2023 Fiscal Year Final Research Report
On embedded resolution of singularities for three dimensional algebraic varieties
| Project/Area Number |
20K03546
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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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| Review Section |
Basic Section 11010:Algebra-related
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| Research Institution | Chubu University |
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Principal Investigator |
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| Project Period (FY) |
2020-04-01 – 2024-03-31
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| Keywords | 特異点解消 / IFP / 代数幾何学 |
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| Outline of Final Research Achievements |
Resolution of singularities is one of the very important problems in algebraic geometry. Existence of resolution in characteristic zero was established by Hironaka in any dimension, while that in positive characteristic is known only for low dimensional cases. To settle this problem, I introduced the approach called IFP, and work on it jointly with Kenji Matsuki in Purdue university. The theme of this project is to establish the existence of embedded resolution of three dimensional varieties, which is still open. We analyzed so-called "monomial cases", which comes as the most essential case after some reduction argument, and obtained some partial results such as giving the invariants in some cases and observing the transitional behavior of them. We also have some results in the area related to the main topic of our project, such as the theory of hyperplane arrangement or the theory of differential equations in positive characteristic.
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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 |
代表者が提案し推進しているIFPというアプローチを用いて3次元多様体の埋め込み特異点解消について解析した．幾つかの場合の解析が終わるなど部分的成果が得られた．IFPの一定の有効性が示された一方で，3次元抽象特異点解消(解決済)と3次元埋め込み特異点解消(未解決)の差が想像以上に大きいことも明らかになった．曲面の埋め込み特異点解消についてはこの間の研究でより良い理解を得られたと言える．近縁の問題についての幾つかの成果も意義がある．B_2型拡大カタラン配置に関する予想の解決はB_n型一般の場合に道を拓く結果である．また正標数の線形微分方程式の解の記述はかなり発展性のある話題であると感じている．
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