Matroid Schubert varieties as equivariant compactifications of affine spaces

Matroid Schubert varieties are algebraic varieties constructed from hyperplane arrangements, which are central to recent developments in matroid theory. We study these varieties through the lens of equivariant compactifications of affine spaces, and give necessary and sufficient conditions to characterize them. We also generalize matroid Schubert varieties to include partial compactifications, and study morphisms between them. Our results resemble the correspondence between toric varieties and polyhedral fans.