2001 Fiscal Year Final Research Report Summary
Elliptic fibre structures and Mordell-Weil groups of Enriques surfaces
| Project/Area Number |
11640047
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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 | Toho University |
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Principal Investigator |
UMEZU Yumiko Toho University, School of Medicine, Assistant Professor, 医学部, 助教授 (70185065)
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| Co-Investigator(Kenkyū-buntansha) |
KOBAYASHI Yuji Toho Unversity, Faculty of Science, Professor, 理学部, 教授 (70035343)
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| Project Period (FY) |
1999 – 2001
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| Keywords | Algebraic surfaces / Fibred surfaces / Mordell-Weil groups / Singularity / Algebraic systems / Monoids / Rewriting methods / Homotopy |
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| Research Abstract |
We studied more generally algebraic surfaces with fibration of curves.
First, we examined Neron's method to construct an infinite family of curves of genus g 【greater than or equal】 2 over Q with high rank. We showed that his method does not give the existence of curves of genus g with rank r 【greater than or equal】 3g + 7, as he claimed, but r 【greater than or equal】 3g + 6. Moreover we improved his method and showed the existance of faimilies of curves with g 【greater than or equal】 3 and rank r 【greater than or equal】 3g + 7.
Next, we considered surfaces with elliptic fibration with Kodaira dimension one who admit normal quintic birational models. To describe their fibre structures we studied their multiple fibres, and obtained the possible number of multiple fibres and their multiplicities. In particular we classified these numbers and multiplicities when the geomertic genus of the surfaces is not zero.
We studied structural and computational problems of algebraic systems (particularly monoids) defined by finite generators and finite relations mainly applying the rewriting methods. We studied the termination problem of one-rule rewriting systems and the homotopy relations on the complex depicting rewriting steps in a monoid, originally introduced by Squier. We studied relationship between homotopy finiteness and homology finiteness, and showed that these properties are undecidable independently on the decidability of the word problem. In particular, we proved that every one-relator monoid has homotopy finiteness property. We studied finite presentability of monoids and gave a finite presentation of the braid inverse monoid.
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