  We compare the deformation theory and the analytic structure of the
Seiberg-Witten moduli spaces of a K\"ahler surface to the corresponding
components of the Hilbert scheme, and show that they are isomorphic. Next we
show how to compute the invariant in case the moduli space is smooth but not of
the expected dimension, and apply this study to elliptic surfaces. Finally we
discuss ruled surfaces, both products and more general ruled surfaces. For
product ruled surfaces we relate the infinitesimal structure of the moduli
spaces to Brill-Noether theory and compute the invariant in special cases. For
more general ruled surfaces, we relate the geometry of the Hilbert scheme to
properties of stable bundles and give more general computations.
