# Iterated Hilb

The workshop at RIMS was excellent: I unexpectedly got to have lunch with Professors/AG heroes Hartshorne, Mori, and Mukai, and had some useful conversation that have supplied much direction for what I’ll be doing this summer. One of the objects that is of current interest to me is the ‘iterated Hilbert scheme’. Recall that the classical $G$-Hilbert scheme is the moduli space of $G$-clusters on $\mathbb{A}^n$, which are essentially the scheme-theoretic version of possibly nonfree group orbits.

The BKR theorem states that, for a finite subgroup $G\subset\text{Aut}{X}$ with $X$ a smooth 3-fold whose canonical bundle is locally trivial as a $G$-equivariant sheaf, the Hilbert-Chow morphism $G\text{-Hilb}\,X\to X/G$ is a crepant resolution with an accompanying equivalence of derived categories. One can ‘iterate’ this theorem in the following situation.

Suppose $N\subset G\subset\text{SL}_3(\mathbb{C})$ is a pair of a normal subgroup of a finite subgroup. You can see from the last condition of BKR why one needs to restrict to $\text{SL}_3(\mathbb{C})$. First form $N\text{-Hilb}\,\mathbb{A}^3$, which is a crepant resolution of $\mathbb{A}^3/G$. The quotient $G/N$ acts on both allowing one to apply BKR again to produce a crepant resolution $G/N\text{-Hilb}\,N\text{-Hilb}\,\mathbb{A}^3\to(\mathbb{A}^3/N)/(G/N)=\mathbb{A}^3/G$. This is the iterated Hilbert scheme, or ‘Hilb of Hilb’, which has been studied most prominently by Ishii-Ito-Nolla de Celis.

This post discusses the results of their paper along with examples of how to compute iterated Hilb when the large group $G$ is abelian, and hence the quotient and iterated Hilb are toric. One of the wider themes that this fits into is the Craw-Ishii conjecture posed in this paper. $G$-Hilb can be famously realised as a moduli space of suitably stable representations of the McKay quiver for $G$, and indeed any chamber inside the stability space produces a crepant resolution of the quotient. Craw-Ishii showed for abelian subgroups of $\text{SL}_3(\mathbb{C})$ and conjectured in general that all crepant resolutions of $\mathbb{A}^3/G$ arise as such moduli spaces for some chamber. Ishii-Ito-Nolla de Celis showed that this is indeed true for iterated Hilb, but the full conjecture is still very much open!