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

本项目研究了K3曲面上的有理曲线理论，作为研究Fano和中间型高维有理曲线的原型和模型。关键问题涉及可数域上K3曲面上无限多条有理曲线的存在性、生成此类曲线的技术及其数值不变量的计算、brir - noether轨迹和有理曲线的枚举几何、混合特征变形理论的各个方面、伽罗瓦表示和Brauer群。术语“K3表面”是由A. Weil在20世纪50年代创造的，并纪念Kummer， Kaehler和Kodaira对其结构的开创性贡献。几十年来，这些曲面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几何、代数几何和算术几何的接口问题。特别是，K3曲面上的曲线结构是怎样的？可以显式地构造它们吗？它们是如何反映环境表面的对称性的？

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?