2006 Fiscal Year Final Research Report Summary
Reconstruction of twistor theory based on combinatorics of Fano varieties
| Project/Area Number |
17540091
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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 |
Geometry
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| Research Institution | Osaka City University |
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Principal Investigator |
YOSHITAKE Hashimoto Osaka City University, Faculty of Science, Assistant Professor, 大学院理学研究科, 助教授 (20271182)
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| Co-Investigator(Kenkyū-buntansha) |
YASUI Yukinori Osaka City University, Faculty of Science, Assistant Professor, 大学院理学研究科, 助教授 (30191117)
SAKAGUCHI Makoto Okayama Institute for Quantum Physics, Researcher, 研究員 (90382027)
OHBA Kiyoshi Ochanomizu Women University, Assistant Professor, 理学部, 助教授 (80242337)
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| Project Period (FY) |
2005 – 2006
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| Keywords | Fano varieties / Mathieu groups / Designs |
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| Research Abstract |
Two-dimensional Fano varieties are called Del Pezzo surfaces. Combinatorics of intersections of exceptional divisors on them is closely related to representation theory. In the case of four-point blowing up of projective planes, the surface is biholomorphic to the moduli of five points on a projective line. Hence it has an action of the symmetric group of degree five. The intersecions of exceptional divisors are described by the Petersen graph. It contains six pairs of complementary pentagons, which are known to be closely related to multiple zeta values.
The action of the symmetric group of degree five on the Petersen graph induces an action on a six-point set without fixed points. This action must preserve a certain structure on the six-point set, which is the bi-icosahedral structure.
The bi-icosahedral structure is defined by A. Grothendieck. In our research, we studied basic theory of bi-icosahedra from the viewpoint "The moduli of bi-icosahedra is the dual of the six-point set, " and applied to construction of the Mathieu groups, some kind of sporadic finite simple groups. We showed that structures called the designs whose automorphism groups are the Mathieu groups can be construted naturally by bi-icosahedra.
The 12-point set, the union of a six-point set and its dual, has a natural 5-(12, 6, 1) design, which is an analogue of symplectic structure. Its automorphism group is the Mathieu group of degree 12.
The 5-(12, 6, 1) design has a dual. The 24-point set, the union of two 12-point sets with 5-(12, 6, 1) designs dual each other, has a natural 5-(24,8,1) design. Its automorphism group is the Mathieu group of degree 24.
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Research Products
(14 results)
