Rational curves via function field analytic number theory

Polynomial equations are the cornerstone of modern mathematics. When viewed over the integers, they define Diophantine equations, and questions about the solubility and density of integer solutions have been studied for millennia. When viewed over the complex numbers, they define algebraic varieties, whose classification is a key part of algebraic geometry today, but dates back to 19th Century and an Italian school of geometers. For many years it has been known that the geometry of a system of equations can have a strong influence on their arithmetic. However, in recent years, the flow of information has been reversed, with a surprising link discovered between the geometry of rational curves on higher-dimensional algebraic varieties and questions about the density of solutions to Diophantine equations over global fields of positive characteristic. The main aim of the proposal is to drive this connection forward, by refining a Fourier-analytic circle method approach to tackle central problems in algebraic geometry about moduli spaces of curves on suitable varieties.