Degenerations and Moduli in Algebraic Geometry

代数几何中的简并和模

• 批准号: 1603604
• 负责人: Valery Alexeev
• 金额: $ 24.9万
• 依托单位: University of Georgia Research Foundation Inc
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2016
• 资助国家: 美国
• 起止时间: 2016-07-01 至 2020-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1603604&HistoricalAwards=false
• 关键词: Degenerations; Moduli; Algebraic; Geometry

This award supports research in algebraic geometry, a central branch of mathematics. It aims to understand, both practically and conceptually, solutions of systems of polynomial equations in many variables. Algebraic geometry has important applications to other fields of mathematics, such as number theory, topology, and analysis, as well as to physics, biology, cryptography, and engineering. The particular questions under study in this research project involve moduli spaces, which are sets that parameterize solutions of geometric classification problems. Some of the topics under study originated in string theory in physics, and results of this work may in turn find application there. Graduate students are involved in the project. The investigator will work on a range of questions in algebraic geometry centering on degenerations, compact moduli spaces of stable varieties and pairs, and connections to the minimal model program. A central project is the study of functorial compactifications of moduli spaces of K3 surfaces and their explicit combinatorial description. The investigator will also study the birational geometry of abelian six-folds.

该奖项支持代数几何（数学的一个核心分支）的研究。 它的目的是从实践和概念上理解多变量多项式方程组的解。代数几何在数论、拓扑和分析等其他数学领域以及物理学、生物学、密码学和工程学中都有重要的应用。本研究项目中研究的特定问题涉及模空间，模空间是参数化几何分类问题的解的集合。正在研究的一些主题起源于物理学中的弦理论，这项工作的结果可能反过来会在那里找到应用。研究生参与了该项目。研究人员将研究代数几何中的一系列问题，重点是退化、稳定簇和对的紧模空间以及与最小模型程序的联系。一个中心项目是研究 K3 曲面模空间的函数紧化及其显式组合描述。研究人员还将研究阿贝尔六重的双有理几何。