On Pencil genus for normal surface singularities (II)

在我们的研究中，我们一直在研究紧致黎曼曲面（即复数域上的紧致代数曲线）的单参数族与法向复曲面奇点之间的关系。我们还研究了法向曲面奇异点的循环覆盖的一些性质。我们的研究结果如下：1。当分支除数减少且循环阶足够高时，我们证明了循环覆盖所得到的奇点是Kodaira奇点。三年前，首席研究员（Tomaru）为法向表面奇点定义了一个属（命名为“铅笔属”）。我们计算了几类表面奇点的铅笔格。当我们考虑由循环覆盖得到的奇点时，我们观察到当改变循环阶数时奇点的例外集的形式的变化现象。研究了具有C^*-作用的紧黎曼曲面的一个参数族与法向曲面奇点C^*-作用的关系。研究紧黎曼曲面单参数族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.