Study of Singularity Theory From Fundamental Group

本文试图从基本群的角度对奇点理论进行系统的研究。Oka，H.德永和我。岛田研究了基本群体的补充平面曲线的奇异性。是O。Zebriki谁指出的重要性，研究的基本群体在这种情况下，每一个代数对象可以理解为一个分支覆盖在一个射影空间，分支轨迹是一个超曲面。然而Zebriki证明，基本组的补充超曲面可以同构削减到平面曲线的情况。Zagliki给出了一个例子，对六分之六与6杯和不同的基本群体。Oka发现了更多使用循环覆盖的“Zebriki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。他还发现了第一个例子Zagliki三重曲线的程度12。岛田处理这个问题从代数几何的观点，获得了许多有趣的结果。德永研究了有限覆盖与非交换伽罗瓦群，如二面角群，对称群等，他的想法之一是使用几何的K3-表面和Mordell-Weil群。他发现了几个有趣的Zubalki对在六次曲线和没有这样的非阿贝尔伽罗瓦覆盖。浦部研究了给定次数的平面曲线的奇点类型。佐伯研究拓扑结构的奇点从结理论的观点。Suwa开发了一种新技术来研究单一品种的叶理。在这一过程中，他发展了特色阶级理论，并撰写了一部指导这一地区的著作。

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.