Explicit Study of Algebraic Varieties
| Project/Area Number |
18540001
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| Research Category |
Grant-in-Aid for Scientific Research (C)
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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 | Hokkaido University |
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Principal Investigator |
SHIMADA Ichiro Hokkaido University, Fac. of .Sci., Associate Professor (10235616)
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| Co-Investigator(Kenkyū-buntansha) |
OKA Mutsuo Tokyo University of Science, Grad. school of sci, Professor (40011697)
ISHIKAAWA Goo Hokkaido University, Fac. of.Sci, Professor (50176161)
KONNO Kazuhiro Osaka University, Grad school of sci., Professor (10186869)
SAITO Mutsuni Hokkaido University, Fac. of.Sci,Asso, Professor (70215565)
TOKUNAGA Hiroo Tokyo Metropolitan University, Grad. school of science and engineering, Professor (30211395)
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| Project Period (FY) |
2006 – 2007
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| Project Status |
Completed (Fiscal Year 2007)
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| Budget Amount *help |
¥3,920,000 (Direct Cost: ¥3,500,000、Indirect Cost: ¥420,000)
Fiscal Year 2007: ¥1,820,000 (Direct Cost: ¥1,400,000、Indirect Cost: ¥420,000)
Fiscal Year 2006: ¥2,100,000 (Direct Cost: ¥2,100,000)
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| Keywords | K3 surface / plane curve / lattices / supersingularity / rational double point / fundamental group / dual variety / K3曲面 / 平面曲線 / 分岐被覆 / 基本群 / グレブナー基底 / 格子 / 6次曲線 / 超特異性 / 正標数 |
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| Research Abstract |
(1) By a joint work with Due Tai Pho, we proved the unirationality of supersingular K3 surfaces in characteristic 5 with Artin invariant 〓 3.
(2) We classified all possible configurations of rational double points on complex normal algebraic K3 surfaces, and discussed the same classification for supersingular K3 surfaces in sufficiently large characteristics.
(3) We investigated the set of isomorphism classes of transcendental lattices of complex algebraic varieties obained from a single algebraic variety defined over a number field by various embeddings of the base field into the complex number field, and produced many examples of non-homeomorphic complex algebraic varieties that are conjugate under the automorphism of the complex number field.
In particular, we investigate the case of singular. K3 surfaces by means of the class field theory of imaginary quadratic fields. As an application, we obtained a lower bound of the degree of the base field of singular K3 surfaces.
(4) Suppose that we obtain a supersingular K3 surface X (P) by reduction at a finite place P of a base field of a singular K3 surface X defined over a number field. We investigated the orthogonal complement of the Neron-Severi lattice of X in that of X (P), and proved an analogous result as the case of the transvendental lattices.
(5) We proved a Lefschetz hyperplane section theorem for the topological fundamental group of the complement of the duial variety in the Grassmannian variety, and invetigate the Zariski-van Kampen type relation of that fundamental group with the barid group of a punctured Riemann surface.
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Report
(3 results)
Research Products
(54 results)
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[Presentation] 数体上定義された特異K3曲面2007
Author(s)
I. Shimada
Organizer
第52回代数学シンポジウム
Place of Presentation
神戸大学
Year and Date
2007-08-08
Description
「研究成果報告書概要(和文)」より
Related Report
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