Derivation modules of orthogonal duals of hyperplane arrangements

Joseph P.S. Kung, Hal Schenck

Research output: Contribution to journalArticlepeer-review

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 languageEnglish (US)
Pages (from-to)253-262
Number of pages10
JournalJournal of Algebraic Combinatorics
Volume24
Issue number3
DOIs
StatePublished - 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

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