2019/2020

## Research Seminar "An Introduction to Stacks"

Language:
English

ECTS credits:
6

### Course Syllabus

#### Abstract

Both in topology and in algebraic geometry quotients are often badly behaved, if they exist at all, which leads to problems when one wishes to construct moduli spaces, i.e. spaces whose points naturally correspond to isomorphism classes of objects of some type, such as vector bundles or algebraic curves. Stack theory offers a way around this problem: one enlarges the category of spaces to include more objects (stacks). They are more complicated than usual spaces or algebraic varieties, but behave essentially in the same way: most constructions that work for spaces (e.g. sheaves or fibrations) have stacky analogues. As the name suggests, this seminar is an introduction to stacks.

#### Learning Objectives

- Students would get to know one of the most powerful tools in algebraic topology and algebraic geometry and would find something of a special interest while giving a talk on a seminar.

#### Expected Learning Outcomes

- We will start with examples and explain the motivation behind the definition of a stack. Then we will take time to digest this definition and see that several familiar constructions (e.g. homotopy quotients in algebraic topology or moduli spaces of curves in algebraic geometry) are in fact stacks in disguise. If time allows, we will cover other examples such as Artin stacks and/or stable curves.

#### Course Contents

- First examples of stacks. Triangles in R^2 . Orbifolds.
- Faithfully flat descent for quasi-coherent sheaves.
- Functor of points view of schemes.
- ∘ Algebraic spaces.
- Categories fibered in groupoids. The definition of a stack.
- Deligne – Mumford stacks. Moduli spaces of curves and universal curves.
- Sheaves on stacks and cohomology.
- Further examples.

#### Interim Assessment

- Interim assessment (4 module)The final mark is 100% accumulated mark. The default way to earn it is to give a talk and prepare the notes in LaTeX.

#### Bibliography

#### Recommended Core Bibliography

- Edidin, D. (2010). Equivariant geometry and the cohomology of the moduli space of curves. Retrieved from http://search.ebscohost.com/login.aspx?direct=true&site=eds-live&db=edsarx&AN=edsarx.1006.2364

#### Recommended Additional Bibliography

- Hollander, S. (2001). A Homotopy Theory for Stacks. Retrieved from http://search.ebscohost.com/login.aspx?direct=true&site=eds-live&db=edsarx&AN=edsarx.math%2f0110247
- Mata-Gutiérrez, O., & Neumann, F. (2016). Geometry of Moduli Stacks of $(k,l)$-stable vector bundles over an algebraic curve. https://doi.org/10.1016/j.geomphys.2016.10.003
- Vistoli, A. (2004). Notes on Grothendieck topologies, fibered categories and descent theory. Retrieved from http://search.ebscohost.com/login.aspx?direct=true&site=eds-live&db=edsarx&AN=edsarx.math%2f0412512