1999 Fiscal Year Final Research Report Summary
Study of Singularity Theory From Fundamental Group
| Project/Area Number |
09440039
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| Research Category |
Grant-in-Aid for Scientific Research (B)
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| Allocation Type | Single-year Grants |
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| Section | 一般 |
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| Research Field |
Geometry
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| Research Institution | TOKYO METOROPOLITAN UNIVERSITY |
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Principal Investigator |
OKA Mutsuo Tokyo Metropolitan University, Department of Mathematics, Prof., 大学院・理学研究科, 教授 (40011697)
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| Co-Investigator(Kenkyū-buntansha) |
URABE Tosuke Ibaragi University, Department of Mathematics, Prof., 理学部, 教授 (70145655)
SAEKI Osamu Hiroshima University, Department of Mathematics, Associate Prof., 理学部, 助教授 (30201510)
SUWA Tatsuo Hokkaido University, Department of Mathematics, Prof., 大学院・理学研究科, 教授 (40109418)
SHIMADA Ichiro Hokkaido University, Department of Mathematics, Associate Prof., 大学院・理学研究科, 助教授 (10235616)
TOKUNAGA Hiroo Kochi University, Department of Mathematics, Associate Prof., 理学部, 助教授 (30211395)
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| Project Period (FY) |
1997 – 1999
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| Keywords | Fundamental group / Zariski Pair / Galois covering / Characteristic Class / Knot / Singularity / Singular foliation / Elliptic Curves |
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| Research Abstract |
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.
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