Combinatorial structure of exceptional sets in resolutions of singularities

TL;DR

This paper investigates the topological properties of dual complexes associated with resolutions of singularities, demonstrating their trivial homotopy type in specific cases and reviewing related results.

Contribution

It proves the homotopy triviality of dual complexes for certain 3D singularities and revisits previous findings on rational and hypersurface singularities.

Findings

Dual complex is homotopy trivial for 3D terminal singularities

Dual complex is homotopy trivial for Brieskorn singularities

Reviews of earlier results on rational and hypersurface singularities

Abstract

The dual complex can be associated to any resolution of singularities whose exceptional set is a divisor with simple normal crossings. It generalizes to higher dimensions the notion of the dual graph of a resolution of surface singularity. The homotopy type of the dual complex does not depend on the choice of a resolution and thus can be considered as an invariant of singularity. In this preprint we show that the dual complex is homotopy trivial for resolutions of 3-dimensional terminal singularities and for resolutions of Brieskorn singularities. We also review our earlier results on resolutions of rational and hypersurface singularities.