Main Page/Reading Seminar Stacks (2025)

Current Room Information: Wednesdays for Summer 2025 over Zoom.

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.

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. Also indicate proof of algebraicity of the stack Coh_X/S. File:QuotSchemes.pdf
05/09/2025, 05/16/2025, 05/23/2025, 05/30/2025	No Talk	Break!
06/05/2025	Kevin D.	Introduction to Bun_G	Introductory talk to some topics regarding Bun_G. Outline main goals for the seminar. Talked about structure groups, defined G-bundle. Formulate the statements of the main theorems. File:BunGTalkNotes1.pdf
Note			Outline below is tentative. Likely need to spread out the talks because 1 hour will not be enough for some of these talks.
06/18/2025	Jeremy N.	Topological Classification and Examples	Discuss topological classification, and explicit examples of Bun_G e.g. Bun_{G_m}^d(X)\cong Pic^d(X)\times BG_m, describe Bun_GL_2(P^1). File:BunGTalkSlides2.pdf
06/25/2025	Jameson A.	Bun_G is an Artin Stack and Uniformization of Bun_G	Explain why, for X a smooth algebraic curve of genus g(X) and G a reductive algebraic group that Bun_G(X) is an algebraic stack. Then discuss Weil's Uniformization of Bun_G(X), connected components of Bun_G(X).
07/09/2025	Kev D.	Deformation Theory of Bun_G	File:SmoothDimBunG.pdfExplain why Bun_G(X) is a smooth algebraic stack of dimension (g(X)-1)\dim(G).
07/16/2025	Jeremy N.	Gr_G and its Line Bundles	Discuss the affine Grassmannian Gr_G and line bundles on Gr_G File:Ind-Schemes, Affine Lie Algebras, and the Affine Grassmannian.pdf
07/23/2025	BREAK!	BREAK!	Due to various conflicts in schedules, we took a break this week!
07/30/2025	Yourong Z.	Line Bundles on Bun_G	Discuss line bundles on Bun_G. In particular, work out some examples and outline the ideas of the proof (don't worry about tedious details).
08/06/2025	Kevin D.	Cohomology of Bun_n^d	File:CohomologyRingOfBunrdAndAtiyahBottClasses.pdfDiscuss the cohomology of Bun_n^d and the answer via Atiyah-Bott Classes.
08/20/2025	Kevin D.	More on Line Bundles on Bun_G	Clean up some details regarding Line Bundles on Bun_G. Some general facts about Gr_G to be used later if we cover the Geometric Satake Correspondence. Determinant bundles and Pfaffian Bundles and Pic(Bun_G) when G is simple and simply connected of various types. Determine when the determinant bundle or the Pfaffian bundle is a generator. File:LineBundlesOnBunG.pdf
08/27/2025	Break	Break	We will skip this week since many people will be at the LSA training for the Fall.
09/03/2025	Break	Break	This is the first week of classes for the Fall Semester. I will work on organizing the continuation for the Fall Semester during this week: https://forms.gle/rG6FF6eNm321BmjA9
09/10/2025			Via an undisclosed democratic process, we might just do Geometric Satake for Fall 2025.
		Other potential topics for talks	Clean up details regarding line bundles on Bun_G plus examples. Stability and Harder-Narasimhan Filtration: Discuss the Harder-Narasimhan ``stratification" of Bun_n^d(X) and the general story. Talk more about Gr_G (some notes are linked below).

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
5. Timm Peerenboom's Thesis on the Affine Grassmannian
6. Xinwen Zhu's Introduction to affine Grassmannians and the geometric Satake equivalence