Main Page/Reading Seminar Stacks (2025): Difference between revisions

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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)
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.
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==
==Tentative schedule==
Line 33: Line 35:
|Jeremy N.
|Jeremy N.
|Descent and Stack Conditions
|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|thumb]]
|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|thumb]]
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|3/28/2025
|3/28/2025
Line 43: Line 46:
|Jameson A.
|Jameson A.
|Algebraic Spaces Part 1  
|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|thumb]]
|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|thumb]]
|-
|-
|04/11/2025
|04/11/2025
|Hairuo X.
|Hairuo X.
|Algebraic Spaces Part 2
|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.  
|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 [https://drive.google.com/file/d/1HXu62XhWF6euTrDDuoeoWt6UNMqPaWfm/view?usp=sharing here].
|-
|-
|04/18/2025
|04/18/2025
|Kevin D.
|Kevin D.
|Stacks: Motivation and Artin Stacks
|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.  
|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|thumb]]
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|04/25/2025
|04/25/2025
|
|No Talk
|Stacks: Deligne-Mumford Stacks and Quasicoherent Sheaves on Algebraic Stacks
|Break!
|Theorem: An algebraic stack is Deligne-Mumford iff \Delta:X->X\times_S X is formally unramified.
|Break! People are either busy this week or will be traveling so a break right now is great!
Examples of Deligne-Mumford stacks which arise in nature. Do some computations with M_g and M_{g,n} e.g. show M_0=BPGL_2. Discuss quasicoherent sheaves on algebraic stacks.
|-
|-
|05/02/2025
|05/02/2025
|Jeremy N.
|Kevin D.
|Expedition: Quot Scheme and Hilbert Scheme
|Expedition: Quot Scheme and Hilbert Scheme
|Introduction to Quot Scheme and Hilbert Scheme. Phrase it using the language introduced thus far.  
|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.  
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|thumb]]
|-
|-
|05/09/2025
|05/09/2025, 05/16/2025, 05/23/2025, 05/30/2025
|No Talk
|Break!
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|Examples: M_g, Bun(C), Quotient Stacks, Classifying Stack.
|Returning to the examples described before -- actually prove / explain why these are algebraic / Deligne-Mumford stacks.
|-
|-
|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|thumb]]
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|Quasicoherent Sheaves on Algebraic Stacks
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|'''Note'''
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|Coarse Moduli Spaces and more Moduli of Curves
|Outline below is tentative. Likely need to spread out the talks because 1 hour will not be enough for some of these talks.
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|Gerbes
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|06/18/2025
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|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).
|-
|06/25/2025
|
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|Expedition: Coarse Moduli Spaces and Geometric Invariant Theory
|Bun_G as a stack
|Explain why, for X a smooth algebraic curve of genus g(X) and G a reductive algebraic group that Bun_G(X) is a smooth algebraic stack of dimension (g(X)-1)dim(G).
|-
|07/02/2025
|
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|Uniformization of Bun_G
|Weil's Uniformization of Bun_G(X), connected components of Bun_G(X).
|-
|-
|07/09/2025
|
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|Gr_G and its Line Bundles
|Discuss the affine Grassmannian Gr_G and line bundles on Gr_G
|-
|07/16/2025
|
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|Coarse Moduli Spaces and The Keel-Mori Theorem
|Line Bundles on Bun_G
|References: Chapter 11 of Olsson has a more general formulation than 4.4.2 of Alper's notes. Applications in next lecture. See also [https://math.stanford.edu/~conrad/papers/coarsespace.pdf Conrad's paper] which has more details and which Olsson follows.
|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).
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|07/23/2025
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|Cohomoloy of Bun_n^d
|Discuss the cohomology of Bun_n^d and the answer via Atiyah-Bott Classes
|-
|07/30/2025
|
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|Local Structure of Algebraic Stacks
|Stability and Harder-Narasimhan Filtration
|Apply Keel-Mori Theorem to give local structure of Deligne-Mumford stacks. Discuss the local structure of Artin stacks afterwards.
|Discuss the Harder-Narasimhan ``stratification" of Bun_n^d(X) and the general story.
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|08/06/2025
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|Luna's Etale Slice Theorem
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|08/13/2025
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|08/20/2025
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|08/27/2025
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|Moduli of Semistable Bundles
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|09/03/2025
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|Future Potential Topics
|09/10/2025
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Latest revision as of 23:17, 5 June 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. 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 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).
06/25/2025 Bun_G as a stack Explain why, for X a smooth algebraic curve of genus g(X) and G a reductive algebraic group that Bun_G(X) is a smooth algebraic stack of dimension (g(X)-1)dim(G).
07/02/2025 Uniformization of Bun_G Weil's Uniformization of Bun_G(X), connected components of Bun_G(X).
07/09/2025 Gr_G and its Line Bundles Discuss the affine Grassmannian Gr_G and line bundles on Gr_G
07/16/2025 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).
07/23/2025 Cohomoloy of Bun_n^d Discuss the cohomology of Bun_n^d and the answer via Atiyah-Bott Classes
07/30/2025 Stability and Harder-Narasimhan Filtration Discuss the Harder-Narasimhan ``stratification" of Bun_n^d(X) and the general story.
08/06/2025
08/13/2025
08/20/2025
08/27/2025
09/03/2025
09/10/2025

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
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