Let $B$ be a projective variety, and $f: \mathbb P^n\to B$ be any rational map; such that the generic fiber is of dimension at least $2$. I guess the following statement is always true:
There exists some fiber of $f$, which contains a rational curve.
I guess this because, I think it is a very strong condition for a variety to contain no rational curve. Basically the only counterexample I can think about, are the abelian varieties, and over $\mathbb C$ they are tori, and I believe this cannot happen in our case. However, I have no idea how to prove the statement above.
Any hint, or counterexample would be very helpful. Thanks in advance.