  Motivated by problems arising in the complex analysis of perturbative quantum
field theory, we investigate the homology of finite unions of certain
non-degenerate quadratic affine hypersurfaces of complex dimension $n$ in
general position. The homology of such unions and the homology of
$\mathbb{C}^{n+1}$ relative to such unions is decomposed into a direct sum of
homology groups of the various possible intersections of these hypersurfaces.
This allows us to compute the homology groups up to isomorphism. Furthermore,
we compute an explicit set of generators for a specific arrangement of these
hypersurfaces by constructing a sufficiently large CW-subcomplex of the union
in question. The intersection indices of all generators with the Borel-Moore
homology class of $(i\cdot\mathbb{R})^{n+1}$ in the complement of the union of
surfaces are computed.
