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

本课题研究光滑射影簇上有理曲线空间的几何，以期了解定义在函数域上的簇的有理点的结构。考虑一个有理连通的变种：哪些同系类包含自由有理曲线？非常自由的有理曲线？这些曲线的空间是相通的吗？不可还原？有联系吗？普通类型的？有没有一个可行的‘理性简单连通性’的概念，这是一种混血属性吗？我们如何区分无意义变种作为合理连通变种的一个子类？这些问题涉及函数域上丢番几何的基本问题：C(T)上的有理连通变量是否满足弱逼近？C(S，t)上的Tsen/Langn定理的假设能用几何形式表示吗？对于C(S，t)上的有理连通变种，上同调障碍在多大程度上支配有理点的存在？这一奖项将支持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。一个方程(或几个独立方程)的情况是在20世纪中叶提出的；求解的可行性取决于方程的阶数、自由变量的数量和变化参数的数量。最近，人们发展了一种综合的几何方法来处理参数变化的问题。然而，对于具有两个可变参数的多个(不一定是独立的)方程，仍有许多有待理解的地方。这项工作还将对研究生和博士后研究员的教育产生更广泛的影响，开发基于网络的协作工具，并促进将全国各地的大学连接起来的强大的学术网络。

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.