In this paper we determine the irreducible components of the Hilbert schemes H4,g of locally Cohen-Macaulay space curves of degree four and arbitrary arithmetic genus g: there are roughly ∼(g2/24) of them, most of which are families of multiplicity structures on lines. We give deformations which show that these Hilbert schemes are connected. For g≤−3 we exhibit a component that is disjoint from the component of extremal curves and use this to give a counterexample to a conjecture of Aït-Amrane and Perrin.
