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 25: Line 27:
|-
|-
|03/14/2025
|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|thumb]]
|-
|3/21/2025
|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. Discuss torsors and principal homogeneous spaces. See exercise 4.G for invertible sheaves on sites.
|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.
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.
[[File:ScribeNotesJeremyDescent.pdf|thumb]]
|-
|3/21/2025
|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.
|-
|-
|3/28/2025
|3/28/2025
|No Speaker
|Spring Break!
|
|
|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.
|-
|-
|04/01/2025
|04/01/2025
|Kevin D.
|Jameson A.
|Stacks: Motivation and Artin Stacks
|Algebraic Spaces Part 1
|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.
|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.
|Stacks: Deligne-Mumford Stacks and Quasicoherent Sheaves on Algebraic Stacks
|Algebraic Spaces Part 2
|Theorem: An algebraic stack is Deligne-Mumford iff \Delta:X->X\times_S X is formally unramified.
|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].
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.  
|-
|-
|04/18/2025
|04/18/2025
|Jeremy N.
|Kevin D.
|Expedition: Quot Scheme and Hilbert Scheme
|Stacks: Motivation and Artin Stacks
|Introduction to Quot Scheme and Hilbert Scheme. Phrase it using the language introduced thus far.  
|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.
Indicate application to identifying M_g=[H'/PGL_{5g-5}] for H' a locally closed subscheme of the Hilbert scheme.
[[File:NotesAlgebraicStacks1.pdf|thumb]]
|-
|-
|04/25/2025
|04/25/2025
|
|No Talk
|Examples: M_g, Bun(C), Quotient Stacks, Classifying Stack.
|Break!
|Returning to the examples described before -- actually prove / explain why these are algebraic / Deligne-Mumford stacks.
|Break! People are either busy this week or will be traveling so a break right now is great!
|-
|-
|05/02/2025
|05/02/2025
|
|Kevin D.
|Quasicoherent Sheaves on Algebraic Stacks
|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|thumb]]
|-
|-
|05/09/2025
|
|Coarse Moduli Spaces and more Moduli of Curves
|
|-
|
|
|Gerbes
|
|-
|
|
|Expedition: Coarse Moduli Spaces and Geometric Invariant Theory
|
|-
|
|
|Coarse Moduli Spaces and 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 [https://math.stanford.edu/~conrad/papers/coarsespace.pdf 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 [https://www.math.sciences.univ-nantes.fr/~sorger/assets/pdf/trieste.pdf Sorger]), Artin Algebraization, Formal Moduli, etc. We'll figure it out when we get there.
|
|
|Kevin D.
|Introduction to Bun_G
|Introductory talk to some topics regarding Bun_G.
|- }
|- }
|}
|}

Latest revision as of 15:34, 9 May 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
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