Xiao's Conjecture on Canonically Fibered Surfaces

TL;DR

This paper proves Xiao's conjecture that the relative genus of a canonically fibered surface cannot exceed 4, refining previous bounds established using the Miyaoka-Yau inequality.

Contribution

The paper provides a proof confirming Xiao's conjecture, establishing a sharper upper bound on the relative genus for canonically fibered surfaces.

Findings

Confirmed that the relative genus is at most 4 for canonically fibered surfaces

Refined previous bounds from the Miyaoka-Yau inequality

Contributed to the classification of fibered algebraic surfaces

Abstract

A canonically fibered surface is a surface whose canonical series maps it to a curve. Using Miyaoka-Yau inequality, A. Beauville proved that a canonically fibered surface has relative genus at most 5 when its geometric genus is sufficiently large. G. Xiao further conjectured that the relative genus cannot exceed 4. We give a proof of this conjecture.