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Abbas Nasrollah Nejad

Publications and source records attributed to Abbas Nasrollah Nejad.

At least 19 recordsLinked to original sources
arXiv2026

The Bourbaki Degree of Line Arrangements

We study the Bourbaki degree of line arrangements in the projective plane and its interaction with the intersection lattice and the syzygies of the gradient ideal. We show that the Bourbaki degree is obtained by evaluating the reduced characteristic polynomial at the initial degree of the first syzygy module. This leads to a sharp upper bound and to refinements involving the two largest intersection multiplicities, with consequences for the global Tjurina number, the freeness defect, and Terao's conjecture, including an improvement of Dimca's numerical criterion when the smaller exponent is at least five. We also establish addition--deletion formulas and show that the defect from the sharp bound is monotone under line addition. Finally, we prove that the Bourbaki degree is determined by the intersection lattice for arrangements with at most eight lines, while examples with isomorphic intersection lattices and different Bourbaki degrees exist for every number of lines at least nine.

math.AC↗
arXiv2026

Numerical invariants for three-generated ideals

We study numerical invariants of homogeneous ideals generated by three forms, with particular emphasis on the interaction between the initial degree of the syzygy module, the multiplicity of the codimension-two part of the ideal, and the Bourbaki degree. We extend the Bourbaki degree from gradient ideals of plane curves to arbitrary three-generated homogeneous ideals and obtain sharp numerical bounds in both the general and the equigenerated settings. In the equigenerated case, these bounds extend the du Plessis--Wall picture and lead to the study of numerical, structural, dimensional, and integrable gaps. We show that numerical admissibility does not in general imply realizability and obtain further restrictions from the minimal graded free resolution. As an application, we determine the possible Bourbaki degrees of reduced singular quartic plane curves and describe them in terms of their singularity configurations; the possible values are 0,1,2,3,4,6,7,8.

math.AC↗
arXiv2026

Closing two recent conjectures related to the Jacobian ideal of hyperplane arrangements

This work is about two conjectures stated by Burity--Simis--Tohăneanu regarding the Jacobian ideal of the defining polynomial of a central arrangement of $m$ hyperplanes. One settles one of these conjectures referring to the Jacobian ideal being a minimal reduction of the ideal of $(m-1)$-fold products. The second conjecture claiming the linear type property of the Jacobian ideal is disproved in rank at least four, by means of an explicit counter-example. In the latter the corresponding Rees algebra admits a torsion defining equation which is a Pfaffian syzygetic obstruction in degree two. One also relates this Pfaffian obstruction to circuits and codimension-two flats of the arrangement.

math.AC↗
arXiv2026

Jacobian algebras and variation of hyperplane sections

We study the variation in moduli of hyperplane sections of a hypersurface $V(f)\subseteq \mathbf P^n$ with at most isolated singularities. Using the Milnor algebra $M(f)$, we give an infinitesimal quotient criterion for the hyperplane-section map $Φ(f):(\mathbf P^n)^*\dashrightarrow M(d,n-1)$ to be generically finite onto its image. The passage from the infinitesimal quotient to the coarse moduli space is justified by a local GIT slice argument. Our approach gives a Jacobian-algebraic extension of the Beauville--Patel--Riedl--Tseng theory from smooth hypersurfaces to hypersurfaces with isolated singularities. In the smooth case it recovers the Lefschetz criterion and, using recent weak Lefschetz results, gives generic finiteness for $n\geq 3$ in the range $d\geq n+2$. In the singular case a new obstruction appears: a linear Jacobian syzygy, equivalently, for non-cones, a positive-dimensional projective automorphism group. After this obstruction is excluded, maximal infinitesimal variation is governed by the injectivity of the critical Lefschetz map $\ell:M(f)_{d-1}\to M(f)_d$. We apply the criterion to plane curves, surfaces in $\mathbf P^3$, and hypersurfaces admitting singular hyperplane sections, obtaining new criteria involving nodal sections and an application to the Schoen quintic threefold.

math.AG↗
arXiv2026

The relation type of point configurations in the projective plane

We study the {\it relation type} of ideals of finite reduced sets of points in the projective plane; for a given ideal, this is the maximal $T$-degree of a minimal generator of the defining ideal of the Rees algebra. Our main focus is on point configurations whose defining ideals are not necessarily linearly presented, with an emphasis on almost collinear configurations. We prove that $\rt(X)\in\{1,3\}$ whenever $X\subseteq \PP^2_k$ is a finite set of at most ten points; and we characterize the configurations of relation type $3$ in this range. We then show that a configuration of eleven points in generic position has relation type $5$, thereby yielding the first occurrence of relation type larger than $3$. Finally, we exhibit a configuration of $17$ points with relation type $4$ and we formulate some questions regarding the spectrum of admissible relation types of point configurations.

math.AC↗
arXiv2026

The Bourbaki degree of the syzygy module of 2 $\times$ 4 matrices

We introduce and study the Bourbaki degree as a numerical invariant for \(2 \times 4\) matrices $Θ$ of homogeneous polynomials over a polynomial ring \(R = k[x_1, \dots, x_n]\). This invariant, defined via a Bourbaki sequence for the syzygy module \(\operatorname{Syz}(Θ)\), generalizes previous constructions for plane curves and Jacobian matrices. Our main result is an explicit formula expressing the Bourbaki degree in terms of the degrees of the rows, the initial degree of a syzygy, and the first two Hilbert coefficients of the cokernel module \(\mathcal{Q} = \operatorname{coker}(Θ)\). We apply this framework to two important cases. First, matrices with constant first row, which are determined by a three-equigenerated ideal \(J = (f_1, f_2, f_3)\), where we show the Bourbaki degree measures how far \(J\) is from being a perfect ideal, and we completely characterize its smaller and larger values. Second, for a linear matrix, we use the Kronecker--Weierstrass classification to determine all possible Bourbaki degrees and homological types. This classification reveals the existence of a linear matrix with Bourbaki degree equal to 2, a value that does not occur for Jacobian matrices. Finally, in the geometric context of \(\mathbb{P}^3\), we provide a sufficient condition for \(\operatorname{Syz}(Θ)\) to define a codimension one distribution and obtain bounds on the Bourbaki degree when the initial degree is small.

math.AC↗
arXiv2026

Veronese Avoiding Hypersurfaces

We introduce Veronese-Avoiding hypersurfaces, inspired by the theory of associated forms of Alper--Isaev. In the smooth case, we reinterpret their criterion via Macaulay inverse systems: the Veronese-Avoiding condition is equivalent to the non-degeneracy of the associated form. In the singular case, our main theorem shows that a reduced hypersurface with exactly $n$ isolated singular points is Veronese-Avoiding if and only if these points are ordinary nodes in general linear position; we also classify singular plane cubics and treat fewer than $n$ nodes via a natural rational map. We then study the parameter space, proving local closedness and identifying a distinguished irreducible nodal locus. Finally, we prove a Lefschetz-type consequence for the Milnor algebra in degree $1$.

math.AG↗
arXiv2025

Quasihomogeneous isolated singularities in terms of syzygies and foliations

One considers quasihomogeneous isolated singularities of hypersurfaces in arbitrary dimensions through the lenses of three apparently quite apart themes: syzygies, singularity invariants, and foliations. In the first of these, one adds to the well-known result of Saito's a syzygy-theoretic characterization of a quasihomogeneous singularity affording an effective computational criterion. In the second theme, one explores the Milnor-Tjurina difference number from a commutative algebra viewpoint. Building on the Briancon-Skoda theorem and exponent, we extend previously known inequalities by Dimca and Greuel to arbitrary dimension and provide algebraic formulas involving the syzygy-theoretic part and reduction exponents. In the last theme one recovers and bring up to an algebraic light a result of Camacho and Movasati by establishing a couple of characterizations of quasihomogeneous isolated singularities in terms of the generators of the module of invariant vector fields.

math.AC↗
arXiv2023

The Bourbaki Degree of Plane Projective Curves

Bourbaki sequences and Bourbaki ideals have been studied by several authors since its inception sixty years ago circa. Generic Bourbaki sequences have been thoroughly examined by the senior author with B. Ulrich and W. Vasconcelos, but due to their nature, no numerical invariant was immediately available. Recently, J. Herzog, S. Kumashiro, and D. Stamate introduced the {\em Bourbaki number} in the category of graded modules as the shifted degree of a Bourbaki ideal corresponding to submodules generated in degree at least the maximal degree of a minimal generator of the given module. The present work introduces the{\em Bourbaki degree} as the algebraic multiplicity of a Bourbaki ideal corresponding to choices of minimal generators of minimal degree. The main intent is a study of plane curve singularities via this new numerical invariant. Accordingly, quite naturally, the focus is on the case where the standing graded module is the first syzygy module of the gradient ideal of a reduced form $f\in k[x,y,z]$ -- i.e., the main component of the module of logarithmic derivations of the corresponding curve. The overall goal of this project is to allow for a facet of classification of projective plane curves based on the behavior of this new numerical invariant, with emphasis on results about its lower and upper bounds.

math.AC↗
arXiv2022

The Relation Type of Varieties

In this paper, we introduce the notion of relation type of analytic and formal algebras and prove that it is well-defined and invariant by describing this notion in terms of the André-Quillen homology and using the Jacobi-Zariski long exact sequence of homology. In particular, the relation type is an invariant of schemes of finite type over a field, analytic varieties, and algebroid varieties.

math.AG↗
arXiv2020

Universal Decomposition Algebras Represent Endomorphisms

The goal of this paper is to supply an explicit description of the universal decomposition algebra of the generic polynomial of degree $n$ into the product of two monic polynomials, one of degree $r$, as a representation of Lie algebras of $n\times n$ matrices with polynomial entries. This is related with the bosonic vertex representation of the Lie algebra $gl_\infty$ due to Date, Jimbo, Kashiwara and Miwa.

math.AG↗
arXiv2020

The Valabrega-Valla modules of monomial ideals

In this paper, we focus on the initial degree and the vanishing of the Valabrega-Valla module of a pair of monomials ideals $J\subseteq I$ in a polynomials ring over a field $\mathbb{K}$. We prove that the initial degree of this module is bounded above by the maximum degree of a minimal generators of $J$. For edge ideals of graphs, a complete characterization of the vanishing of the Valabrega-Valla module is given. For higher degree ideals, we find classes which the Valabrega-Valla module vanishes. For the case that $J$ is the facet ideal of a clutter $\mathcal{C}$ and $I$ is the defining ideal of singular subscheme of $J$, the non-vanishing of this module is investigated in terms of the combinatorics of $\mathcal{C}$. Finally, we describe the defining ideal of the Rees algebra of $I/J$ provided that the Valabrega-Valla module is zero.

math.AC↗
arXiv2019

The module of Valabrega-Valla of the Jacobian ideal of points in projective plane

The module of Valabrega-Valla of the Jacobian ideal of a reduced projective variety $V$ is the torsion of the Aluffi algebra. One considers the problem of its vanishing in the case of where $V$ is a reduced set of points in the projective plane. It is shown that the module is nonzero for several cases of a special configuration class therein -- called $(s-r)$-{fold collinear configuration}. A complete classification of types is given for $5$ and $6$ points in regard to this problem.

math.AC↗
arXiv2019

Hypersurfaces with linear type singular loci

In this paper, necessary and sufficient criteria for the Jacobian ideal of a reduced hypersurface with isolated singularity to be of linear type, are presented. We prove that the gradient ideal of a reduced projective plane curve with simple singularities ($\mathrm{ADE}$) is of linear type. We show that any reduced projective quartic curve is of gradient linear type.

math.AC↗
arXiv2018

On the Gauss algebra of toric algebras

Let $A$ be a $K$-subalgebra of the polynomial ring $S=K[x_1,\ldots,x_d]$ of dimension $d$, generated by finitely many monomials of degree $r$. Then the Gauss algebra $\GG(A)$ of $A$ is generated by monomials of degree $(r-1)d$ in $S$. We describe the generators and the structure of $\GG(A)$, when $A$ is a Borel fixed algebra, a squarefree Veronese algebra, generated in degree $2$, or the edge ring of a bipartite graph with at least one loop. For a bipartite graph $G$ with one loop, the embedding dimension of $\GG(A)$ is bounded by the complexity of the graph $G$.

math.AG↗
arXiv2017

Torsion-free Aluffi Algebras

A pair of ideals $J\subseteq I\subseteq R$ has been called Aluffi torsion-free if the Aluffi algebra of $I/J$ is isomorphic with the corresponding Rees algebra. We give necessary and sufficient conditions for the Aluffi torsion-free property in terms of the first syzygy module of the form ideal $J^*$ in the associated graded ring of $I$. For two pairs of ideals $J_1,J_2\subseteq I$ such that $J_1-J_2\in I^2$, we prove that if one pair is Aluffi torsion-free the other one is so if and only if the first syzygy modules of $J_1$ and $J_2$ have the same form ideals. We introduce the notion of strongly Aluffi torsion-free ideals and present some results on these ideals.

math.AC↗
arXiv2017

The Aluffi Algebra of a hypersurface with isolated singularity

The Aluffi algebra is algebraic definition of characteristic cycles of a hypersurface in intersection theory. In this paper we focus on the Aluffi algebra of quasi-homogeneous and locally Eulerian hypersurface with isolated singularities. We prove that the Jacobian ideal of an affine hypersurfac with isolated singularities is of linear type if and only if it is locally Eulerian. We show that the gradient ideal of a projective hypersurface is of linear type if and only if the corresponding affine curve in the affine chart associated to singular points is locally Eulerian. We prove that the gradient ideal of the Nodal and Cuspidal projective plane curves are of linear type.

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