  Hyperplane arrangements form the geometric counterpart of combinatorial
objects such as matroids. The shape of the sequence of Betti numbers of the
complement of a hyperplane arrangement is of particular interest in
combinatorics, where they are known, up to a sign, as Whitney numbers of the
first kind, and appear as the coefficients of chromatic, or characteristic,
polynomials. We show that certain combinations, some nonlinear, of these Betti
numbers satisfy Schur positivity. At the same time, we study the higher degree
resonance varieties of the arrangement. We draw some consequences, using
homological algebra results and vector bundles techniques, of the fact that all
resonance varieties are determinantal.
