Classifying sections of del Pezzo fibrations, II

TL;DR

This paper advances the understanding of rational points on del Pezzo fibrations by developing a new inductive classification method and proving cases of Geometric Manin's Conjecture for specific split del Pezzo surfaces.

Contribution

It introduces the Movable Bend and Break Lemma for classifying sections and proves Geometric Manin's Conjecture for certain split del Pezzo surfaces.

Findings

Established bounds on the counting function for rational points.

Proved Geometric Manin's Conjecture for some split del Pezzo surfaces.

Developed an inductive approach to classify relatively free sections.

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 $\geq 2$ admitting a birational morphism to $\mathbb P^2$ over the ground field.