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 $H^2$.


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".

| cite | improve this answer | |
  • $\begingroup$ 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 Jan 25 '13 at 18:10
  • 1
    $\begingroup$ Mohammad: see arxiv.org/pdf/math/0504020.pdf $\endgroup$ – Mahdi Majidi-Zolbanin Jan 25 '13 at 19:02

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.