Hilbert scheme of rational cubic curves via stable maps

TL;DR

This paper compares three different compactifications of the space of rational cubic curves in projective space, analyzing their geometric relations and computing the cohomology of one of these compactifications.

Contribution

It establishes explicit geometric relations among the compactifications and provides a detailed cohomology calculation for the moduli space of stable sheaves.

Findings

$H$ is the blow-up of $S$ along a smooth subvariety.

$S$ is obtained from $M$ by three blow-ups and three blow-downs.

Cohomology of $S$ is explicitly computed.

Abstract

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 $\bS$.