On Pencil genus for normal surface singularities (II)

在我们的研究中，我们一直在研究紧Riemann曲面族(即复数域上的紧代数曲线)与正规复曲面奇点之间的关系。我们还研究了正常曲面奇点的循环覆盖的一些性质。我们的结果如下：1.当分支因子降低且循环阶足够高时，我们证明了由循环覆盖得到的奇点是Kodaira奇点。2.三年前，首席研究员Tomaru定义了正规曲面奇点的亏格(称为“铅笔亏格”)。我们计算了几类曲面奇点的铅笔亏格。3.当考虑由循环覆盖得到的奇点时，我们观察到当改变循环顺序时奇点例外集的形式变化的一些现象。4.研究了具有C^*作用的紧致黎曼曲面族与正规曲面奇点C^*作用之间的关系。5.研究了紧黎曼曲面族周围的全纯微分形式的情况。

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.