Balazs Szendroi; Virtual motives of the Hilbert scheme of points on threefolds

The Hilbert scheme of points of affine three-space has a description as a representation of a quiver with relations of a very special type, which can be used to realize the Hilbert scheme as a global degeneracy locus. This allows one to associate virtual motives to the Hilbert schemes, whose generating series can be computed using a calculation involving the motive of the space of pairs of commuting matrices. Generalizing this, the generating function of virtual motives of Hilbert schemes of points on an arbitrary smooth and quasi-projective threefold can be expressed as a motivic exponential of the generating series of motives of projective spaces, resulting in a formula resembling Goettsche's formula for an algebraic surface. A consequence is a formula for the generating series of the corresponding weight polynomials in terms of deformed MacMahon functions.
This is all joint work with Kai Behrend and Jim Bryan.