Related papers: Uniform Harbourne-Huneke Bounds via Flat Extension…

We prove a uniform bound on the growth of symbolic powers of arbitrary (not necessarily radical) ideals in arbitrary (not necessarily excellent) regular rings of all characteristics. This gives a complete answer to a question of Hochster…

Let $P_N(R)$ be the space of all real polynomials in $N$ variables with the usual inner product $<, >$ on it, given by integrating over the unit sphere. We start by deriving an explicit combinatorial formula for the bilinear form…

This paper is concerned with the tight closure of an ideal $I$ in a commutative Noetherian ring $R$ of prime characteristic $p$. The formal definition requires, on the face of things, an infinite number of checks to determine whether or not…

Let $S = \mathsf{k}[x_1, \ldots, x_n]$, $I$ be an ideal of $S$, and $\bar{I}$ denote its integral closure. A conjecture of K\"{u}ronya and Pintye states that for any homogeneous ideal $I$ of $S$, the inequality $\operatorname{reg}(\bar{I})…

Suppose $A=k[X_1, X_2, \ldots, X_n]$ is a polynomial ring over a field $k$ and $I$ is an ideal in $A$. Then M. P. Murthy conjectured that $\mu(I)=\mu(I/I^2)$, where $\mu$ denotes the minimal number of generators. Recently, Fasel \cite{F}…

We investigate the higher divisorial ideal $D(I):= Ann(Ext^g_R(R/I,R))$ associated to an ideal I of grade g. Our main focus is the containment problem $D(I) \subseteq \overline{I}$. We show that this inclusion holds for broad classes of…

The space of unitary local systems of rank one on the complement of an arbitrary divisor in a complex projective algebraic variety can be described in terms of parabolic line bundles. We show that multiplier ideals provide natural…

We study ideals $\mathcal{I}$ on $\mathbb{N}$ satisfying the following Baire-type property: if $X$ is a complete metric space and $\{X_{A} \colon A \in \mathcal{I} \}$ is a family of nowhere dense subsets of $X$ with $X_{A} \subset X_{B}$…

In this paper, we investigate containment statements between symbolic and ordinary powers and bounds on the Waldschmidt constant of defining ideals of points in projective spaces. We establish the stable Harbourne conjecture for the…

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…

An ideal $I$ in a Noetherian ring is called \textit{normal} if $I^n$ is integrally closed for all $n \geq 1$. Zariski proved that in two-dimensional regular local rings, every integrally closed ideal is normal. However, in dimension three…

Let $K$ be a field and let $R = K[X_1, \ldots, X_m]$ with $m \geq 2$. Give $R$ the standard grading. Let $I$ be a homogeneous ideal of height $g$. Assume $1 \leq g \leq m -1$. Suppose $H^i_I(R) \neq 0$ for some $i \geq 0$. We show (1)…

Let $R$ be a commutative ring with identity, $S$ a multiplicative subset of $R$ and $I$ an ideal of $R$ disjoint from $S$. In this paper, we introduce the notion of an $S$-$n$-absorbing ideal which is a generalization of both the $S$-prime…

Let $S$ be a polynomial ring over a field and $I\subseteq S$ a homogeneous ideal containing a regular sequence of forms of degrees $d_1, \ldots, d_c$. In this paper we prove the Lex-plus-powers Conjecture when the field has characteristic 0…

Let $R$ be a Noetherian ring, $I$ and $J$ two ideals of $R$ and $t$ an integer. Let $S$ be the class of Artinian $R$-modules, or the class of all $R$-modules $N$ with $\dim_RN\leq k$, where $k$ is an integer. It is proved that $\inf\{i:…

We consider ideals arising in the context of conditional independence models that generalize the class of ideals considered by Fink [7] in a way distinct from the generalizations of Herzog-Hibi-Hreinsdottir-Kahle-Rauh [13] and Ay-Rauh [1].…

Let $R$ be a finite commutative ring with identity, and let $P$ be a proper prime ideal of $R$. The prime ideal graph $\Gamma_P(R)$ has vertex set of $R\setminus\{0\}$, where two distinct vertices $x$ and $y$ are adjacent if and only if…

Let $R$ be a standard graded polynomial ring that is finitely generated over a field of characteristic $0$, let $\mathfrak{m}$ be the homogeneous maximal ideal of $R$, and let $I$ be a homogeneous prime ideal of $R$. Dao and Monta\~{n}o…

A homogeneous ideal $I$ of a polynomial ring $S$ is said to have the Rees property if, for any homogeneous ideal $J \subset S $ which contains $I$, the number of generators of $J$ is smaller than or equal to that of $I$. A homogeneous ideal…

A finite ring R and a weight w on R satisfy the Extension Property if every R-linear w-isometry between two R-linear codes in R^n extends to a monomial transformation of R^n that preserves w. MacWilliams proved that finite fields with the…