  In this paper we study the deformation problem of pairs consisting of a
Riemann surface and a holomorphic line bundle over that surface, and also
sections thereof. We emphasize a constructive approach throughout and work and
use covering space techniques. In particular, we also describe the limits of
such degenerations as the boundary of Teichm\"uller space is approached, and
review the construction of augmented Teichm\"uller space in great detail.
