Smooth curves specialize to extremal curves

Let denote the Hilbert scheme of locally Cohen-Macaulay curves of degree and genus in projective three space. We show that, given a smooth irreducible curve of degree and genus , there is a rational curve in such that for is projectively equivalent to , while the special fibre is an extremal curve. It follows that smooth curves lie in a unique connected component of . We also determine necessary and sufficient conditions for a locally Cohen-Macaulay curve to admit such a specialization to an extremal curve.