arXiv subjects

Mousumi Mandal

Publications and source records attributed to Mousumi Mandal.

At least 19 recordsLinked to original sources

arXiv2026

Asymptotic $\mathrm{v}$-number of graded families of ideals and the Newton-Okounkov region

In this paper, we prove that for Noetherian graded families $\mathcal{I} = \{I_k\}_{k \ge 0}$ of homogeneous ideals in a Noetherian $\mathbb{N}$-graded Noetherian domain, $\lim\limits_{k \to \infty} \frac{\mathrm{v}(I_k)}{k}$ exists, and is given by $\frac{α(I_r)}{r}$ for some $r \ge 1$, where $α(I)$ denotes the initial degree. Extending these results to integral closures, we show that $ \lim\limits_{k\to\infty}\frac{\mathrm{v}(\overline{I_k})}{k} = \lim\limits_{k\to\infty}\frac{α(\overline{I_k})}{k}=\lim\limits_{k\to\infty}\frac{\mathrm{v}(I_k)}{k}=\lim\limits_{k\to\infty}\frac{α(I_k)}{k} $. For a graded family of monomial ideals in a polynomial ring, we provide a combinatorial interpretation of these limits via Newton--Okounkov regions $Δ(\mathcal{I})$. This connection is further generalized to arbitrary homogeneous ideals using good valuations. We also establish that both $\operatorname{reg}(I_k)$ and $\mathrm{v}(I_k)$ are eventually quasi-linear functions of $k$ for any Noetherian graded family. For a stable monomial ideal $I$ we show that $\mathrm{v}(I) < \operatorname{reg}(I)$. Finally, for a zero-dimensional homogeneous ideal $I$ in a polynomial ring $S$, we prove that $\mathrm{v}(I) < e(S/I)$, where $e(S/I)$ denotes the multiplicity.

math.AC↗

arXiv2026

A comparison of the v-number of a monomial ideal and its integral closure

Let $I$ be a monomial ideal in a standard graded polynomial ring and let $\overline{I}$ denote its integral closure. We study the relationship between $\mathrm{v}(I)$ and $\mathrm{v}(\overline{I})$. We prove that $\mathrm{v}(\overline{I}) \leq \mathrm{v}(I)$ for monomial ideals in two variables, for equigenerated monomial ideals in three variables and for several special classes of monomial ideals, while providing examples showing that this inequality does not hold in general. For the edge ideal $I(G)$ of a connected graph $G$, we show that $\mathrm{v}(I(G)^k)=\mathrm{v}(\overline{I(G)^k}) = 2k-1$ for all $k \geq 1+|E(G)|$. Moreover, when $G$ is disconnected, we prove that $\mathrm{v}(\overline{I(G)^k})\leq\mathrm{v}({I(G)^k})$ for all sufficiently large $k$.

math.AC↗

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

A note on Ratliff-Rush filtration, reduction number and postulation number of $\mathfrak m$-primary ideals

Let $(R,\mathfrak m)$ be a Cohen-Macaulay local ring of dimension $d\geq 2$ and $I$ an $\mathfrak m$-primary ideal. Let rd$(I)$ be the reduction number of $I$ and n$(I)$ the postulation number. We prove that for $d=2,$ if n$(I)=ρ(I)-1,$ then rd$(I) \leq$n$(I)+2$ and if n$(I)\neq ρ(I)-1,$ then rd$(I)\geq$n$(I)+2.$ For $d \geq 3$, if $I$ is integrally closed, depth gr$(I) = d-2$ and n$(I)=-(d-3).$ Then we prove that rd$(I)\geq$n$(I)+d$. Our main result is to generalize a result of T. Marley on the relation between the Hilbert-Samuel function and the Hilbert-Samuel polynomial by relaxing the condition on the depth of the associated graded ring with the good behaviour of the Ratliff-Rush filtration with respect to $I$ mod a superficial element. From this result, it follows that for a Cohen-Macaulay ring of dimension $d\geq2$, if $P_{I}(k)=H_{I}(k)$ for some $k \geq ρ(I)$, then $P_{I}(n)=H_{I}(n)$ for all $n \geq k.$

math.AC↗

arXiv2026

Bounds on the Ratliff-Rush Index and the Castelnuovo-Mumford Regularity

Let $(R, \mathfrak m)$ be a Cohen-Macaulay local ring of dimension $d \geq 2,$ and $I$ an $\mathfrak m$-primary ideal of $R.$ Denote $r_{J}(I)$ as the reduction number of $I$ with respect to a minimal reduction $J$ of $I,$ and $ρ(I)$ as the Ratliff-Rush index of $I$. We establish upper bounds on $ρ(I)$ in terms of Hilbert coefficients $e_{i}(I)$ for $0 \leq i \leq d+1,$ and $r_{J}(I).$ Suppose $\widetilde{I^{r_{J}(I)}} \neq I^{r_{J}(I)}.$ We prove that $ρ(I) \leq r_{J}(I)-1+(-1)^{d+1}(e_{d+1}(I)-\widetilde{e}_{d+1}(I)).$ When $d=2,$ we prove that $ρ(I) \leq r_{J}(I) -1 +(e_{2}(I)-1)e_{2}(I)-e_{3}(I).$ This established bound on $ρ(I)$ consequently leads to a bound on the Castelnuovo-Mumford regularity of the associated graded ring of $I.$ We also determine bound on $ρ(I)$ in two-dimensional Buchsbaum rings with positive depth.

math.AC↗

arXiv2026

Lattice points arising from regularity and $\mathrm{v}$-number of Graphs: Whisker and Cameron-Walker

Let $G$ be a simple graph on $n$ vertices and $I(G)\subseteq R$ be its edge ideal. In this paper, we initiate the study of determining lattice points in $\mathbb{N}^2$ that appear as a pair $(\mathrm{reg}(R/I(G)), \mathrm{v}(I(G)))$, where $G$ ranges over all connected graphs on $n$ vertices, and we denote this set by $\mathcal{RV}(n)$. Here `$\mathrm{reg}$' denotes the (Castelnuovo-Mumford) regularity and `$\mathrm{v}$' denotes the $\mathrm{v}$-number. We establish general bounds for $\mathcal{RV}(n)$ by identifying two sets $A(n)$ and $B(n)$ satisfying $A(n)\subseteq \mathcal{RV}(n)\subseteq B(n)$. Furthermore, we explicitly determine the subsets of $\mathcal{RV}(n)$ consisting of all possible pairs $(\mathrm{reg}(R/I(G)), \mathrm{v}(I(G)))$ arising from whisker graphs and Cameron-Walker graphs on $n$ vertices. Finally, we propose a conjecture on the subset of $\mathcal{RV}(n)$ arising from connected chordal graphs.

math.AC↗

arXiv2024

Asymptotic behaviour and stability index of v-numbers of graded ideals

Recently, Ficarra and Sgroi initiated the study of v-numbers of powers of graded ideals. They proved that for a graded ideal $I$ in a polynomial ring $S$, $\mathrm{v}(I^k)$ is a linear function in $k$ for $k>>0$. Later, Ficarra conjectured that if $I$ is a monomial ideal with linear powers, then $\mathrm{v}(I^k)=α(I)k-1$ for all $k\geq 1$, where $α(I)$ denotes the initial degree of $I$. In this paper, we generalize this conjecture for graded ideals. We prove this conjecture for several classes of graded ideals: principal ideals, ideals $I$ with $\mathrm{depth}(S/I)=0$, cover ideals of graphs, $t$-path ideals, monomial ideals generated in degree $2$, edge ideals of weighted oriented graphs. We reduce the conjecture for several classes of graded ideals (including square-free monomial ideals) by showing it is enough to prove the conjecture for $k=1$ only. We define the stability index of the $\mathrm{v}$-number for graded ideals and investigate the stability index for edge ideals of graphs.

math.AC↗

arXiv2023

A study of $v$-number for some monomial ideals

In this paper, we give formulas for $v$-number of edge ideals of some graphs like path, cycle, 1-clique sum of a path and a cycle, 1-clique sum of two cycles and join of two graphs. For an $\mathfrak{m}$-primary monomial ideal $I\subset S=K[x_1,\ldots,x_t]$, we provide an explicit expression of $v$-number of $I$, denoted by $v(I)$, and give an upper bound of $v(I)$ in terms of the degree of its generators. We show that for a monomial ideal $I$, $v(I^{n+1})$ is bounded above by a linear polynomial for large $n$ and for certain classes of monomial ideals, the upper bound is achieved for all $n\geq 1$. For $\mathfrak m$-primary monomial ideal $I$ we prove that $v(I)\leq$ reg$(S/I)$ and their difference can be arbitrarily large.

math.AC↗

arXiv2023

Properties of symbolic powers of edge ideals of weighted oriented graphs

Let $D$ be a weighted oriented graph and $I(D)$ be its edge ideal. We provide one method to find all the minimal generators of $ I_{\subseteq C} $, where $ C $ is a maximal strong vertex cover of $D$ and $ I_{\subseteq C} $ is the intersections of irreducible ideals associated to the strong vertex covers contained in $C$. If $ D^{\prime} $ is an induced digraph of $D$, under certain condition on the strong vertex covers of $ D^{\prime} $ and $D$, we show that $ {I(D^{\prime})}^{(s)} \neq {I(D^{\prime})}^s $ for some $s \geq 2$ implies $ {I(D)}^{(s)} \neq {I(D)}^s $. We characterize all the maximal strong vertex covers of $D$ such that at most one edge is oriented into each of its vertex and $w(x) \geq 2$ if $°_D(x)\geq 2 $ for all $x \in V(D)$. If $ D $ is a weighted rooted tree with degree of root is $ 1 $ and $ w(x) \geq 2 $ when $ °_D(x) \geq 2 $ for all $ x \in V(D) $, we show that $ {I(D)}^{(s)} = {I(D)}^s $ for all $s \geq 2$

math.AC↗

arXiv2023

Bounds for the reduction number of primary ideal in dimension three

Let $(R,\mathfrak{m})$ be a Cohen-Macaulay local ring of dimension $d\geq 3$ and $I$ an $\mathfrak{m}$-primary ideal of $R$. Let $r_J(I)$ be the reduction number of $I$ with respect to a minimal reduction $J$ of $I$. Suppose depth $G(I)\geq d-3$. We prove that $r_J(I)\leq e_1(I)-e_0(I)+λ(R/I)+1+(e_2(I)-1)e_2(I)-e_3(I)$, where $e_i(I)$ are Hilbert coefficients. Suppose $d=3$ and depth $G(I^t)>0$ for some $t\geq 1$. Then we prove that $r_J(I)\leq e_1(I)-e_0(I)+λ(R/I)+t$.

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

Symbolic defects of edge ideals of unicyclic graphs

We introduce the concept of minimum edge cover for an induced subgraph in a graph. Let $G$ be a unicyclic graph with a unique odd cycle and $I=I(G)$ be its edge ideal. We compute the exact values of all symbolic defects of $I$ using the concept of minimum edge cover for an induced subgraph in a graph. We describe one method to find the quasi-polynomial associated with the symbolic defects of edge ideal $I$. We classify the class of unicyclic graphs when some power of maximal ideal annihilates $ I^{(s)}/I^s $ for any fixed $ s $. Also for those class of graphs, we compute the Hilbert function of the module $I^{(s)}/I^s$ for all $s.$

math.AC↗

arXiv2022

Regularity of symbolic powers of certain graphs

Let $G_{n,r}$ denote the graph with $n$ vertices $\{x_1,\ldots,x_n\}$ in cyclic order and for each vertex $x_i$ consider the set $A_i=\{x_{i-r},\ldots,x_{i-1},x_{i+1},x_{i+2},\ldots, x_{i+r}\},$ where $x_{i-j}$ is the vertex $x_{n+i-j}$, whenever $i<j$ and $0\leq r\leq \Bigl\lfloor\dfrac{n}{2}\Bigr\rfloor -1$. In $G_{n,r}$, every vertex $x_i$ is adjacent to all the vertices of $V(G_{n,r})\backslash A_i$. Let $I=I(G_{n,r})$ be the edge ideal of $G_{n,r}$. We show that Minh's conjecture is true for $I,$ i.e. regularity of ordinary powers and symbolic powers of $I$ are equal. We compute the Waldschmidt constant and resurgence for the whole class.

math.AC↗

arXiv2021

Comparing symbolic powers of edge ideals of weighted oriented graphs

Let $D$ be a weighted oriented graph and $I(D)$ be its edge ideal. If $D$ contains an induced odd cycle of length $2n+1$, under certain condition we show that $ {I(D)}^{(n+1)} \neq {I(D)}^{n+1}$. We give necessary and sufficient condition for the equality of ordinary and symbolic powers of edge ideal of a weighted oriented graph having each edge in some induced odd cycle of it. We characterize the weighted naturally oriented unicyclic graphs with unique odd cycles and weighted naturally oriented even cycles for the equality of ordinary and symbolic powers of their edge ideals. Let $ D^{\prime} $ be the weighted oriented graph obtained from $D$ after replacing the weights of vertices with non-trivial weights which are sinks, by trivial weights. We show that the symbolic powers of $I(D)$ and $I(D^{\prime})$ behave in a similar way. Finally, if $D$ is any weighted oriented star graph, we show that $ {I(D)}^{(s)} = {I(D)}^s $ for all $s \geq 2.$

math.AC↗

arXiv2021

Regularity in weighted oriented graphs

Let $D$ be a weighted oriented graph with the underlying graph $G$ and $I(D), I(G) $ be the edge ideals corresponding to $D$ and $G$ respectively. We show that the regularity of edge ideal of a certain class of weighted oriented graph remains same even after adding certain kind of new edges to it. We also establish the relationship between the regularity of edge ideal of weighted oriented path and cycle with the regularity of edge ideal of their underlying graph when vertices of $V^+$ are sinks.

math.CO↗

arXiv2021

Symbolic powers in weighted oriented graphs

Let $D$ be a weighted oriented graph with the underlying graph $G$ when vertices with non-trivial weights are sinks and $I(D), I(G) $ be the edge ideals corresponding to $D$ and $G,$ respectively. We give explicit description of the symbolic powers of $I(D)$ using the concept of strong vertex covers. We show that the ordinary and symbolic powers of $I(D)$ and $I(G)$ behave in a similar way. We provide a description for symbolic powers and Waldschmidt constant of $I(D)$ for certain classes of weighted oriented graphs. When $D$ is a weighted oriented odd cycle we compute $\reg (I(D)^{(s)}/I(D)^s)$ and prove $\reg I(D)^{(s)}\leq\reg I(D)^s$ and show that equality holds when there is only one vertex with non-trivial weight.

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↗

arXiv2019

Invariants of the symbolic powers of edge ideals

Let $G$ be a graph and $I=I(G)$ be its edge ideal. When $G$ is the clique sum of two different length odd cycles joined at single vertex then we give an explicit description of the symbolic powers of $I$ and compute the Waldschmidt constant. When $G$ is complete graph then we describe the generators of the symbolic powers of $I$ and compute the Waldschmidt constant and the resurgence of $I$. Moreover for complete graph we prove that the Castelnuvo-Mumford regularity of the symbolic powers and ordinary powers of the edge ideal coincide.

math.AC↗