2004 Fiscal Year Final Research Report Summary
Many-sided researches on rationally connected projective varieties-Fano varieties are unirational ?
| Project/Area Number |
14340014
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| Research Category |
Grant-in-Aid for Scientific Research (B)
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| Allocation Type | Single-year Grants |
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| Section | 一般 |
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| Research Field |
Algebra
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| Research Institution | Kyushu University |
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Principal Investigator |
SATO Eiichi Kyushu University, Faculty of Mathematics, Professor, 大学院・数理学研究院, 教授 (10112278)
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| Co-Investigator(Kenkyū-buntansha) |
CHO Koji Kyushu University, Faculty of Mathematics, Associate Professor, 大学院・数理学研究院, 助教授 (10197634)
INABA Michi-aki Kyushu University, Faculty of Mathematics, Research Associate, 大学院・数理学研究院, 助手 (80359934)
TAKAYAMA Sigeharu Tokyo University, Faculty of Mathematical Science, Associate Professor, 大学院・数理科学研究科, 助教授 (20284333)
FURUSHIMA Mikio Kumamoto university, Faculty of Science, Professor, 理学部, 教授 (00165482)
MORIWAKI Atsushi Kyoto University, Faculty of Science, Associate Professor, 大学院・理学研究科, 助教授 (70191062)
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| Project Period (FY) |
2002 – 2004
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| Keywords | rational curve / unirationality / Fano variety / deformation / conic bundle / tangent bundle / Brauer group / rationally connectedness |
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| Research Abstract |
The aim of the scientific research is to study rationally connected projective varieties and problems related with them and to develope them.
1.Reseach results.
・The following results were obtained as for families of rational curves : When a projective variety X contains an ample divisor A which has fiber structure whose general fiber is a projective line, the fiber structure of A is extended to X. This yields that X inherits the property of blowing-down from A. Furthermore it has an application to the procedure for constructing conclete minimal model.(Title : Hyperplane section principle of Lefschetz about ${bf P}^1$-fiber space and blowing-down).
・We studied the structure of infinite sequence of rationally connected varieties. These varieties are closely related with unirational, toric and abelian varieties. Particularly we determined the structure of such varieties with Picard number 2 「Title : Tower theorem on smooth projective varieties」。
・We are now studying the birational group of cubic 3-folds and the unirationality of quartic 3-folds.
・I had a talk on the behavior of rational curves on infinite dimensional projective varieties in 2003 at Kochi Univ.
2.Meeting: We organized a meeting for algebraic geometry "higher dimensional varieties" at Kyushu Univ in 2003 and published the report.
3.Discussions. Takagi(in Rims of Kyoto Univ) had lectures of 3-dimensional Q-primary Fano varieties -and discussion with us and Nakayama did the one on endmorphisms of varieties. Abe and Aoki (Kyoto Univ) had discussions about the constructions of vector bundles and algebraic stacks respectively.
4.An investigator Takayama published "Iitaka fibrations via multiplier ideals", Furushima "non-normal Del Pesso surfaces" "compactification of $C^3$" and Hanamura "relative Chow-Kuneth projectors for modular varieties" and so on respectively.
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