Séminaire champêtre (stacky seminar)

Institut Fourier. Université Grenoble Alpes, Grenoble, France.

Organizers:

• Louis-Clément LEFÈVRE - http://louisclement.lefevre.perso.math.cnrs.fr/
• Pedro MONTERO - http://pmontero.mat.utfsm.cl/

Description

The goal of this seminar is to learn about stacks, both in the context of algebraic geometry (moduli spaces, quotients) and of topology (orbifolds, algebraic topology).

We do not have one unique source to follow. For the first talks we worked on the general theory of stacks using the book of F. Neumann below, for which the prerequisite are mainly classical category theory and schemes. Then everyone should work on the problems he is interested in, with his own point view, and is welcome explain it to the others. Some have a background in algebraic geometry over any field, others prefer algebraic topology or complex algebraic geometry or differential geometry, and none of these points of view should be excluded.

Talks

• --/--/--: ???
• 8/02/2017: Louis-Clément. Orbifolds. Notes (by Pedro).
• 20/01/2017: Pedro. Deligne-Mumford compactification. Notes.
• 10/01/2017: Louis-Clément. Fibered categories, algebraic stacks. Notes (by Pedro).
• 3/01/2017: Pedro. The 2-category of stacks. Notes.
• 13/12/2016: Louis-Clément. Sites, 2-categories. Notes (by Pedro).
• 2/12/2016: Louis-Clément. Generalities on representable functors and moduli problems. Notes (by Pedro).

References

• Our main source for the first talks: F. Neumann, Algebraic Stacks and Moduli of Vector Bundles. PDF.
• Another course to be read, more precise, lots of interesting ideas: A. Kresch, Course on algebraic stacks (Universität Zürich). link.
• A nice introduction to orbifolds and groupoids: I. Moerdijk, Orbifolds as Groupoids: an Introduction. arXiv:math/0203100.
• Two more complete articles on orbifolds and stacks: E. Lerman, Orbifolds as stacks?. arXiv:0806.4160. G. Ginot, Introduction to Differentiable Stacks. PDF.
• For a much more precise treatment of fibered categories and descent, but no geometric intuition: A. Vistoli, Notes on Grothendieck topologies, fibered categories and descent theory. arXiv:math/0412512.
• The Stacks Project is a good complement for several parts where our main sources are not formal enough. Note that it separates strongly between the general theory of stacks (chapters 2-8), algebraic spaces, and algebraic stacks. See in particular, for the moment, the following chapters that are very well-written: 4 (categories, fibered categories, 2-categories), 7 (sites and sheaves), 8 (general theory of stacks), 79 (examples of stacks), and you are encouraged to read the short chapter 89 which is an accessible introduction to the general ideas.
• It might be interesting to learn separately the theory of algebraic spaces. For this the book of D. Knutson Algebraic spaces is well-written. link.