Main Page/Reading Seminar Stacks (2025)
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Current Room Information: Fri Feb 28 2025 12:00 pm - 1:30 pm in Ingraham 225 (Repeats every week on Friday through 5/2)
Disclaimer: Notes from this page were scribed by an audience member. Mistakes, typos, and so forth are possible and likely. No originality is claimed.
Tentative schedule
| date | speaker | title | topics |
|---|---|---|---|
| 02/28/2025 | Hairuo X. | Grothendieck Topologies / Sites | Introduction to Grothendieck Toplogies /sites. More information can be found in Notes on a Seminar by Michael Artin. If one wishes to present more on the \'etale site, Milne's Lecture Notes has far more details. |
| 03/07/2025 | Kevin D. | Fibred Categories | Introduction to fibred categories. Describe correspondence between fibred categories /C and presheaves on C. Groupoids in C, fibre products of fibred categories, Yoneda Lemma, and discussion of categories fibred in groupoids. Discuss examples. The following exercises in Olsson are relevant for the future: 3.A, 3.B, 3.C, 3.D, 3.F, and 3.G. |
| 03/14/2025 | Jeremy N. | Descent and Stack Conditions | Discuss generalities on descent. Explain why fppf descent and fpqc descent work. Applications of descent e.g. closed subschemes, open embeddings, affine morphisms, polarized schemes. Discuss torsors and principal homogeneous spaces. See exercise 4.G for invertible sheaves on sites.
Definition of stack using fibred categories. Fibred products of stacks, explanation of the stackification functor. Examples of stacks: Olsson's examples are found in the exercises 4.C, 4.E, 4.H. If exercises are too hard, see Laumon-Moret-Bailly. |
| 3/21/2025 | Kevin D. (can give up slot) | Algebraic Spaces Part 1 | TBD. Olsson's presentation spans three chapters. Alper's notes are really good for this. Note the Stacks project uses the fppf topology instead of the etale topology. |
| 3/28/2025 | Kevin D. (can give up slot) | Algebraic Spaces Part 2 | TBD |
| 04/01/2025 | Kevin D. (can give up slot) | Algebraic Spaces Part 3 | TBD |
| 04/11/2025 | Kevin D. (can give up slot) | Algebraic Stacks | TBD |
| 04/18/2025 | Jeremy | Expedition: Quot Scheme and Hilbert Scheme | Introduction to Quot Scheme and Hilbert Scheme. Phrase it using the language introduced thus far.
Indicate application to identifying M_g=[H'/PGL_{5g-5}] for H' a locally closed subscheme of the Hilbert scheme. |
| 04/25/2025 | Examples: M_g, Bun(C), Quotient Stacks, Classifying Stack. | ||
| 05/02/2025 | Quasicoherent Sheaves on Algebraic Stacks | ||
| 05/09/2025 | Coarse Moduli Spaces and more Moduli of Curves | ||
| Gerbes | |||
| Expedition: Coarse Moduli Spaces and Geometric Invariant Theory | |||
| Kevin (willing to give up slot) | The Keel-Mori Theorem | References: Chapter 11 of Olsson has a more general formulation than 4.4.2 of Alper's notes. Applications in next lecture. See also Conrad's paper which has more details and which Olsson follows. | |
| Local Structure of Algebraic Stacks | Apply Keel-Mori Theorem to give local structure of Deligne-Mumford stacks. Discuss the local structure of Artin stacks afterwards. | ||
| Luna's Etale Slice Theorem | |||
| Moduli of Semistable Bundles | |||
| Future Potential Topics | Bun_G (perhaps following Sorger), Artin Algebraization, Formal Moduli, etc. We'll figure it out when we get there. |
References
- Martin Olsson's Algebraic Spaces and Stacks
- Laumon-Moret-Bailly Champs Algebriques
- Alper's Stacks and Moduli