Related papers: Symbolic Powers of Monomial Ideals

Let $R$ be a commutative Noetherian ring and let ${\bf x} :=x_1,\ldots,x_d$ be a regular $R$-sequence contained in the Jacobson radical of $R$. An ideal $I$ of $R$ is said to be a monomial ideal with respect to ${\bf x}$ if it is generated…

We introduce and study rational symbolic powers of ideals in Noetherian rings. We give membership criteria for rational symbolic powers and discuss settings where they agree with integer symbolic powers. We investigate the binomial…

In this paper, we investigate the componentwise linearity and the Castelnuovo-Mumford regularity of symbolic powers of polymatroidal ideals. For a polymatroidal ideal $I$, we conjecture that every symbolic power $I^{(k)}$ is componentwise…

The symbolic powers $I^{(n)}$ of a radical ideal $I$ in a polynomial ring consist of the functions that vanish up to order $n$ in the variety defined by $I$. These do not necessarily coincide with the ordinary algebraic powers $I^n$, but it…

Let $I$ and $J$ be nonzero ideals in two Noetherian algebras $A$ and $B$ over a field $k$. Let $I+J$ denote the ideal generated by $I$ and $J$ in $A\otimes_k B$. We prove the following expansion for the symbolic powers: $$(I+J)^{(n)} =…

We develop tools to study the problem of containment of symbolic powers $I^{(m)}$ in powers $I^r$ for a homogeneous ideal $I$ in a polynomial ring $k[{\bf P}^N]$ in $N+1$ variables over an algebraically closed field $k$. We obtain results…

Let $R = \mathbb{K}[x_1, \ldots, x_n]$ be a polynomial ring over a field $\mathbb{K}$, and let $I \subseteq R$ be a monomial ideal of height $h$. We provide a formula for the multiplicity of the powers of $I$ when all the primary ideals of…

B. Harbourne and C. Huneke conjectured that for any ideal $I$ of fat points in $P^N$ its $r$-th symbolic power $I^{(r)}$ should be contained in $M^{(N-1)r}I^r$, where $M$ denotes the homogeneous maximal ideal in the ring of coordinates of…

Given a monomial ideal $I$, we study two functions that quantify ways to measure the difference between symbolic powers and usual powers of $I$. In many cases we determine the asymptotic growth rate of these two functions. We also perform…

We introduce the concept of matching powers of monomial ideals. Let $I$ be a monomial ideal of $S=K[x_1,\dots,x_n]$, with $K$ a field. The $k$th matching power of $I$ is the monomial ideal $I^{[k]}$ generated by the products $u_1\cdots u_k$…

We survey research relating algebraic properties of powers of squarefree monomial ideals to combinatorial structures. In particular, we describe how to detect important properties of (hyper)graphs by solving ideal membership problems and…

In this paper, we propose a uniform approach to tackle problems about squarefree monomial ideals whose powers have good properties. We employ this approach to achieve a twofold goal: (i) recover and extend several well--known results in the…

This paper investigates the v-number of various classes of monomial ideals. First, we considers the relationship between the v-number and the regularity of the mixed product ideal $I$, proving that $\mathrm{v}(I) \leq \mathrm{reg}(S/I)$.…

Let $I$ be a monomial ideal in a polynomial ring. In this paper, we study the asymptotic behavior of the set of associated radical ideals of the (symbolic) powers of $I$. We show that both $\asr(I^s)$ and $\asr(I^{(s)})$ need not stabilize…

The aim of this paper is to study the Stanley depth of symbolic powers of a squarefree monomial ideal. We prove that for every squarefree monomial ideal $I$ and every pair of integers $k, s\geq 1$, the inequalities ${\rm sdepth}…

Let $A = K[X_1,\ldots, X_d]$ and let $I$, $J$ be monomial ideals in $A$. Let $I_n(J) = (I^n \colon J^\infty)$ be the $n^{th}$ symbolic power of $I$ \wrt \ $J$. It is easy to see that the function $f^I_J(n) = e_0(I_n(J)/I^n)$ is of…

We study the symbolic powers of square-free monomial ideals via symbolic Rees algebras and methods in prime characteristic. In particular, we prove that the symbolic Rees algebra and the symbolic associated graded algebra are split with…

The aim of this work is to compare symbolic and ordinary powers of monomial ideals using commutative algebra and combinatorics. Monomial ideals whose symbolic and ordinary powers coincide are called Simis ideals. Weighted monomial ideals…

Two-dimensional squarefree monomial ideals can be seen as the Stanley-Reisner ideals of graphs. The main results of this paper are combinatorial characterizations for the Cohen-Macaulayness of ordinary and symbolic powers of such an ideal…

We study the regularity of symbolic powers of square-free monomial ideals. We prove that if $I = I_\Delta$ is the Stanley-Reisner ideal of a simplicial complex $\Delta$, then $\reg(I^{(n)}) \leqslant \delta(n-1) +b$ for all $n\geqslant 1$,…