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 $\Bbb{P}^1$. 



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By [Cohomology and base change](http://stacks.math.columbia.edu/tag/02KH), 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.



Vanishing of higher direct image of a morphism with generic fiber $\Bbb{P}^1$
No. Take $Y=\mathbb{A}^2\_\mathbb{C}$ and $X=U\times \mathbb{P}^1$ where $j:U\hookrightarrow Y$ is the complement of the origin. One immediately finds $R^1 f\_\*\mathscr{O}\_X=R^1 j\_\*\mathscr{O}\_U$, which is nonzero.


