  The space of smooth rational cubic curves in projective space $\PP^r$ ($r\ge
3$) is a smooth quasi-projective variety, which gives us an open subset of the
corresponding Hilbert scheme, the moduli space of stable maps, or the moduli
space of stable sheaves. By taking its closure, we obtain three
compactifications $\bH$, $\bM$, and $\bS$ respectively. In this paper, we
compare these compactifications. First, we prove that $\bH$ is the blow-up of
$\bS$ along a smooth subvariety which is the locus of stable sheaves which are
planar (i.e. support is contained in a plane). Next we prove that $\bS$ is
obtained from $\bM$ by three blow-ups followed by three blow-downs and the
centers are described explicitly. Using this, we calculate the cohomology of
$\bS$.
