Multiple Lines of Maximum Genus in P³

We introduce a notion of good cohomology for multiple lines in P³ 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 .