Families of Line Bundles over Riemann Surfaces, their Sections, and their Degenerations -- A Constructive Approach using Automorphic Forms

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.