Study of Singularity Theory From Fundamental Group

在本研究中，我们尝试以一种特殊的观点对奇点理论进行系统的研究。Oka， H. Tokunaga和I. Shimada研究了奇异平面曲线补的基本群。O. Zariski指出了在这种情况下研究基本群的重要性，因为每个代数对象都可以被理解为投影空间上的分支覆盖，分支轨迹是一个超曲面。然而，Zariski证明了超曲面补的基群可以同构地切到平面曲线的情形。Zariski给出了一个有6个杯子和不同基本群的一对性别的例子。Oka发现了更多使用循环覆盖物的“Zariski对”的例子。事实上，他的循环覆盖变换方法产生了无限多这样的例子。他还在12度曲线中发现了扎里斯基三重曲线的第一个例子。岛田从代数几何的角度探讨了这个问题，得到了许多有趣的结果。Tokunaga研究了非阿贝尔伽罗瓦群的有限覆盖，如二面体群、对称群等。他的想法之一是利用k3曲面和莫德尔-韦尔群的几何特性。他在两性学中发现了几对有趣的扎里斯基配对，有或没有这种非阿贝尔伽罗瓦封面。Urabe研究了给定次平面曲线上的奇点类型。Saeki从结理论的角度研究了奇点的拓扑结构。Suwa开发了一种新的技术来研究单一品种的叶理。在这个过程中，他发展了特征类理论，并写了一本指导这一领域的书。

In this research, we tried to proceed a systematical study for Singularity theory, with a special viewpoint from fundamental group.M. Oka, H. Tokunaga and I. Shimada studied the fundamental groups of the complement of plane curves with singularities. It was O. Zariski who pointed out the importance of the study of the fundamental group in this situation as every algebraic object can be understood as a branched covering over a projective space, with branching locus to be a hypersurface. However Zariski proved that the fundamental group of the complement of a hypersurface can be isomorphically cut down to the plane curve situation. Zariski gave an example of pair of sextics with 6 cups and with different fundamental groups. Oka found more examples of "Zariski pairs" using cyclic coverings. In fact, his cyclic covering transformation method produces infinitely many such examples. He found also a first example of Zariski triple in curves of degree 12. Shimada approached this problem from algebraic geometrical viewpoint, obtaining many interesting results. Tokunaga studied finite covering with non-abelian Galois groups, like dihedral groups, symmetric groups etc. One of his idea is to use the geometry of K3-surface and Mordell-Weil group. He found several interesting Zariski pairs in sextics with and without such non-abelian Galois covers. Urabe studied type of singularities in a plane curve of given degree. Saeki studied topology of singularities from the knot theory point of view. Suwa developed a new technique to study foliations on a singular varieties. In the process, he developed the theory of characteristic classes and he wrote a book which is a guide line of this region.