Linear degenerations of flag varieties
In 2011, Evgeny Feigin introduced the degenerate flag variety, which is a flat degeneration of the flag variety inspired by the representation theory of SL_n. In the years to follow, several desription of this variety have been provided, similarly to the classical case of the flag variety: as highest weight orbit using a Kodaira embedding, as Schubert variety, as quiver Grassmannian, as 'degenerate' flags, and as vanishing locus of degenerate Plücker relations.
In a joint work with G. Cerulli-Irelli, X.Fang, E.Feigin, M.Reineke, we generalized this notion of degenerate flags and introduced the universal linear degenerate flag variety. We studied the fibres over the space of degeneration parameters, e.g. a product of endomorphism rings, and provided a finite set (rank sequences) parametrizing their isomorphism classes.
We classified the rank sequences such that the fibre is a) irreducible, b) a Schubert variety, c) a PBW degeneration (see M.Laninis talk next week on their cohomology), d) a flat degeneration of the flag variety. Moreover, we identify two special fibres, namely the degenerate flag variety and the mf-degenerate flag variety, and I will explain why these two might be the most interesting ones to study.
I will finish with an outlook on how to generalize and degenerate the other descriptions of the flag variety.