On the Xiao conjecture for plane curves

Let f: S⟶ B be a non-trivial fibration from a complex projective smooth surface S to a smooth curve B of genus b. Let cf the Clifford index of the general fibre F of f. In Barja et al. (Journal für die reine und angewandte Mathematik, 2016) it is proved that the relative irregularity of f, qf= h1 , 0(S) - b is less or equal than or equal to g(F) - cf. In particular this proves the (modified) Xiao’s conjecture: qf≤g(F)2+1 for fibrations of general Clifford index. In this short note we assume that the general fiber of f is a plane curve of degree d≥ 5 and we prove that qf≤ g(F) - cf- 1. In particular we obtain the conjecture for families of quintic plane curves. This theorem is implied for the following result on infinitesimal deformations: let F a smooth plane curve of degree d≥ 5 and let ξ be an infinitesimal deformation of F preserving the planarity of the curve. Then the rank of the cup-product map H0(F, ωF) ⟶ · ξH1(F, OF) is at least d- 3. We also show that this bound is sharp.