A note on discriminantal arrangements

Abstract

Let A 0 {\mathcal {A}_0} be a fixed affine arrangement of n hyperplanes in general position in K k {{\mathbf {K}}^k} . Let U ( n , k ) U(n,k) denote the set of general position arrangements whose elements are parallel translates of the hyperplanes of A 0 {\mathcal {A}_0} . Then U ( n , k ) U(n,k) is the complement of a central arrangement B ( n , k ) \mathcal {B}(n,k) . These are the well-known discriminantal arrangements introduced by Y. I. Manin and V. V. Schechtman. In this note we give an explicit description of B ( n , k ) \mathcal {B}(n,k) in terms of the original arrangement A 0 {\mathcal {A}_0} . In terms of the underlying matroids, B ( n , k ) \mathcal {B}(n,k) realizes an adjoint of the dual of the matroid associated with A 0 {\mathcal {A}_0} . Using this description we show that, contrary to the conventional wisdom, neither the intersection lattice of B ( n , k ) \mathcal {B}(n,k) nor the topology of U ( n , k ) U(n,k) is independent of the original arrangement A 0 {\mathcal {A}_0} .