2016 Fiscal Year Annual Research Report
Birational geometry: subgroups of the Cremona groups and their generators
| Project/Area Number |
15F15751
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| Research Institution | Kyoto University |
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Principal Investigator |
向井 茂 京都大学, 数理解析研究所, 教授 (80115641)
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| Co-Investigator(Kenkyū-buntansha) |
HEDEN ISAC 京都大学, 数理解析研究所, 外国人特別研究員
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| Project Period (FY) |
2015-10-09 – 2018-03-31
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| Keywords | 代数学 / クレモナ変換群 / アフィン代数幾何学 |
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| Outline of Annual Research Achievements |
Given a plane projective curve C, we can define its decomposition group as the subgroup of the 2-dimensional Cremona group that consists of birational maps which maps C to C, and for which the restriction to C gives a birational map of C. I (Heden) have been studying the decomposition group of a line L in the projective plane, i.e. the subgroup of birational transformations of the plane that send L to itself birationally. In a joint project with S. Zimmermann (Toulouse Univ.), we have been studying the decomposition group of rational plane curves C. We have proven that this group is generated by its linear and quadratic elements if C is of degree 1, 2 or 3. We have also shown that this result is not in general true when the degree of C is 4 or higher.
In a joint project with A. Dubouloz (Bourgogne Univ.) and T. Kishimoto (Saitama Univ.), we study affine extensions: Given an additive principal bundle P on the punctured affine plane (for example P = SL2 with projection on the 1st column), an affine extension P’ is an affine smooth variety which is obtained from P by adding a fibre over the origin of the affine plane. We have obtained classification results for the case where this exceptional fibre is isomorphic to the affine plane and the additive action on P extends to a proper action on the exceptional fibre.
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| Current Status of Research Progress |
Current Status of Research Progress
2: Research has progressed on the whole more than it was originally planned.
Reason
The joint project with A. Dubouloz and T. Kishimoto has turned out to be surprisingly complicated, and it seems that the answer to our original question is out of reach at the moment. Therefore we are studying a subcase. However, in this subcase, our results are satisfactory and this project is coming close to an end. There are also satisfactory results that have been obtained in the joint work with S. Zimmermann, as described above. My research-in-aid grant has allowed me to participate in conferences and meeting all of my collaborators, which has been very beneficial for my reserach.
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| Strategy for Future Research Activity |
My main focus in the coming months will be to write up the above mentioned two projects so that they can be submitted for publicaiton. I (Heden) have constructed examples of Sarkisov links, but only at an experimental stage so far; I plan to use my remaining time to try to construct Sarkisov links in a more systematic way, and apply such links to the study of Cremona transformations as described in my research plan.
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Research Products
(11 results)
