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Branched Galois Covers of Curves: Lifting and Reduction

曲线的分支伽罗瓦覆盖：提升和归约

基本信息

批准号：
1900396
负责人：
Andrew Obus
金额：
$ 2.88万
依托单位：
CUNY Baruch College
依托单位国家：
美国
项目类别：
Standard Grant
财政年份：
2018
资助国家：
美国
起止时间：
2018-09-01 至 2020-07-31
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=1900396&HistoricalAwards=false
关键词：
Branched Galois Covers Curves Lifting

项目摘要

This research project concerns algebraic geometry, the study of systems of polynomial equations in many variables. The project focuses on equations formulated in characteristic p for some prime number p. The "characteristic p world" is one where, every time you take p steps forward, you wind up back where you start. Far from being esoteric, such mathematical systems describe the procession of the days of the week (p = 7), the fundamentals of computer architecture (p = 2), and also the setting for important cryptosystems. Algebraic geometry in characteristic p is the study of geometric objects ("varieties") given by solutions to polynomial equations in this setting. This project investigates the relationship between varieties in characteristic p and in characteristic zero ("standard" algebraic geometry), with the goal of shedding light on both worlds. Graduate students will take a major part in the research project. The research will be complemented by educational activities involving the undergraduate math club.The first part of this research project is devoted to understanding when branched Galois covers of curves lift from characteristic p to characteristic zero, and to obtaining information about the geometry and arithmetic of the lifts. In particular, the project aims to obtain a classification of the so-called "local Oort groups" and "weak local Oort groups". The second part of the project involves analyzing integral models of Galois covers of curves over p-adic fields. New tools involving deformation data shed light on stable models of Galois covers of curves with complicated Galois groups. These tools will be exploited further in order to understand stable models of Galois covers as explicitly as possible.

这个研究项目涉及代数几何，研究多元多项式方程组。该项目的重点是方程制定的特征p为一些素数p。“特征p的世界”是一个，每次你采取p步骤前进，你风回到你开始的地方。这样的数学系统远不是深奥的，它描述了一周中几天的进程（p = 7），计算机体系结构的基本原理（p = 2），以及重要密码系统的设置。特征p的代数几何学是研究由多项式方程的解给出的几何对象（“变种”）。这个项目研究了特征p和特征零（“标准”代数几何）中的变量之间的关系，目的是阐明这两个世界。研究生将主要参加这个研究项目。这项研究将补充教育活动，涉及本科生数学俱乐部。本研究项目的第一部分是致力于了解分支伽罗瓦覆盖的曲线升降机时，从特征p到特征零，并获得有关的几何和算术升降机的信息。特别是，该项目旨在获得所谓的“当地奥尔特群体”和“弱当地奥尔特群体”的分类。该项目的第二部分涉及分析p-adic域上曲线的伽罗瓦覆盖的积分模型。涉及变形数据的新工具揭示了具有复杂伽罗瓦群的曲线的伽罗瓦覆盖的稳定模型。这些工具将被进一步利用，以尽可能明确地理解伽罗瓦覆盖的稳定模型。