# Fortnightly links (136)

Alastair Craw: An introduction to Hilbert schemes of points on ADE singularities are great lecture notes on $\operatorname{Hilb}^n\mathbb{A}^2/\Gamma$ for $\Gamma\subset\operatorname{SL}_2$ finite, explaining how Nakajima quiver varieties shed light on classical results when $\Gamma$ is actually trivial, and what happens when $\Gamma$ becomes non-trivial. This is a really pretty story, and one of the ways that I can understand what happens with Nakajima quiver varieties.

Francis Bischoff, Marco Gualtieri: Brane quantization of toric Poisson varieties is not a paper I can do justice to right now. But it seems to be the closest that exists to a dictionary between noncommutative projective geometry (due to Artin, the Antwerp school, and many others) and generalised complex geometry (due to Hitchin and Gualtieri), and I hope to understand it one day!

Lev Borisov, Vernon Chan, Chengxi Wang: Derived partners of Enriques surfaces constructs a derived equivalence between an Enriques surface and a Brauer-twisted root stack obtained from $\mathbb{P}^2$. This completes the picture started in Theorem 6.16 of Alexander Kuznetsov, Alexander Perry: Categorical joins where the orthogonal to the structure sheaf in the derived category of an Enriques surface was described using a similar construction.

### Noncommutative root stacks?

Seeing noncommutative projective geometry and root stacks in one installment of fortnightly links, I'm led to the question whether it makes sense to consider root stacks of noncommutative projective planes (and more general noncommutative varieties, under suitable conditions). These all have a "commutative divisor", and given that one can blow up noncommutative surfaces (à la Van den Bergh) in points on this divisor, maybe root stacks also have an incarnation in this setting.

When the input is a homogeneous coordinate ring together with a normal element such that the quotient ring describes the commutative divisor, I would hope it is possible to define a $\mathbb{Z}\oplus\mathbb{Z}$-graded ring (à la the scaled Rees rings discussed by Van Oystaeyen) so that a suitable qgr gives the abelian category of the root stack. Its derived category should have the expected semiorthogonal decomposition, which for blowups is in fact known.

If the noncommutative projective plane is finite over its center I've been expecting for a few years that this construction (or something similar) leads to something interesting but I haven't worked out the details unfortunately. But I now hope some construction might work in sweeping generality.

Let me know if you are interested in this!