Ricardo Burity

Bio: Ricardo Burity is an academic researcher from Federal University of Paraíba. The author has contributed to research in topics: Rees algebra & Ideal (order theory). The author has an hindex of 1, co-authored 8 publications receiving 7 citations. Previous affiliations of Ricardo Burity include Universidade Federal Rural de Pernambuco.

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

Abstract: Given $\Sigma\subset R:=\mathbb K[x_1,\ldots,x_k]$, where $\mathbb K$ is a field of characteristic 0, any finite collection of linear forms, some possibly proportional, and any $1\leq a\leq |\Sigma|$, we prove that $I_a(\Sigma)$, the ideal generated by all $a$-fold products of $\Sigma$, 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 $\Sigma$ defining such a line arrangement, the ideal $I_{|\Sigma|-2}(\Sigma)$ is of fiber type. We also prove several conjectures of symbolic powers for defining ideals of star configurations of any codimension $c$.

On a conjecture of Vasconcelos

Abstract: 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

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

Abstract: . 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

The Sally Modules of Ideals: A Survey

Abstract: The Sally module of a Rees algebra $\BB$ relative to one of its Rees subalgebras $\AA$ is a construct that can be used as a mediator for the trade-off of cohomological (e.g. depth) information between $\BB$ and the corresponding associated graded ring for several types of filtrations. While originally devised to deal with filtrations of finite colength, here we treat aspects of these developments for filtrations in higher dimensions as well.

On the conjecture of vasconcelos for artinian almost complete intersection monomial ideals

Abstract: In this short note, we confirm a conjecture of Vasconcelos which states that the Rees algebra of any Artinian almost complete intersection monomial ideal is almost Cohen–Macaulay.

De Jonquières transformations in arbitrary dimension. An ideal theoretic view

Abstract: A generalization of the plane de Jonquières 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 Jonquières transformation is the identity map. In this case we give some results on the homological behavior of the graph of the transformation.

The Rees Algebra of a monomial plane parametrization

Abstract: We compute a minimal bigraded resolution of the Rees Algebra associated to a proper rational parametrization of a monomial plane curve We describe explicitly both the bigraded Betti numbers and the maps of the resolution in terms of a generalized version of the Euclidean Algorithm We also explore the relation between pencils of adjoints of the monomial plane curve and elements in a suitable piece of the defining ideal of the Rees Algebra

Bounds for the degree and Betti sequences along a graded resolution

Abstract: The main goal of this paper is to size up the minimal graded free resolution of a homogeneous ideal in terms of its generating degrees. By and large, this is too ambitious an objective. As understood, sizing up means looking closely at the two available parameters: the shifts and the Betti numbers. Since, in general, bounds for the shifts can behave quite steeply, we filter the difficulty by the subadditivity of the syzygies. The method we applied is hopefully new and sheds additional light on the structure of the minimal free resolution. For the Betti numbers, we apply the Boij-S¨oderberg techniques in order to get polynomial upper bounds for them. It is expected that the landscape of hypersurface singularities already contains most of the difficult corners of the arbitrary case. In this regard, we treat some facets of this case, including an improvement of one of the basic results of a recent work by Bus´e–Dimca–Schenck–Sticlaru.