The following question was asked on math.stackexchange.com with no reply for the past week or so. Let $f : X \to Y$ be a morphism of smooth (integral) varieties over $\Bbb{C}$ with generic fiber equal to $\Bbb{P}^1$.

Is it true that $R^if_\ast \mathcal{O}_X = 0$ for all $i > 0$?

By Cohomology and base change, we know that $R^if_\ast \mathcal{O}_X|_{\eta_Y} = 0$. However, because $f$ is not assumed proper, we do not know if $R^if_\ast \mathcal{O}_X$ is coherent. This leads me to suspect that the statement may be false, but I cannot come up with a counterexample.