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、物理和其他科学中无处不在。 人们可以从几何学的角度研究多项式方程组，通过思考这些解形成的形状，以及算术，通过考虑解中出现的数字类型。 本计画研究一维代数曲线之几何与算术之关系。 从广义上讲，这个项目将研究一个抽象的代数曲线的多项式方程（所谓的Brill-Noether理论）的可能的显式实现，这既通知曲线的几何形状，以及它的算法的解决方案在有理数的有界扩展。该项目包括培训本科生和研究生，并与代表性不足的群体的成员合作。具体而言，PI将启动算术Brill-Noether理论程序，以阐明曲线的Picard簇中Brill-Noether轨迹上的有理点的结构。 这可以应用于确定数域上定义的曲线上的低阶点。 PI还将研究经典Brill-Noether定理的类似物，当曲线在模中是特殊的，因为在射影空间中实现了低程度。 特别是，当曲线与投影线的映射度较低时，PI将研究与仿射置换和仿射格拉斯曼的关系。该奖项反映了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。