If $S$ is a smooth projective surface, it is well-known (I believe first proven by Fogarty?) that the Hilbert scheme of points $\text{Hilb}^{n}(S)$ is a smooth irreducible variety of dimension $2n$. Moreover, there exists a morphism
$$\pi: \text{Hilb}^{n}(S) \to \text{Sym}^{n}(S)$$
basically taking a length $n$ subscheme to it's cycle-support. It turns out that $\pi$ is a crepant resolution of the singular variety $\text{Sym}^{n}(S)$. That is, the canonical bundle upstairs is simply the pullback of the canonical bundle downstairs. There are many nice resources outlining the proof of irreducibility and smoothness, but it is incredibly hard to find anything proving the crepant resolution statement!
Can someone shed some light on the crepant resolution proof? Maybe some intuition for it, or a link to a resource.
Just for historical understanding, who is this statement due to? I don't think it came from Fogarty's landmark work. Maybe it's too simple to attribute to anyone? I don't know.