Rationality, Rationality Index, and Rational Points

理性、理性指数和理性点

• 批准号: 2101434
• 负责人: Bianca Viray
• 金额: $ 30.73万
• 依托单位: University of Washington
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2021
• 资助国家: 美国
• 起止时间: 2021-09-01 至 2024-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2101434&HistoricalAwards=false
• 关键词: Rationality; Index; Rational; Points

This research project focuses on questions within arithmetic geometry. As the name suggests, this research area sits at the interface of two fields: arithmetic (the study of whole numbers and fractions); and geometry (the study of curves, surfaces, and higher-dimensional shapes). These distinct areas are brought together through the study of solutions to polynomial equations, that is, the study of varieties. A guiding philosophy of arithmetic geometry is that "geometry controls arithmetic," in other words, that the geometric properties of a variety (those that can be studied in terms of the complex numbers) influence the arithmetic behavior of the variety. The broad goals of this project are to better understand the influence of geometry on two arithmetic properties: rationality over the ground field, and the existence of isolated points. The research involves several projects for students at both the graduate and undergraduate levels.The project concerns a systematic study of an arithmetic measure of the failure of k-rationality, focusing on geometrically rational threefolds that have a conic bundle structure or that have a fibration into high degree del Pezzo surfaces. These two types of varieties together cover a large swath of all geometrically rational threefolds. In another direction, the project aims to gather more information on isolated points on curves. The aims in this direction are twofold: 1) determine whether the Bombieri-Lang conjecture and the torsion conjecture imply that the number of isolated points on a curve is bounded depending solely on its genus, and 2) compute all isolated points on a collection of modular curves.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.

该研究项目重点关注算术几何中的问题。顾名思义，这个研究领域位于两个领域的交汇处：算术（整数和分数的研究）；和几何（曲线、曲面和高维形状的研究）。这些不同的领域通过多项式方程解的研究（即簇的研究）聚集在一起。算术几何的指导思想是“几何控制算术”，换句话说，簇的几何性质（可以用复数来研究的几何性质）影响簇的算术行为。该项目的总体目标是更好地理解几何对两个算术属性的影响：地面场的合理性和孤立点的存在。该研究涉及研究生和本科生的几个项目。该项目涉及对 k 有理性失败的算术测量的系统研究，重点关注具有圆锥束结构或具有纤维化的几何有理三重。高度 del Pezzo 表面。这两种类型一起涵盖了所有几何有理三元组的一大片区域。另一方面，该项目旨在收集有关曲线上孤立点的更多信息。这个方向的目标有两个：1) 确定 Bombieri-Lang 猜想和扭转猜想是否意味着曲线上孤立点的数量仅取决于其亏格，并且 2) 计算集合上的所有孤立点该奖项反映了 NSF 的法定使命，并通过使用基金会的智力价值和更广泛的影响审查标准进行评估，被认为值得支持。