Classifying sections of del Pezzo fibrations, II

Lehmann, Brian¹ ; Tanimoto, Sho²

¹ Department of Mathematics, Boston College, Chestnut Hill, MA, United States
² Graduate School of Mathematics, Nagoya University, Nagoya, Japan

Geometry & topology, Tome 26 (2022) no. 6, pp. 2565-2647.

Résumé

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 ${\mathbb{P}}^2$ over the ground field.

Détail

Keywords: section, del Pezzo fibration, Fujita invariant, geometric Manin's conjecture

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