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

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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.