arXiv subjects

Alessandra Costantini

Publications and source records attributed to Alessandra Costantini.

10 recordsLinked to original sources

arXiv2025

Algebraic invariants of the special fiber ring of ladder determinantal modules

We provide explicit formulas for key invariants of special fiber rings of ladder determinantal modules, that is, modules that are direct sums of ideals of maximal minors of a ladder matrix. Our results are given in terms of the combinatorial data of the associated ladder matrix. In particular, we compute its dimension, regularity, $a$-invariant, and multiplicity, which via \textsc{Sagbi} degeneration coincide with those of Hibi rings associated to a distributive lattice. Then, via Gröbner degeneration these calculations are reduced to quotients of polynomial rings by monomial ideals. Our formula for the multiplicity of the special fiber ring of these ladder determinantal modules is obtained by counting the number of standard skew Young tableaux associated to a certain skew partition, and so provides a natural generalization of the classical formula for the degree of the Grassmannian.

math.AC↗

arXiv2024

On Rees algebras of ideals and modules with weak residual conditions

Let $E$ be a module of projective dimension one over $R=k[x_1,\ldots,x_d]$. If $E$ is presented by a matrix $φ$ with linear entries and the number of generators of $E$ is bounded locally up to codimension $d-1$, the Rees ring $\mathcal{R}(E)$ is well understood. In this paper, we study $\mathcal{R}(E)$ when this generation condition holds only up to codimension $s-1$, for some $s<d$. Moreover, we provide a generating set for the ideal defining this algebra by employing a method of successive approximations of the Rees ring. Although we employ techniques regarding Rees rings of modules, our findings recover and extend known results for Rees algebras of perfect ideals with grade two in the case that $\mathrm{rank} \, E=1$.

math.AC↗

arXiv2023

On Rees algebras of linearly presented ideals and modules

Let $I$ be a perfect ideal of height two in $R=k[x_1, \ldots, x_d]$ and let $φ$ denote its Hilbert-Burch matrix. When $φ$ has linear entries, the algebraic structure of the Rees algebra $\mathcal{R}(I)$ is well-understood under the additional assumption that the minimal number of generators of $I$ is bounded locally up to codimension $d-1$. In the first part of this article, we determine the defining ideal of $\mathcal{R}(I)$ under the weaker assumption that such condition holds only up to codimension $d-2$, generalizing previous work of P.~H.~L.~Nguyen. In the second part, we use generic Bourbaki ideals to extend our findings to Rees algebras of linearly presented modules of projective dimension one.

math.AC↗

arXiv2023

Rees algebras and generalized depth-like conditions in prime characteristic

In this article we address a question concerning nilpotent Frobenius actions on Rees algebras and associated graded rings. We prove a nilpotent analog of a theorem of Huneke for Cohen-Macaulay singularities. This is achieved by introducing a depth-like invariant which captures as special cases Lyubeznik's F-depth and the generalized F-depth from Maddox-Miller and is related to the generalized depth with respect to an ideal. We also describe several properties of this new invariant and identify a class of regular elements for which weak F-nilpotence deforms.

math.AC↗

arXiv2022

The combinatorial structure of symmetric strongly shifted ideals

Symmetric strongly shifted ideals are a class of monomial ideals which come equipped with an action of the symmetric group and are analogous to the well-studied class of strongly stable monomial ideals. In this paper we focus on algebraic and combinatorial properties of symmetric strongly shifted ideals. On the algebraic side, we elucidate properties that pertain to behavior under ideal operations, primary decomposition, and the structure of their Rees algebra. On the combinatorial side, we develop a notion of partition Borel generators which leads to connections to discrete polymatroids, convex polytopes, and permutohedral toric varieties.

math.AC↗

arXiv2022

Residual Intersections and Core of Modules

We introduce the notion of residual intersections of modules and prove their existence. We show that projective dimension one modules have Cohen-Macaulay residual intersections, namely they satisfy the relevant Artin-Nagata property. We then establish a formula for the core of orientable modules satisfying certain homological conditions, extending previous results of Corso, Polini, and Ulrich on the core of projective one modules. Finally, we provide examples of classes of modules that satisfy our assumptions.

math.AC↗

arXiv2021

Rees algebras of ideals of star configurations

In this article we study the defining ideal of Rees algebras of ideals of star configurations. We characterize when these ideals are of linear type and provide sufficient conditions for them to be of fiber type. In the case of star configurations of height two, we give a full description of the defining ideal of the Rees algebra, by explicitly identifying a minimal generating set.

math.AC↗

arXiv2021

Cohen-Macaulay fiber cones and defining ideal of Rees algebras of modules

Generic Bourbaki ideals were introduced by Simis, Ulrich and Vasconcelos to study the Cohen-Macaulay property of Rees algebras of modules. In this article we prove that the same technique can sometimes be used to investigate the Cohen-Macaulay property of fiber cones of modules and to study the defining ideal of Rees algebras. This is possible as long as the Rees algebra of a given module $E$ is a deformation of the Rees algebra of a generic Bourbaki ideal $I$ of $E$. Our main technical result provides a deformation condition that in fact extends the applicability of generic Bourbaki ideals to situations not covered by previous work.

math.AC↗

arXiv2020

Residual intersections and modules with Cohen-Macaulay Rees algebra

In this paper, we consider a finite, torsion-free module $E$ over a Gorenstein local ring. We provide sufficient conditions for $E$ to be of linear type and for the Rees algebra $\mathcal{R}(E)$ of $E$ to be Cohen-Macaulay. Our results are obtained by constructing a generic Bourbaki $I$ ideal of $E$ and exploiting properties of the residual intersections of $I$.

math.AC↗

arXiv2020

On the Cohen-Macaulay property of the Rees algebra of the module of differentials

Let $R$ be an algebra essentially of finite type over a field $k$ and let $Ω_k(R)$ be its module of Kähler differentials over $k$. If $R$ is a homogeneous complete intersection and $\mathrm{char}(k)=0$, we prove that $Ω_k(R)$ is of linear type whenever its Rees algebra is Cohen-Macaulay and locally at every homogeneous prime $\mathfrak{p}$ the embedding dimension of $R_{\mathfrak{p}}$ is at most twice its dimension.

math.AC↗