摘要
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 $4$. In this talk, I will give a sketch of my proof of this conjecture.