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Let $f:X \rightarrow Y$ be a flat surjective morphism of complex projective varieties with connected fibers, $X$ normal, $Y$ smooth and $dim (Y) < dim(X)$. Suppose $L \in Pic (X)$ such that $L|_{X_{y}} \cong \mathcal{O}_{X_{y}}$ for all $y \in Y$. Then does there exist $L_Y \in Pic(Y)$ such that $ L \cong f^*(L_Y)$? If not, does this hold after possibly replacing $X$ and $Y$ by higher birational models?

We can see that $f_*(L) \in Pic(Y)$ by Grauert's theorem in case all the fibers are integral, it is easy to show that $L \cong f^*f_*(L)$. However in general, integrality of all fibers need not be satisfied and I'm not sure how to argue.

In case anyone has any good ideas or references, please let me know.

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  • $\begingroup$ Are you assuming the fibres are geometrically reduced? In positive characteristic, it is not necessarily true that $f_*\mathscr L$ is invertible (think about the relative Frobenius morphism and $\mathscr L = \mathcal O_X)$. But if the geometric fibres are reduced and connected, then $f_*\mathcal O_X = \mathcal O_Y$ holds universally, and this still gives $\mathscr L \cong f^*f_* \mathscr L$. $\endgroup$
    – R. van Dobben de Bruyn
    Commented Jun 3, 2021 at 19:49
  • $\begingroup$ Thanks for your comment. I'm interested in the the char 0 case here (So I've edited the question). You're right, even if all the fibers are connected and reduced, then also the proof works. But I don't know what to do in case non-reduced fibers show up (for example, pencil of smooth conics degenerating to a double line). $\endgroup$
    – anonymous
    Commented Jun 3, 2021 at 21:33
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    $\begingroup$ Note that the problem you are facing is the possible non-separatedness of the Picard scheme. If $f$ is cohomologically flat in degree $0$, the answer to your question is positive if the relative Picard scheme is separated (nothing new here but it might help research the problem). $\endgroup$
    – Damian Rössler
    Commented Jun 5, 2021 at 12:51
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    $\begingroup$ You can use normalized base change and norms to prove that your invertible sheaf is "relatively $n$-torsion" for some positive integer $n$, i.e., the $n$-fold self tensor product is a pullback from the base. $\endgroup$
    – Jason Starr
    Commented Jun 7, 2021 at 18:34
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    $\begingroup$ You do not need to use alterations or resolution of singularities. Because of $S2$ extension / Hartog's phenomenon, you only need to prove triviality of an invertible sheaf away from codimension $2$. The normalization of a scheme is regular away from codimension $2$. That is why the normalization of a finite base change suffices (combined with the simple analysis of ramification in characteristic $0$ at codimension $1$ points). $\endgroup$
    – Jason Starr
    Commented Jun 8, 2021 at 17:13

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