Xiao's conjecture for general fibred surfaces

TL;DR

This paper proves an inequality relating genus, irregularity, and Clifford index for certain fibred surfaces, confirming Xiao's conjecture in cases with maximal Clifford index.

Contribution

The authors establish a new inequality connecting key invariants of fibred surfaces, providing a proof of Xiao's conjecture for fibrations with maximal Clifford index.

Findings

Proved the inequality q_f ≤ g - c_f for non-isotrivial fibrations.

Confirmed Xiao's conjecture for fibrations with maximal Clifford index.

Enhanced understanding of the relationship between surface invariants.

Abstract

We prove that the genus $g$, the relative irregularity $q_f$ and the Clifford index $c_f$ of a non-isotrivial fibration $f$ satisfy the inequality $q_f \leq g - c_f$. This gives in particular a proof of Xiao's conjecture for fibrations whose general fibres have maximal Clifford index.