2004 Fiscal Year Final Research Report Summary
Moduli spaces and arithmetic geometry
| Project/Area Number |
13440008
|
|---|
| Research Category |
Grant-in-Aid for Scientific Research (B)
|
| Allocation Type | Single-year Grants |
|---|
| Section | 一般 |
|---|
| Research Field |
Algebra
|
|---|
| Research Institution | Kyoto University |
|---|
Principal Investigator |
MORIWAKI Atushi Kyoto University, Mathematics, Professor, 大学院・理学研究科, 教授 (70191062)
|
|---|
| 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)
|
|---|
| Project Period (FY) |
2001 – 2004
|
| Keywords | Logarithmic Geometry / Diophantine Geometry / Rational point / Moduli space |
|---|
| 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.
|
Research Products
(9 results)
-
-
-
-
-
-
-
-
[Book] Algebraic geometry in East Asia2002
Author(s)
Akira Ohbuchi, Kazuhiro Konno, Sampei Usui, Atsushi Moriwaki, Noboru Nakayama
Total Pages
263
Publisher
World Scientific Publishing Co., Inc.
Description
「研究成果報告書概要(和文)」より
-
[Book] Algebraic geometry in East Asia2002
Author(s)
Akira Ohbuchi, Kazuhiro Konno, Sampei Usui, Atsushi Moriwaki, Noboru Nakayama
Total Pages
263
Publisher
World Scientific Publishing Co., Inc
Description
「研究成果報告書概要(欧文)」より
