Classifying sections of del Pezzo fibrations, II

Abstract

Let $X$ be a del Pezzo surface over the function field of a complex curve. We study the behavior of rational points on $X$ leading to bounds on the counting function in the geometric Manin conjecture. A key tool is the movable bend-and-break lemma, which yields an inductive approach to classifying relatively free sections for a del Pezzo fibration over a curve. Using this lemma we prove the geometric Manin conjecture for certain split del Pezzo surfaces of degree $\ge 2$ admitting a birational morphism to ${\mathbb{P}}^2$ over the ground field.