Elliptic fibre structures and Mordell-Weil groups of Enriques surfaces

我们研究了更一般的具有曲线纤维的代数曲面。首先，我们研究了Nelon构造高阶亏格为g[大于或等于]2的曲线族的方法。我们证明了他的方法并不像他所说的那样给出了具有阶r[大于等于]3g+7的亏格g的曲线的存在性，而是给出了具有r[大于等于]3g+6阶的亏格g的曲线的存在性。此外，我们改进了他的方法，证明了阶为g[大于等于]3且阶为r[大于等于]3g+7的曲线存在缺陷。为了描述它们的纤维结构，我们研究了它们的多重纤维，并得到了可能的多重纤维数目和它们的多重性。特别地，当曲面的几何亏格不为零时，我们对这些数和重数进行了分类。我们主要应用重写方法研究了由有限生成元和有限关系定义的代数系统(特别是么半群)的结构和计算问题。我们研究了单规则重写系统的终止性问题以及由Squier首先提出的刻画么半群中重写步骤的复数上的同伦关系。我们研究了同伦有限性和同调有限性之间的关系，证明了这些性质是独立于字问题的可判定性而不可判定的。特别地，我们证明了每个单相关么半群都具有同伦有限性质。研究了么半群的有限可表示性，给出了辫子逆么半群的有限表示。

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.