# Linear degenerations of flag varieties: partial flags, defining equations, and group actions

## TL;DR

This paper explores linear degenerations of flag varieties, realizing them as quiver Grassmannians, and identifies the deepest flat degenerations, providing explicit descriptions and conjectures about their structure and representation theory.

## Contribution

It generalizes previous work by constructing specific degenerations of flag varieties via quiver representations and characterizes their geometric and algebraic properties.

## Key findings

- Existence of deepest flat degeneration analogous to mf-degenerate flag variety
- Explicit description of the reduced scheme structure on degenerations
- Proved an analogue of the Borel-Weil theorem for the flat irreducible locus

## Abstract

We continue, generalize and expand our study of linear degenerations of flag varieties from [G. Cerulli Irelli, X. Fang, E. Feigin, G. Fourier, M. Reineke, Math. Z. 287 (2017), no. 1-2, 615-654]. We realize partial flag varieties as quiver Grassmannians for equi-oriented type A quivers and construct linear degenerations by varying the corresponding quiver representation. We prove that there exists the deepest flat degeneration and the deepest flat irreducible degeneration: the former is the partial analogue of the mf-degenerate flag variety and the latter coincides with the partial PBW-degenerate flag variety. We compute the generating function of the number of orbits in the flat irreducible locus and study the natural family of line bundles on the degenerations from the flat irreducible locus. We also describe explicitly the reduced scheme structure on these degenerations and conjecture that similar results hold for the whole flat locus. Finally, we prove an analogue of the Borel-Weil theorem for the flat irreducible locus.