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 or written up by the speaker. Mistakes, typos, and so forth are possible and very likely. No originality is claimed.

Future plans: For plans after 5/2, please see the email thread or email ktdao@wisc.edu if you are interested in attending. We will likely go for an online modality for the summer.

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. File:StacksNotesHairuoKevinScribe.pdf
03/14/2025	Kevin D.	Moduli, Representability, and Motivation for the Future	Representable functors, moduli functors of interest, why the \'etale topology?, schemes vs. algebraic spaces vs. Deligne-Mumford stacks vs. algebraic stacks. Sketch of future goals. File:ScribeNotesKevinMotivationTalk.pdf
3/21/2025	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. (Optional: Discuss torsors and principal homogeneous spaces.) Definition of stack using fibred categories and the stackification functor. Examples of stacks. File:ScribeNotesJeremyDescent.pdf
3/28/2025	No Speaker	Spring Break!
04/01/2025	Jameson A.	Algebraic Spaces Part 1	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. File:ScribeNotesJamesonAlgSpace.pdf
04/11/2025	Hairuo X.	Algebraic Spaces Part 2	Define quasicoherent sheaves on algebraic spaces. Discuss more examples of algebraic spaces and where they might arise naturally. Explain why algebraic spaces are not enough for many moduli problems. Notes can be found here.
04/18/2025	Kevin D.	Stacks: Motivation and Artin Stacks	Definition of an algebraic (Artin) stack. Define what is a Deligne-Mumford stack. Define properties of morphisms of stacks (for representable morphisms only). Define M_g, quotient stacks, and classifying stacks as examples (to be verified later). Discuss separation axioms. Theorem: An algebraic stack is Deligne-Mumford iff \Delta:X->X\times_S X is formally unramified. File:NotesAlgebraicStacks1.pdf
04/25/2025	No Talk	Break!	Break! People are either busy this week or will be traveling so a break right now is great!
05/02/2025	Kevin D.	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. File:QuotSchemes.pdf
	Kevin D.	Introduction to Bun_G	Introductory talk to some topics regarding Bun_G.

References

1. Martin Olsson's Algebraic Spaces and Stacks. See the errata here.
2. Laumon-Moret-Bailly Champs Algebriques
3. Alper's Stacks and Moduli
4. Dan Edidin Notes on the Construction of the Moduli Space of Curves