On the Xiao conjecture for plane curves

Let \(f: S\longrightarrow B\) be a non-trivial fibration from a complex projective smooth surface S to a smooth curve B of genus b. Let \(c_f\) the Clifford index of the general fibre F of f. In Barja et al. (Journal fur die reine und angewandte Mathematik, 2016) it is proved that the relative irregularity of f, \(q_f=h^{1,0}(S)-b\) is less or equal than or equal to \(g(F)-c_f\). In particular this proves the (modified) Xiao’s conjecture: \(q_f\le \frac{g(F)}{2} +1\) for fibrations of general Clifford index. In this short note we assume that the general fiber of f is a plane curve of degree \(d\ge 5\) and we prove that \(q_f\le g(F)-c_f-1\). 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 \(d\ge 5\) and let \(\xi \) be an infinitesimal deformation of F preserving the planarity of the curve. Then the rank of the cup-product map \(H^0(F,\omega _F) {\overset{ \cdot \xi }{\longrightarrow }} H^1(F,O_F)\) is at least \(d-3\). We also show that this bound is sharp.