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Let $f:X\to Y$ be the morphism of smooth varieties over $\mathbb{C}$ with generic fiber equal to $\mathbb{P}^1$. How to prove that $R^if_*\mathcal{O}_X=0$ for $i>0$?

(I do not need the complete solution, just an idea)

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  • $\begingroup$ Doesn't this work by proving the vanishing stalkwise? $\endgroup$ – user40276 Jun 27 '15 at 0:38
  • $\begingroup$ It's possible that this lemma from the Stacks Project might help. I was able to use it to prove that all the higher direct images are at least torsion irrespective of whether $X$ or $Y$ is smooth, but I don't know if that will help. $\endgroup$ – Avi Steiner Jun 27 '15 at 3:15
  • $\begingroup$ Looks to me to be an application of cohomology and base change. $\endgroup$ – John Brevik Jun 27 '15 at 9:54
  • $\begingroup$ If $X=Y\times\mathbb{P}^1$ then indeed it is easy to deduce this from the flat base change formula. But why is it true in general case? $\endgroup$ – vitaliy Jun 27 '15 at 14:45
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    $\begingroup$ @user178979: leaving aside the issue of coherent versus quasicoherent, you've got your semicontinuity the wrong way round. Upper semicontinuous means the value can "jump up" at special points. So a coherent sheaf whose stalk is zero at the generic point need not be the zero sheaf! $\endgroup$ – Relapsarian Jun 29 '15 at 23:14

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