Theory of resolution of singularities in positive characteristic
| Project/Area Number |
14540005
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| Research Category |
Grant-in-Aid for Scientific Research (C)
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| Allocation Type | Single-year Grants |
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| Section | 一般 |
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| Research Field |
Algebra
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| Research Institution | Ibaraki University |
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Principal Investigator |
URABE Tohsuke Ibaraki University, Department of Mathematical Sciences, Professor, 理学部, 教授 (70145655)
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| Co-Investigator(Kenkyū-buntansha) |
OSHIMA Hideaki Ibaraki University, the college of Mathematical sciences, Professor, 理学部, 教授 (70047372)
OHTSUKA Fumiko Ibaraki University, the college of Mathematical Sciences, Assistant Professor, 理学部, 助教授 (90194208)
SHIMOMURA Katsunori Ibaraki University, the college of Mathematical Sciences, Assistant Professor, 理学部, 助教授 (00201559)
AIBA Akira Ibaraki University, the college of Mathematical Sciences, Assistant Professor, 理学部, 助教授 (90202457)
松田 隆輝 茨城大学, 理学部, 教授 (10006934)
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| Project Period (FY) |
2002 – 2004
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| Project Status |
Completed (Fiscal Year 2004)
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| Budget Amount *help |
¥2,400,000 (Direct Cost: ¥2,400,000)
Fiscal Year 2004: ¥800,000 (Direct Cost: ¥800,000)
Fiscal Year 2003: ¥800,000 (Direct Cost: ¥800,000)
Fiscal Year 2002: ¥800,000 (Direct Cost: ¥800,000)
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| Keywords | singularity / resolution / positive characteristic / blowing-up / monoidal transformation / polynomial / hypersurface / normal crossing / normal crosing / power series |
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| Research Abstract |
This research subject is one of subjects of the world-wide mathematical society for long years. According to our common sense, it must be a very difficult subject. I know that it takes by far longer time to study this subject than standard ones. It is because we need very exact arguments to manipulate complicated situation avoiding hidden traps.
During the term of the project I considered various aspects of the subject from various view points. Several years ago I succeeded to construct the theory of three variables. (However, it seems to be essentially the same result by Hironaka several tens years ago. Hironaka's result was not published formally. It is an appendix of some book.) Here, I would like to explain the latest result for the case of hypersurface singularity defined by a power series with four variables, which is the simplest case with no definite positive results. It explains whole scheme of the subject. Using ideas in the case of three variables, I constructed the theory of
… More
the more complicated case of four variables. In positive characteristic cases induction on the dimension or on the number of variables does not work. Needless to say, induction on the number of variables is the key of Hironaka's success in characteristic zero. Therefore, there is essential difference between the case of four variables and the case of three variables. In positive characteristic case we define the same number of Newton polygons in order as the number of variables, and we use induction on the order of Newton polygons instead of induction on the number of variables. We can apply this method also in characteristic zero, and we can conclude that resolution of singularities is always possible in characteristic zero. By this method one knows that also in positive characteristic almost the same phenomena as in characteristic zero occur in almost all cases. However, there exist few cases where very pathological phenomena occur in characteristic positive. It is our essential subject to make these pathological cases dearer. I surprised to know that there may exist several pathological cases in addition to three cases I found some years ago. I found three cases further. It is dear that analyzing only six cases, we can reach the final result, and great advance has been achieved. Less
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Report
(4 results)
Research Products
(26 results)
