Reduced and non-reduced linear spaces: Lines and points

In this paper we consider the problem of determining the Hilbert function of schemes X of the proiective space P^n which are the generic union of s lines and one m-multiple point. We completely solve this problem for any s and m when n > 3. When n=3 we find several defective such schemes and conjecture that they are the only ones. We verify this conjecture in several cases.