On the Hilbert Function of General Unions of Curves in Projective Spaces

Let X = X1 ∪···∪ Xs ⊂ Pn, n ≥ 4, be a general union of smooth non-special curves with Xi of degree di and genus gi and di ≥ max {2gi − 1, gi + n} if gi 0. We prove that X has maximal rank, i.e., for any t ∈ N either h0 (IX (t)) = 0 or h1 (IX (t)) = 0 if it is so in a few explicit cases in P4. We also prove an unconditional weaker result, maximal rank up to a positive integer δn.