Related papers: Symbolic powers: Simis and weighted monomial ideal…

We use the correspondence between hypergraphs and their associated edge ideals to study the minimal graded free resolution of squarefree monomial ideals. The theme of this paper is to understand how the combinatorial structure of a…

This article investigates under which conditions the symbolic powers of the extension of an ideal is the same as the extension of the symbolic powers. Our result generalizes the known scenarios. As an application, we prove formulas for the…

In this paper we consider monomial localizations of monomial ideals and conjecture that a monomial ideal is polymatroidal if and only if all its monomial localizations have a linear resolution. The conjecture is proved for squarefree…

In this work, we explore the relationship between free resolution of some monomial ideals and Generalized Hamming Weights (GHWs) of binary codes. More precisely, we look for a structure smaller than the set of codewords of minimal support…

We show that a monomial ideal $I$ has projective dimension $\leq$ 1 if and only if the minimal free resolution of $S/I$ is supported on a graph that is a tree. This is done by constructing specific graphs which support the resolution of the…

Let $S=K[x_1,\ldots,x_n]$ be the polynomial ring over a field and $A$ a standard graded $S$-algebra. In terms of the Gr\"obner basis of the defining ideal $J$ of $A$ we give a condition, called the x-condition, which implies that all graded…

We prove that second and higher powers of the edge ideals of anticycles admit linear quotient orderings, although the edge ideals themselves do not, thus resolving an open question of Hoefel and Whieldon in the affirmative and providing the…

In this paper we provide some exact formulas for the projective dimension and the regularity of edge ideals associated to three special types of vertex-weighted oriented $m$-partite graphs. These formulas are functions of the weight and…

The shedding vertices of simplicial complexes are studied from an algebraic point of view. Based on this perspective, we introduce the class of ass-decomposable monomial ideals which is a generalization of the class of Stanley-Reisner…

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 $\Delta$ be a simplicial complex of a matroid $M$. In this paper, we explicitly compute the regularity of all the symbolic powers of a Stanley-Reisner ideal $I_\Delta$ in terms of combinatorial data of the matroid $M$. In order to do…

Linear resolutions and the stronger notion of linear quotients are important properties of monomial ideals. In this paper, we fully characterize linear quotients in terms of the lcm-lattice of monomial ideals. We also formulate an analogous…

Several relations and bounds for the dimension of principal ideals in group algebras are determined by analyzing minimal polynomials of regular representations. These results are used in the two last sections. First, in the context of…

One of the fundamental invariants connecting algebra and geometry is the degree of an ideal. In this paper we derive the probabilistic behavior of degree with respect to the versatile Erd\H{o}s-R\'enyi-type model for random monomial ideals…

This paper gives exact formulas for the regularity of edge ideals of edge-weighted integrally closed trees. In addition, we provide some linear upper bounds on the regularity of powers of such ideals.

Let $G$ be a simple graph on $n$ vertices. We introduce the notion of bipartite connectivity of $G$, denoted by $\operatorname{bc}(G)$ and prove that $$\lim_{s \to \infty} \operatorname{depth} (S/I(G)^{(s)}) \le \operatorname{bc}(G),$$…

Let $G$ be a simple graph and $I(G)$ be its edge ideal. In this article, we study the Castelnuovo-Mumford regularity of symbolic powers of edge ideals of join of graphs. As a consequence, we prove Minh's conjecture for wheel graphs,…

In this paper, we find a criterion to answer when the second symbolic and ordinary powers of the edge ideal of a 3-partite hypergraph are equal. Also, we give the formula to compute the Waldschmidt constant of the path ideals of a cycle.

Let $S=K[x_1,\ldots,x_n]$ be the polynomial ring over a field $K$, and let $A$ be a finitely generated standard graded $S$-algebra. We show that if the defining ideal of $A$ has a quadratic initial ideal, then all the graded components of…

A Cameron-Walker graph is a graph for which the matching number and the induced matching number are the same. Assume that $G$ is a Cameron-Walker graph with edge ideal $I(G)$, and let $\ind-match(G)$ be the induced matching number of $G$.…