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Clare D'Cruz

Publications and source records attributed to Clare D'Cruz.

At least 19 recordsLinked to original sources
arXiv2026

Bounds on the second Hilbert coefficient and the depth of the associated graded ring

Let $(R, \mathfrak m)$ be a Noetherian local ring of dimension $d \geq 1$ with $\mathrm{depth} R \geq d-1,$ and let $I$ be an $\mathfrak m$-primary ideal. In this paper, we study bounds on the second Hilbert coefficient of $I$, denoted by $e_{2}(I)$. Under the assumption that the associated graded ring $G(I)$ has depth at least $d-1,$ we first establish a lower bound for $e_{2}(I).$ We then extend several known results from the Cohen-Macaulay case to this general setting and obtain upper bounds for $e_{2}(I)$ in terms of the sectional genus denoted by $\mathrm{g}_{s}(I)$ and the Hilbert coefficients of $I$ and those of a minimal reduction $Q$ of $I$. We further analyze the extremal case when $e_{2}(I)$ attains this bound and relate it to the depth of $G(I)$. In addition, for Buchsbaum local rings, we establish a sharp upper bound for $e_{2}(\mathfrak m)$ using the technique of $S_{2}$-fication. Finally, in the Cohen-Macaulay case, we give sufficient conditions to ensure good properties on the depth of $G(I)$ and of $G(I^n)$ under the assumption that $e_{2}(I)=0$.

math.AC↗
arXiv2026

Curves in ${\mathbb P}^n$ of analytic spread at most $n$

We study closed subschemes $X$ in ${\mathbb P}^n$ of dimension one, locally defined at any point by at most $n$ equations such that the analytic spread of $I_{\mathfrak{m}}$ is at most $n$, where $I \subseteq \Bbbk[x_0, \ldots, x_n] $ is the defining ideal of $X$ and ${\mathfrak{m}} = (x_0, \ldots, x_n)$. In this situation, we show that, under mild conditions, all the powers of $I_{\mathfrak{m}}$ have positive depth, hence the limit depth of $I_{\mathfrak{m}}$ is $1$ unless $I$ is a complete intersection. Moreover, the regularity of the Rees ring is at most one and the fiber cone is Cohen-Macaulay. This applies to every ideal defining a monomial curve in ${\mathbb P}^3$.

math.AC↗
arXiv2022

Symbolic Rees algebras and set-theoretic complete intersections

In this paper we extend a result of Cowsik on set-theoretic complete intersection and a result Huneke, Morales and Goto and Nishida about Noetherian symbolic Rees algebras of ideals. As applications, we show that the symbolic Rees algebras of the following ideals are Noetherian and the ideals are set-theoretic complete intersections: (a) the edge ideal of a complete graph, (b) the Fermat ideal and (c) the Jacobian ideal of a certain hyperplane arrangement.

math.AC↗
arXiv2022

An extension of Rees theorem and two interpretations of a vector in the joint reduction lattice

In \cite{rees} Rees gave a characterization for the normal joint reduction number zero of two $\m$-primary ideals in an analytically unramified Cohen-Macaulay local ring of dimension two. Rees' result is a generalization of Zariski's product theorem for complete ideals in a regular local ring of dimension two. The aim of this paper is to extend Rees' theorem for the ordinary powers of $\m$-primary ideals $I$ and $J$ in a Cohen-Macaulay local ring of dimension two. Following Rees' approach, we define the modified Koszul homology modules $M^1_{r,s}(a^k,b^k)$ for a joint reduction $(a,b)$ of $I$ and $J$. Under the additional assumption that the associated graded rings of $I$ and $J$ have positive depth, we obtain a characterization of the joint reduction number zero of $I$ and $J$ in terms of the vanishing of the module $M^1_{0,0}(a,b)$, as well as in terms of the Hilbert coefficients and the bigraded Hilbert coefficients. More generally, we introduce the joint reduction lattice and study the vanishing of $M^1_{r,s}(a,b)$ for any $r, s \geq 0$. This gives a characterization for a vector $(r,s)$ to be in the joint reduction lattice of $I$ and $J$. We also give a cohomological interpretation of these theorems by investigating the local cohomology modules of the bigraded extended Rees algebra. This gives another characterization for a vector $(r,s)$ to be in the joint reduction lattice and also extends a recent result of Masuti and Verma in \cite{masuti-verma} for ordinary powers of ideals.

math.AC↗
arXiv2020

Symbolic powers, set-theoretic complete intersection and certain invariants

In this survey article we give a brief history of symbolic powers and its connection with the interesting problem of set-theoretic complete intersection. We also state a few problems and conjectures. Recently, in connection to symbolic powers is the containment problem. We list a few interesting results and related problems on the resurgence, Waldschmidt constant and Castelnuovo-Mumford regularity.

math.AC↗
arXiv2020

Symbolic blowup algebras and invariants associated to certain monomial curves in ${\mathbb P}^3$

In this paper we explicitly describe the symbolic powers of curves ${\mathcal C}(q,m)$ in ${\mathbb P}^3$ parametrized by $( x^{d+2m}, x^{d+m} y^m, x^{d} y^{2m}, y^{d+2m})$, where $q,m$ are positive integers, $d=2q+1$ and $\gcd(d,m)=1$. The defining ideal of these curves is a set-theoretic complete intersection. We show that the symbolic blowup algebra is Noetherian and Gorenstein. An explicit formula for the resurgence and the Waldschmidt constant of the prime ideal ${\mathfrak p}:={\mathfrak p}_{ { \mathcal C}(q,m) }$ defining the curve ${\mathcal C}(q,m)$ is computed. We also give a formula for the Castelnuovo-Mumford regularity of the symbolic powers ${\mathfrak p}^{(n)}$ for all $n \geq 1$.

math.AC↗
arXiv2020

Resurgence and Castelnuovo-Mumford regularity of certain monomial curves in ${\mathbb A}^3$

Let ${\mathfrak p}$ be the defining ideal of the monomial curve ${\mathcal C}(2q+1, 2q+1+m, 2q+1+2m)$ in the affine space ${\mathbb A}_k^3$ parameterized by $(x^{2q +1}, x^{2q +1 + m}, x^{2q +1 +2 m})$ where $gcd( 2q+1,m)=1$. In this paper we compute the resurgence of ${\mathfrak p}$, the Waldschmidt constant of ${\mathfrak p}$ and the Castelnuovo-Mumford regularity of the symbolic powers of ${\mathfrak p}$.

math.AC↗
arXiv2019

Symbolic Blowup algebras and invariants of certain monomial curves in an affine space

Let $d \geq 2$ and $m\geq 1$ be integers such that $\gcd (d,m)=1.$ Let ${\mathfrak p}$ be the defining ideal of the monomial curve in ${\mathbb A}_{ \Bbbk{k}}^d$ parametrized by $(t^{n_1}, \ldots, t^{n_d})$ where $n_i = d + (i-1)m$ for all $i = 1, \ldots, d$. In this paper, we describe the symbolic powers ${\mathfrak p}^{(n)} $ for all $n \geq 1$. As a consequence we show that the symbolic blowup algebras ${\mathcal R}_s{({\mathfrak p})}$ and $G_{s}({\mathfrak p}) $ are Cohen-Macaulay. This gives a positive answer to a question posed by S.~Goto in \cite{goto}. We also discuss when these blowup algebras are Gorenstein. Moreover, for $d=3$, considering ${\mathfrak p}$ as a weighted homogeneous ideal, we compute the resurgence, the Waldschmidt constant and the Castelnuovo-Mumford regularity of ${\mathfrak p}^{(n)}$ for all $n \geq 1$. The techniques of this paper for computing ${\mathfrak p}^{(n)}$ are new and we hope that these will be useful to study the symbolic powers of other prime ideals.

math.AC↗
arXiv2017

Symbolic Blowup algebras of monomial curves in ${\mathbb A}^3$ defined by arithmetic sequence

In this paper, we consider monomial curves in ${\mathbb A}_k^3$ parameterized by $t \rightarrow (t^{2q +1}, t^{2q +1 + m}, t^{2q +1 +2 m})$ where $gcd( 2q+1,m)=1$. The symbolic blowup algebras of these monomial curves is Gorenstein (\cite{goto-nis-shim}, \cite{goto-nis-shim-2}). We give a simple proof for the the Gorenstein property for the symbolic blowup algebras of these curves.

math.AC↗
arXiv2010

On the depth of blow-up rings of ideals of minimal mixed multiplicity

We show that if $(R, \m)$ is a Cohen-Macaulay local ring and $I$ is an ideal of minimal mixed multiplicity, then $\depth G(I) \geq d- 1$ implies that $\depth F(I) \geq d-1$. We use this to show that if $I$ is a contracted ideal in a two dimensional regular local ring then $\depth R[It]-1= \depth G(I) = \depth F(I)$. We also give an infinite class of ideals where $R[It]$ is Cohen-Macaulay but $F(I)$ is not.

math.AC↗
arXiv2004

Multigraded extended Rees algebras of $m$-primary ideals

In this paper we consider multi-graded extended Rees algebras of zero dimensional ideals which are Cohen-Macaulay (CM) with minimal multiplicity. We show that the minimal multiplicity property can occur only for the ordinary extended Rees algebra and the bigraded extended Rees algebra. For the bigraded extended Rees algebra we find necessary conditions for it to be CM with minimal multiplicity. We also produce bigraded Rees algebras which are Cohen-Macaulay with minimal multiplicity.

math.AC↗