Collaborative Research: FRG: Geometry of moduli spaces of rational curves with applications to Diophantine problems over function fields

合作研究：FRG：有理曲线模空间的几何及其在函数域上丢番图问题的应用

• 批准号: 0553921
• 负责人: Jason Starr
• 金额: $ 23.88万
• 依托单位: Massachusetts Institute of Technology
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2006
• 资助国家: 美国
• 起止时间: 2006-07-01 至 2007-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0553921&HistoricalAwards=false
• 关键词: Collaborative; Research; FRG; Geometry; moduli

This project addresses the geometry of spaces of rational curveson smooth projective varieties, with a view toward understanding thestructure of rational points for varieties defined over functionfields. Consider a rationally-connected variety: Which homologyclasses contain free rational curves? Very free rational curves? Isthe space of such curves connected? Irreducible? Rationallyconnected? Of general type? Is there a workable notion of `rationalsimple connectedness' and is this a birational property? How can wedistinguish unirational varieties as a subclass of rationally-connectedvarieties? These questions are related to fundamental problems in Diophantinegeometry over function fields: Does a rationally-connected varietyover C(t) satisfy weak approximation? Can the hypothesis of the Tsen/LangTheorem over C(s,t) be formulated geometrically? For rationally-connectedvarieties over C(s,t), to what extent do cohomological obstructionsgovern the existence of rational points?This award will support research on systems of polynomialequations with coefficents varying in parameters. Our goal is tosolve these equations with rational functions that depend on theseparameters. The case of a single equation (or of several independentequations) was addressed in the mid 20th century; the feasibilityof finding a solution depends on the degree of the equation, the numberof free variables, and the number of varying parameters. Recently,a comprehensive geometric approach was developed when there is justone varying parameter. However, for multiple (not necessarilyindependent) equations in two varying parameters much remains to beunderstood. This work will also have broader impacts on the education ofgraduate students and postdoctoral fellows, the development of web-basedcollaboration tools, and the promotion of robust academic networkslinking universities across the country.

该项目解决了光滑射影簇上有理曲线空间的几何形状，以期理解在函数域上定义的簇的有理点的结构。 考虑一个有理连接的簇：哪些同源类包含自由有理曲线？ 非常自由的有理曲线？这些曲线的空间是连通的吗？ 不可约？ 理性联系？ 属于一般型的吗？ 是否存在“理性简单关联性”的可行概念，并且这是一种双理性属性吗？ 我们如何将非理性品种区分为理性关联品种的子类？ 这些问题与函数域上的丢番图几何中的基本问题相关：C(t) 上的有理连通簇是否满足弱近似？ C(s,t) 上的 Tsen/Lang 定理的假设可以用几何形式表示吗？ 对于 C(s,t) 上的有理连通簇，上同调障碍在多大程度上控制有理点的存在？该奖项将支持对系数随参数变化的多项式方程组的研究。 我们的目标是用依赖于这些参数的有理函数来求解这些方程。 单个方程（或几个独立方程）的情况在 20 世纪中叶得到了解决； 找到解的可行性取决于方程的次数、自由变量的数量以及变化参数的数量。 最近，当只有一个变化参数时，开发了一种综合几何方法。 然而，对于两个不同参数的多个（不一定是独立的）方程，还有很多事情有待理解。 这项工作还将对研究生和博士后的教育、基于网络的协作工具的开发以及促进连接全国大学的强大学术网络产生更广泛的影响。