Smooth curves specialize to extremal curves

TL;DR

The paper proves that smooth curves in projective space can be specialized to extremal curves within the Hilbert scheme, showing they lie in a unique connected component and providing conditions for such specializations.

Contribution

It establishes the existence of rational curves in the Hilbert scheme connecting smooth and extremal curves, and characterizes when such specializations are possible.

Findings

Smooth curves can be specialized to extremal curves via rational curves in the Hilbert scheme.

Smooth curves belong to a unique connected component of the Hilbert scheme.

Necessary and sufficient conditions are provided for a curve to admit extremal specialization.

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.