# Extending abelian schemes

Let $R$ be a regular local ring of dimension at least 2, and let $U$ be the complement of the closed point in $\mathrm{Spec} R$. Given a polarized abelian scheme over $U$, under what hypotheses can it be extended over the entire base?

In the mixed characteristic or equicharacteristic $p$ setting, some conditions are needed - an example over $W[[x,y]]/((xy)^{p-1}-p)$ due to Raynaud-Ogus-Gabber is described in a paper of de Jong and Oort, "On extending families of curves", Journal of Algebraic Geometry, 6 (1997), pp. 545--562, apparently illustrating some errors in Faltings-Chai. Is there some standard fix that makes such extensions possible?

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## 1 Answer

A false theorem was given in Faltings and Chai in the end of the section V of their book, but as you said Raynaud found a counter-example. Vasiu and Zink have since worked on the subject :

http://www.mathematik.uni-bielefeld.de/~zink/ValidusJ.pdf

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Great - do you know of other earlier papers along similar lines as well? –  anon Apr 20 '11 at 17:13