Finiteness theorems for the Picard objects of an algebraic stack

TL;DR

This paper establishes finiteness properties for the Picard functor of algebraic stacks, extending classical results and providing criteria for the structure and components of the Picard group with applications.

Contribution

It generalizes finiteness theorems for Picard functors to algebraic stacks, including a stacky version of Raynaud's theorem and criteria for Picard group components.

Findings

Finiteness theorems for Picard functor of algebraic stacks

Criteria for the existence of torsion components in Picard functor

Semicontinuity theorem for algebraic stacks

Abstract

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.