Rationality, Rationality Index, and Rational Points
理性、理性指数和理性点
基本信息
- 批准号:2101434
- 负责人:
- 金额:$ 30.73万
- 依托单位:
- 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:
- 关键词:
项目摘要
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.
这个研究项目的重点是算术几何中的问题。顾名思义,这个研究领域位于两个领域的交汇处:算术(研究整数和分数);还有几何学(研究曲线、曲面和高维形状)。这些不同的领域通过对多项式方程的解的研究,也就是对变量的研究,结合在一起。算术几何的指导思想是“几何控制算术”,换句话说,各种几何性质(那些可以用复数研究的几何性质)影响各种的算术行为。该项目的总体目标是更好地理解几何对两个算术性质的影响:地面场的合理性和孤立点的存在。这项研究涉及几个面向研究生和本科生的项目。该项目涉及对k-合理性失效的算术度量的系统研究,重点关注具有圆锥束结构或具有高度del Pezzo曲面的纤维化的几何理性三倍。这两种类型的变种一起覆盖了所有几何上有理的三倍的大片。另一方面,该项目旨在收集曲线上孤立点的更多信息。在这个方向上的目标是双重的:1)确定Bombieri-Lang猜想和扭转猜想是否暗示曲线上孤立点的数量仅取决于其属,2)计算模曲线集合上的所有孤立点。该奖项反映了美国国家科学基金会的法定使命,并通过使用基金会的知识价值和更广泛的影响审查标准进行评估,被认为值得支持。
项目成果
期刊论文数量(0)
专著数量(0)
科研奖励数量(0)
会议论文数量(0)
专利数量(0)
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Bianca Viray其他文献
Bianca Viray的其他文献
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{{ truncateString('Bianca Viray', 18)}}的其他基金
CAREER: Rational Points via Asymptotics and Geometry
职业:通过渐近学和几何学有理点
- 批准号:
1553459 - 财政年份:2016
- 资助金额:
$ 30.73万 - 项目类别:
Continuing Grant
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