Choose a finite subgroup $\Gamma\subset SL_2(\mathbb C)$. We introduce a class of projective noncommutative surfaces $\mathbb P^2_I$, indexed by sets $I$ of irreducible $\Gamma$-representations. Extending the action of $\Gamma$  from $\mathbb C^2$ to $\mathbb P^2$ , we show that this class of noncommutative surfaces includes both the stack quotient $[\mathbb P^2 /\Gamma]$ and the scheme quotient $\mathbb P^2/\Gamma$. The Hilbert schemes of points on $[\mathbb C^2 /\Gamma]$ and $\mathbb C^2 /\Gamma$ can be interpreted as spaces of framed sheaves of rank 1 on $[\mathbb P^2 /\Gamma]$ and $\mathbb P^2/\Gamma$. With this in mind, we can show that  sets of isomorphism classes of framed torsion-free sheaves on any $\mathbb P^2_I$ carry a canonical bijection to the closed points of an appropriate Nakajima quiver variety. This partially generalises several previous results on such quiver varieties, including the construction of Hilbert schemes of points on the Kleinian singularity $\mathbb C^2 /\Gamma$ as quiver varieties.

This is joint work with Ádám Gyenge. (arXiv:2406.00709)