Multiple Lines of Maximum Genus in ${\mathbb{P}}^3$

We introduce a notion of good cohomology for multiple lines in ${\mathbb{P}}^3$ and we classify multiple lines with good cohomology up to multiplicity 4. In particular, we show that the family of space curves of degree d, not lying on a surface of degree less than d, and of maximal arithmetic genus is not irreducible already for d=4 and d=5.