Towards the classification of atoms of degenerations, I---Splitting criteria via configurations of singular fibers

Abstract

Motivated by the classification problem of atomic degenerations, in our series of papers, we make a systematic study for splitting deformations of degenerations of complex curves. We provide various new methods to construct splitting deformations, and deduce many splitting criteria of degenerations, which will be applied to the classification of atomic degenerations. Roughly, our criteria are separated into two types; in the first type the criteria are expressed in terms of the configuration of a singular fiber, and in the second type, in terms of sub-divisors of a singular fiber. In both types, our constructions are 'visible', in that we can view how the singular fiber is deformed. In the present paper, we demonstrate splitting criteria of the first type.