Interpolation for normal bundles of general curves

Given n general points p_1, p_2, \ldots , p_n \in \mathbb P^r, it is natural to ask when there exists a curve C \subset \mathbb P^r, of degree d and genus g, passing through p_1, p_2, \ldots , p_n. In this paper, the authors give a complete answer to this question for curves C with nonspecial hyperplane section. This result is a consequence of our main theorem, which states that the normal bundle N_C of a general nonspecial curve of degree d and genus g in \mathbb P^r (with d \geq g + r) has the property of interpolation (i.e. that for a general effective divisor D of any degree on C, either H^0(N_C(-D)) = 0 or H^1(N_C(-D)) = 0), with exactly three exceptions.