Hilbert scheme of rational cubic curves via stable maps

Abstract

The space of smooth rational cubic curves in projective space ${\Bbb P}^r$ ($r\ge 3$) 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 ${\bf H}$, ${\bf M}$, and ${\bf S}$ respectively. In this paper, we compare these compactifications. First, we prove that ${\bf H}$ is the blow-up of ${\bf S}$ along a smooth subvariety parameterizing planar stable sheaves. Next we prove that ${\bf S}$ is obtained from ${\bf M}$ by three blow-ups followed by three blow-downs and the centers are described explicitly. Using this, we calculate the cohomology of ${\bf S}$.