Narrow Results

Let $D$ be a weighted oriented graph with the underlying graph $G$ when vertices with non-trivial weights are sinks and $I(D),I(G)$ be the edge ideals corresponding to $D$ and $G,$ respectively. We give an explicit description of the symbolic powers of $I(D)$ using the concept of strong vertex covers. We show that the ordinary and symbolic powers of $I(D)$ and $I(G)$ behave in a similar way. We provide a description for symbolic powers and Waldschmidt constant of $I(D)$ for certain classes of weighted oriented graphs. When $D$ is a weighted oriented odd cycle, we compute $\mathrm{\text{reg}}(I{(D)}^{(s)}/I{(D)}^s)$ and prove $\mathrm{\text{reg}}I{(D)}^{(s)}\le \mathrm{\text{reg}}I{(D)}^s$ and show that equality holds when there is only one vertex with non-trivial weight.

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 $G$ 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.

The purpose of this note is to generalize a result of [M. Dumnicki, T. Szemberg and H. Tutaj-Gasińska, Symbolic powers of planar point configurations II, J. Pure Appl. Alg.220 (2016) 2001–2016] to higher-dimensional projective spaces and classify all configurations of $(N-2)$-planes $Z$ in ${\mathbb{P}}^N$ with the Waldschmidt constants less than two. We also determine some numerical and algebraic invariants of the defining ideals $I(Z)$ of these classes of configurations, i.e. the resurgence, the minimal free resolution and the regularity of $I(Z)$, as well as the Hilbert function of $Z$.

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 $G$.

In this paper, the containment problem for the defining ideal of a special type of zero-dimensional subscheme of ℙ², the so-called quasi star configuration, is investigated. Some sharp bounds for the resurgence of these types of ideals are given. As an application of this result, for every real number $0<\epsilon <\frac{1}{2}$, we construct an infinite family of homogeneous radical ideals of points in 𝕂[ℙ²] such that their resurgences lie in the interval [2−ε, 2). Moreover, the Castelnuovo-Mumford regularity of all ordinary powers of defining ideal of quasi star configurations are determined. In particular, it is shown that all of these ordinary powers have a linear resolution.