## Abstract

Let A be an n × d matrix having full rank n. An orthogonal dual A ^{⊥} of A is a (d-n) × d matrix of rank (d-n) such that every row of A ^{⊥} is orthogonal (under the usual dot product) to every row of A. We define the orthogonal dual for arrangements by identifying an essential (central) arrangement of d hyperplanes in n-dimensional space with the n × d matrix of coefficients of the homogeneous linear forms for which the hyperplanes are kernels. When n ≥ 5, we show that if the matroid (or the lattice of intersection) of an n-dimensional essential arrangement A contains a modular copoint whose complement spans, then the derivation module of the orthogonally dual arrangement A ^{⊥} has projective dimension at least ⌈ n(n+2)/4 ⌉ - 3.

Original language | English (US) |
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Pages (from-to) | 253-262 |

Number of pages | 10 |

Journal | Journal of Algebraic Combinatorics |

Volume | 24 |

Issue number | 3 |

DOIs | |

State | Published - Nov 2006 |

## Keywords

- Hyperplane arrangement
- Matroid
- Module of derivations
- Orthogonal duality
- Projective dimension

## ASJC Scopus subject areas

- Algebra and Number Theory
- Discrete Mathematics and Combinatorics