On the Xiao conjecture for plane curves

Abstract

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 für 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.