We work over an algebraically closed field of characteristic zero. Let \(C \subseteq \mathbb{P}^3\) be a space curve. M. Martin-Deschamps and D. Perrin show [C. R. Acad. Sci., Paris, Sér. I 317, No. 12, 1159-1162 (1993; Zbl 0796.14029)] how to give bounds on \(h^1({\mathcal I}_C(n))\) in function of \(d\) (the degree of \(C)\) and \(g\) (the arithmetic genus of \(C)\). The method essentially amounts to look at the general plane section of \(C\) and the bound is obtained by considering the “worst” (from the Hilbert function point of view) plane section. In this paper, which is just a revisitation of the note by M. Martin-Deschamps and D. Perrin (loc. cit.) we pursue this approach further by “stratifying” by the possible Hilbert functions. This allows us to classify the “extremal” curves, i.e. the curves whose cohomology is extremal, and hence to improve the bounds for those curves which are not extremal. In the last section we give some applications to the study of Hilbert schemes. We show that if \(d\geq 5\) and \((d-2) (d-3)/2\geq g>1 +(d-3) (d-4)/2\), then any space curve of degree \(d\), arithmetic genus \(g\) is extremal. Then we describe the Hilbert scheme of space curves of maximal and submaximal genus, and some Hilbert schemes of curves of low degree.