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

FRG：协作研究：K3 曲面上有理曲线的算术和几何

• 批准号: 0968318
• 负责人: Yuri Tschinkel
• 金额: $ 47.05万
• 依托单位: New York University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2010
• 资助国家: 美国
• 起止时间: 2010-07-01 至 2014-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0968318&HistoricalAwards=false
• 关键词: FRG; Collaborative; Research; Arithmetic; geometry

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?

该项目介绍了K3表面上的理性曲线理论，作为对FANO和中间类型的高维品种的理性曲线进行研究的原型和模型。 关键问题涉及在可数领域的K3表面上存在许多理性曲线的存在，这种曲线的生成技术以及其数值不变性的计算，Brill-noether boci基因座以及理性曲线的枚举几何形状以及对混合性变形理论的各个方面的表面上的表达方式，以及Braiis of the Braiis'k3 k3 k3 k3 1950年代，并尊重Kummer，Kaehler和Kodaira对他们的结构的开创性贡献。 几十年来，这些表面一直是复杂几何形状的核心，但是最近它们的算术特性受到了越来越多的关注。 该项目解决了复杂，代数和算术几何形状的界面上的问题。 特别是，K3表面上曲线的结构是什么？ 它们可以明确构造吗？ 它们如何反映环境表面的对称性？