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Let us work in the "nice" situation where $X,Y,Z$ are smooth complex algebraic varieties, not necessarily compact. Assume that the fiber product $W:= X \times_Z Y$ is also smooth. What can we say about the Picard group of $W$?

More precisely, assume that $Pic(X)=Pic(Z)=0$.

1) Can we deduce that $Pic(W)=Pic(Y)$?

2) If 1) is not true in general, can we draw the conclusion if $Z$ is a point and thus $W=X \times Y$?

3) If 1) is not true in general, can we draw the conclusion if $Y \to Z$ is a finite étale cover?

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# Picard groups of (fiber) products

Let us work in the "nice" situation where $X,Y,Z$ are smooth complex algebraic varieties, not necessarily compact. Assume that the fiber product $W:= X \times_Z Y$ is also smooth. What about the Picard group of $W$?

More precisely, assume that $Pic(X)=Pic(Z)=0$.

1) Can we deduce that $Pic(W)=Pic(Y)$?

2) If 1) is not true in general, can we draw the conclusion if $Z$ is a point and thus $W=X \times Y$?

3) If 1) is not true in general, can we draw the conclusion if $Y \to Z$ is a finite étale cover?