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  • 1.
    Abdallah, Nancy
    et al.
    University of Borås, Faculty of Textiles, Engineering and Business.
    Altafi, Nasrin
    Department of Mathematics and Statistics, Queen’s University, Kingston, ON, Canada; Institutionen för Matematik, Kungliga Tekniska högskolan (KTH), Stockholm, Sweden.
    De Poi, Pietro
    Dipartimento di Scienze Matematiche, Informatiche e Fisiche, Università degli Studi di Udine, Via delle Scienze 206, 33100, Udine, Italy.
    Fiorindo, Luca
    Dipartimento di Matematica, Università degli Studi di Genova, Via Dodecaneso 35, 16146, Genova, Italy.
    Iarrobino, Anthony
    Department of Mathematics, Northeastern University, Boston, MA, 02115, USA.
    Macias Marques, Pedro
    Departamento de Matemática, ECT, CIMA, IFA, Universidade de Évora, Rua Romão Ramalho, 59, 7000-671, Évora, Portugal.
    Mezzetti, Emilia
    Dipartimento di Matematica, Informatica e Geoscience, Università degli Studi di Trieste, Via Valerio 12/1, 34127, Trieste, Italy.
    Miró-Roig, Rosa M.
    Facultat de Matemàtiques i Informàtica, Universitat de Barcelona, Gran Via des les Corts Catalanes 585, 08007, Barcelona, Spain.
    Nicklasson, Lisa
    Avdelningen för matematik och fysik, Mälardalens universitet, 883, 721 23, Västerås, Sweden.
    Hilbert Functions and Jordan Type of Perazzo Artinian Algebras2024In: Lefschetz Properties: Current and New Directions / [ed] Uwe Nagel, Karim Adiprasito, Roberta Di Gennaro, Sara Faridi, Satoshi Murai, Springer Nature, 2024, p. 59-80Conference paper (Refereed)
    Abstract [en]

    We study Hilbert functions, Lefschetz properties, and Jordan type of Artinian Gorenstein algebras associated to Perazzo hypersurfaces in projective space. The main focus lies on Perazzo threefolds, for which we prove that the Hilbert functions are always unimodal. Further we prove that the Hilbert function determines whether the algebra is weak Lefschetz, and we characterize those Hilbert functions for which the weak Lefschetz property holds. By example, we verify that the Hilbert functions of Perazzo fourfolds are not always unimodal. In the particular case of Perazzo threefolds with the smallest possible Hilbert function, we give a description of the possible Jordan types for multiplication by any linear form. 

  • 2.
    Abdallah, Nancy
    et al.
    University of Borås, Faculty of Textiles, Engineering and Business.
    Emsalem, Jacques
    Northeastern Univ, Dept Math, Boston.
    Iarrobino, Anthony
    Northeastern Univ, Dept Math, Boston.
    Yaméogo, Joachim
    Univ Cote Azur, CNRS, LJAD, Nice, France.
    Limits of graded Gorenstein algebras of Hilbert function $$(1,3^k,1)$$2024In: European Journal of Mathematics, ISSN 2199-675X, E-ISSN 2199-6768, Vol. 10, no 1, article id 9Article in journal (Refereed)
    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).

  • 3.
    Abdallah, Nancy
    et al.
    University of Borås, Faculty of Textiles, Engineering and Business.
    Altafi, Nasrin
    Department of Mathematics, KTH Royal Institute of Technology, S-100 44 Stockholm, Sweden; Department of Mathematics and Statistics, Queen's University, Kingston, Ontario, K7L 3N6, Canada.
    Iarrobino, Anthony
    Department of Mathematics, Northeastern University, Boston, MA 02115, USA.
    Seceleanu, Alexandra
    Department of Mathematics, University of Nebraska-Lincoln, Lincoln, NE 68588, USA.
    Yaméogo, Joachim
    Université Côte d'Azur, CNRS, LJAD, France.
    Lefschetz properties of some codimension three Artinian Gorenstein algebras2023In: Journal of Algebra, ISSN 0021-8693, E-ISSN 1090-266X, Vol. 625, p. 28-45Article in journal (Refereed)
    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.

  • 4.
    Abdallah, Nancy
    et al.
    University of Borås, Faculty of Textiles, Engineering and Business.
    Schenck, Hal
    Department of Mathematics, Auburn University, Auburn, AL, 36849, USA.
    Nets in P^2 and Alexander Duality2023In: Discrete & Computational Geometry, ISSN 0179-5376, E-ISSN 1432-0444Article in journal (Refereed)
    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.

  • 5.
    Abdallah, Nancy
    et al.
    University of Borås, Faculty of Textiles, Engineering and Business.
    Emsalem, Jacques
    Paris, France.
    Iarrobino, Anthony
    Department of Mathematics, Northeastern University, Boston, MA, 02115, USA.
    Nets of conics and associated Artinian algebras of length 72023In: European Journal of Mathematics, ISSN 2199-675X, E-ISSN 2199-6768, Vol. 9, no 2, article id 22Article in journal (Refereed)
    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.

  • 6.
    Abdallah, Nancy
    et al.
    Linköpings Universitet.
    Hansson, Mikael
    Linköpings Universitet.
    Hultman, Axel
    Linköpings Universitet.
    Topology of posets with special partial matchings2019In: Advances in Mathematics, ISSN 0001-8708, E-ISSN 1090-2082, Vol. 348, p. 255-276Article in journal (Refereed)
    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.

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  • 7.
    Abdallah, Nancy
    et al.
    Linköpings Universitet.
    Hultman, Axel
    Linköpings Universitet.
    Combinatorial invariance of Kazhdan–Lusztig–Vogan polynomials for fixed point free involutions2017In: Journal of Algebraic Combinatorics, ISSN 0925-9899, E-ISSN 1572-9192, Vol. 47, no 4, p. 543-560Article in journal (Refereed)
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  • 8.
    Abdallah, Nancy
    University of Nice Sophia Antipolis, CNRS, LJAD, UMR 7351, 06100 Nice, Franc.
    On Hodge Theory of Singular Plane Curves2016In: Canadian mathematical bulletin, ISSN 0008-4395, Vol. 59, no 3, p. 449-460Article in journal (Refereed)
    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).

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  • 9.
    Abdallah, Nancy
    University of Nice.
    On Plane Curves with Double and Triple Points2016In: Mathematica Scandinavica, ISSN 0025-5521, E-ISSN 1903-1807, Vol. 119, no 1, p. 60-72Article in journal (Refereed)
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