Title: Hilbert scheme of twisted cubics via stable maps
Abstract: The space of smooth rational cubic curves in projective space
$\mathbb{P}^r$ ($r\ge 3$) is a smooth quasi-projective variety, which gives
us an open subset of the corresponding Hilbert scheme, or the moduli
space of stable maps, or the quasi-map space, or the moduli space of stable sheaves. By
taking its closure, we obtain compactifications $\mathbf{H}$, $\mathbf{M}$, $\mathbf{X}$,
and $\mathbf{S}$ respectively. I will explain how these compactifications are related by explicit blow-ups and -downs.