2018 Fiscal Year Final Research Report
Research on singularities on an algebraic variety
| Project/Area Number |
16K05089
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| Research Category |
Grant-in-Aid for Scientific Research (C)
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| Allocation Type | Multi-year Fund |
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| Section | 一般 |
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| Research Field |
Algebra
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| Research Institution | The University of Tokyo (2018)
Tokyo Woman's Christian University (2016-2017) |
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Principal Investigator |
Ishii Shihoko 東京大学, 大学院数理科学研究科, 名誉教授 (60202933)
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| Research Collaborator |
Watanabe Kei-ichi
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| Project Period (FY) |
2016-04-01 – 2019-03-31
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| Keywords | singularities / arc space / minimal log discrepancy |
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| Outline of Final Research Achievements |
A non-smooth point on a variety is called a singularity. In order to study those singularities, we need good invariants. In this research, we introduce an invariant; Mather-Jacobian log discrepancy and study the properties of this invariant.
This invariant is interpreted in terms of the arc space on singularities of a variety. Therefore, making use of it, we obtain Inversion of Adjunction for the base field of positive characteristic and finite determinacy for
the M-J log discrepancy for surfaces of characteristic 0.
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| Free Research Field |
代数幾何学
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| Academic Significance and Societal Importance of the Research Achievements |
多様体が滑らかであれば，コホモロジーの消滅定理をはじめとして，色々な良い性質が成立し，多様体全体の様相がわかりやすいのであるが，特異点があるとそれがわかりにくくなる．特異点をよく知ることで，多様体の理解を深めようというのが本研究の意義である．この研究は数学的に意義があることはもちろん，特異点の不変数の一つである log canonical threshold が学習理論において，重要な役割を果たすことが知られていることからもわかるように，社会にとっても意義のあることである．
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