Modèles semi-factoriels et modèles de Néron

Abstract

Let S be the spectrum of a discrete valuation ring with function field K. Let X be a scheme over S. We will say that X is semi-factorial over S if any invertible sheaf on the generic fiber XK can be extended to an invertible sheaf on X. Here we show that any proper geometrically normal scheme over K admits a proper, flat, normal and semi-factorial model over S. We also construct some semi-factorial compactifications of regular S-schemes, such as Neron models of abelian varieties. The semi-factoriality property for a scheme X/S corresponds to the Neron property of its Picard functor. In particular, one can recover the Neron model of the Picard variety \({{\rm Pic}_{X_K/K,{\rm red}}^0}\) of XK from the Picard functor PicX/S, as in the known case of curves. This provides some information on the relative algebraic equivalence on the S-scheme X.