  Consider a linear realization of a matroid over a field. One associates with
it a configuration polynomial and a symmetric bilinear form with linear
homogeneous coefficients. The corresponding configuration hypersurface and its
non-smooth locus support the respective first and second degeneracy scheme of
the bilinear form. We show that these schemes are reduced and describe the
effect of matroid connectivity: for (2-)connected matroids, the configuration
hypersurface is integral, and the second degeneracy scheme is reduced
Cohen-Macaulay of codimension 3. If the matroid is 3-connected, then also the
second degeneracy scheme is integral. In the process, we describe the behavior
of configuration polynomials, forms and schemes with respect to various matroid
constructions.
