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Nov 2, 2017 at 19:59 comment added Emiliano Ambrosi @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).
Nov 2, 2017 at 11:42 comment added Jason Starr 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/…
Nov 2, 2017 at 10:56 history asked Emiliano Ambrosi CC BY-SA 3.0