On the graded resolution of ideals of a few general fat points of image

We compute the graded Betti numbers of the ideal of “few” (at most n) fat points of image with support in general position, under the assumption that the multiplicities are all equal except one, which has to be at least twice as much. We do so by performing a sequence of splittings and applying the graded version given by Fatabbi (J. Algeb. 242 (2001) 92) of a result of Eliahou and Kervaire (J. Algeb. 129 (1990) 1). In doing so, we also prove that the ideal of at most n general fat points of image is always splittable and we compute the graded Betti numbers of the product of two ideals whose generators involve disjoint sets of indeterminates.