ℂ^*-equivariant degenerations of curves and normal surface singularities with ℂ^*-action

Abstract

This paper presents a definition of ℂ^*-equivariant degeneration families of compact complex curves over ℂ. Those families are called ℂ^*-pencils of curves. We give the canonical method to construct them and prove some results on relations between them and normal surface singularities with ℂ^*-action. We also define ℂ^*-equivariant degeneration families of compact complex curves over ℙ¹. From this, it is possible to introduce a notion of dual ℂ^*-pencils of curves naturally. Associating it, we prove a duality for cyclic covers of normal surface singularities with ℂ^*-action.