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Let $f:X \to Y$ be a smooth projective morphism between smooth quasi-projective complex varieties such that fibers of $f$ over closed points are connected.

Assume that $S\subset Y$ is a non-empty complement of a union of countably many closed subvarieties of $Y$ such that $Pic(f^{-1}(y))$ is of rank one for any closed point $y\in S$.

Question: Is $Pic(f^{-1}(o))$ also of rank one, where $o\in S\subset Y$ is the generic point of $Y$? And is it true that $\mathrm{rk}(Pic(X))\leq \mathrm{rk}(Pic(Y))+1$?

The example in my mind is when $f$ is a smooth family of abelian varieties.

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The answer to the first question is yes. As explained at

Geometric generic fibre

the geometric generic fibre $f^{-1}(o) \times_{k(Y)} \overline{k(Y)}$ is isomorphic as a scheme to a very general fibre, hence has Picard number 1. But then the generic fibre itself must have Picard number 1.

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    $\begingroup$ I forgot to say something about the second question. The answer to that is "no"; if $Y$ is an elliptic curve and $X=Y \times Y$ then its Picard group has rank at least 3 (cf. Hartshorne Ex IV.4.10.) $\endgroup$
    – Lazzaro Campeotti
    Commented Aug 5 at 10:28

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