Finiteness theorems for the Picard objects of an algebraic stack

Details Finiteness theorems for the Picard objects of an algebraic stack Finiteness theorems for the Picard objects of an algebraic stack

2009-04-07

arxiv.org/abs/0904.1166v3

Mathematics

Algebraic Geometry

29 pages. Final version, including the remarks of the referee. To appear in Adv. Math

Scientific paper

10.1016/j.aim.2011.12.011

We prove some finiteness theorems for the Picard functor of an algebraic stack, in the spirit of SGA 6, exp. XII and XIII. In particular, we give a stacky version of Raynaud's relative representability theorem, we give sufficient conditions for the existence of the torsion component of the Picard functor, and for the finite generation of the Neron-Severi groups or of the Picard group itself. We give some examples and applications. In an appendix, we prove the semicontinuity theorem for a (non necessarily tame) algebraic stack.

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