TITLE: Homotopy type of Hilbert schemes of points of $\mathbb C^2$
QUESTION [14 upvotes]: Let $X=\mathbb C^2$, let $X^{[n]}$ be the Hilbert scheme of length $n$ 0-cycles in $X$, and let $X^{[n]}_0$ be the closed subscheme formed by the 0-cycles supported at 0. As far as I know $X^{[n]}_0$ and $X^{[n]}$ have the same homotopy type. Can anybody suggest a proof? (according to Nakajima this can be proved by adapting an argument of Slodowy (Four lectures on simple groups and singularities, Section 4.3) but I am unable to do it...)

REPLY [6 votes]: Here's how I understand the argument in Slodowy's book.  Sticking with Tamas' notation, let $X$ denote the Hilbert scheme and $D$ the punctual Hilbert scheme.  You'd like to define a deformation retraction from $X$ to $D$ by sending every point to its limit under the dilation action.  Unfortunately, as Ben points out, $D$ is not fixed by the dilation action, and this function isn't even continuous!  So that doesn't work.  
Instead, you choose a closed tubular neighborhood $A$ of $D$ such that $D$ is a deformation retract of $A$; this is possible by a basic result in algebraic topology (Slodowy cites Spanier's book).  You then use the dilation action to define a deformation retraction from $X$ to $A$.  More precisely, send every element $x\in X$ to the first element of $A$ that it hits when you dilate it inward.  This is a perfectly well-behaved map, and it does the trick.