Families of space curves with large cohomology

We investigate space curves with large cohomology. To this end we introduce curves of subextremal type. This class includes all subextremal curves. Based on geometric and numerical characterizations of curves of subextremal type, we show that, if the cohomology is "not too small," then they can be parameterized by the union of two generically smooth irreducible families; one of them corresponds to the subextremal curves. For curves of negative genus, the general curve of each of these families is also a smooth point of the support of an irreducible component of the Hilbert scheme. The two components have the same (large) dimension and meet in a subscheme of codimension one.