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Let $k$ be an algebraically closed field of positive characteristic $p$ and fix a prime $\ell\neq p$. Let $X$ be a smooth connected $k$-variety and $f:Y\rightarrow X$ a smooth projective morphism. Assume that there is an open embedding with dense image $Y\rightarrow Y^{cmp}$ of $Y$ inside a smooth projective $k$-variety $Y^{cmp}$.

Is it true that the natual map $$H^{i}(Y^{cmp},\mathbb Q_{\ell}(j))\rightarrow H^{0}(X,R^{i}f_*\mathbb Q_{\ell}(j))$$ is surjective?

It seems like some cases are dealt by Deligne in Weil II (6.2.11), but I don't see if it's possible to extend it to this situation.

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  • $\begingroup$ If you allow "non-general" pencils of hyperplane sections, some pretty strange things can happen in positive characteristic, cf. the following MathOverflow question: mathoverflow.net/questions/115361/… $\endgroup$
    – Jason Starr
    Commented Nov 2, 2017 at 11:42
  • $\begingroup$ @JasonStarr Thank you for your answer. I don't see exactly if that post gives a counterexample. When $X$ is a curve, it should be true (it is claimed in "Deformation et specialisation de cycles motives" by Y.André as a conseguence of Weil II 6.2.12). $\endgroup$
    – Emiliano Ambrosi
    Commented Nov 2, 2017 at 19:59

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