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In this paper, the containment problem for the defining ideal of a specialtype of zero dimensional subschemes of $\mathbb{P}^2$, so called quasi starconfigurations, is investigated. Some sharp bounds for the resurgence of thesetypes of ideals are given. As an application of this result, for every realnumber $0 < \varepsilon < \frac{1}{2}$, we construct an infinite family ofhomogeneous radical ideals of points in $\mathbb{K}[\mathbb{P}^2]$ such thattheir resurgences lie in the interval $[2- \varepsilon ,2)$. Moreover, theCastelnuovo-Mumford regularity of all ordinary powers of defining ideal ofquasi star configurations are determined. In particular, it is shown that, %thedefining ideal of a quasi star configuration, and all of them have linearresolution.

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