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.

该项目介绍了Ronication Curveson平滑射击品种的几何形状，以了解在功能场上定义的品种的理性点的影响。 考虑一个合理连接的品种：哪些同源类包含自由有理曲线？ 非常自由的理性曲线？这种曲线的空间是否连接？ 不可约吗？ 合理联系？ 一般类型？ 是否有“理性简单连接”的可行概念，这是一个男性属性吗？ 如何将Unirational品种作为合理联系的子类别？ 这些问题与功能领域的二聚体几何测量学的基本问题有关：合理连接的c（t）是否满足弱近似？ 在c（s，t）上的TSEN/LANGTHEOREM的假设是否可以几何形式进行表述？ 对于c（s，t）的合理连接变化，同时障碍物的存在在多大程度上是理性点的存在吗？该奖项将支持对参数系数变化的多项式序列系统的研究。 我们的目标是用依赖于eSeparameter的理性函数来解决这些方程。 20世纪中叶解决了单个方程式（或几个独立方程）的情况； 找到解决方案的可行性取决于方程的程度，自由变量的数量和不同参数的数量。 最近，当存在变化的参数时，开发了一种全面的几何方法。 但是，对于两个不同参数中的多个（不一定独立的）方程式，还有很多值得一提的。 这项工作还将对研究生和博士后研究员的教育，基于Web的Collaboration工具的开发以及促进全国强大的学术网络链接大学的促进。