Fat Points a grid in P^2

We study homogeneous schemes of fat points in P2 whose supportis either a complete intersection (CI for short) constructed on an a × bgrid or a CI minus a point, i.e. Xgrid = {CIgrid (a, b); m} and Ygrid ={CIgrid (a, b) \ Pab; m} respectively.We study the connections between the above fat point schemes andparticular varieties of simple points called partial intersections (p.i. forshort). We prove that a homogeneous fat point scheme of type Xgrid ={CIgrid (a, b); m} has the same graded Betti numbers, and hence, the sameHilbert function of a particular p.i. depending only on a, b, m. Moreover, ascheme of double points of type Ygrid = {CIgrid (a, b)\ Pab; 2} has the sameHilbert function of another particular p.i. depending on a, b, m.We also describe an alternative approach to the problem by consideringthe Grobner basis of I_Ygrid .