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

该项目研究了 K3 曲面上的有理曲线理论，作为研究 Fano 和中间类型的高维变体上有理曲线的原型和模型。 关键问题涉及可数域上 K3 曲面上无限多个有理曲线的存在、生成此类曲线的技术及其数值不变量的计算、有理曲线的 Brill-Noether 轨迹和枚举几何、混合特征变形理论、伽罗瓦表示和布劳尔群的各个方面。术语“K3 曲面”由 A. Weil 在 1950 年代创造，并纪念 Kummer 的开创性贡献， Kaehler 和 Kodaira 介绍了他们的结构。 几十年来，这些曲面一直是复杂几何的核心，但最近它们的算术特性受到越来越多的关注。 该项目解决复杂、代数和算术几何接口的问题。 特别是，K3 曲面上的曲线结构是什么？ 它们可以显式构造吗？ 它们如何反映环境表面的对称性？

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?