Homological Properties of Determinantal Arrangements

Abstract

We explore a natural extension of braid arrangements in the context of determinantal arrangements. We show that these determinantal arrangements are free divisors. Additionally, we prove that free determinantal arrangements defined by the minors of $2\times n$ matrices satisfy nice combinatorial properties. We also study the topology of the complements of these determinantal arrangements, and prove that their higher homotopy groups are isomorphic to those of $S^3$. Furthermore, we find that the complements of arrangements satisfying those same combinatorial properties above have Poincar\'e polynomials that factor nicely.