1
$\begingroup$

Apologies in advance for the naive and rather speculative question.

In this blog post by John Baez, he paints a (perhaps not original) picture of how one might expect that the minimal resolution of a Kleinian singularity $\mathbb{C}^2/G$ is given by Nakajima's equivariant Hilbert scheme of points. At the end though, he admits that its not clear how one could interpret the exceptional divisor, indeed the end of the post reads:

As I hope you see, these are certain ‘limits’ of 600-cells that have ‘shrunk to the origin’… or in other words, highly symmetrical ways for 120 points in C2 to collide at the origin, with some highly symmetrical conditions on their velocities, accelerations, etc.

That’s what I need to understand.

I am looking for some resource or a pointer to how one could formalize this further. While both the Hilbert scheme and the blow up are crepant resolutions, he seems to be trying to understand the exceptional divisor from the viewpoint that the resolution is a "parameter space" (informally) for various positions of the polyhedra that the finite subgroup acts on. My interest is exactly this viewpoint; rather then blowing up, as computing the intersection graph of the exceptional divisor seems like a "coincidence" to me, I think this viewpoint could help me understand the McKay correspondence better.

Thanks in advance.

Crosspost on StackExchange

$\endgroup$
  • 1
    $\begingroup$ The paper Kleinian singularities, derived categories and Hall algebras by Kapranov and Vasserot (Math. Annalen 2000, arXiv version 9812016) discusses this at length, with references to Kronheimer & Nakajima, Gonzalez-Springberg & Verdier and some others $\endgroup$ – მამუკა ჯიბლაძე Jan 17 '19 at 23:03
  • $\begingroup$ Yea I've been meaning to read that paper, but at a quick glance it doesn't seem to say anything about how you can understand the exceptional divisor from this viewpoint detailed above? $\endgroup$ – DKS Jan 18 '19 at 11:47
  • 1
    $\begingroup$ Their starting point (in 1.1) to employ equivariant sheaves is the identification (referred to Kronheimer & Nakajima) of the resolution with the closure of the lift of the variety of free orbits to the Hilbert scheme. All the rest rests on that. $\endgroup$ – მამუკა ჯიბლაძე Jan 18 '19 at 13:03
  • $\begingroup$ Analysis of the inverse image of 0 in the resolution is done in section 2 using vector bundles introduced by Gonzalez-Springberg & Verdier. $\endgroup$ – მამუკა ჯიბლაძე Jan 18 '19 at 13:10
  • $\begingroup$ Ahh I see, I’ll look at the paper by Kronheimer and Nakajima then. Thanks $\endgroup$ – DKS Jan 18 '19 at 15:39
3
$\begingroup$

Exceptional divisors are realized by Hecke correspondences, pairs $(I_1, I_2)$ of $G$ invariant ideals with $I_1\subset I_2$ such that $\mathbb C[x,y]/I_1$ is the regular representation, and $\mathbb C[x,y]/I_2$ is the regular minus an irreducible representation $\rho_i$. Hecke correspondences were originally used to define actions of $E_i$, $F_i$, generators of the simple Lie algebra, in more general contexts of quiver varieties. In this special case of the minimal resolution, other factors of Hecke correspondences, the moduli spaces for $I_2$ are points, hence Hecke correspondences are regarded as subvarieties in the minimal resolution.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.