2014 Fiscal Year Final Research Report
Resolution of singularities of an algebraic variety over an algebraically closed field in positive characteristic
| Project/Area Number |
23740016
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| Research Category |
Grant-in-Aid for Young Scientists (B)
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| Allocation Type | Multi-year Fund |
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| Research Field |
Algebra
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| Research Institution | Kyoto University |
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Principal Investigator |
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| Project Period (FY) |
2011-04-28 – 2015-03-31
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| Keywords | 代数幾何学 / 特異点解消 / IFP |
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| Outline of Final Research Achievements |
For a given variety, a proper birational morphism from some nonsingular variety to it is called its resolution of singularities. The existence of resolution of singularities for any variety is one of the most important problem in algebraic geometry. In characteristic 0, it is solved by Professor Hironaka in any dimension. However, it is only known up to 3 dimensional case in positive characteristic.
I proposed an approach "Idealistic Filtration Program (IFP)" to attack this problem in positive characteristic and in any dimension, and develop it jointly with Professor Matsuki (Purdue University). By this grant, we had further development in the theory of IFP. Namely, we established several fundamental properties in IFP such as the algebraization of the operations in resolution process or the nonsingularity principle in monomial cases. Also, we implemented IFP into an exact algorithm in the case of embedded resolution for surfaces, which gives an alternative proof of its existence.
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| Free Research Field |
代数幾何学
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