Semi-factorial models and Néron models

2011/03/02 by Pépin, Cédric
#11G10 #14A15 #14G20 #14K30 #Algebraic Geometry (math.AG) #FOS: Mathematics

paper · doi:10.48550/arxiv.1103.0565

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 each invertible sheaf on the generic fiber X_K can be extended to an invertible sheaf on X. Here we show that any proper geometrically normal scheme over K admits a model over S which is proper, flat, normal and semi-factorial. We also construct some semi-factorial compactifications of regular S-schemes, such as Néron models of abelian varieties. Moreover, the semi-factoriality property for a scheme X/S corresponds to the Néron property of its Picard functor. In particular, one can recover the Néron model of the Picard variety of X_K from the Picard functor of X/S, as in the case of curves. This provides some information about relative algebraic equivalence on the S-scheme X.

Related

• Néron's pairing and relative algebraic equivalence
• Relative compactifications of semiabelian Néron models, I
• Canonical extensions of Néron models of Jacobians
• Relative compactification of semiabelian Néron models, II
• Néron models of pseudo-Abelian varieties