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

We introduce a notion of good cohomology for multiple lines in \({\mathbb {P}}^3\) ${\mathbb{P}}^3$ 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 \(<d\) $<d$ , and of maximal arithmetic genus is not irreducible already for \(d=4\) $d=4$ and \(d=5\) $d=5$ .