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

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 $\overline{{\mathcal{C}}_r}$. For $r\ge 3$, we verify that the subvariety of curves in $\overline{{\mathcal{C}}_r}$ 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 $\overline{\text{Eff}({\mathcal{C}}_3)}$.