| Project/Area Number |
13440008
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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 | Kyoto University |
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Principal Investigator |
MORIWAKI Atushi Kyoto University, Mathematics, Professor, 大学院・理学研究科, 教授 (70191062)
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| Co-Investigator(Kenkyū-buntansha) |
MARUYAMA Masaki Kyoto University, Mathematics, Professor, 大学院・理学研究科, 教授 (50025459)
UENO Kenji Kyoto University, Mathematics, Professor, 大学院・理学研究科, 教授 (40011655)
FUKAYA Kenji Kyoto University, Mathematics, Professor, 大学院・理学研究科, 教授 (30165261)
NAKAJIMA Hiraku Kyoto University, Mathematics, Professor, 大学院・理学研究科, 教授 (00201666)
KATO Fumiharu Kyoto University, Mathematics, Associate Professor, 大学院・理学研究科, 助教授 (50294880)
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| Project Period (FY) |
2001 – 2004
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| Project Status |
Completed (Fiscal Year 2004)
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| Budget Amount *help |
¥14,700,000 (Direct Cost: ¥14,700,000)
Fiscal Year 2004: ¥3,200,000 (Direct Cost: ¥3,200,000)
Fiscal Year 2003: ¥3,200,000 (Direct Cost: ¥3,200,000)
Fiscal Year 2002: ¥3,200,000 (Direct Cost: ¥3,200,000)
Fiscal Year 2001: ¥5,100,000 (Direct Cost: ¥5,100,000)
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| Keywords | Logarithmic Geometry / Diophantine Geometry / Rational point / Moduli space / 代数的サイクル / ゼータ函数 / 数論的多様体 / モジュラー的高さ / 数学的函数体 / ゼータ関数 / 数論的関数体 / 安定曲線 / ピカール群 / モジュラル空間 / 代数スタッフ / 算術的多様体 / サイクル |
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| Research Abstract |
During this research project, we mainly studied the following three materials :
(1)Picard group of the moduli space of curves
(2)Counting problem of algebraic cycles
(3)Kobayashi-Ochiai's theorem in the category of log schemes
In the following, we explain the details of each material.
(1)We did not know the problem of algebraic cycles on the moduli space of curves in positive characteristic even for the divisor case. We justify this problem. Namely, we showed that the Picard group of the moduli space of stable n-pointed curves is generated by the tautological line bundles and the boundary classes. By this theorem, several results in characteristic zero were generalized to the case of positive characteristic by Gibney-Keel-Morrison and Schroeer. Besides them, we obtained the results concerning the Mori cone.
(2)We estimated the order of growth of the number of algebraic cycles with bounded arithmetic degree on an arithmetic variety. By this, we can introduce a new kind of zeta functions in terms of the number of algebraic cycles. Similarly, we obtained the same result on an algebraic variety over a finite field.
(3)Kabayashi-Ochiai‘s theorem states that the number of dominant rational maps onto a compact complex manifold of general type is finite. From the view-point of Diophantine geometry, this theorem means that the number of rational points of a compact manifold of general type is finite for a big function field. This gives rise to an evidence for Lang's conjecture. Kazuya Kato conjectured a similar result in the category of log schemes. We proved the conjecture by the joint work with Dr.Iwanari. In the proof of this result, the crucial points are the local structure theorem and the rigidity theorem, which were generalized to the case of semistable schemes over a locally noetherian scheme.
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