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 \({\mathcal A}\) contains a modular copoint whose complement spans, then the derivation module of the orthogonally dual arrangement \({\mathcal A}\) ⊥ has projective dimension at least ⌈ n(n+2)/4 ⌉ - 3.
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Hal Schenck partially supported by NSF DMS 03-11142, NSA MDA 904-03-1-0006, and ATP 010366-0103.
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Kung, J.P.S., Schenck, H. Derivation modules of orthogonal duals of hyperplane arrangements. J Algebr Comb 24, 253–262 (2006). https://doi.org/10.1007/s10801-006-0023-6
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DOI: https://doi.org/10.1007/s10801-006-0023-6