ncts

Manin's conjecture predicts the asymptotic formula for the counting function of rational points on a smooth Fano variety, and it predicts an explicit asymptotic formula in terms of geometric invariants of the underlying variety. When you count rational points, it is important to exclude some contribution of rational points from an exceptional set so that the asymptotic formula reflects global geometry of the underlying variety. In Part I of my talk, I will describe birational geometry of these exceptional sets using higher dimensional algebraic geometry such as the minimal model program. In Part II, I will discuss applications of the study of exceptional sets to moduli spaces of sections of Fano fibrations. Then in Part III I will discuss the case of del Pezzo fibrations in more details. Part I is based on joint work with Brian Lehmann and Akash Kumar Sengupta. Part II is based on joint work with Brian Lehmann and Eric Riedl. Part III is based on joint work with Brian Lehmann.