Let $H$ be a very ample linear system on a smooth compact complex surface $X$ whose Kodaira dimension is $\geq 0$. A general element of $H$ is smooth and has genus $\geq 2$.

Let $L\subset H$ be a pencil. a general element of $L$ can be a union $C_1 \cup \dots \cup C_r \cup F$, $F$ the fixed part. The curves $C_i$ can be wildly singular on the base points of $L$ but all have geometric genus at least one, by the Kodaira dimension of $X$. Suppose that they all have geometric genus equal to 1.

Removing $F$, blowing up the base points and taking Stein's factorization, we get a fibration $f : X \longrightarrow \mathbb{P}^1$ (the existence of base points imples that the base is rational).

Can one infer which singular fibers can appear in this situation?

Can one calculate (or bound) the number of singular fibers that are multiple of smooth elliptic curves?

I was thinking that the rational components of the singular fibers would come mostly from rational curves with positive self-intersection...

I've been thinking about this but I didn't find any clue yet.