Families of curves in P3 and Zeuthen's problem

We make a detailed study of families of smooth curves in P-3 which lie on cubic surfaces? and their possible degenerations as the cubic surface containing the curve becomes reducible. in particular, we investigate the conditions under which such a family can have as its limit a stick figure, which is a union of lines with at most two meeting at a point. These conditions allow us to give a negative solution to the problem of Zeuthen, by showing that there are values of d,g for which the Hilbert scheme H-d,g(0) of smooth curves of degree d and genus g in P-3 has no stick figures in its closure.