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A note on discriminantal arrangements

Author: Michael Falk
Journal: Proc. Amer. Math. Soc. 122 (1994), 1221-1227
MSC: Primary 52B30; Secondary 52B40
MathSciNet review: 1209098
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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}$.

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Keywords: Manin-Schechtman arrangement, Grassmann stratum, dual matroid, adjoint
Article copyright: © Copyright 1994 American Mathematical Society

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