Rational curves and arithmetic

该建议提出了几个问题有关的几何代数簇和他们的模。第一个研究目标是研究稳定有理曲线的Grothendieck-Knudsen模空间的双有理几何，特别是通过算术技术;一个子项目是发展这个空间的Arakelov理论。第二个目标是证明2-Fano流形是有理单连通的。这是著名的Kollár-Miyaoka-Mori结果的一个类比，即Fano流形是有理连通的。利用德容和斯塔尔的结果，一个结论将是一个推广的森定理，即，一个2-法诺流形上定义的功能领域的一个表面有一个合理的点。这个项目的更广泛的背景是代数几何领域，目前数学中最活跃的分支之一，在整个数学中有广泛的应用，并深入到物理和工程领域。代数几何是研究代数簇的学科，代数簇是由多项式方程组的解给出的几何对象。代数簇的变化被所谓的模空间所捕获，模空间本身就是具有非常丰富结构的代数簇。在这个项目的两个主要主题中，模空间起着核心作用，既是我们研究的几何空间，也是回答某些多项式方程组是否有解的工具。期望这些项目将对数学的其他领域产生重大影响，特别是算术几何。

The proposal presents several problems related to the geometry of algebraic varieties and their moduli. A first research objective is to study the birational geometry of the Grothendieck-Knudsen moduli space of stable rational curves, in particular, via arithmetic techniques; a sub-project is to develop the Arakelov theory of this space. A second objective is to prove that 2-Fano manifolds are rationally simply connected. This is an analogue of the celebrated Kollár-Miyaoka-Mori result that Fano manifolds are rationally connected. Using results of de Jong and Starr, a consequence will be a generalization of Tsen's theorem, namely, that a 2-Fano manifold defined over the function field of a surface has a rational point. This would give a natural geometric condition for the existence of rational points.The broader context of the project is the area of algebraic geometry, currently one of the most active branches of mathematics, with widespread applications throughout mathematics and reaching into physics and engineering. Algebraic geometry is the study of algebraic varieties, which are geometric objects given by the solutions of systems of polynomial equations. The variation of algebraic varieties is captured by the so-called moduli spaces, which are themselves algebraic varieties with a very rich structure. In the two main themes of this project, moduli spaces play a central role, both as spaces whose geometry we investigate, and as tools for answering questions about whether certain systems of polynomial equations have solutions or not. The expectation is that the projects will significantly impact other areas of mathematics, especially arithmetic geometry.