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

本项目研究光滑投影变异上有理曲线空间的几何，以期理解在函数域上定义的变异的有理点的结构。考虑一个有理连接的变种：哪些同构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?Rationallyconnected吗?一般类型的？是否存在一种可行的“理性简单连通性”的概念，这是否是一种与生俱来的属性？我们如何区分非民族品种作为理性联系品种的一个子类？这些问题与函数域上丢芬蒂几何的基本问题有关：C(t)上的理性连接变分是否满足弱近似？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）上的有理点，上同调障碍在多大程度上控制了有理点的存在？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解这些方程。单个方程（或多个独立方程）的情况在20世纪中期得到了解决；找到解的可行性取决于方程的程度、自由变量的数量和变化参数的数量。近年来，一种综合的几何方法在只有一个变参数的情况下得到了发展。然而，对于两个不同参数下的多个（不一定是独立的）方程，仍有许多有待了解的地方。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全国各大学之间强大的学术网络产生更广泛的影响。

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.