Description
Many crepant resolutions of Gorenstein quotient singularities can be realised as moduli of quiver representations, which depends on a stability condition. The space of stability conditions has a wall-and-chamber structure that captures much, and in some cases all, of the birational geometry of the singularity. We study this chamber decomposition for abelian subgroups of SL(3) and give an explicit combinatorial / representation-theoretic description of the walls for the chamber corresponding to the G-Hilbert scheme. This description includes the geometry of each wall-crossing, and makes heavy use of Reid’s recipe. We outline some work in progress using these results to compare the G-Hilbert scheme with another crepant resolution ‘Hilb of Hilb’ introduced by Ishii—Ito—Nolla de Celis.
