Towards the Classification of Atoms of Degenerations, II: Quotient Singularities and Degenerations of Riemann Surfaces

This is the second of our series of works that constitute a systematic study of degenerations of complex curves (or Riemann surfaces) and their splitting deformations. The present volume focuses on the construction of degenerations, the description of topological monodromies, and related topics. It begins with a detailed study of holomorphic maps on the resolution spaces of singularities associated with degenerations, determining their smooth and singular fibers and topological monodromies. It then applies these results to the construction of degenerations whose topological monodromies are prescribed homeomorphisms. The relationship between (cyclic and dihedral) quotient singularities and topological monodromies is established. This process also determines the correspondence between degenerations and their topological monodromies, which amounts to a reconstruction of the Matsumoto–Montesinos theory from the viewpoint of algebraic geometry: Our approach connects algebraic geometry, topology and singularity theory. We also show various results and introduce concepts needed for our subsequent volumes.