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

Reduced and non-reduced linear spaces: Lines and points