Containment Problem for Quasi Star Configurations of Points in ℙ²

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.