Derivation Modules of Orthogonal Duals of Hyperplane Arrangements

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.