### Abstract

Let A_{0}be 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 A_{0}. 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 A_{0}. In terms of the underlying matroids, B(n, k) realizes an adjoint of the dual of the matroid associated with A_{0}. 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 A_{0}.

Original language | English (US) |
---|---|

Pages (from-to) | 1221-1227 |

Number of pages | 7 |

Journal | Proceedings of the American Mathematical Society |

Volume | 122 |

Issue number | 4 |

DOIs | |

State | Published - 1994 |

### Fingerprint

### Keywords

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

### ASJC Scopus subject areas

- Mathematics(all)
- Applied Mathematics

### Cite this

**A note on discriminantal arrangements.** / Falk, Michael J.

Research output: Contribution to journal › Article

*Proceedings of the American Mathematical Society*, vol. 122, no. 4, pp. 1221-1227. https://doi.org/10.1090/S0002-9939-1994-1209098-1

}

TY - JOUR

T1 - A note on discriminantal arrangements

AU - Falk, Michael J

PY - 1994

Y1 - 1994

N2 - 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.

AB - 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.

KW - Adjoint

KW - Dual matroid

KW - Grassmann stratum

KW - Manin-Schechtman arrangement

UR - http://www.scopus.com/inward/record.url?scp=0037954928&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0037954928&partnerID=8YFLogxK

U2 - 10.1090/S0002-9939-1994-1209098-1

DO - 10.1090/S0002-9939-1994-1209098-1

M3 - Article

VL - 122

SP - 1221

EP - 1227

JO - Proceedings of the American Mathematical Society

JF - Proceedings of the American Mathematical Society

SN - 0002-9939

IS - 4

ER -