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Suppose you have a one parameter family of algebraic varieties over unit disk, such that the central fiber is singular and is a union (normal-crossing) of two varieties and the rest are smooth.

Is it true that every line bundle over a smooth fiber, after some base-change, extends to the whole family? Do you know any counter example?

This is the true (although not very obvious) for a family of curves, since line bundles are just bunch of points.

Extra assumption: $h^{2,0}=0$ for the smooth fibers, and they are simply connected, so that the ample cone does not move within $H^2$.

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This is not even true when all fibers are smooth. For example, see Example 5.22 in Kleiman's paper on the Picard scheme in "FGA explained".

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  • $\begingroup$ I dont have access to the book, can you recall what the example is. In the setting I have, this is true for the smooth ones. So may be i need to put some restrictions on my family. I will edit my question then. $\endgroup$
    – Mohammad Farajzadeh-Tehrani
    Commented Jan 25, 2013 at 18:10
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    $\begingroup$ Mohammad: see arxiv.org/pdf/math/0504020.pdf $\endgroup$
    – Mahdi Majidi-Zolbanin
    Commented Jan 25, 2013 at 19:02

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