8 October 2025 — Padova

Quiver schemes and sheaves on noncommutative projective partial resolutions

Søren Gammelgaard, University of Ferrara

Abstract

Consider a finite subgroup \(G\) of \(SL_2(\mathbb{C})\). Then the singular variety \(\mathbb{C}^2/G\) is a Kleinian singularity, also known as an ADE singularity or a canonical surface singularity. Associated to \(G\), we can construct a particular quiver, the McKay quiver. We shall use this quiver to construct a class of schemes, the Nakajima quiver schemes, and indicate how these provide constructions of several interesting moduli spaces associated to \(G\): for instance, the Hilbert schemes of points on \(\mathbb{C}^2/G\), the “equivariant Hilbert schemes” \(nG\mbox{-}\mathsf{Hilb}(\mathbb{C}^2)\), and moduli spaces of framed sheaves on \(P^2/G\). We generalise this construction to noncommutative geometry, introducing “noncommutative projective partial resolutions” of the singular scheme \(P^2/G\), and considering moduli spaces of sheaves on these noncommutative spaces. This is based on joint work with Ádám Gyenge.