  Let X be a zero-dimensional scheme in P1 \times P1. Then X has a minimal free
resolution of length 2 if and only if X is ACM. In this paper we determine a
class of reduced schemes whose resolutions, similarly to the ACM case, can be
obtained by their Hilbert functions and depends only on their distributions of
points in a grid of lines. Moreover, a minimal set of generators of the ideal
of these schemes is given by curves split into the union of lines.
