A Non-Archimedean Approach to the Geometry of General Curves

代数几何研究多项式方程组的解，是数学中的一个中心课题，在许多其他学科中都有应用。物理学、生物学和计算机科学中的许多自然现象都可以用多项式来建模，这使得代数几何成为科学界的一个有用工具。本研究项目旨在研究代数几何中最简单对象的几何性质，即代数曲线。虽然有些曲线可能具有奇异的或病态的特性，但大多数曲线都没有。这个项目的目标是利用最近发展起来的非阿基米德解析几何领域的技术，证明“典型”曲线在几何上表现良好。布里尔-诺特理论的目的是通过检查曲线到投影空间的所有映射来研究曲线的几何，或者等价地研究其线性序列的存在性和行为。在20世纪，随着该领域从研究固定曲线转向研究一般曲线，即曲线模空间中的一般点，视角发生了重大变化。研究一般曲线的标准方法是退化论证。一种是考虑一组退化为奇异曲线的光滑曲线，并试图利用奇异纤维的信息推断出一般纤维的几何特性。最近，人们观察到这一理论适用于非阿基米德分析技术的更广泛框架。本项目主要关注一般曲线的Brill-Noether理论的突出问题，以及非阿基米德解析几何中的新技术如何揭示这些问题。

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.