Smooth curves specialize to extremal curves

Abstract

Let $H_{d,g}$ 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 $\{[C_t]: t \in \mathbb{A}^1\}$ in $H_{d,g}$ such that $C_t$ for $t \neq 0$ is projectively equivalent to $C$, while the special fibre $C_0$ is an extremal curve. It follows that smooth curves lie in a unique connected component of $H_{d,g}$. We also determine necessary and sufficient conditions for a locally Cohen-Macaulay curve to admit such a specialization to an extremal curve.