Thanks, rita! I think the original statement should be in the Xiao's Lecture Notes. But I can not find the statement in the book. I knew it from papers written by other people where they used "PENCIL" not "fibration". So this is why I was confused with the precise statement. Do you know the precise statement of Xiao's result?

I have another quick question, rita. It is about what I said before. I said that minimal surfaces of general type with $p_g=0$ and $K^2 \ge 3$ have no pencils of genus 2. Actually I do not know what the pencil means here. Does it mean a \emph{morphism} of fibration or just a \emph{rational map}?

Right. I should assume that $L$ is not canonical bundle. Thank you!

Thanks, rita. I remember that minimal surfaces of general type with $p_g=0$ and $K^2 \ge 3$ do not have genus two pencil. I just want to know the cases that when $K^2$ is smaller, which correspond the Numerical Godeaux or Campedelli surfaces. I will check the paper you mentioned.

You are right. But I really want to see this proof and I can not find the "another paper".

Is this result independent of characteristic of the base field?

I have not yet. But I will try to check it soon. Thanks!

Agree with Sandor's comment!

I GUESS in some sense you can make it. For example, you can consider the canonical map of the surface, which will work if $K_S^2$ is not so small. Assume that the fiber of the Albanese map is $F$, then consider whether $\mathscr{O}(K_S-F)$ has global section or not. If it has a global section, then after computing some intersection number, maybe you can have a bound of $g(F)$.

@inkspot: What you mean is that there ARE only finitely many smooth rational curve on surfaces of general type?

@rita: right. But I think using the technique of the proof of Noether inequality, maybe $M^2 \ge 2p_g-4$.