Let $f:U\to V$ be a separable dominant morphism of irreducible positive-dimensional varieties. Let $F$ be a perverse sheaf on $U$. Are there infinitely many closed points $p\in V$ such that $F|_{U_p}$ is perverse up to a shift?
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$\begingroup$ If I understand the hypotheses correctly, $f$ is generically etale and $F$ is generically a local system. So there is an open dense subset $V'$ of $V$ such that the pullback of $F$ to $U'=f^{-1}(V')$ is a local system. Does this answer your question? $\endgroup$– Sam GunninghamOct 11, 2020 at 13:06
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$\begingroup$ @SamGunningham do you mean generically smooth? I don't think that's true, consider the quasi-elliptic fibrations. $\endgroup$– user166192Oct 11, 2020 at 13:12
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$\begingroup$ Ah ok, I see the sort of thing you have in mind. I read separable as finite separable. Thanks for clarifying! $\endgroup$– Sam GunninghamOct 11, 2020 at 13:28
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