FRG: Collaborative Research: Arithmetic and geometry of rational curves on K3 surfaces

该项目致力于K3曲面上的有理曲线理论，作为Fano和中间类型高维品种上有理曲线研究的原型和模型。 其关键问题涉及可数域上K3曲面上无穷多条有理曲线的存在性、这类曲线的生成技术及其数值不变量的计算、有理曲线的Brill-Noether轨迹和计数几何、混合特征形变理论、Galois表示和Brauer群等方面。韦尔在20世纪50年代，并荣誉的开创性贡献的库默，凯勒，和科代拉，他们的结构。 这些曲面几十年来一直是复杂几何的核心，但最近它们的算术性质受到越来越多的关注。 这个项目解决了复杂的，代数的和算术几何的接口问题。 特别是，K3曲面上的曲线结构是什么？ 它们可以被显式地构造吗？ 它们如何反映周围表面的对称性？

This project addresses the theory of rational curves on K3 surfaces, as a prototype and model for investigations of rational curves on higher-dimensional varieties of Fano and intermediate type. The key problems concern the existence of infinitely many rational curves on K3 surfaces over countable fields, techniques for the generation of such curves and computation of their numerical invariants, Brill-Noether loci and enumerative geometry of rational curves, aspects of mixed-characteristic deformation theory, Galois representations, and Brauer groups.The term `K3 surface' was coined by A. Weil in the 1950's, and honors the seminal contributions of Kummer, Kaehler, and Kodaira to their structure. These surfaces have been central to complex geometry for decades, but recently their arithmetic properties have received increasing attention. This project addresses problems at the interface of complex, algebraic, and arithmetic geometry. In particular, what is the structure of the curves on a K3 surface? Can they be constructed explicitly? And how do they reflect symmetries of the ambient surface?