2022 Fiscal Year Final Research Report
Study of universal families over moduli spaces based on geometry of group actions
| Project/Area Number |
20K03533
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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 | Kyoto University |
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Principal Investigator |
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| Project Period (FY) |
2020-04-01 – 2023-03-31
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| Keywords | モジュライ空間 / 普遍族 / 滑層分解 / 固定化部分群ポセット / 群作用 |
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| Outline of Final Research Achievements |
In order to describe the stratifications of singular loci on moduli spaces of algebraic curves as well as those on the universal families over them, we pushed forward the theory of linear quotient families, and established two kinds of algorithms to explicitly determine the stablizer posets corresponding to the stratifications via geometric Galois correspondence. These algorithms can be run on a computer, and powerful for giving local descriptions of the above stratifications.
Besides, we introduced ``higher order structures'' of groups, which consist of subgroup products and coset products. For finite groups, we introduced the concept of bifurcations of these structures, and constructed semi-simplicial complexes that reflect the bifurcations. These complexes are regarded as geometric invariants of finite groups.
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| Free Research Field |
幾何学
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| Academic Significance and Societal Importance of the Research Achievements |
モジュライ空間上の普遍族の局所的描写を計算機上で実行可能なアルゴリズムとして実装でき，様々な具体例の計算・比較が容易となった．これは，今後のモジュライ空間の研究に大いに役立つと期待される．
群の高次構造の構成要素である部分群積やコセット積の分岐現象は，これら高次対象の導入で初めて意味を持つもので，単に部分群やコセットなどの古典的対象（これらは，いわば「一次」の対象）からは導出されえない．
したがって分岐複体は，古典的対象を超えたところにある，群の深い性質を反映していると言え，今後，分野横断的な研究対象となることが期待される．
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