The relative irregularity q_f of a fibered surface f:S↦B is the relative dimension of the induced surjection Alb(S)↦Alb(B). A conjecture of Xiao Gang predicts that the relative irregularity of f is related to the genus g of a generic fiber through the inequality qf≤g/2+1. This conjecture has been verified if the general fiber of f is Clifford general or hyperelliptic. I will present a proof of this conjecture when the generic fiber is trigonal. It will arise as a consequence of much stronger rigidity statements for trigonal curves on abelian varieties.

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