Algebraic Varieties, Birational Geometry and the Structure of the Galois Groups

代数簇、双有理几何和伽罗瓦群的结构

• 批准号: 0100837
• 负责人: Fedor Bogomolov
• 金额: $ 24.15万
• 依托单位: New York University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2001
• 资助国家: 美国
• 起止时间: 2001-08-01 至 2004-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0100837&HistoricalAwards=false
• 关键词: Algebraic; Varieties; Birational; Geometry; Structure

The purpose of the proposal is to study the geometry of algebraic varietiesand their invariants from different points of view. In particular theinvestigator and his colleagues studythe structure of fundamental groups and universal coverings of algebraicvarieties, the problemof placement of singularities, restrictions on the concentration of singularpoints in special fibers of the fibrations of algebraic varieties. In thelatter case the goal is to develop a sufficiently simple method which can begeneralized to the case of arithmetic family of curves. The study of theGalois groups of functional fields is going to provide with an alternativeapproach to the problems of birational geometry.The investigator suggests to continue a broad scope of research in the areaof algebraic geometry with potential applications to number theory andtheoretical physics. Of special interest is the study of a formulathat provides nontrivial estimates for a number of solutions in equationssimilar in form to the famous Fermat equation..

该建议的目的是从不同的角度研究代数变量及其不变量的几何。特别是，这位研究者和他的同事们研究了代数簇的基本群和泛覆盖的结构，奇点的放置问题，对代数簇的原纤维中奇点集中的限制。在后一种情况下，目标是发展一种足够简单的方法，该方法可以推广到算术曲线族的情况。对函数域伽罗华群的研究将为二次几何问题提供另一种方法。研究者建议继续在代数几何领域进行广泛的研究，并将其潜在地应用于数论和理论物理。特别令人感兴趣的是对一个公式的研究，该公式为方程中的许多解提供了非平凡的估计，其形式类似于著名的费马方程。