Linear degenerations of flag varieties

TL;DR

This paper investigates linear degenerations of flag varieties, characterizing their geometric properties and connecting them to Schubert varieties and degenerate Lie algebra orbits.

Contribution

It provides new characterizations of flatness, irreducibility, and normality of degenerate flag varieties using rank tuples, and links some degenerations to Schubert varieties and Lie algebra orbits.

Findings

Some degenerations are isomorphic to Schubert varieties

Normality is studied via cell decompositions of quiver Grassmannians

Degenerate flag varieties relate to highest weight orbits of degenerate Lie algebras

Abstract

Linear degenerate flag varieties are degenerations of flag varieties as quiver Grassmannians. For type A flag varieties, we obtain characterizations of flatness, irreducibility and normality of these degenerations via rank tuples. Some of them are shown to be isomorphic to Schubert varieties and can be realized as highest weight orbits of partially degenerate Lie algebras, generalizing the corresponding results on degenerate flag varieties. To study normality, cell decompositions of quiver Grassmannians are constructed in a wider context of equioriented quivers of type A.