2005 Fiscal Year Final Research Report Summary
Root system construction of a compactification of the moduli space of rational surfaces
| Project/Area Number |
14540023
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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 | Kyoto University |
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Principal Investigator |
MATSUZAWA Jun-ichi Kyoto University, Graduate School of Engineering, Lecturer, 工学研究科, 講師 (00212217)
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| Co-Investigator(Kenkyū-buntansha) |
ISHII Akira Hiroshima University, Graduate School of Science, Associate Professor, 大学院・理学研究科, 助教授 (10252420)
NARUKI Isao Ritsumeikan University, Graduate School of Science and Engineering, Professor, 理工学部, 教授 (90027376)
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| Project Period (FY) |
2002 – 2005
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| Keywords | cubic surface / moduli / compactification / root system / Weyl group / configuration space |
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| Research Abstract |
Matsuzawa and Naruki :
The aim of our research is to study the geometry of surfaces and its moduli from the point of view of Lie group, root systems and Weyl group. We constructed the universal family of marked cubic surfaces from the maximal torus of adjoint group of simple Lie group of type E6. Also we gave defining equation of a cubic surface in terms of root systems. Furthermore we constructed a smooth compactification of the universal family of marked cubic surfaces and gave a Weyl group equivariant mapping to Naruki's compactification of the moduli space of marked cubic surfaces. These constructions enable us to study the geometry of cubic surfaces from the point of view of root systems and Weyl groups. The family of cubic surfaces can be regarded as the configuration space of seven points of projective plane or mojuli space of algebrac curve of genus 3. We found interesting relationship among the geometry of cubic surface, that of algebraic curve of genus 2 and the structure of root system and Weyl groups of type E7, E6, D4.
Ishii :
He generalized the Mckay correspondence for simple singularities to general quotient surface singularities via Hilbert scheme of G-orbits. He studied the case for 3-dimensional quotient singularities when the group is abelian and gave a local coordinates of a crepant resolution of the singularity as the representation moduli of the McKay quiver. He also gave explicit description of the groups of self-equivalences of derived category on the minimal resolutions.
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