A note on discriminantal arrangements

Let A₀be a fixed affine arrangement of n hyperplanes in general position in K^k. Let U(n, k) denote the set of general position arrangements whose elements are parallel translates of the hyperplanes of A₀. Then U(n, k) is the complement of a central arrangement 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) in terms of the original arrangement A₀. In terms of the underlying matroids, B(n, k) realizes an adjoint of the dual of the matroid associated with A₀. Using this description we show that, contrary to the conventional wisdom, neither the intersection lattice of B(n, k) nor the topology of U(n, k) is independent of the original arrangement A₀.