A note on discriminantal arrangements

Abstract:

Let ${\mathcal {A}_0}$ be a fixed affine arrangement of n hyperplanes in general position in ${{\mathbf {K}}^k}$. Let $U(n,k)$ denote the set of general position arrangements whose elements are parallel translates of the hyperplanes of ${\mathcal {A}_0}$. Then $U(n,k)$ is the complement of a central arrangement $\mathcal {B}(n,k)$. 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 $\mathcal {B}(n,k)$ in terms of the original arrangement ${\mathcal {A}_0}$. In terms of the underlying matroids, $\mathcal {B}(n,k)$ realizes an adjoint of the dual of the matroid associated with ${\mathcal {A}_0}$. Using this description we show that, contrary to the conventional wisdom, neither the intersection lattice of $\mathcal {B}(n,k)$ nor the topology of $U(n,k)$ is independent of the original arrangement ${\mathcal {A}_0}$.