%

Squeezed complexes

Juhnke-Kubitzke

Nagel

2019

J. London Math. Soc.

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“…It is noteworthy that we do not find any general results on the unimodality of the strong/weak shape of a semigroup action on a valuated poset. We are wondering if the theory on the Lefschetz properties of commutative graded algebras [27] will be useful in this line of research. Penttila and Siciliano [47, Lemma 3.1] suggested a machinery (Lemma 3.6) to remove certain finiteness assumption.…”

Section: Discussionmentioning

confidence: 99%

Top-heavy phenomena for transformations

Zhu

2022

Ars Math. Contemp.

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Let S be a transformation semigroup acting on a set Ω. The action of S on Ω can be naturally extended to be an action on all subsets of Ω. We say that S is ℓ-homogeneous provided it can send A to B for any two (not necessarily distinct) ℓ-subsets A and B of Ω. On the condition that k ≤ ℓ < k + ℓ ≤ |Ω|, we show that every ℓ-homogeneous transformation semigroup acting on Ω must be k-homogeneous. We report other variants of this result for Boolean semirings and affine/projective geometries. In general, any semigroup action on a poset gives rise to an automaton and we associate some sequences of integers with the phase space of this automaton. When this poset is a geometric lattice, we propose to investigate various possible regularity properties of these sequences, especially the so-called top-heavy property. In the course of this study, we are led to a conjecture about the injectivity of the incidence operator of a geometric lattice, generalizing a conjecture of Kung.

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“…Finally, recall the following result, saying that the Jacobian module N (f ) enjoys a weak Lefschetz type property; see [12] for this result and [18,17,24] for Lefschetz properties of Artinian algebras.…”

Section: Preliminariesmentioning

confidence: 99%

On the jumping lines of bundles of logarithmic vector fields along plane curves

Dimca¹,

Sticlaru²

2020

Publicacions Matemàtiques

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For a reduced curve C : f = 0 in the complex projective plane P 2 , we study the set of jumping lines for the rank two vector bundle T C on P 2 whose sections are the logarithmic vector fields along C. We point out the relations of these jumping lines with the Lefschetz type properties of the Jacobian module of f and with the Bourbaki ideal of the module of Jacobian syzygies of f . In particular, when the vector bundle T C is unstable, a line is a jumping line if and only if it meets the 0-dimensional subscheme defined by this Bourbaki ideal, a result going back to Schwarzenberger. Other classical general results by Barth, Hartshorne, and Hulek resurface in the study of this special class of rank two vector bundles.

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The Lefschetz Properties

Cited by 102 publications

References 0 publications

Squeezed complexes

Squeezed complexes

Top-heavy phenomena for transformations

On the jumping lines of bundles of logarithmic vector fields along plane curves

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