McKayCorr Research Group

In general there are many different minimal resolutions of a given singularity in three dimensions, which are related by flops. Mary Barker & Ben Standaert are working towards interpreting flops as mutations of combinatorial objects associated to resolutions in the three dimensional McKay correspondence, with a view to approaching wilder nonabelian cases. The project has yielded an applet to compute mutations of quivers with potential.

References

Crepant resolutions, mutations, and the space of potentials, M. Barker, B. Standaert and B. Wormleighton (arXiv)

Flops and mutations for crepant resolutions of polyhedral singularities, A. Nolla de Celis and Y. Sekiya (arXiv)

Wall-crossing for iterated Hilbert schemes (or 'Hilb of Hilb'), B. Wormleighton (arXiv)

All minimal resolutions of a three dimensional toric quotient singularity can be realised as moduli spaces of stable quiver representations as a stability parameter varies in some large vector space. This vector space decomposes into chambers, reflecting the geometry of each resolution. Joseph Fluegemann is working to complete the classification of walls for a distinguished chamber and study the numerics that arise.

References

Walls for G-Hilb via Reid's recipe, B. Wormleighton (SIGMA)

Flops of G-Hilb and equivalences of derived categories by variation of GIT quotient, A. Craw and A. Ishii (arXiv)

Beyond the toric case we have little understanding of the prototypical resolution, the equivariant Hilbert scheme, for the singularities in McKay. Some resolutions that can be more accessible are iterated Hilbert schemes introduced by Ishii-Ito-Nolla de Celis. Ben Wormleighton is working to understand how the chambers for these resolutions relate to that of the equivariant Hilbert scheme and to further access their geometry.

References

Wall-crossing for iterated Hilbert schemes (or 'Hilb of Hilb'), B. Wormleighton (arXiv)

On G/N-Hilb of N-Hilb, A. Ishii, Y. Ito and A. Nolla de Celis (arXiv)