Rationality, Rationality Index, and Rational Points

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界面：算术（研究整数和分数）;几何（研究曲线，曲面和高维形状）。这些不同的领域是通过研究多项式方程的解决方案，即研究品种。算术几何的一个指导思想是“几何控制算术”，换句话说，一个变量的几何性质（那些可以用复数来研究的）会影响这个变量的算术行为。这个项目的主要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。该研究涉及研究生和本科生的几个项目。该项目涉及k-合理性失败的算术度量的系统研究，重点关注具有圆锥束结构或具有纤维化成高次del Pezzo曲面的几何合理三重。这两种类型的变种一起覆盖了所有几何理性三重的大片。在另一个方向上，该项目旨在收集更多关于曲线上孤立点的信息。这方面的目标有两个：1）确定Bézieri-Lang猜想和挠猜想是否意味着曲线上孤立点的个数仅依赖于其亏格而有界，和2）计算一组模数曲线上的所有孤立点。该奖项反映了NSF的法定使命，并被认为值得通过使用基金会的智力价值和更广泛的影响进行评估来支持审查标准。

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.