${\Bbb C}^*$-equivariant degenerations of curves and normal surface singularities with ${\Bbb C}^*$-action

Abstract

This paper presents a definition of ${\Bbb C}^*$-equivariant degeneration families of compact complex curves over ${\Bbb C}$. Those families are called ${\Bbb C}^*$-pencils of curves. We give the canonical method to construct them and prove some results on relations between them and normal surface singularities with ${\Bbb C}^*$-action. We also define ${\Bbb C}^*$-equivariant degeneration families of compact complex curves over ${\Bbb P}^1$. From this, it is possible to introduce a notion of dual ${\Bbb C}^*$-pencils of curves naturally. Associating it, we prove a duality for cyclic covers of normal surface singularities with ${\Bbb C}^*$-action.