I am looking for an explaination or an reference for the following fact:
Let $\pi:X\rightarrow Z$ be a contraction from a smooth surface $X$ to a curve $Z$. Assume that the geometric generic fiber of $\pi$ is a rational curve. Then we may run a $K_X$-MMP over $Z$ and reach a minimal ruled surface $\pi^\prime:X^\prime \rightarrow Z$.
Additional question: Why is one interested in geometric generic fibers instead of the usual generic fibers? Do they have "nicer" properties? For example, in the above question, does the geometric generic fiber of $\pi$ is a rational curve implies the generic fiber of $\pi$ is also a rational curve? Or, does the above statement still hold if we replace "geometric generic fiber" with "generic fiber"?