Geometry and Arithmetic of Brill--Noether Loci and Brill--Noether curves

布里尔-诺特轨迹和布里尔-诺特曲线的几何与算术

• 批准号: 2200655
• 负责人: Isabel Vogt
• 金额: $ 21万
• 依托单位: Brown University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2022
• 资助国家: 美国
• 起止时间: 2022-07-01 至 2025-06-30
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2200655&HistoricalAwards=false
• 关键词: Geometry; Arithmetic; Brill; Noether; Loci

Polynomial equations are ubiquitous in mathematics, physics, and other sciences. One can study a system of polynomial equations geometrically, by thinking about the shape formed by these solutions, as well as arithmetically, by considering what types of numbers arise in solutions. This project studies the relationship between the geometry and arithmetic in the one-dimensional case of algebraic curves. Broadly, this project will investigate the possible explicit realizations of an abstract algebraic curve by polynomial equations (so-called Brill-Noether theory), which informs both the geometry of the curve, as well as its arithmetic of solutions over bounded extensions of the rational numbers. The project includes the training of undergraduate and graduate students and work with members of underrepresented groups. Specifically, the PI will initiate the arithmetic Brill-Noether theory program to elucidate the structure of the rational points on Brill-Noether loci in the Picard variety of a curve. This has applications to determining the low degree points on curves defined over number fields. The PI will also investigate analogues of the classic Brill-Noether theorem when the curve is special in moduli due to a low degree realization in projective space. In particular, when the curve has a low degree map to the projective line, the PI will study the relationship with affine permutations and the affine Grassmannian.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.

多项式方程在数学，物理和其他科学方面无处不在。 通过考虑这些溶液形成的形状以及算术，可以通过考虑解决方案中出现了哪些类型的数字来研究多项式方程的系统。 该项目研究了代数曲线的一维情况下的几何形状和算术之间的关系。 从广义上讲，该项目将通过多项式方程（所谓的Brill-Noether理论）研究抽象代数曲线的明确实现，该方程式既可以告知曲线的几何形状及其对解决方案的算术而不是合理数量的界面扩展。该项目包括对本科生和研究生的培训，以及与代表性不足的团体成员合作。具体而言，PI将启动算术Brill-Noether理论计划，以阐明Brill-Noether loci在PICARD曲线品种中的理性点的结构。 这具有确定在数字字段上定义的曲线上的低度点的应用。 当曲线在模量特殊时，PI还将调查经典的Brill-Noether定理的类似物，这是由于投影空间的低度实现。 特别是，当曲线具有低度图与投影线的较低度图时，PI将研究与仿射排列和格拉斯曼尼亚人的关系。该奖项反映了NSF的法定任务，并被认为是值得通过基金会的知识分子优点和更广泛影响的审查标准通过评估来进行评估的。