A Non-Archimedean Approach to the Geometry of General Curves

University Of Kentucky Research Foundation, Lexington KY

Investigators

Abstract

Algebraic geometry, which studies solutions to systems of polynomial equations, is a central topic in mathematics with applications to many other disciplines. Many natural phenomena of interest in physics, biology, and computer science can be modeled by polynomials, making algebraic geometry a useful tool for the scientific community at large. This research project aims to study the geometric properties of the simplest objects in algebraic geometry, known as algebraic curves. While some curves may have exotic or pathological properties, it is expected that most curves do not. This goal of this project is to show that "typical" curves are geometrically well-behaved, using techniques from the recently developed field of non-Archimedean analytic geometry. Brill-Noether theory aims to study the geometry of a curve by examining all of its maps to projective space, or equivalently the existence and behavior of its linear series. A major change in perspective occurred during the twentieth century, as the field shifted from studying fixed to general curves -- that is, general points in the moduli space of curves. A standard approach to the study of general curves is via degeneration arguments. One considers a family of smooth curves degenerating to a singular curve, and attempts to deduce geometric properties of the general fiber using information about the singular fiber. Recently, it has been observed that this theory fits within a broader framework of non-Archimedean analytic techniques. This project focuses on outstanding problems concerning the Brill-Noether theory of general curves, and how new techniques in non-Archimedean analytic geometry may shed light on these problems.