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A Non-Archimedean Approach to the Geometry of General Curves

一般曲线几何的非阿基米德方法

基本信息

批准号：
1601896
负责人：
David Jensen
金额：
$ 13.63万
依托单位：
University of Kentucky Research Foundation
依托单位国家：
美国
项目类别：
Standard Grant
财政年份：
2016
资助国家：
美国
起止时间：
2016-09-01 至 2020-08-31
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=1601896&HistoricalAwards=false
关键词：
Non Archimedean Approach Geometry General

项目摘要

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.

代数几何学研究多项式方程组的解，它是数学的中心课题，并应用于许多其他学科。许多物理学、生物学和计算机科学中感兴趣的自然现象都可以用多项式建模，这使得代数几何成为科学界的一个有用工具。该研究项目旨在研究代数几何中最简单对象的几何特性，称为代数曲线。虽然一些曲线可能具有奇异或病理性质，但预期大多数曲线不具有。 Brill-Noether理论的目的是通过研究曲线在射影空间中的所有映射，或者等价地研究曲线的线性级数的存在性和行为，来研究曲线的几何性质。世纪，随着研究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，观点发生了重大变化。研究一般曲线的标准方法是通过退化论证。考虑一族退化为奇异曲线的光滑曲线，并试图利用奇异纤维的信息来推导一般纤维的几何性质。最近，人们观察到，这一理论适合在一个更广泛的框架内的非阿基米德分析技术。这个项目的重点是突出的问题有关的Brill-Noether理论的一般曲线，以及如何在非阿基米德解析几何的新技术可能揭示这些问题。