# On the Xiao conjecture for plane curves

## TL;DR

This paper proves a new bound on the relative irregularity of fibrations with plane curve fibers of degree at least five, confirming a special case of Xiao's conjecture and analyzing related infinitesimal deformations.

## Contribution

It establishes a specific inequality for fibrations with plane curve fibers of degree ≥ 5, confirming the conjecture for quintic plane curves and analyzing infinitesimal deformations.

## Key findings

- Proves $q_f \,\le\, g(F) - c_f - 1$ for plane curve fibers of degree ≥ 5.
- Confirms Xiao's conjecture for families of quintic plane curves.
- Shows the rank of the cup-product map is at least $d-3$, and this bound is sharp.

## 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 generic fibre $F$ of $f$. In [arXiv:1401.7502v4] it is proved that the relative irregularity of $f$, $q_f=h^{1,0}(S)-b$ is less than or equal to $g(F)-c_f$. In particular this proves the (modified) Xiao's conjecture: $q_f\le 1+g(F)/2$ for fibrations of general Clifford index. In this short note we assume that the generic 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 $\cdot \xi: H^0(F,\omega_F) \rightarrow H^1(F,O_F)$ is at least $d-3$. We also show that this bound is sharp.