I will introduce and discuss residual flags and their connection with Hilbert schemes. The space of residual flags of a given type is a smooth and projective variety. It is a bundle over a flag variety, and also a subvariety of the Hilbert scheme. Closed subschemes of projective m-space that have specific geometric properties are residual flags. One obtains a classification of the smooth Hilbert schemes by exploring the geometry of the residual flags.
The talk is based on a joint work with Greg Smith.