On the intersection of the curves through a set of points in P2

Given a set of points in P2, we consider the common zeros of the set of curves of a given degree passing through those points. For general sets of points, these zero sets have the expected dimension and are smooth. In fact, given graded Betti numbers, for any arrangement of points whose ideal has those graded Betti numbers, general among such arrangements, the zero sets have the expected dimension and are smooth.