Related papers: Depth and regularity of monomial ideals via polari…

Let $S=K[x_1, \ldots,x_n]$ denote the polynomial ring in $n$ variables over a field $K$ and $I(G) \subset S$ the edge ideal of a finite graph $G$ on $n$ vertices. Given a vector $\mathfrak{c}\in\mathbb{N}^n$ and an integer $q\geq 1$, we…

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…

This paper concerns the exponentiation of monomial ideals. While it is customary for the exponentiation operation on ideals to consider natural powers, we extend this notion to powers where the exponent is a positive real number. Real…

In this paper, we study Cstelnuovo-Mumford regularity of square-free monomial ideals generated in degree 3. We define some operations on the clutters associated to such ideals and prove that the regularity is conserved under these…

Let $G$ be a bipartite graph and $I=I(G)$ be its edge ideal. The aim of this note is to investigate different aspects of the Rees algebra $\mathcal{R}(I)$ of $I$. We compute its regularity and the universal Gr\"obner basis of its defining…

We count the numbers of associated primes of powers of ideals as defined by Bandari, Hibi, and Herzog in 2014. We generalize those ideals to monomial ideals $\operatorname{BHH}(m,r,s)$ for $r \ge 2$, $m$, $s \ge 1$; we establish partially…

Let $G$ be a finite simple graph and let $I(G)$ be its edge ideal. In this article, we deeply investigate the squarefree powers of $I(G)$ by means of Betti splittings. When $G$ is a forest, it is shown that the normalized depth function of…

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…

Let K be a finite field and let X be a subset of a projective space, over the field K, which is parameterized by monomials arising from the edges of a clutter. We show some estimates for the degree-complexity, with respect to the revlex…

We obtain an improved lower bound for the regularity of the binomial edge ideals of trees. We prove an upper bound for the regularity of the binomial edge ideals of certain subclass of block-graphs. As a consequence we obtain sharp upper…

Let $K$ be a field, $R=K[X_1, ..., X_n]$ be the polynomial ring and $J \subsetneq I$ two monomial ideals in $R$. In this paper we show that $\mathrm{sdepth}\ {I/J} - \mathrm{depth}\ {I/J} = \mathrm{sdepth}\ {I^p/J^p}-\mathrm{depth}\…

We survey recent studies on the Castelnuovo-Mumford regularity of edge ideals of graphs and their powers. Our focus is on bounds and exact values of $\text{reg} I(G)$ and the asymptotic linear function $\text{reg} I(G)^q$, for $q \geq 1,$…

We generalize some results of $\mathrm{v}$-number for arbitrary monomial ideals by showing that the $\mathrm{v}$-number of an arbitrary monomial ideal is the same as the $\mathrm{v}$-number of its polarization. We prove that the…

We present combinatorial characterizations for the associated primes of the second power of squarefree monomial ideals and criteria for this power to have positive depth or depth greater than one.

For each $i \geq 0$, we study the trace ideal of the $i$-th exterior power of the module of differentials. We show that these ideals characterize the polynomial rank of graded rings and the formal power series rank of complete local rings,…

We show how multiplier ideals can be used to obtain uniform multiplicative bounds for certain families of ideals on a smooth complex algebraic variety. In particular we prove a quick but rather surprising result about symbolic powers of…

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 $\mathrm{v}(I) \leq \mathrm{reg}(S/I)$.…

In this paper, we relate combinatorial conditions for polarizations of powers of the graded maximal ideal with rank conditions on submodules generated by collections of Young tableaux. We apply discrete Morse theory to the hypersimplex…

We study when blowup algebras are $F$-split or strongly $F$-regular. Our main focus is on algebras given by symbolic and ordinary powers of ideals of minors of a generic matrix, a symmetric matrix, and a Hankel matrix. We also study ideals…

A tame ideal is an ideal $I$ such that the blowup of the affine space $\mathbb{A}_k^n$ along $I$ is regular. In this paper, we give a combinatorial characterization of tame squarefree monomial ideals. More precisely, we show that a square…