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Home > Vol 24, No 3 (2006)

Joseph P.S. Kung¹ and Hal Schenck²

¹University of North Texas Department of Mathematics Denton TX USA 76203
²Texas A\&M University Department of Mathematics College Station TX USA 77843

Let A be an n \times  d matrix having full rank n. An orthogonal dual A ^\perp  of A is a (d-n) \times  d matrix of rank ( d -  n) such that every row of A ^\perp  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 \times  d matrix of coefficients of the homogeneous linear forms for which the hyperplanes are kernels. When n \geq  5, we show that if the matroid (or the lattice of intersection) of an n-dimensional essential arrangement A {\mathcal A} contains a modular copoint whose complement spans, then the derivation module of the orthogonally dual arrangement A {\mathcal A} ^\perp  has projective dimension at least \lceil  n(n+2)/4 \rceil  - 3.

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