# Higher direct image of morphism with generic fiber $\mathbb{P}^1$

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)

• Doesn't this work by proving the vanishing stalkwise? – user40276 Jun 27 '15 at 0:38
• 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. – Avi Steiner Jun 27 '15 at 3:15
• Looks to me to be an application of cohomology and base change. – John Brevik Jun 27 '15 at 9:54
• 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? – vitaliy Jun 27 '15 at 14:45
• @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! – Relapsarian Jun 29 '15 at 23:14