Paucity of rational points on fibrations with multiple fibres

TL;DR

This paper investigates the scarcity of rational points on fibrations with multiple fibers, introducing new conjectures and criteria using log geometry, and providing evidence through number-theoretic methods.

Contribution

It formulates a new sparsity criterion for locally soluble fibers in fibrations with multiple fibers and proposes generalized conjectures involving geometric invariants.

Findings

New sparsity criterion for locally soluble fibers

Formulation of generalized conjectures involving orbifold invariants

Evidence supporting conjectures using Chebotarev's theorem and sieve methods

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-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 using Chebotarev's theorem and sieve methods.