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Just to be sure that I have understood completely your answer: you are using the fact that the analytic Picard group of $f^{-1}(U)$ is the invariant Picard group in $E\times \tilde U$ by the involution, right?
For simplicity asume $X$ to be complete. Then $F$ defines a class in $H^{2n}(X,\mathbb Z)$ and $F^2$ corresponds to a number via $H^{2n}(X,\mathbb Z)\times H^{2n}(X,\mathbb Z)\to H^{4n}(X,\mathbb Z)\cong\mathbb Z$.
Thank you. I was looking for a linear algebra method, like in characteristic 0 case, where it suffice to take partial derivatives. Is there some analogue in positive characteristic?
