The Hilbert function and the graded Betti numbers of the 2-nd symbolic power of a 𝕜-configuration in ℙ2

In 2011, Cooper, Harbourne, and Teitler pointed out that the Hilbert function of a [Formula: see text]-dimensional subscheme (not necessarily reduced) in [Formula: see text] having a GMS reduction vector is uniquely determined, but the graded Betti numbers are not uniquely determined. In this paper, we find how to calculate the Hilbert function or a graded minimal free resolution of the [Formula: see text]-nd symbolic power [Formula: see text] of a [Formula: see text]-configuration [Formula: see text] in [Formula: see text] of type [Formula: see text] with [Formula: see text]. For [Formula: see text], we find a complete answer for the graded Betti numbers for [Formula: see text] when the number of lines containing the maximum possible number of points is greater than [Formula: see text], and the Hilbert function when [Formula: see text] is linear, namely, only one line contains [Formula: see text] points, and the reduction vector for [Formula: see text] is NOT GMS. For [Formula: see text], we find the graded Betti numbers for [Formula: see text] when [Formula: see text] is linear, i.e., the reduction vector for [Formula: see text] is NOT GMS. For [Formula: see text], we find the Hilbert function for [Formula: see text] when [Formula: see text] is linear, i.e., the reduction vector for [Formula: see text] is NOT GMS. We also define a complete intersection [Formula: see text]-configuration in [Formula: see text], which is a useful tool to find the Hilbert function or a graded minimal free resolution of the [Formula: see text]-nd symbolic power of a [Formula: see text]-configuration in [Formula: see text]. We also find a graded minimal free resolution of a non-linear [Formula: see text]-configuration in [Formula: see text] of type [Formula: see text] with [Formula: see text], whose reverse reduction vector [Formula: see text] is NOT GMS.