arXiv subjects

Vivek Mukundan

Publications and source records attributed to Vivek Mukundan.

At least 19 recordsLinked to original sources

arXiv2025

Optimal Parametrizations and Valuations

This article discusses a way for uniquely setting up the valuations for the minimal generators of the maximal ideal of a one dimensional complete reduced and irreducible local algebra over an algebraically closed field, when treated as a subring of its integral closure. Our observations are a generalization of the more well-studied case of a numerical semigroup ring. These results provide completion to some missing arguments in certain proofs present in the existing literature, including some results concerning a long-standing conjecture of R. Berger.

math.AC↗

arXiv2025

Symbolic Powers of Classical Varieties

Let $R=\mathbb{K}[x_1,\dots,x_n]$ and let $\mathfrak{a}_1,\dots,\mathfrak{a}_m$ be homogeneous ideals satisfying certain properties, which include a description of the Noetherian symbolic Rees algebra. We give a solution to a question of Harbourne and Huneke for this set of ideals. We also compute the Waldschmidt constant and resurgence and show that it exhibits a stronger version of the Chudnovsky and Demailly-type bounds. We further show that these properties are satisfied for classical varieties such as the generic determinantal ideals, minors of generic symmetric matrices, generic extended Hankel matrices, and ideal of pfaffians of skew-symmetric matrices.

math.AC↗

arXiv2024

Two Criteria For Quasihomogeneity

Let $(R,\mathfrak{m}_R,k)$ be a one-dimensional complete local reduced $k$-algebra over a field of characteristic zero. The ring $R$ is said to be quasihomogeneous if there exists a surjection $Ω_R\twoheadrightarrow \mathfrak{m}$ where $Ω_R$ denotes the module of differentials. We present two characterizations of quasihomogeneity of $R$ in the situation when $R$ is a domain: the first one on the valuation semigroup of $R$ and the other on the trace ideal of the module $Ω_R$.

math.AC↗

arXiv2023

Computing mixed multiplicities, mixed volumes, and sectional Milnor numbers

This is an expository version of our paper [arXiv:1902.07384]. Our aim is to present recent Macaulay2 algorithms for computation of mixed multiplicities of ideals in a Noetherian ring which is either local or a standard graded algebra over a field. These algorithms are based on computation of the equations of multi-Rees algebras of ideals that generalises a result of Cox, Lin and Sosa. Using these equations we propose efficient algorithms for computation of mixed volumes of convex lattice polytopes and sectional Milnor numbers of hypersurfaces with an isolated singularity.

math.AC↗

arXiv2023

Extremal behavior of reduced type of one dimensional rings

Let $R$ be a domain that is a complete local $\mathbb{k}$ algebra in dimension one. In an effort to address the Berger's conjecture, a crucial invariant reduced type $s(R)$ was introduced by Huneke et. al. In this article, we study this invariant and its max/min values separately and relate it to the valuation semigroup of $R$. We justify the need to study $s(R)$ in the context of numerical semigroup rings and consequently investigate the occurrence of the extreme values of $s(R)$ for the Gorenstein, almost Gorenstein, and far-flung Gorenstein complete numerical semigroup rings. Finally, we study the finiteness of the category $\text{CM}(R)$ of maximal Cohen Macaulay modules and the category $\text{Ref}(R)$ of reflexive modules for rings which are of maximal/minimal reduced type and provide many classifications.

math.AC↗

arXiv2022

Valuations and Nonzero Torsion in Module of Differentials

Let $(R,\mathfrak{m}_R,k)$ be a one-dimensional complete local reduced $k$-algebra over a field of characteristic zero. R. Berger conjectured that $R$ is regular if and only if the universally finite module of differentials $Ω_R$ is torsion free. When $R$ is a domain, we prove the conjecture in several cases. Our techniques are primarily reliant on making use of the valuation semi-group of $R$. First, we establish a method of verifying the conjecture by analyzing the valuation semi-group of $R$ and orders of units of the integral closure of $R$. We also prove the conjecture in the case when certain monomials are missing from the monomial support of the defining ideal of $R$. These monomials are based on the smallest power of $\mathfrak{m}_R$ that is contained within the conductor ideal. This also generalizes a previous result of Cortiñas, Geller and Weibel.

math.AC↗

arXiv2022

On the resurgence and asymptotic resurgence of homogeneous ideals

Let $\mathbb{K}$ be a field and $R = \mathbb{K}[x_1, \ldots, x_n]$. We obtain an improved upper bound for asymptotic resurgence of squarefree monomial ideals in $R$. We study the effect on the resurgence when sum, product and intersection of ideals are taken. We obtain sharp upper and lower bounds for the resurgence and asymptotic resurgence of cover ideals of finite simple graphs in terms of associated combinatorial invariants. We also explicitly compute the resurgence and asymptotic resurgence of cover ideals of several classes of graphs. We characterize a graph being bipartite in terms of the resurgence and asymptotic resurgence of edge and cover ideals. We also compute explicitly the resurgence and asymptotic resurgence of edge ideals of some classes of graphs.

math.AC↗

arXiv2022

Tight closure of powers of parameter ideals in hypersurface rings and their tight Hilbert polynomials

In this paper we find the tight closure of powers of parameter ideals of certain diagonal hypersurface rings. In many cases the associated graded ring with respect to tight closure filtration turns out to be Cohen-Macaulay. This helps us find the tight Hilbert polynomial in these diagonal hypersurfaces. We determine the tight Hilbert polynomial in the following cases: (1) F-pure diagonal hypersurfaces where number of variables is equal to the degree of defining equation, (2) diagonal hypersurface rings where characteristic of the ring is one less than the degree of defining equation and (3) quartic diagonal hypersurface in four variables.

math.AC↗

arXiv2022

Algorithms for computing mixed multiplicities, mixed volumes and sectional Milnor numbers

We present a package 'MixedMultiplicity' for computing mixed multiplicities of ideals in a Noetherian ring which is either local or a standard graded algebra over a field. This enables us to find mixed volumes of convex lattice polytopes and sectional Milnor numbers of hypersurfaces with an isolated singularity. The algorithms make use of the defining equations of the multi-Rees algebra of ideals, which are obtained by generalising a result of D. Cox. K.-N. Lin, and G. Sosa.

math.AC↗

arXiv2022

Torsion in Differentials and Berger's Conjecture

Let $(R,\mathfrak{m},\mathbb{k})$ be an equicharacteristic one-dimensional complete local domain over an algebraically closed field $\mathbb{k}$ of characteristic 0. R. Berger conjectured that R is regular if and only if the universally finite module of differentials $Ω_R$ is a torsion-free $R$-module. We give new cases of this conjecture by extending works of Güttes (Arch Math 54:499-510, 1990) and Cortiñas et al. (Math Z 228:569-588, 1998).This is obtained by constructing a new subring $S$ of $\operatorname{Hom}_R(\mathfrak{m},\mathfrak{m})$ and constructing enough torsion in $Ω_S$, enabling us to pull back a nontrivial torsion to $Ω_R$.

math.AC↗

arXiv2020

Expected resurgence of ideals defining Gorenstein rings

Building on previous work by the same authors, we show that certain ideals defining Gorenstein rings have expected resurgence, and thus satisfy the stable Harbourne Conjecture. In prime characteristic, we can take any radical ideal defining a Gorenstein ring in a regular ring, provided its symbolic powers are given by saturations with the maximal ideal. While this property is not suitable for reduction to characteristic $p$, we show that a similar result holds in equicharacteristic $0$ under the additional hypothesis that the symbolic Rees algebra of $I$ is noetherian.

math.AC↗

arXiv2020

Multiplicity of the saturated special fiber ring of height three Gorenstein ideals

Let $R$ be a polynomial ring over a field and $I \subset R$ be a Gorenstein ideal of height three that is minimally generated by homogeneous polynomials of the same degree. We compute the multiplicity of the saturated special fiber ring of $I$. The obtained formula depends only on the number of variables of $R$, the minimal number of generators of $I$, and the degree of the syzygies of $I$. Applying results from arXiv:1805.05180, we get a formula for the $j$-multiplicity of $I$ and an effective method to study a rational map determined by a minimal set of generators of $I$.

math.AC↗

arXiv2020

Expected resurgences and symbolic powers of ideals

We give explicit criteria that imply the resurgence of a self-radical ideal in a regular ring is strictly smaller than its codimension, which in turn implies that the stable version of Harbourne's conjecture holds for such ideals. This criterion is used to give several explicit families of such ideals, including the defining ideals of space monomial curves. Other results generalize known theorems concerning when the third symbolic power is in the square of an ideal, and a strong resurgence bound for some classes of space monomial curves.

math.AC↗

arXiv2019

Equations defining certain graphs

Consider the rational map $ϕ: \mathbb{P}^{n-1}_{\mathbf k} \stackrel{[f_0:\cdots: f_n]}{\longrightarrow} \mathbb{P}^{n}_{\mathbf k}$ defined by homogeneous polynomials $f_0,\dots,f_n$ of the same degree $d$ in a polynomial ring $R=\mathbf k [x_1,\dots,x_n]$ over a field $\mathbf k$. Suppose $I=(f_0,\dots,f_n)$ is a height two perfect ideal satisfying $μ(I_p)\leq\dim R_p$ for $p\in \operatorname{Spec} (R) \setminus V(x_1,\dots, x_n)$. We study the equations defining the graph of $ϕ$ whose coordinate ring is the Rees algebra $R[It]$. We provide new methods to construct these equations using work of Buchsbaum and Eisenbud. Furthermore, for certain classes of ideals satisfying the conditions above, our methods lead to explicit equations defining Rees algebras of the ideals in these classes. These classes of examples are interesting, in that, there are no known methods to compute the defining ideal of the Rees algebra of such ideals. These new methods also give rise to effective criteria to check that $ϕ$ is birational onto its image.

math.AC↗

arXiv2019

On the Vanishing of the normal Hilbert coefficients of ideals

Using vanishing of graded components of local cohomology modules of the Rees algebra of the normal filtration of an ideal, we give bounds on the normal reduction number. This helps to get necessary and sufficient conditions in Cohen-Macaulay local rings of dimension $d\geq 3$, for the vanishing of the normal Hilbert coefficients $\overline{e}_k(I)$ for $k\leq d,$ in terms of the normal reduction number.

math.AC↗

arXiv2019

Tight Closure of powers of ideals and tight Hilbert polynomials

Let $(R,\mathfrak m)$ be an analytically unramified local ring of positive prime characteristic $p.$ For an ideal $I$, let $I^*$ denote its tight closure. We introduce the tight Hilbert function $H^*_I(n)=\ell(R/(I^n)^*)$ and the corresponding tight Hilbert polynomial $P_I^*(n)$ where $I$ is an $\mathfrak m$-primary ideal. It is proved that $F$-rationality can be detected by the vanishing of the first coefficient of $P_I^*(n).$ We find the tight Hilbert polynomial of certain parameter ideals in hypersurface rings and Stanley-Reisner rings of simplicial complexes.

math.AC↗

arXiv2019

Diagonal Subalgebras of Residual Intersections

Let ${\sf k}$ be a field, $S$ be a bigraded ${\sf k}$-algebra, and $S_Δ$ denote the diagonal subalgebra of $S$ corresponding to $Δ= \{ (cs,es) \; | \; s \in \mathbb{Z} \}$. It is know that the $S_Δ$ is Koszul for $c,e \gg 0$. In this article, we find bounds for $c,e$ for $S_Δ$ to be Koszul, when $S$ is a geometric residual intersection. Furthermore, we also study the Cohen-Macaulay property of these algebras. Finally, as an application, we look at classes of linearly presented perfect ideals of height two in a polynomial ring, show that all their powers have a linear resolution, and study the Koszul, and Cohen-Macaulay property of the diagonal subalgebras of their Rees algebras.

math.AC↗

arXiv2017

Detecting Koszulness and related homological properties from the algebra structure of Koszul homology

Let $k$ be a field and $R$ a standard graded $k$-algebra. We denote by $\operatorname{H}^R$ the homology algebra of the Koszul complex on a minimal set of generators of the irrelevant ideal of $R$. We discuss the relationship between the multiplicative structure of $\operatorname{H}^R$ and the property that $R$ is a Koszul algebra. More generally, we work in the setting of local rings and we show that certain conditions on the multiplicative structure of Koszul homology imply strong homological properties, such as existence of certain Golod homomorphisms, leading to explicit computations of Poincaré series. As an application, we show that the Poincaré series of all finitely generated modules over a stretched Cohen-Macaulay local ring are rational, sharing a common denominator.

math.AC↗