Narrow Results

This paper describes the primary decomposition of the edge ideal of the join of weighted oriented graphs in terms of the information from each of those graphs. Using this decomposition and induction, we derive formulas for the depth and regularity of ordinary powers of the edge ideal of the join of two graphs consisting of isolated vertices. Furthermore, we provide upper bounds for the regularity of symbolic powers of such an edge ideal. For the edge ideal of the join of two graphs with at least one oriented edge for each graph, we provide exact formulas for their depth and regularity, as well as upper bounds for the regularity of ordinary powers of such an edge ideal. Some examples demonstrate that these upper bounds are sometimes attainable and sometimes strict.

We show that attaching a whisker (or a pendant) at the vertices of a cycle cover of a graph results in a new graph with the following property: all symbolic powers of its cover ideal are Koszul or, equivalently, componentwise linear. This extends previous work where the whiskers were added to all vertices or to the vertices of a vertex cover of the graph.

In this paper, an alternative proof is presented of the following result on symbolic powers due to Ein, Lazarsfeld and Smith [3] (for the affine case over ) and to Hochster and Huneke [4] (for the general case). Let A be a regular ring containing a field K. Let be a radical ideal of A and let h be the maximum of the heights of its minimal primes. Then for all n, we have an inclusion , where the first ideal denotes the hnth symbolic power of . In prime characteristic, this result admits an easy tight closure proof due to Hochster and Huneke. In this paper, the characteristic zero version is obtained from this by an application of the Lefschetz Principle.

We study the symbolic powers of the Stanley–Reisner ideal $I_{B_n}$ of a bipyramid $B_n$ over a $n$-gon $Q_n$. Using a combinatorial approach, based on analysis of subtrees in $Q_n$ we compute the Waldschmidt constant of $I_{B_n}$.

Let $I$ denote an ideal of a Noetherian ring $R$ and $N$ a nonzero finitely generated $R$-module. It is shown that if the $𝔭$-symbolic topology is equivalent to the $𝔭$-adic topology on $N$, for all $𝔭\in {mAss}_{R}N/IN$, then the $I$-symbolic topology on $N$ is equivalent to the $I$-adic topology on $N$. Moreover, we show that if ${Ass}_{\widehat{R_{𝔭}}}\widehat{N_{𝔭}}$ consists of a single prime ideal, for all $𝔭\in A^{\ast }(I,N)$, then the $I$-adic and the $I$-symbolic topologies on $N$ are equivalent. Finally, it is shown that if for every one-dimensional prime ideal $𝔭$ in $Supp(N)$, the $𝔭$-adic and the $𝔭$-symbolic topologies are equivalent on $N$, then $N$ is unmixed and ${Ass}_{R}N$ has only one element.

In the present note, we study configurations of codimension 2 flats in projective spaces and classify those with the smallest rate of growth of the initial sequence. Our work extends those of Bocci, Chiantini in ${\mathbb{P}}^2$ and Janssen in ${\mathbb{P}}^3$ to projective spaces of arbitrary dimension.

Let $G$ be a graph and $I=I(G)$ be its edge ideal. When $G$ is the clique sum of two different length odd cycles joined at single vertex then we give an explicit description of the symbolic powers of $I$ and compute the Waldschmidt constant. When $G$ is complete graph then we describe the generators of the symbolic powers of $I$ and compute the Waldschmidt constant and the resurgence of $I$. Moreover for complete graph we prove that the Castelnuovo–Mumford regularity of the symbolic powers and ordinary powers of the edge ideal coincide.

We investigate the rational powers of ideals. We find that in the case of monomial ideals, the canonical indexing leads to a characterization of the rational powers yielding that symbolic powers of squarefree monomial ideals are indeed rational powers themselves. Using the connection with symbolic powers techniques, we use splittings to show the convergence of depths and normalized Castelnuovo–Mumford regularities. We show the convergence of Stanley depths for rational powers, and as a consequence of this, we show the before-now unknown convergence of Stanley depths of integral closure powers. Additionally, we show the finiteness of asymptotic associated primes, and we find that the normalized lengths of local cohomology modules converge for rational powers, and hence for symbolic powers of squarefree monomial ideals.

In this paper, we study the connection between the odd cycles of a finite, simple graph $G$ and the symbolic powers of its edge ideal. When the odd cycles of $G$ are disjoint, we give a decomposition of the symbolic powers of the edge ideal based on the number and size of the odd cycles of $G$. This shows that the symbolic powers of the edge ideal are completely dependent on the odd cycles of $G$.