Smooth curves specialize to extremal curves

Let Hd,g denote the Hilbert scheme of locally Cohen–Macaulay curves of degree d and genus g in projective three space. We show that, given a smooth irreducible curve C of degree d and genus g, there is a rational curve {[Ct] : t ∈ A1} in Hd,g such that Ct for t = 0 is projectively equivalent to C, while the special fibre C0 is an extremal curve. It follows that smooth curves lie in a unique connected component of Hd,g.We also determine necessary and sufficient conditions for a locally Cohen–Macaulay curve to admit such a specialization to an extremal curve.