Rationality, Rationality Index, and Rational Points

这个研究项目集中在算术几何中的问题上。顾名思义，这个研究领域位于两个领域的交界处：算术(研究整数和分数)和几何(研究曲线、曲面和高维形状)。这些不同的领域是通过对多项式方程的解的研究，即对变种的研究而结合在一起的。算术几何的指导思想是“几何控制算术”，换句话说，一个变种的几何性质(那些可以用复数来研究的几何性质)影响该变种的算术行为。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：地面场上的合理性和孤立点的存在。这项研究涉及研究生和本科生的几个项目。该项目涉及对k-有理失效的算术度量的系统研究，重点是具有二次圆锥丛结构或具有高次Del Pezzo曲面纤维的几何有理三重。这两种类型的变种一起覆盖了所有几何有理三重数中的一大部分。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一条曲线上的孤立点的数量仅取决于它的亏格是有界的，以及2)计算一组模曲线上的所有孤立点。这个奖项反映了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。

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.