Minimal Free Resolutions of Zero-dimensional Schemes in ℙ¹ × ℙ¹

Let X be a zero-dimensional scheme in ℙ¹ × 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 depend 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.