Rational curves and arithmetic

该提案提出了与代数簇及其模的几何有关的几个问题。第一个研究目标是研究稳定有理曲线的Grothendieck-Knudsen模空间的双曲面几何，特别是通过算术技巧；子项目是发展该空间的Arakelov理论。第二个目标是证明2-Fano流形是有理单连通的。这类似于著名的Kolár-Miyaoka-Mori结果，即Fano流形是有理连通的。利用De Jong和Starr的结果，一个结果将是Tsen定理的推广，即定义在曲面的函数域上的2-Fano流形有一个有理点。这将为有理点的存在提供一个自然的几何条件。该项目的更广泛的背景是代数几何领域，这是目前数学中最活跃的分支之一，在整个数学中有着广泛的应用，并延伸到物理和工程领域。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捉，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，模空间扮演着中心角色，既是我们研究其几何的空间，也是回答某些多项式方程组是否有解的问题的工具。预计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.