# Smooth curves specialize to extremal curves

Research paper by Robin Hartshorne, Paolo Lella, Enrico Schlesinger

Indexed on: 03 Aug '14Published on: 03 Aug '14Published in: Mathematische Annalen

#### 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 $$t \ne 0$$ 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.