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

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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.