This project addresses the geometry of spaces of rational curves on smooth projective varieties, with a view toward understanding the structure of rational points for varieties defined over function fields. Consider a rationally-connected variety: Which homology classes contain free rational curves? Very free rational curves? Is the space of such curves connected? Irreducible? Rationally connected? Of general type? Is there a workable notion of `rational simple connectedness' and is this a birational property? How can we distinguish unirational varieties as a subclass of rationally-connected varieties? These questions are related to fundamental problems in Diophantine geometry over function fields: Does a rationally-connected variety over C(t) satisfy weak approximation? Can the hypothesis of the Tsen/Lang Theorem over C(s,t) be formulated geometrically? For rationally-connected varieties over C(s,t), to what extent do cohomological obstructions govern the existence of rational points?
This award will support research on systems of polynomial equations with coefficents varying in parameters. Our goal is to solve these equations with rational functions that depend on these parameters. The case of a single equation (or of several independent equations) was addressed in the mid 20th century; the feasibility of finding a solution depends on the degree of the equation, the number of free variables, and the number of varying parameters. Recently, a comprehensive geometric approach was developed when there is just one varying parameter. However, for multiple (not necessarily independent) equations in two varying parameters much remains to be understood. This work will also have broader impacts on the education of graduate students and postdoctoral fellows, the development of web-based collaboration tools, and the promotion of robust academic networks linking universities across the country.
