Whilst it might look like I am hyperproductive these few weeks, with a new blogpost coming up almost every day about something I have done, I am mostly just putting things online that were finished before, in preparation of the summer holidays.

This post is about Fano 4-fold quiver moduli from subspace quivers, a new paper joint with Markus Reineke. Let's discuss the origin story of this paper in a way that is not appropriate for the paper's introduction, but that is perfect for a blog post.

Origin

It all started from Laurent Manivel's A four-dimensional cousin of the Segre cubic. In this paper, he studies a Fano fourfold, defined as the zero locus of a vector bundle on the product of two Grassmannians. Looking at the properties he finds for this Fano fourfold, my mind was applying the duck test in the following form:

If it looks like a quiver moduli space, swims like a quiver moduli space, and quacks like a quiver moduli space, then it probably is a quiver moduli space.

Quiver moduli are after all very special varieties: rational, Hodge–Tate, rigid. Laurent's Segre cubic cousin had all these properties. Through its relation to the Segre cubic, I also quickly guessed the quiver and dimension vector that should do the trick: the 6-subspace quiver, and $\mathbf{d}=(1,1,1,1,1,2;3)$. However, I couldn't pin down an isomorphism.

Then in February Markus came to Utrecht as the Springer visiting chair. Markus realized there is actually a fun classification problem hiding here: which other subspace quivers give rise to Fano 4-folds? The reason to be interested in subspace quivers is that, besides their moduli spaces being rigid, their moduli spaces also have no infinitesimal automorphisms, making them extra special.

It turns out that up to natural identifications, there are precisely 4 such Fano 4-fold subspace quiver moduli: one being the (expected) Segre cousin, another being the Fano model of $\mathop{\rm Bl}_6\mathbb{P}^4$, which is also $(\mathbb{P}^1)^7//\mathrm{PGL}_2$, the moduli space of 7 points on $\mathbb{P}^1$ (let's not spell out stability).

The other two are also very interesting: one admits a natural map to $\mathbb{P}^2$, exhibiting it as something called an involution surface bundle (the first time I encountered one in the wild), and another which looks a lot like the Segre cubic cousin.

And in the end, we also managed to prove that Manivel's Segre cubic cousin was indeed the subspace quiver moduli space we expected it to be all along. All's well that ends well.

Working with quiver moduli

What I like about this paper is that it is so very explicit with quiver moduli, and applies so many of the tools that have been developed for them. I consider it a bit of an advertisement, and I hope that it gets picked up by people who, a priori, would not work with quiver moduli, but now realize how useful this perspective can be!

Related to this is also the use of QuiverTools, which makes it possible to compute many invariants of quiver moduli. For the 4 cases in this paper, the code is available as FanoFourfoldSubspaceQuiverModuli.jl. And the tools we have used in this paper were also the inspiration for some of the new features for QuiverTools, more about those later.

Note that the duck test for quiver moduli doesn't always work so well! We are also working on a paper that explains how well the duck test works for Fano 3-folds. Stay tuned for that.

Also note that there is another work-in-progress (don't expect either of them before the start of my summer break, though!) where we take the involution surface bundle Fano 4-fold subspace quiver moduli, and take its connection to the extended Dynkin quiver $\widetilde{\mathrm{D}}_4$, to find 5 infinite series of even-dimensional Fano varieties attached to extended Dynkin quivers, with lots of interesting properties. Stay tuned for that too!