Various aspects of pencils of algebraic curves

本课题主要从两个方面对代数曲线束进行研究。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。我们特别关注以下主题。(1)代数曲面的斜率，（2）签名的局部化，（3）Mordell-Weil格。讨论了一般纤维的Clifford指标与Mordell-Weil秩的不等式，具有极值Mordell-Weil秩的有理曲面的完全刻划，将相对正则线性系统的固定部分收缩为法向曲面奇点和非法向代数曲面奇点所得到的奇点的合理性或椭圆性，从奇异纤维芽的eta不变量和半稳定约化获得的局部签名之间的比较。在研究期间，我们的研究人员总共撰写了57篇论文并发表了121份研究报告。我们组织（或共同组织）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家之间的研究互动。我们发表了会议记录，以展示研究项目的活动情况。我们还邀请了6位国外研究人员，就各种问题进行了富有成效的讨论，总之，由于我们的联合研究人员的积极贡献和合作，我们的目的已经取得了很大的成功。

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.