arXiv subjects

Ricardo Burity

Publications and source records attributed to Ricardo Burity.

9 recordsLinked to original sources

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

Betti numbers of fold products of linear forms

The ideals generated by fold products of linear forms are generalizations of powers of defining ideals of star configurations, or of Veronese type ideals, and in this paper we study their Betti numbers. In earlier work, the authors together with Yu Xie showed that these ideals have linear graded free resolution, and in this paper we take the study of the homological information about these ideals to the next natural level.

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↗

arXiv2023

Free divisors, blowup algebras of Jacobian ideals, and maximal analytic spread

Free divisors form a celebrated class of hypersurfaces which has been extensively studied in the past fifteen years. Our main goal is to introduce four new families of homogeneous free divisors and investigate central aspects of the blowup algebras of their Jacobian ideals. For instance, for all families the Rees algebra and its special fiber are shown to be Cohen-Macaulay -- a desirable feature in blowup algebra theory. Moreover, we raise the problem of when the analytic spread of the Jacobian ideal of a (not necessarily free) polynomial is maximal, and we characterize this property with tools ranging from cohomology to asymptotic depth. In addition, as an application, we give an ideal-theoretic homological criterion for homaloidal divisors, i.e., hypersurfaces whose polar maps are birational.

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

Ideals generated by $a$-fold products of linear forms have linear graded free resolution

Given $Σ\subset R:=\mathbb K[x_1,\ldots,x_k]$, where $\mathbb K$ is a field of characteristic 0, any finite collection of linear forms, some possibly proportional, and any $1\leq a\leq |Σ|$, we prove that $I_a(Σ)$, the ideal generated by all $a$-fold products of $Σ$, has linear graded free resolution. This allows us to determine a generating set for the defining ideal of the Orlik-Terao algebra of the second order of a line arrangement in $\mathbb P_{\mathbb{K}}^2$, and to conclude that for the case $k=3$, and $Σ$ defining such a line arrangement, the ideal $I_{|Σ|-2}(Σ)$ is of fiber type. We also prove several conjectures of symbolic powers for defining ideals of star configurations of any codimension $c$.

math.AC↗

arXiv2020

Logarithmic derivations associated to line arrangements

In this paper we give full classification of rank 3 line arrangements in $\mathbb P^2$ (over a field of characteristic 0) that have a minimal logarithmic derivation of degree 3. The classification presents their defining polynomials, up to a change of variables, with their corresponding affine pictures. We also analyze the shape of such a logarithmic derivation, towards obtaining criteria for a line arrangement to possess a cubic minimal logarithmic derivation.

math.CO↗

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↗

arXiv2015

On a conjecture of Vasconcelos

One studies the structure of the Rees algebra of an almost complete intersection monomial ideal of finite co-length in a polynomial ring over a field, assuming that the least pure powers of the variables contained in the ideal have the same degree. It is shown that the Rees algebra has a natural quasi-homogeneous structure and its presentation ideal is generated by explicit Sylvester forms. A consequence of these results is a proof that the Rees algebra is almost Cohen--Macaulay, thus answering affirmatively an important case of a conjecture of W. Vasconcelos.

math.AC↗