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Elliptic fibre structures and Mordell-Weil groups of Enriques surfaces

椭圆纤维结构和 Enriques 曲面的 Mordell-Weil 群

基本信息

批准号：
11640047
负责人：
UMEZU Yumiko
金额：
$ 0.83万
依托单位：
Toho University
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (C)
财政年份：
1999
资助国家：
日本
起止时间：
1999 至 2001
项目状态：
已结题

项目摘要

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.

我们研究了更一般的具有曲线颤动的代数曲面。首先，我们检验了Neron构造一个高秩的g【大于或等于】2 / Q的无穷曲线族的方法。我们证明了他的方法并没有给出秩r大于等于3g + 7的g属曲线的存在性，而是r大于等于3g + 6。进一步改进了该方法，证明了g【大于等于】3且秩r【大于等于】3g + 7的曲线族的存在性。其次，我们考虑了具有Kodaira维数为1的椭圆型纤颤曲面，其允许正规五次双胞模型。为了描述它们的纤维结构，我们研究了它们的多纤维，得到了可能的多纤维数量及其多重度。特别地，当曲面的几何属不为零时，我们对这些数和多重性进行了分类。我们主要应用重写方法研究了由有限生成器和有限关系定义的代数系统（特别是独群）的结构和计算问题。研究了由Squier提出的单规则重写系统的终止问题和描述重写步骤的复上的同伦关系。我们研究了同伦有限性与同伦有限性的关系，并证明了这些性质在字问题的可判定性上是不可判定的。特别地，我们证明了每一个单相关模群都具有同伦有限性。研究了单群的有限可表示性，给出了辫逆单群的有限表示。