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Given a family of varieties over the projective line, we study the density offibres that are everywhere locally soluble in the case that components ofhigher multiplicity are allowed. We use log geometry to formulate a newsparsity criterion for the existence of everywhere locally soluble fibres andformulate new conjectures that generalise previous work of Loughran-Smeets.These conjectures involve geometric invariants of the associated multiplicityorbifolds on the base of the fibration in the spirit of Campana. We giveevidence for the conjectures using Chebotarev's theorem and sieve methods.

Paucity of rational points on fibrations with multiple fibres