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Publications (8 of 8) Show all publications
Abdallah, N., Emsalem, J., Iarrobino, A. & Yaméogo, J. (2024). Limits of graded Gorenstein algebras of Hilbert function $$(1,3^k,1)$$. European Journal of Mathematics, 10(1), Article ID 9.
Open this publication in new window or tab >>Limits of graded Gorenstein algebras of Hilbert function $$(1,3^k,1)$$
2024 (English)In: European Journal of Mathematics, ISSN 2199-675X, E-ISSN 2199-6768, Vol. 10, no 1, article id 9Article in journal (Refereed) Published
Abstract [en]

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, Abdallah et al. (Eur J Math 9(2), Art. No. 22, 2023). We here show that when k is algebraically closed, and considering the Hilbert function sequence T =(1,3(k),1), k >= 2 (i.e. T = (1, 3, 3, ... , 3, 1) where k is the multiplicity of 3), then the family GT parametrizing graded Artinian algebra quotients A = R/I of R having Hilbert function T is irreducible, and G(T) is the closure of the family Gor(T) of Artinian Gorenstein algebras of Hilbert function T. We then classify up to isomorphism the elements of these families Gor(T) and of G(T). Finally, we give examples of codimension 3 Gorenstein sequences, such as (1, 3, 5, 3, 1), for which G(T) has several irreducible components, one being the Zariski closure of Gor(T).

Keywords
Artinian Gorenstein algebra, Closure, Deformation, Hilbert function, Irreducible component, Isomorphism class, Limits, Nets of conics, Normal form, Parametrization
National Category
Algebra and Logic
Identifiers
urn:nbn:se:hb:diva-31422 (URN)10.1007/s40879-023-00714-0 (DOI)001142227000001 ()2-s2.0-85182220162 (Scopus ID)
Available from: 2024-01-24 Created: 2024-01-24 Last updated: 2024-04-09Bibliographically approved
Abdallah, N., Altafi, N., Iarrobino, A., Seceleanu, A. & Yaméogo, J. (2023). Lefschetz properties of some codimension three Artinian Gorenstein algebras. Journal of Algebra, 625, 28-45
Open this publication in new window or tab >>Lefschetz properties of some codimension three Artinian Gorenstein algebras
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2023 (English)In: Journal of Algebra, ISSN 0021-8693, E-ISSN 1090-266X, Vol. 625, p. 28-45Article in journal (Refereed) Published
Abstract [en]

Codimension two Artinian algebras have the strong and weak Lefschetz propertiesprovided the characteristic is zero or greater than the socle degree. It is open to whatextent such results might extend to codimension three Artinian Gorenstein algebras. De-spite much work, the strong Lefschetz property for codimension three Artinian Gorensteinalgebra has remained largely mysterious; our results build on and strengthen some of theprevious results. We here show that every standard-graded codimension three ArtinianGorenstein algebra A having maximum value of the Hilbert function at most six has thestrong Lefschetz property, provided that the characteristic is zero. When the characteris-tic is greater than the socle degree of A, we show that A is almost strong Lefschetz, theyare strong Lefschetz except in the extremal pair of degrees.

National Category
Algebra and Logic
Identifiers
urn:nbn:se:hb:diva-29568 (URN)10.1016/j.jalgebra.2023.03.005 (DOI)000973002400001 ()2-s2.0-85150809568 (Scopus ID)
Available from: 2023-03-23 Created: 2023-03-23 Last updated: 2024-02-01Bibliographically approved
Abdallah, N. & Schenck, H. (2023). Nets in P^2 and Alexander Duality. Discrete & Computational Geometry
Open this publication in new window or tab >>Nets in P^2 and Alexander Duality
2023 (English)In: Discrete & Computational Geometry, ISSN 0179-5376, E-ISSN 1432-0444Article in journal (Refereed) Published
Abstract [en]

A net in P^2 is a configuration of lines A and points X satisfying certain incidence properties. Nets appear in a variety of settings, ranging from quasigroups to combinatorial design to classification of Kac–Moody algebras to cohomology jump loci of hyperplane arrangements. For a matroid M and rank r, we associate a monomial ideal (a monomial variant of the Orlik–Solomon ideal) to the set of flats of M of rank ≤r. In the context of line arrangements in P^2, applying Alexander duality to the resulting ideal yields insight into the combinatorial structure of nets.

National Category
Mathematics
Identifiers
urn:nbn:se:hb:diva-29866 (URN)10.1007/s00454-023-00504-1 (DOI)000977047600001 ()2-s2.0-85153409489 (Scopus ID)
Available from: 2023-06-05 Created: 2023-06-05 Last updated: 2024-02-01Bibliographically approved
Abdallah, N., Emsalem, J. & Iarrobino, A. (2023). Nets of conics and associated Artinian algebras of length 7. European Journal of Mathematics, 9(2), Article ID 22.
Open this publication in new window or tab >>Nets of conics and associated Artinian algebras of length 7
2023 (English)In: European Journal of Mathematics, ISSN 2199-675X, E-ISSN 2199-6768, Vol. 9, no 2, article id 22Article in journal (Refereed) Published
Abstract [en]

We classify the orbits of nets of conics under the action of the projective linear group and we determine the specializations of these orbits, using geometric and algebraic methods. We study related geometric questions, as the parametrization of planar cubics. We show that Artinian algebras of Hilbert function H=(1,3,3,0) determined by nets, can be smoothed—deformed to a direct sum of fields; and that algebras of Hilbert function H=(1,r,2,0), determined by pencils of quadrics, can also be smoothed. This portion is a translation and update of a 1977 version, a typescript by the second two authors that was distributed as a preprint of University of Paris VII. In a new Historical Appendix A we describe related work prior to 1977. In an Update Appendix B we survey some developments since 1977 concerning nets of conics, related geometry, and deformations of Artinian algebras of small length.

National Category
Geometry
Identifiers
urn:nbn:se:hb:diva-29575 (URN)10.1007/s40879-023-00600-9 (DOI)000959966000001 ()2-s2.0-85150776591 (Scopus ID)
Available from: 2023-03-28 Created: 2023-03-28 Last updated: 2024-02-01Bibliographically approved
Abdallah, N., Hansson, M. & Hultman, A. (2019). Topology of posets with special partial matchings. Advances in Mathematics, 348, 255-276
Open this publication in new window or tab >>Topology of posets with special partial matchings
2019 (English)In: Advances in Mathematics, ISSN 0001-8708, E-ISSN 1090-2082, Vol. 348, p. 255-276Article in journal (Refereed) Published
Abstract [en]

Special partial matchings (SPMs) are a generalisation of Brenti's special matchings. Let a pircon be a poset in which every non-trivial principal order ideal is finite and admits an SPM. Thus pircons generalise Marietti's zircons. We prove that every open interval in a pircon is a PL ball or a PL sphere. It is then demonstrated that Bruhat orders on certain twisted identities and quasiparabolic W-sets constitute pircons. Together, these results extend a result of Can, Cherniaysky, and Twelbeck, prove a conjecture of Hultman, and confirm a claim of Rains and Vazirani.

Keywords
Topology of pircons Special partial matching Twisted identities
National Category
Other Mathematics
Identifiers
urn:nbn:se:hb:diva-28686 (URN)10.1016/j.aim.2019.02.031 (DOI)000466835800008 ()2-s2.0-85063074385 (Scopus ID)
Available from: 2022-10-01 Created: 2022-10-01 Last updated: 2022-11-16Bibliographically approved
Abdallah, N. & Hultman, A. (2017). Combinatorial invariance of Kazhdan–Lusztig–Vogan polynomials for fixed point free involutions. Journal of Algebraic Combinatorics, 47(4), 543-560
Open this publication in new window or tab >>Combinatorial invariance of Kazhdan–Lusztig–Vogan polynomials for fixed point free involutions
2017 (English)In: Journal of Algebraic Combinatorics, ISSN 0925-9899, E-ISSN 1572-9192, Vol. 47, no 4, p. 543-560Article in journal (Refereed) Published
Keywords
Kazhdan–Lusztig–Vogan polynomials, Special partial matching, Combinatorial invariance
National Category
Other Mathematics
Identifiers
urn:nbn:se:hb:diva-28685 (URN)10.1007/s10801-017-0785-z (DOI)000435125100002 ()2-s2.0-8502834174 (Scopus ID)
Available from: 2022-10-01 Created: 2022-10-01 Last updated: 2022-11-24Bibliographically approved
Abdallah, N. (2016). On Hodge Theory of Singular Plane Curves. Canadian mathematical bulletin, 59(3), 449-460
Open this publication in new window or tab >>On Hodge Theory of Singular Plane Curves
2016 (English)In: Canadian mathematical bulletin, ISSN 0008-4395, Vol. 59, no 3, p. 449-460Article in journal (Refereed) Published
Abstract [en]

The dimensions of the graded quotients of the cohomology of a plane curve complement U = P-2/C with respect to the Hodge filtration are described in terms of simple geometrical invariants. The case of curves with ordinary singularities is discussed in detail. We also give a precise numerical estimate for the difference between the Hodge filtration and the pole order filtration on H-2(U, C).

Keywords
plane curvesHodge and pole order filtrations
National Category
Geometry
Identifiers
urn:nbn:se:hb:diva-28684 (URN)10.4153/cmb-2016-010-4 (DOI)000381328900001 ()2-s2.0-84991255377 (Scopus ID)
Available from: 2022-10-01 Created: 2022-10-01 Last updated: 2022-11-16Bibliographically approved
Abdallah, N. (2016). On Plane Curves with Double and Triple Points. Mathematica Scandinavica, 119(1), 60-72
Open this publication in new window or tab >>On Plane Curves with Double and Triple Points
2016 (English)In: Mathematica Scandinavica, ISSN 0025-5521, E-ISSN 1903-1807, Vol. 119, no 1, p. 60-72Article in journal (Refereed) Published
Keywords
Singularity theory, Hodge theory
National Category
Geometry
Identifiers
urn:nbn:se:hb:diva-28683 (URN)
Available from: 2022-10-01 Created: 2022-10-01 Last updated: 2022-11-16Bibliographically approved
Organisations
Identifiers
ORCID iD: ORCID iD iconorcid.org/0000-0002-8001-6787

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