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 $\overline{ \text{ Eff}(\mathscr{ C}_{ 3})}$.
Categories: math.AG