Main Page/Reading Seminar Stacks (2025)

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. File:StacksNotesHairuoScribe.pdf
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

1. Martin Olsson's Algebraic Spaces and Stacks
2. Laumon-Moret-Bailly Champs Algebriques
3. Alper's Stacks and Moduli