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

In 2011, Cooper, Harbourne, and Teitler pointed out that the Hilbert function of a 0-dimensional subscheme (not necessarily reduced) in P2 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 2nd symbolic power 2X of a k-configuration X in P2 of type (1, 2,…,s − 1,s) with s = 4, 5, 6. For s = 4, we find a complete answer for the graded Betti numbers for 2X when the number of lines containing the maximum possible number of points is greater than 1, and the Hilbert function when X is linear, namely, only one line contains 4 points, and the reduction vector for 2X is NOT GMS. For s = 5, we find the graded Betti numbers for 2X when X is linear, i.e. the reduction vector for 2X is NOT GMS. For s = 6, we find the Hilbert function for 2X when X is linear, i.e. the reduction vector for 2X is NOT GMS. We also define a complete intersection k-configuration in P2, which is a useful tool to find the Hilbert function or a graded minimal free resolution of the 2nd symbolic power of a k-configuration in P2. We also find a graded minimal free resolution of a nonlinear k-configuration in P2 of type (d1,d2,d3) with d1 < d2 = 2d1 < d3 = d2 + 1, whose reverse reduction vector (d1,d2,d2,d2, 2d2 + 1, 2d3) is NOT GMS.