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Hilbert scheme of rational cubic curves via stable maps
- American Journal of Mathematics
- Johns Hopkins University Press
- Volume 133, Number 3, June 2011
- pp. 797-834
- 10.1353/ajm.2011.0021
- Article
- Additional Information
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
${\bf S}$.