A note on discriminantal arrangements

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Abstract

Let A0be a fixed affine arrangement of n hyperplanes in general position in Kk. 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.

Original languageEnglish (US)
Pages (from-to)1221-1227
Number of pages7
JournalProceedings of the American Mathematical Society
Volume122
Issue number4
DOIs
StatePublished - 1994

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Keywords

  • Adjoint
  • Dual matroid
  • Grassmann stratum
  • Manin-Schechtman arrangement

ASJC Scopus subject areas

  • Mathematics(all)
  • Applied Mathematics

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