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.