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

TL;DR

This paper investigates how families of line bundles and their sections over Riemann surfaces deform, especially near degenerations, using automorphic forms and covering space techniques to describe boundary limits in Teichmüller space.

Contribution

It provides a constructive method for analyzing degenerations of line bundles and sections over Riemann surfaces via automorphic forms, with detailed descriptions of boundary behavior in augmented Teichmüller space.

Findings

Describes limits of degenerating line bundles and sections.

Uses covering space techniques for constructive analysis.

Details the construction of augmented Teichmüller space.

Abstract

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.