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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

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 2 admitting a birational morphism to 2 over the ground field.

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section, del Pezzo fibration, Fujita invariant, geometric Manin's conjecture
Mathematical Subject Classification
Primary: 14H10
Received: 15 July 2020
Revised: 10 June 2021
Accepted: 8 July 2021
Published: 13 December 2022
Proposed: Dan Abramovich
Seconded: Mark Gross, Benson Farb
Brian Lehmann
Department of Mathematics
Boston College
Chestnut Hill, MA
United States
Sho Tanimoto
Graduate School of Mathematics
Nagoya University