Foncteur de Picard d’un champ algébrique

Brochard, Sylvain

doi:10.1007/s00208-008-0282-8

Cited by 24 publications

(38 citation statements)

References 24 publications

(15 reference statements)

Supporting

Mentioning

Contrasting

Unclassified

Order By: Relevance

“…That Pic C/S is algebraic is a standard application of Artin's criterion for verifying that a stack is algebraic [5], and can also be seen as a Homstack. See [6] for details.…”

Section: Twisted Curvesmentioning

confidence: 99%

Twisted stable maps to tame Artin stacks

2010

View full text Add to dashboard Cite

We develop the theory of twisted stable maps into a tame Artin stack M \mathcal {M} . We show that the stacks K g , n ( M ) \mathcal {K}_{g,n}(\mathcal {M}) of twisted stable maps are algebraic, and proper and quasi-finite over the corresponding stacks K g , n ( M ) \mathcal {K}_{g,n}(M) of stable maps of the coarse moduli space M M of M \mathcal {M} . In the special case where M = B G \mathcal {M}=\mathcal {B}G , the classifying stack of a linearly reductive group scheme G G , we show that K g , n ( B G ) → M ¯ g , n \mathcal {K}_{g,n}(\mathcal {B}G)\to \overline {\mathcal {M}}_{g,n} is a flat morphism with local complete intersection fibers.

show abstract

“…That Pic C/S is algebraic is a standard application of Artin's criterion for verifying that a stack is algebraic [5], and can also be seen as a Homstack. See [6] for details.…”

Section: Twisted Curvesmentioning

confidence: 99%

Twisted stable maps to tame Artin stacks

2010

View full text Add to dashboard Cite

show abstract

“…This result on torsors under Picard S-2-stacks allows us to obtain the two categorical dimensions higher generalization of Grothendieck's study of extensions via torsors done in [12]. In this setting of translating between algebrogeometric information and categorical information we can cite also the paper [18, p. 64] where Mumford introduced the notion of invertible sheaves on a S-stack (categorical side) and the paper [9,Prop. 2.1.2] where Brochard computed the homological interpretation of such invertible sheaves (algebro-geometric side).…”

Section: Introductionmentioning

confidence: 79%

Higher-dimensional study of extensions via torsors

Bertolin

Tatar

2017

Annali di Matematica

View full text Add to dashboard Cite

Let S be a site. First we define the 3-category of torsors under a Picard S-2-stack and we compute its homotopy groups. Using calculus of fractions we define also a pure algebraic analogue of the 3-category of torsors under a Picard S-2-stack. Then we describe extensions of Picard S-2-stacks as torsors endowed with a group law on the fibers. As a consequence of such a description, we show that any Picard S-2-stack admits a canonical free partial left resolution that we compute explicitly. Moreover we get an explicit right resolution of the 3-category of extensions of Picard S-2-stacks in terms of 3-categories of torsors. Using the homological interpretation of Picard S-2-stacks, we rewrite this three categorical dimensions higher right resolution in the derived category D(S) of abelian sheaves on S.

show abstract

“…For the proof, as in [1] XII we treat separately the case of a nilpotent immersion, and then we use non-flat descent. However, the proof given here is much simpler than the proof given (for the particular case of schemes) in [1], due to the use of the representability theorem by an algebraic space for Pic X /S (see [4] and [7]). We do not need to prove the representability of f * , but only its quasi-affineness and its quasi-compactness.…”

Section: A Relative Representability Theoremmentioning

confidence: 99%

Finiteness theorems for the Picard objects of an algebraic stack

Brochard

2012

Advances in Mathematics

Self Cite

View full text Add to dashboard Cite

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.Comment: 29 pages. Final version, including the remarks of the referee. To appear in Adv. Mat

show abstract

Foncteur de Picard d’un champ algébrique

Cited by 24 publications

References 24 publications

Twisted stable maps to tame Artin stacks

Twisted stable maps to tame Artin stacks

Higher-dimensional study of extensions via torsors

Finiteness theorems for the Picard objects of an algebraic stack

Contact Info

Product

Resources

About