  We study the Hadamard product of the linear forms defining a hyperplane
arrangement with those of its dual, which we view as generating an ideal in a
certain polynomial ring. We use this ideal, which we call the ideal of pairs,
to study logarithmic derivations and critical set varieties of arrangements in
a way which is symmetric with respect to matroid duality. Our main result
exhibits the variety of the ideal of pairs as a subspace arrangement whose
components correspond to cyclic flats of the arrangement. As a corollary, we
are able to give geometric explanations of some freeness and projective
dimension results due to Ziegler and Kung--Schenck.
