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.

该研究项目的重点是算术几何形状中的问题。顾名思义，该研究领域位于两个字段的界面：算术（整数和分数的研究）；和几何形状（曲线，表面和高维形状的研究）。这些不同的领域通过研究多项式方程的解决方案，即品种研究。算术几何形状的指导哲学是，“几何控制算术”，换句话说，一种多样性的几何特性（可以根据复数进行研究的几何特性）会影响品种的算术行为。该项目的广泛目标是更好地了解几何学对两种算术特性的影响：地面上的合理性以及孤立点的存在。这项研究涉及研究生和本科级别的学生的几个项目。该项目涉及对K理性失败的算术测量的系统研究，重点是具有圆锥形束结构的几何理性三倍或具有纤维化为高级Pezzo表面的几何理性三倍。这两种类型的品种一起覆盖了所有几何理性三倍的大量。在另一个方向上，该项目旨在收集有关曲线隔离点的更多信息。在这个方向上的目的是双重的：1）确定Bombieri-lang的猜想和扭力猜想是否意味着曲线上的孤立点的数量仅取决于其属，而2）2）2）计算所有孤立的点，在模块化曲线的集合上，这一奖项反映了NSF的范围，这是NSF的法定任务及其范围的范围。 标准。