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Classifying sections of del Pezzo fibrations, II

### Brian Lehmann and Sho Tanimoto

Geometry & Topology 26 (2022) 2565–2647
DOI: 10.2140/gt.2022.26.2565
##### 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 ${ℙ}^{2}$ over the ground field.

##### Keywords
section, del Pezzo fibration, Fujita invariant, geometric Manin's conjecture
Primary: 14H10