Combinatorial and nonarchimedean methods in algebraic geometry

代数几何中的组合和非阿基米德方法

• 批准号: 1068689
• 负责人: Sam Payne
• 金额: $ 26.2万
• 依托单位: Yale University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2011
• 资助国家: 美国
• 起止时间: 2011-07-01 至 2014-12-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1068689&HistoricalAwards=false
• 关键词: Combinatorial; nonarchimedean; methods; algebraic; geometry

The investigator will pursue several lines of research in algebraic geometry involving the application of combinatorial and nonarchimedean methods to study algebraic curves and their moduli, plus intersection theory. In particular, this proposal deals with nonarchimedean approaches to the Gieseker-Petri Theorem and Maximal Rank Conjectures, the weight filtration on cohomology of moduli of curves, metric properties of tropicalizations and analytifications, and the development of a functorial tropicalization of intersection theory.Algebraic geometry studies solution sets of systems of polynomial equations. Over a nonarchimedean field, one can split the problem of understanding such a solution set into two parts. What are the possible valuations of solutions? And what are the solutions with a given valuation? The set of valuations of solutions has a rich combinatorial and polyhedral structure, and is the primary object of study in tropical geometry. Recent developments in this field make it possible to resolve subtle questions about the geometry of the actual solution set using the geometry of these sets of valuations. The current proposal aims to refine these new methods and explore deeper applications to open problems in algebraic geometry.

研究人员将进行代数几何的多项研究，涉及应用组合和非阿基米德方法来研究代数曲线及其模以及交集理论。 特别是，该提案涉及 Gieseker-Petri 定理和最大等级猜想的非阿基米德方法、曲线模上同调的权重过滤、热带化和分析的度量属性，以及交集理论的函数热带化的发展。 代数几何研究多项式方程组的解集。 在非阿基米德域上，我们可以将理解这样一个解决方案集的问题分为两部分。 解决方案的可能估值是多少？ 在给定估值的情况下，有哪些解决方案？ 解的评估集具有丰富的组合和多面体结构，是热带几何研究的主要对象。 该领域的最新发展使得使用这些评估集的几何形状来解决有关实际解决方案集的几何形状的微妙问题成为可能。 当前的提案旨在完善这些新方法并探索代数几何中开放问题的更深入应用。