arXiv subjects

Aron Simis

Publications and source records attributed to Aron Simis.

At least 19 recordsLinked to original sources

arXiv2026

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

This work is about two conjectures stated by Burity--Simis--Toh\u{a}neanu 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

The structure of almost Cohen-Macaulay $3$-generated ideals of codimension $2$ in terms of matrix theory

Let $R$ be a standard graded polynomial ring over a field $k$. The paper focuses on homogeneous ideals $J \subset R$ of codimension $2$ generated by three forms of the same degree $d \geq 2$ that are almost Cohen--Macaulay, i.e., of homological dimension $2$. Based on the structure of the minimal graded free resolution of $J$ and numerical data encoded in certain \emph{latent data}, one introduces the notion of \emph{level matrices} associated with these data. The main result provides a complete characterization of an almost Cohen--Macaulay $3$-generated ideal $J$ of codimension $2$ in terms of the existence of a related level matrix for which $J$ arises as the ideal of its maximal minors that fix a submatrix. One provides algebraic and geometric examples illustrating the results.

math.AC↗

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↗

arXiv2025

Additions to a theorem of Morey-Ulrich

Let $R = k[x_1,\ldots, x_d]$ denote a standard graded polynomial ring over an algebraically closed field $k$, and let $I \subset R$ be a perfect ideal of codimension $2$ with an $n\times (n-1)$ linear presentation matrix $\phi$. We prove an extended formulation of a theorem of Morey and Ulrich in the case where $I$ satisfies condition $G_{d-1}$, but not condition $G_d$.

math.AC↗

arXiv2025

Cohen--Macaulay ideals of codimension two and the geometry of plane points

We consider classes of codimension two Cohen--Macaulay ideals over a standard graded polynomial ring over a field. We revisit Vasconcelos' problem on $3\times 2$ matrices with homogeneous entries and describe the homological details of Geramita's work on plane points. An additional topic is the homological discussion of minors fixing a submatrix in the context of a perfect codimension two ideal. A combinatorial outcome of the results is a proof of the conjecture on the Jacobian ideal of a hyperplane arrangement stated by Burity, Simis and Toh\v{a}neanu. The basic drive behind the present landscapes is a thorough analysis of the related Hilbert--Burch matrix, often without assuming equigeneration, linear presentation or even the popular $G_d$ condition of Artin--Nagata.

math.AC↗

arXiv2024

Rose-Terao-Yuzvinsky theorem for reduced forms

Yuzvinsky and Rose-Terao have shown that the homological dimension of the gradient ideal of the defining polynomial of a generic hyperplane arrangement is maximum possible. In this work one provides yet another proof of this result, which in addition is totally different from the one given by Burity-Simis-Tohaneanu. Another main drive of the paper concerns a version of the above result in the case of a product of general forms of arbitrary degrees (in particular, transverse ones). Finally, some relevant cases of non general forms are also contemplated.

math.AC↗

arXiv2024

Matrices over polynomial rings approached by commutative algebra

The main goal of the paper is the discussion of a deeper interaction between matrix theory over polynomial rings over a field and typical methods of commutative algebra and related algebraic geometry. This is intended in the sense of bringing numerical algebraic invariants into the picture of determinantal ideals, with an emphasis on non-generic ones. In particular, there is a strong focus on square sparse matrices and features of the dual variety to a determinantal hypersurface. Though the overall goal is not exhausted here, one provides several environments where the present treatment has a degree of success.

math.AC↗

arXiv2024

Bounds on the degrees of vector fields

In this article, we study the generalized Poincare problem from the opposite perspective, by establishing lower bounds on the degree of the vector field in terms of invariants of the variety.

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↗

arXiv2021

De Jonqui\`eres transformations in arbitrary dimension. An ideal theoretic view

A generalization of the plane de Jonqui\`eres transformation to arbitrary dimension is studied, with an eye for the ideal theoretic side. In particular, one considers structural properties of the corresponding base ideal and of its defining relations. Useful throughout is the idea of downgraded sequences of forms, a tool considered in many sources for the rounding-up of ideals of defining relations. The emphasis here is on the case where the supporting Cremona transformation of the de Jonqui\`eres transformation is the identity map. In this case we establish aspects of the homological behavior of the graph of the transformation.

math.AC↗

arXiv2021

On the Jacobian ideal of central arrangements

Let $\mathcal{A}$ denote a central hyperplane arrangement of rank $n$ in affine space $\mathbb{K}^n$ over an infinite field $\mathbb{K}$ and let $l_1,\ldots, l_m\in R:= \mathbb K[x_1,\ldots,x_n]$ denote the linear forms defining the corresponding hyperplanes, along with the corresponding defining polynomial $f:=l_1\cdots l_m\in R$. Let $J_f$ denote the ideal generated by the partial derivatives of $f$ and let $\mathbb{I}$ designate the ideal generated by the $(m-1)$-fold products of $l_1,\ldots, l_m$. This paper is centered on the relationship between the two ideals $J_f, \mathbb{I}\subset R$, their properties and two conjectures related to them. Some parallel results are obtained in the case of forms of higher degrees provided they fulfill a certain transversality requirement.

math.AC↗

arXiv2020

Equigenerated ideals of analytic deviation one

The overall goal is to approach the Cohen--Macaulay property of the special fiber $\mathcal{F}(I)$ of an equigenerated homogeneous ideal $I$ in a standard graded ring over an infinite field. When the ground ring is assumed to be local, the subject has been extensively looked at. Here, with a focus on the graded situation, one introduces two technical conditions, called respectively, {\em analytical tightness} and {\em analytical adjustment}, in order to approach the Cohen--Macaulayness of $\mathcal{F}(I)$. A degree of success is obtained in the case where $I$ in addition has analytic deviation one, a situation looked at by several authors, being essentially the only interesting one in dimension three. Naturally, the paper has some applications in this case.

math.AC↗

arXiv2020

Equigenerated Gorenstein ideals of codimension three

We focus on the structure of a homogeneous Gorenstein ideal $I$ of codimension three in a standard polynomial ring $R=\kk[x_1,\ldots,x_n]$ over a field $\kk$, assuming that $I$ is generated in a fixed degree $d$. For such an ideal $I$ this degree comes along with the minimal number of generators of $I$ and the degree of the entries of the associated skew-symmetric matrix in a simple formula. We give an elementary characteristic-free argument to the effect that, for any such data linked by this formula, there exists a Gorenstein ideal $I$ of codimension three filling them. We conjecture that, for arbitrary $n\geq 2$, an ideal $I\subset \kk[x_1,\ldots,x_n]$ generated by a general set of $r\geq n+2$ forms of degree $d\geq 2$ is Gorenstein if and only if $d=2$ and $r= {{n+1}\choose 2}-1$. We prove the `only if' implication of this conjecture when $n=3$. For arbitrary $n\geq 2$, we prove that if $d=2$ and $r\geq (n+2)(n+1)/6$ then the ideal is Gorenstein if and only if $r={{n+1}\choose 2}-1$, which settles the `if' assertion of the conjecture for $n\leq 5$. Finally, we elaborate around one of the questions of Fr\"oberg--Lundqvist. In a different direction, we reveal a connection between the Macaulay inverse and the so-called Newton dual, a matter so far not brought out to our knowledge. Finally, we consider the question as to when the link $(\ell_1^m,\ldots,\ell_n^m):\mathfrak{f}$ is equigenerated, where $\ell_1,\ldots,\ell_n$ are independent linear forms and $\mathfrak{f}$ is a form, is given a solution in some important cases.

math.AC↗

arXiv2020

Generic freeness of local cohomology and graded specialization

The main focus is the generic freeness of local cohomology modules in a graded setting. The present approach takes place in a quite nonrestrictive setting, by solely assuming that the ground coefficient ring is Noetherian. Under additional assumptions, such as when the latter is reduced or a domain, the outcome turns out to be stronger. One important application of these considerations is to the specialization of rational maps and of symmetric and Rees powers of a module.

math.AC↗

arXiv2019

Degree of rational maps via specialization

One considers the behavior of the degree of a rational map under specialization of the coefficients of the defining linear system. The method rests on the classical idea of Kronecker as applied to the context of projective schemes and their specializations. For the theory to work one is led to develop the details of rational maps and their graphs when the ground ring of coefficients is a Noetherian integral domain.

math.AG↗

arXiv2018

Symmetry preserving degenerations of the generic symmetric matrix

One considers certain degenerations of the generic symmetric matrix over a field $k$ of characteristic zero and the main structures related to the determinant $f$ of the matrix, such as the ideal generated by its partial derivatives, the polar map defined by these derivatives and its image $V(f)$, the Hessian matrix, the ideal and the map given by the cofactors, and the dual variety of $V(f)$.

math.AC↗

arXiv2017

The depth of the Rees algebra of three general binary forms

One proves that the Rees algebra of an ideal generated by three general binary forms of same degree $\geq 5$ has depth one. The proof hinges on the behavior of the Ratliff-Rush filtration for low powers of the ideal and on establishing that certain large matrices whose entries are quadratic forms have maximal rank. One also conjectures a shorter result that implies the main theorem of the paper.

math.AC↗

arXiv2017

Linearly presented perfect ideals of codimension $2$ in three variables

The goal of this paper is the fine structure of the ideals in the title, with emphasis on the properties of the associated Rees algebra and the special fiber. The watershed between the present approach and some of the previous work in the literature is that here one does not assume that the ideals in question satisfy the common generic properties. One exception is a recent work of N. P. H. Lan which inspired the present work. Here we recover and extend his work. We strongly focus on the behavior of the ideals of minors of the corresponding so-called Hilbert--Burch matrix and on conjugation features of the latter. We apply the results to three important models: linearly presented ideals of plane fat points, reciprocal ideals of hyperplane arrangements and linearly presented monomial ideals.

math.AC↗