Moduli Stacks

Time and location

Meeting 0: Thursday 10 September, 10:30-11:00, common room

Meeting 1: Thursday 24 September, 09:45-11:30, room B03

Meeting 2: Friday 9 October, 10:00-12:00, room B03

Meeting 3: Thursday 15 October, 09:45-11:30, room B03

Meeting 4: Friday 23 October, 11:00-13:00, room B03

Meeting 5: Friday 30 October, 11:00-13:00, room 302-304

Meeting 6: Wednesday 4 November, 11:00-13:00, room B03

Meeting 7: Friday 20 November, 11:00-13:00, room B03

Meeting 8: Wednesday 2 December, 15:00-17:00, room B03

Outline

See program below. We will treat the algebraic stack \(\mathcal{M}_g\), look at the definition, properties, some applications and maybe its compactification.

Program

• Categories fibered in groupoids, descent, \(\mathcal{M}_g\) as a stack
• \(\mathcal{M}_g\) as algebraic stack
• Deformation theory, relative dimension of \(\mathcal{M}_g\) over \(\mathrm{Spec}(\mathbb{Z})\), smoothness of \(\mathcal{M}_g\)

We will also aim to treat at least one (but hopefully multiple) of the following topics.

• Applications of stack theory
• Compactification of \(\mathcal{M}_g\)

References

• To learn about Grothendieck topologies and more: Étale cohomology seminar→ Jinbi Jin
• Stacks Project→ colombia.edu
• A. Vistoli, Notes on Grothendieck topologies, fibered categories and descent theory→ arxiv.org
• Harris, Moduli of curves
• Vakil, Olsson, Lieblich et al., Deformation theory and moduli

Schedule