Hyperplane Arrangements and Compactifications of Vector Groups

Abstract Arrangement Schubert varieties have recently played an essential role in the proof of the Dowling–Wilson conjecture and in Kazhdan–Lusztig theory for matroids. We study these varieties as equivariant compactifications of affine spaces, and give necessary and sufficient conditions to characterize them. We also generalize the theory to include partial compactifications and morphisms between them. Our results resemble the correspondence between toric varieties and polyhedral fans.