Various aspects of pencils of algebraic curves

研究项目的目的是主要从两个方面研究代数曲线的铅笔。一个是代数几何，另一个是拓扑学。两个数学领域之间的相互作用是至关重要的。我们特别关注了以下主题。(1)纤维代数曲面的斜率；(2)特征的局部化；(3)modell - weil格。我们的成就包括：纤维有理曲面斜率的下界，一般纤维的Clifford指数与Mordell-Weil秩有关的不等式，具有Mordell-Weil秩极值的纤维有理曲面的完整描述，通过压缩法向曲面奇点与纤维代数曲面的相对标准线性系统的固定部分得到奇点的合理性或椭圆性，单纤维胚芽的不变约简与半稳定约简得到的局部特征的比较。在研究期间，研究者共发表论文57篇，做研究报告121次。我们组织（或联合组织）了6次会议。最重要的是，我们于2005年10月在河内（越南）举办了“东亚代数几何II”国际会议，以促进该地区代数几何学者之间的研究互动。我们出版了会议记录，以显示研究项目的活动情况。我们还邀请了6位国外的研究人员，对各种问题进行了富有成果的讨论。总之，由于我们联合调查人员的积极贡献和合作，我们的目的已经取得了巨大的成功。

The purpose of the research project is to study pencils of algebraic curves from mainly two sides. One is algebraic geometry and the other is topology. The interaction between two fields of mathematics has been crucial .We paid our special attention to the following topics.(1) The slope of fibred algebraic surfaces,(2) The localization of signature,(3) The Mordell-Weil lattices.Our achievements include: the lower bounds of the slopes for fibred rational surfaces, the inequality involving the Clifford index of a general fibre and the Mordell Weil rank, the complete description of fibred rational surfaces with extremal Mordell-Weil rank, the rationality or ellipticity of the singularity obtained by contracting the fixed part of the relative canonial linear system for a normal surface singularity and a fibred algebraic surface, the comparison between local signatures obtained from the eta invariant and semi-stable reduction of a singular fibre germ. During the research period, in total, 57 papers are written and 121 research presentations are given by our investigators. We organized (or co-organized) 6 conferences. Above all, we held the international conference "Algebraic Geometry in East Asia, II" at Hanoi(Vietnam), October 2005, in order to promote research interaction among algebraic geometers in the area. We published the proceedings of the meetings to show the activity of the research project. Also, we invited 6 researchers in abroad and discussed verious problems fruitfully.In sum, our purpose has been achieved m great success, thanks to the active contributions and cooperation of our joint investigators.