The Hilbert scheme of points on Kleinian Singularities

This thesis will explore a celebrated geometric space, called the Hilbert scheme of points - associated to some much studied singular surfaces.
The Hilbert scheme of points associated to a smooth surface is one of the most beautiful geometric objects to be studied in algebraic geometry and geometric representation theory. Many of the strongest results about this space go back to the work of Fogarty from the late 60s and early 70s. Put simply, the Hilbert scheme, which on the face of it can be arbitrary complicated, turns out to be a smooth space when the surface itself is smooth. That work underpinned many advances in the subsequent decades.
On the other hand, remarkably little is known for the Hilbert scheme of points associated to singular surfaces. Recent work of Craw and collaborators [Craw-Gamelgaard-Gyenge-Szendroi, Alg. Geom. 2021] established many key properties for the Hilbert scheme of points associated to the simplest family of singular surfaces; these surfaces, whose study goes back to the 1930s, have many names, including Kleinian singularities. Their breakthrough came in realising these Hilbert schemes as examples of quiver varieties by applying relatively recent work of [Bellamy-Craw, Invent. Math. 2018].
The concrete goal of the thesis is to generalise these techniques to study a family of singular surfaces, known as the crepant partial resolutions of the Kleinian singularities. The expectation is that one can formulate a common statement that describes the Hilbert scheme of points on the Kleinian singularities, all of their partial crepant resolutions, and a natural smooth surface associated to the Kleinian surface; put simply, we aim to unify the classical description of Fogarty with the recent work of Craw et al., leading to a complete and uniform description of the Hilbert schemes of points on the family of all partial crepant resolutions of Kleinian singularities.
Initially, Ruth with get to grips with the study of Mori Dream Spaces and quiver varieties, putting her in a position to come to terms with the work of [Craw-Gamelgaard-Gyenge-Szendroi, Alg. Geom. 2021]. The challenge then is to take a parameter introduced in that work and allow it to vary more freely, thereby probing not only the Hilbert scheme of points on the Kleinian singularity, but more generally, all partial crepant resolutions of that singular space.
A key goal that Ruth will consider in parallel is to decide whether or not the Hilbert scheme of points on a Kleinian singularity is normal (which means roughly that it is not too badly singular). This question is still open, and there are some natural thought experiments that one can carry out to help settle this question.