Derivation modules of orthogonal duals of hyperplane arrangements

Abstract

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