On Pencil genus for normal surface singularities (II)

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In our research, we have been researching the relation between one parameter families of compact Riemann surfaces (i.e., compact algebraic curves over the complex number field) and normal complex surface singularities. Also we investigated some properties of cyclic coverings of normal surface singularities. Our results are as follows :1.When the branched divisor is reduced and cyclic order is enoughly higher, we prove that the singularity obtained by cyclic cover is a Kodaira singularity.2.Three years ago the head investigator (Tomaru) had defined a genus (named "pencil genus") for normal surface singularities. We compute pencil genus for some classes of surface singularities.3.When we consider singularities obtained by cyclic covering, we observed some phenomena about the change of the forms of exceptional sets of the singularities when we change the cyclic order.4.We investigated the relation between one parameter families of compact Riemann surfaces with C^*-action and normal surface singularities C^*-action.5.We investigated the situation of holomorphic differential forms around one parameter families of compact Riemann surfaces.