Liaison theory and the birational geometry of the Hilbert scheme of curves in $\mathbb{P}^{3}$

TL;DR

This paper investigates the structure of the Hilbert scheme of certain algebraic curves in projective 3-space, proving the uniqueness of a Cohen-Macaulay component, analyzing subvarieties with Rao modules, and describing the effective cone of divisors.

Contribution

It establishes the uniqueness of a Cohen-Macaulay component in the Hilbert scheme and characterizes the effective cone of divisors for specific curve families.

Findings

Unique Cohen-Macaulay component exists in the Hilbert scheme.

Subvariety with Rao module of rank one contains a reducible divisor.

Divisors are linearly independent and generate extremal rays of the effective cone.

Abstract

In the Hilbert scheme of curves of degree $d_{r}=\frac{r(r+1)}{2}$ and arithmetic genus $g_{r}=\frac{r(r+1)(2r-5)}{6}+1$ in $\mathbb{P}^{3}$ we prove that there exists a unique component of arithmetically Cohen-Macaulay curves denoted by $\overline{\mathscr{C}_{r}}$. For $r\geq 3$, we verify that the subvariety of curves in $\overline{\mathscr{C}_{r}}$ with Rao module of rank one always contains a reducible divisor. In particular, in the case of curves of degree $6$ and genus $3$ we prove that this subvariety is a reducible divisor. Furthermore, the components of such divisor are linearly independent and each component generates an extremal ray of the effective cone $\overline{\text{Eff}(\mathscr{C}_{3})}$.