Xiao's conjecture for general fibred surfaces

We prove that the genus g, the relative irregularity q(f) and the Clifford index c(f) of a non-isotrivial fibration f satisfy the inequality q(f) <= g - c(f). This gives in particular a proof of Xiao's conjecture for fibrations whose general fibres have maximal Clifford index.We prove that the genus g, the relative irregularity q(f) and the Clifford index c(f) of a non-isotrivial fibration f satisfy the inequality q(f) <= g - c(f). This gives in particular a proof of Xiao's conjecture for fibrations whose general fibres have maximal Clifford index.