Hyperplane Arrangements and Compactifications of Vector Groups

TL;DR

This paper investigates hyperplane arrangement Schubert varieties as equivariant compactifications of affine spaces, providing criteria for their characterization and extending the theory to partial compactifications and morphisms.

Contribution

It introduces a characterization of hyperplane arrangement Schubert varieties as equivariant compactifications and generalizes the framework to include partial compactifications and morphisms.

Findings

Criteria for characterizing these varieties

Extension to partial compactifications

Analogy with toric varieties and polyhedral fans

Abstract

Schubert varieties of hyperplane arrangements, also known as matroid Schubert varieties, play 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.