I'm super-stoked to have Projectivity and effective global generation of determinantal line bundles on quiver moduli on the arXiv now! This is a joint project with Chiara Damiolini, Hans Franzen, Victoria Hoskins, Svetlana Makarova and Tuomas Tajakka, and it is truly a pandemic project:

  • the idea was conceived during Moduli problems beyond geometric invariant theory, which happened almost a year into the pandemic,
  • the question was inspired by the expository write-up Tuomas and I wrote together with Jarod, Dan and Jason on a GIT-free proof of the projectivity of the moduli space of vector bundles on a curve, which was started during the Stacks project workshop in 2020 which was also fully online,
  • the authors have never met in-person,
  • at least for me it was great to have some sense of purpose during the pandemic, having weekly meetings with a group of great people!

So what is this all about? One powerful tool to construct moduli spaces is geometric invariant theory. If you can write down a variety which "over-parametrises" your moduli space, together with a group action that allows you to identify exactly the things that need to be identified, you can just apply a machine called GIT and construct the moduli space.

This was introduced by Mumford in the first half of the 1960s, and used in 1965 by Narasimhan and Seshadri to construct the moduli space of stable bundles on a curve. It was extended to semistable bundles by Seshadri, thus obtaining a projective moduli space.

Curves are one-dimensional, and quivers are also one-dimensional in a homological sense. There are many parallels between curves and quivers (nicely explained in Vicky's Parallels between moduli of quiver representations and vector bundles over curves). Now that we are convinced we are interested in moduli spaces of quiver representations, and these being linear algebra data from the get-go, it is very natural to use geometric invariant theory for their construction.

But in the case of moduli of vector bundles Faltings introduced in the early 1990s a method to construct the moduli space as a projective variety which does not use geometric invariant theory! Rather (although not quite written in this language) he shows that there exists an ample line bundle on the moduli stack which descends to an ample line bundle on the moduli space. An expository account of this result (although not using the technique of Faltings to show ampleness) was given in Projectivity of the moduli space of vector bundles on a curve.

So what did the six of us do? We unleashed our inner Faltings, and showed the projectivity of the moduli space of semistable quiver representations on an acyclic quiver without GIT, and explicitly prove ampleness of a line bundle constructed using on the moduli space. The construction of this line bundle is completely parallel to what happens for curves (it is a determinantal construction), and was already studied (although not explicitly in this language) by Schofield around 1990.

Why is it interesting to do this? That of course depends on who's asking (or listening). I can tell you that my personal reasons were:

  • it is a great way to understand further parallels between curves and quivers,
  • sometimes you can't use GIT to construct moduli spaces, so having "elementary" cases like this to guide you the way in proving projectivity is very useful,
  • it is a great test case and exposition for the theory of good and adequate moduli spaces (we work over rather general bases!),
  • it actually gives you effective bounds on when certain line bundles become globally generated: a question which has seen great interest in the case of moduli of vector bundles on curves, and which now becomes tractable to study for moduli of quiver representations.

If you want to know more, I happily invite you to:

  • read the introduction of the paper
  • read a related Oberwolfach report
  • read the main body of the paper (if you are ambitious)
  • ask me questions you might have
  • invite one of my coauthors (or me) to give a talk about it