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 2 over the ground field.

An introduction to Geometric Manin's conjecture

“…• del Pezzo fibrations ([LT19a] and [LT21a]). Many of the above results are based on the method pioneered in [HRS04] which used two main tools in algebraic geometry, i.e., the moduli space of stable maps of genus 0 with n marked points M 0,n (X) and Mori's Bend and Break lemma.…”