Universal degeneration of Riemann surfaces and fibered complex surfaces

TL;DR

This paper constructs a universal degenerating family of Riemann surfaces over the Deligne-Mumford compactification, providing a framework to understand fibered complex surfaces with unstable fibers.

Contribution

It introduces a universal degenerating family over the orbifold moduli space that captures all fibered complex surfaces with unstable fibers.

Findings

Construction of a degenerating family over the orbifold moduli space

Universal property for fibered complex surfaces with unstable fibers

Application of Teichmüller theory to degeneration phenomena

Abstract

Based on Teichm\"uller theory, we construct a degenerating family $\overline{Y}_g^{orb} \rightarrow \overline{M}_g^{orb}$ over the Deligne-Mumford compactification of the moduli space with the natural orbifold structure such that any fibered complex surface admitting unstable fibers can be pulled back from this family.