Paucity of rational points on fibrations with multiple fibres

Browning, Tim; Lyczak, Julian; Smeets, Arne

doi:10.2140/ant.2025.19.2049

Download this article
	For screen
	For printing

Recent Issues

The Journal

About the journal

Ethics and policies

Peer-review process

Submission guidelines

ISSN 1944-7833 (online)

ISSN 1937-0652 (print)

Author index

To appear

Other MSP journals

Paucity of rational points on fibrations with multiple fibres

Tim Browning, Julian Lyczak and Arne Smeets

Vol. 19 (2025), No. 10, 2049–2090

DOI: 10.2140/ant.2025.19.2049

Abstract

Given a family of varieties over the projective line, we study the density of fibres that are everywhere locally soluble in the case that components of higher multiplicity are allowed. We use log geometry to formulate a new sparsity criterion for the existence of everywhere locally soluble fibres and formulate new conjectures that generalise previous work of Loughran and Smeets. These conjectures involve geometric invariants of the associated multiplicity orbifolds on the base of the fibration in the spirit of Campana. We give evidence for the conjectures by providing an assortment of bounds using Chebotarev’s theorem and sieve methods, with most of the evidence involving upper bounds.

Keywords

local solubility, fibration, sieve methods, multiple fibres, Halphen surface

Mathematical Subject Classification

Primary: 14G05

Secondary: 11G35, 11N36, 14A21, 14D10

Milestones

Received: 19 January 2024

Revised: 14 October 2024

Accepted: 12 November 2024

Published: 5 September 2025

Authors

Tim Browning
Institute of Science and Technology Austria Klosterneuburg Austria

Julian Lyczak
Department of Mathematics University of Antwerp Antwerpen Belgium

Arne Smeets
Department of Mathematics KU Leuven Heverlee Belgium

Open Access made possible by participating institutions via Subscribe to Open.

Vol. 19, No. 10, 2025