## Estuary

I push past the banks of trees, the

Spindly trunks whose upward cargo

Barely seems worthy fruition

For what’s beneath, as topsoil

Makes way for sand pressed closer

To learn lessons from the water.

The sight is amplified to

A solitary scope: the

Far shore becomes the horizon,

The fleeting movement beneath

The surface the denizens

Of this kingdom of perception.

I’m reminded to heed the

Canonical horizon,

Acknowledge the screen of trees

As the interval curtain,

And with unprepared eyes see

It’s only an estuary.

## Moduli of constellations/representations

This is more an aside to the previous post than an independent account; the objective was to fill in some of the details of the construction and manipulation of the moduli spaces involved. These spaces parameterise $G$-constellations – equivariant coherent sheaves on $\mathbb{A}^n$ whose global sections are isomorphic to the regular representation as $G$-modules – or, equivalently, representations of the McKay quiver of $G$ with a natural choice of dimension vector and relations.

It appears to me that there are two main approaches to constructing and parameterising $G$-constellations. I call them ‘combinatorial’ and ‘algebraic’, although there are clearly combinatorial elements in the algebraic version, and vice versa.

Combinatorial: start with the coordinate ring $k[\mathbb{A}^n]$ on which $G$ acts and find quotients of it that are isomorphic to $k[G]$. This approach was initiated by Ito, Nakamura, and Reid most notably via the technology of $G$-graphs. This is good for a local description of $G\text{-Hilb}$ among other things.

Algebraic: start with the regular representation $k[G]$ and find $k[\mathbb{A}^n]$-actions on it that are compatible with the action of $G$. This is primarily the approach taken by people like Craw and Ishii especially when using representations of quivers, which very visually document the action of $k[\mathbb{A}^n]$.

The nonabelian case presents challenges for whichever approach is taken (usually a mixture of the two). For example, more complicated relations and the sheer number of variables defining $G$-constellations are computational difficulties that one meets almost immediately.

These rough notes also take the opportunity to discuss stability of equivariant sheaves and, again equivalently, quiver representations and the role it plays in constructing and studying the moduli spaces at work in the McKay correspondence.

## 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!

## A dimer primer

I’m now mostly settled in Japan – so far in my downtime I’ve learned hiragana and am up to タ in katakana, but shopping can still be rather challenging.. – though I haven’t been working too seriously before a week-long conference in Kyoto starting on Monday. Here’s my poster for it if you’re interested.

I went to a seminar talk by Alvaro Nolla de Celis this week on dimer models in geometry. Much like seeing Michael Wemyss speak (that I finally managed at BrAG), managing to see a dimer talk has been eluding me for a while; there’s always been some schedule clash meaning that I couldn’t go. I defied my usual practice by taking notes and so used them to base this post.

It seems that using moduli spaces of quiver representations with some stability condition to resolve singularities has been gaining momentum in algebraic geometry for a while. One of the problems with this method is that it is not always clear how to associate a quiver to a singularity. In the McKay correspondence there is a natural choice – the McKay quiver of the group giving the quotient singularity – and one can view all of this as a generalisation of that procedure.

Dimer models are bipartite graphs drawn on an oriented 2-torus. That’s a compact, real 2-torus rather than an algebraic torus.. Lifting them to the universal cover of the torus produces a tiling of the plane by (possibly different) polygons. The first piece of data comes from ‘zigzag paths’ through the dimer. These are 1-cycles defining classes in $H_1(T^2)\cong\mathbb{Z}^2$. By collecting them all together, one obtains some list of lattice points in the plane. This list is quite special though: there are exactly two zigzag paths through any edge of the dimer – one in each direction – and so the sum of these classes is zero. That means that there’s a polygon for which the outward normals to its sides are the listed vectors. This is the ‘zigzag polygon’. The second piece of data is what is essentially the oriented dual graph, which is a quiver. For a dimer $\Gamma$, let $P_\Gamma$ be its zigzag polygon and $Q_\Gamma$ be its quiver. There are a slew of interesting results in commutative and noncommutative geometry going between these two objects in various guises, but the one I focused on is due to Ishii-Ueda…

Theorem: The moduli space of $\theta$-semistable representations of $Q_\Gamma$ is a crepant (or Calabi-Yau) resolution of the toric singularity $X_{\mathcal{C}P_\Gamma}$ for generic $\theta$.

One constructs the singularity – which is actually affine Gorenstein – by inserting $P_\Gamma$ at height 1 in $\mathbb{R}^3$ and taking the cone over it to get a polyhedral cone, then consider the corresponding affine toric variety. If you can’t see why this is Gorenstein, go back to my notes from a few months ago!

This theorem incorporates some versions of the McKay correspondence when $P_\Gamma$ is a triangle and $Q_\Gamma$ is the McKay quiver as discussed. Interestingly, it was also proved without appealing to the original results of Bridgeland-King-Reid. Anyway, I hope that you have as much fun as I did toying with dimers.