# Containment problem for the quasi star configurations of points in $\mathbb{P}^2$

## TL;DR

This paper investigates the containment problem for ideals of quasi star configurations of points in the projective plane, providing bounds on resurgence, constructing examples with specific resurgence values, and determining regularity of their powers.

## Contribution

It introduces sharp bounds for the resurgence of ideals of quasi star configurations and constructs infinite families with prescribed resurgence values.

## Key findings

- Sharp bounds for the resurgence of quasi star configuration ideals.
- Construction of infinite families with resurgence in [2 - ε, 2).
- Determination of Castelnuovo-Mumford regularity for all powers.

## Abstract

In this paper, the containment problem for the defining ideal of a special type of zero dimensional subschemes of $\mathbb{P}^2$, so called quasi star configurations, 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 $0 < \varepsilon < \frac{1}{2}$, we construct an infinite family of homogeneous radical ideals of points in $\mathbb{K}[\mathbb{P}^2]$ such that their resurgences lie in the interval $[2- \varepsilon ,2)$. Moreover, the Castelnuovo-Mumford regularity of all ordinary powers of defining ideal of quasi star configurations are determined. In particular, it is shown that, %the defining ideal of a quasi star configuration, and all of them have linear resolution.