  The space of smooth rational curves of degree $d$ in a projective variety $X$
has compactifications by taking closures in the Hilbert scheme, the moduli
space of stable sheaves or the moduli space of stable maps respectively. In
this paper we compare these compactifications by explicit blow-ups and -downs
when $X$ is a projective homogeneous variety and $d\leq 3$. Using the
comparison result, we calculate the Betti numbers of the compactifications when
$X$ is a Grassmannian variety.
