The subject of my PhD thesis is crepant resolutions and A-Hilbert schemes for four dimensional orbifold quotient singularities. It is structured as follows:
Chapter 1: Introduction and background material
I give a historical introduction to the McKay correspondence and the study of crepant resolutions and A-Hilbert schemes. The chapter also contains basic definitions and notation which will be required later in the thesis.
Chapter 2: Resolutions
In dimension three, a crepant resolution always exists, however this is not true for dimensions four. Terminal singularities are an obvious example of a type of singularity where this fails. This chapter contains a discussion of various attempts to find (crepant) resolutions of quotient singularities and why they fail.
Chapter 3: Resolution algorithm
I have written some Magma code to construct crepant resolutions of cyclic four dimensional quotient singularities. This chapter is an explanation of this algorithm.
Chapter 4: Crepant resolutions and A-Hilbert schemes
This chapter builds on work of Craw and Reid (2002). We construct A-Hilbert schemes for certain families of cyclic quotient singularities, and show this leads to a crepant resolution.
My thesis is available to download pdf file
Magma code described in Chapter 3 can be run for small examples on the online Magma calculator. Comments at the end of the code indicate some test cases and expected output.