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

多项式方程在数学、物理和其他科学中普遍存在。人们可以通过考虑由这些解形成的形状来几何地研究多项式方程组，以及通过考虑在解中出现什么类型的数字来在算术上研究这些多项式方程组。本课题研究一维代数曲线的几何与算术之间的关系。广泛地说，这个项目将研究抽象代数曲线可能通过多项式方程(所谓的Brill-Noether理论)的显式实现，该理论既可以告知曲线的几何形状，也可以知道其在有理数的有界扩张上的解的算法。该项目包括培训本科生和研究生，并与代表性不足群体的成员合作。具体地说，PI将启动算术Brill-Noether理论程序，以阐明曲线Picard变种中Brill-Noether轨迹上有理点的结构。这对于确定在数域上定义的曲线上的低次点具有应用。PI还将研究经典的Brill-Noether定理的类似之处，当曲线由于在射影空间中的低度实现而在模上特殊时。特别是，当曲线具有到投影线的低度映射时，PI将研究与仿射排列和仿射Grassmannian的关系。这一奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

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.