Artinian algebras and Jordan type

Iarrobino, Anthony; Marques, Pedro Macías; McDaniel, Chris

doi:10.1216/jca.2022.14.365

Cited by 14 publications

(4 citation statements)

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“…The Jordan type of a linear form ℓ for an Artinian algebra A determines whether or not ℓ is a weak or strong Lefschetz element of A. In order to explain the connection we need the following definition, and for more details on Artinian algebras and Jordan type the reader is invited to look at [IMMM22].…”

Section: Jordan Typementioning

confidence: 99%

Lefschetz properties of some codimension three Artinian Gorenstein algebras

et al. 2023

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Section: Jordan Typementioning

confidence: 99%

Lefschetz properties of some codimension three Artinian Gorenstein algebras

et al. 2023

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“…There are several refinements of this invariant. We refer to [29] in this volume for their definitions and a discussion of their properties. In particular, when M is a graded module over an Artinian algebra A we define the generic linear Jordan type P 1,M as the Jordan type P ℓ,M for a generic linear form ℓ ∈ A 1 .…”

Section: Lefschetz Properties and Jordan Types Of Artinian Algebrasmentioning

confidence: 99%

“…More generally, under which conditions on a graded module M over a graded Artinian algebra A do their generic and generic linear Jordan types satisfy P M = P 1,M ? (See Questions 1.1 and 2.56 in [30]. )…”

Section: Lefschetz Properties and Jordan Types Of Artinian Algebrasmentioning

confidence: 99%

Intersections of Amoebas

Juhnke-Kubitzke

Wolff²

2020

Discrete Mathematics &Amp; Theoretical Computer Science

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International audience Amoebas are projections of complex algebraic varieties in the algebraic torus under a Log-absolute value map, which have connections to various mathematical subjects. While amoebas of hypersurfaces have been inten- sively studied during the last years, the non-hypersurface case is barely understood so far. We investigate intersections of amoebas of n hypersurfaces in (C∗)n, which are genuine supersets of amoebas given by non-hypersurface vari- eties. Our main results are amoeba analogs of Bernstein's Theorem and Be ́zout's Theorem providing an upper bound for the number of connected components of such intersections. Moreover, we show that the order map for hypersur- face amoebas can be generalized in a natural way to intersections of amoebas. We show that, analogous to the case of amoebas of hypersurfaces, the restriction of this generalized order map to a single connected component is still 1-to-1.

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“…The Jordan degree type of multiplication m ℓ on a graded Artinian algebra A, adds to the information of the lengths of Jordan strings, their initial degrees. Note that [AAIY] has a table with the classification of Jordan degree types for graded AG algebras of Hilbert function (1, 3 k , 1): these are all symmetric, by a general result of T. Harima and J. Watanabe [HW] (see also [IMM,Prop. 2.38]).…”

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confidence: 99%

Limits of graded Gorenstein algebras of Hilbert function $(1,3^k,1)$

Abdallah¹,

Emsalem²,

Iarrobino³

et al. 2023

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Let R = k[x, y, z], the polynomial ring over a field k. Several of the authors previously classified nets of ternary conics and their specializations over an algebraically closed field [AEI]. Building on this work, we here show that when k is algebraically closed, and the * Our friend and colleague Jacques Emsalem passed on 12 July, 2022, after an illness; he had participated actively and enthusiastically in our work.

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Artinian algebras and Jordan type

Cited by 14 publications

References 98 publications

Lefschetz properties of some codimension three Artinian Gorenstein algebras

Lefschetz properties of some codimension three Artinian Gorenstein algebras

Intersections of Amoebas

Limits of graded Gorenstein algebras of Hilbert function $(1,3^k,1)$

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