  Let $H$ be the Hilbert scheme of curves in complex projective $3$-space, with
$d\geq 3$ and genus $g \leq (d-2)^2/4$. A complete, explicit description of the
cone of curves and the ample cone of $H$ is given. From this, partial results
on the group $\mathop{Aut}(H)$ are deduced.
