Locally non-trivial fibred surfaces with maximal unitary rank

Let f:S→B a locally non-trivial fibred surface with fibres of genus g. Let uf be its unitary rank, i.e. the rank of the flat unitary part in the second Fujita decomposition. We study in detail the case when uf is maximal, i.e. uf=g-1. In this case necessarily g≤6, but examples in genus 5 and 6 are not known, and conjecturally do not exist. We prove a strong slope inequality for these extremal cases. We then use this inequality, together with results on trigonal curves, to give new constraints on the case g=6, uf=5. In particular, we prove that the index of the surface is always strictly positive and give strong limitations on the possible classes of the relative canonical divisor.