In the post 'New paper: On Chow rings of quiver moduli' I explained a conjectural identification between two varieties, for which we used our newly gained understanding of Chow rings to give more evidence.

Shortly after posting it to the arXiv I got an email by Laurent Manivel, pointing out that the zero section of $\operatorname{Gr}(2,8)$ makes an appearance Atanas Iliev and Laurent Manivel: Severi varieties and their varieties of reductions, and that it might be useful to prove the identification.

And indeed, after reading up a bit, I realised that the proof of the conjecture can be given by simply stringing together various results, avoiding any actual computations. Namely:

  1. The Kronecker quiver moduli space $X$ is also the moduli space of bundles $Z$ on $\mathbb{P}^2$ with $(r,\mathrm{c}_1,\mathrm{c}_2)=(4,1,3)$, which is of height zero in the sense of Drezet.
  2. The moduli space $Z$ is the image of the second contraction of $\operatorname{Hilb}^3\mathbb{P}^2$ (the other one being $\operatorname{Sym}^3\mathbb{P}^2$).
  3. The zero section $Y$ is the variety $Y_2$ in Iliev–Manivel, this might be clear from Iliev–Manivel itself but a detailed proof is given in Theorem 3.8 of Pietro De Poi, Daniele Faenzi, Emilia Mezzetti and Kristian Ranestad: Fano congruences of index 3 and alternating 3-forms.
  4. Finally, $Y$ and $Z$ are identified by Theorem 4.2 in Iliev–Manivel.

We will update the paper accordingly, but as Hans and I will be on (separate) vacations for the next few weeks, I wanted to write this blog post so that no-one is wasting time on a proof if one is so readily available from the literature already. Of course, if you have something else of interest to tell us, please do!

I would like to thank Laurent Manivel for the suggestion, and Jieao Song and Fabian Reede for related discussions.