arXiv subjects

Nancy Abdallah

Publications and source records attributed to Nancy Abdallah.

13 recordsLinked to original sources

arXiv2024

Jordan degree type for codimension three Gorenstein algebras of small Sperner number

The Jordan type $P_{A,\ell}$ of a linear form $\ell$ acting on a graded Artinian algebra $A$ over a field $\sf k$ is the partition describing the Jordan block decomposition of the multiplication map $m_\ell$, which is nilpotent. The Jordan degree type $\mathcal S_{A,\ell}$ is a finer invariant, describing also the initial degrees of the simple submodules of $A$ in a decomposition of $A$ as ${\sf k}[\ell]$-modules. The set of Jordan types of $A$ or Jordan degree types (JDT) of $A$ as $\ell$ varies, is an invariant of the algebra. This invariant has been studied for codimension two graded algebras. We here extend the previous results to certain codimension three graded Artinian Gorenstein (AG) algebras - those of small Sperner number. Given a Gorenstein sequence $T$ - one possible for the Hilbert function of a codimension three AG algebra - the irreducible variety $\mathrm{Gor}(T)$ parametrizes all Gorenstein algebras of Hilbert function $T$. We here completely determine the JDT possible for all pairs $(A,\ell), A\in \mathrm{Gor}(T)$, for Gorenstein sequences $T$ of the form $T=(1,3,s^k,3,1)$ for Sperner number $s=3,4,5$ and arbitrary multiplicity $k$. For $s=6$ we delimit the prospective JDT, without verifying that each occurs.

math.AC↗

arXiv2024

Lattice Paths, Lefschetz Properties, and Almkvist's Conjecture in Two Variables

We study a certain two-parameter family of non-standard graded complete intersections $A(m,n)$. In case $n=2$, we show that $A(m,2)$ has the strong Lefschetz property and the complex Hodge-Riemann property if and only if $m$ is even. This supports a strengthening of a conjecture of Almkvist on the unimodality of the Hilbert function of $A(m,n)$.

math.AC↗

arXiv2023

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

Let $R={\sf k}[x,y,z]$, the polynomial ring over a field $\sf k$. Several of the authors previously classified nets of ternary conics and their specializations over an algebraically closed field. We here show that when $\sf k$ is algebraically closed, and the Hilbert function sequence $T=(1,3^k,1), k\ge 2$ (i.e. $T=(1,3,3,\ldots,3,1)$ where $k$ is the multiplicity of $3$) then the family $G_T$ 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 $\mathrm{Gor}(T)$ of Artinian Gorenstein algebras of Hilbert function $T$. We then classify up to isomorphism the elements of these families $\mathrm{Gor}(T)$ and of $G_T$. Finally, we give examples of codimension three Gorenstein sequences, such as $(1,3,5,3,1)$, for which $G_T$ has several irreducible components, one being the Zariski closure of $\mathrm{Gor}(T)$.

math.AC↗

arXiv2023

Free resolutions and Lefschetz properties of some Artin Gorenstein rings of codimension four

In 1978, Stanley constructed an example of an Artinian Gorenstein (AG) ring $A$ with non-unimodal $H$-vector $(1,13,12,13,1)$. Migliore-Zanello later showed that for regularity $r=4$, Stanley's example has the smallest possible codimension $c$ for an AG ring with non-unimodal $H$-vector. The weak Lefschetz property (WLP) has been much studied for AG rings; it is easy to show that an AG ring with non-unimodal $H$-vector fails to have WLP. In codimension $c=3$ it is conjectured that all AG rings have WLP. For $c=4$, Gondim showed that WLP always holds for $r \le 4$ and gives a family where WLP fails for any $r \ge 7$, building on an earlier example of Ikeda of failure of WLP for $r=5$. In this note we study the minimal free resolution of $A$ and relation to Lefschetz properties (both weak and strong) and Jordan type for $c=4$ and $r \le 6$.

math.AC↗

arXiv2023

A note on Artin Gorenstein algebras with Hilbert function (1,4,k,k,4,1)

We study the free resolutions of some Artin Gorenstein algebras of Hilbert function $(1,4,k,k,4,1)$ and we prove that all such algebras have the Strong Lefschetz property if they have the Weak Lefschetz property. In the case $k=4$ we prove that the Hilbert function alone fixes the betti table. For higher $k$ stronger conditions on the algebras are needed to fix the betti table. In particular, if the algebra is a complete intersection or if it is defined by an equigenerated ideal then the betti table is unique.

math.AC↗

arXiv2023

Hilbert functions and Jordan type of Perazzo Artinian algebras

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.

math.AC↗

arXiv2022

Nets in $\mathbb P^2$ and Alexander Duality

A net in $\mathbb{P}^2$ is a configuration of lines $\mathcal A$ 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 $\le r$. In the context of line arrangements in $\mathbb{P}^2$, applying Alexander duality to the resulting ideal yields insight into the combinatorial structure of nets.

math.CO↗

arXiv2022

Nets of Conics and associated Artinian algebras of length 7

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. In a 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.

math.AG↗

arXiv2022

Lefschetz properties of some codimension three Artinian Gorenstein algebras

Codimension two Artinian algebras $A$ have the strong and weak Lefschetz properties provided the characteristic is zero or greater than the socle degree. It is open to what extent such results might extend to codimension three AG algebras - the most promising results so far have concerned the weak Lefschetz property for such algebras. We here show that every standard-graded codimension three Artinian Gorenstein algebra $A$ having low maximum value of the Hilbert function - at most six - has the strong Lefschetz property, provided that the characteristic is zero. When the characteristic is greater than the socle degree of $A$, we show that $A$ is almost strong Lefschetz. This quite modest result is nevertheless arguably the most encompassing so far concerning the strong Lefschetz property for graded codimension three AG algebras.

math.AC↗

arXiv2017

Topology of posets with special partial matchings

Special partial matchings (SPMs) are a generalisation of Brenti's special matchings. Let a \emph{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, Cherniavsky, and Twelbeck, prove a conjecture of Hultman, and confirm a claim of Rains and Vazirani.

math.CO↗

arXiv2016

Combinatorial Invariance of Kazhdan-Lusztig-Vogan Polyomials for Fixed Point Free Involutions

When $Sp(2n,\mathbb{C})$ acts on the flag variety of $SL(2n,\mathbb{C})$, the orbits are in bijection with fixed point free involutions in the symmetric group $S_{2n}$. In this case, the associated Kazhdan-Lusztig-Vogan polynomials $P_{v,u}$ can be indexed by pairs of fixed point free involutions $v\geq u$, where $\geq$ denotes the Bruhat order on $S_{2n}$. We prove that these polynomials are combinatorial invariants in the sense that if $f: [u, w_0 ] \rightarrow [u , w_0]$ is a poset isomorphism of upper intervals in the Bruhat order on fixed point free involutions, then $P_{v,u} = P_{f(v),u}$ for all $v \geq u$.

math.CO↗

arXiv2015

On Hodge Theory of Singular Plane Curves

The dimensions of the graded quotients of the cohomology of a plane curve complement 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.

math.AG↗

arXiv2014

On Plane Curves with Double and Triple Points

We describe in simple geometric terms the Hodge filtration on the cohomology groups of the complement U in the projective plane of a curve C with ordinary double and triple points. Relations to Milnor algebra, syzygies of the Jacobian ideal and pole order filtration on the second cohomology of U are given.

math.AG↗