Fat Points on a generic almost complete intersection

We study homogeneous schemes of fat points in P2 whose supportis either a complete intersection (CI for short) generated by two genericforms or a CI minus a point, i.e., Xgen = {CIgen (a, b); m} and Ygen ={CIgen (a, b) \ P; m}.We prove that Xgrid = {CIgrid (a, b); m} whose support is on an a × bgrid and Xgen = {CIgen (a, b); m} have the same graded Betti numbers,and hence, the same Hilbert function. Moreover, if m = 2, then Ygen ={CIgen (a, b)\P; 2} and Ygrid = {CIgrid (a, b)\Pab; 2} have the same Hilbertfunction, but they may not have the same graded Betti numbers.