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 Geometric Manin’s 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 Geometric Manin’s Conjecture for certain split del Pezzo surfaces of degree ≥ 2 admitting a birational morphism to P over the ground field.