Lecture notes: https://drive.google.com/drive/folders/1Rqq2RHdnJQvP-flWNj_Y1wj3E1IqILS3?usp=drive_link.
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References:
J. Alper, Stacks and Moduli. Available at https://sites.math.washington.edu/~jarod/moduli.pdf.
D. Halpern-Leistner, The Moduli Space. Available at https://book.themoduli.space.
P. E. Newstead, Introduction to moduli problems and orbit spaces.
R. Vakil, The Rising Sea: Foundations Of Algebraic Geometry. Available at https://math.stanford.edu/~vakil/216blog/.
Schedule:
Monday in HN 1.37: 2–3 lecture, 3-4 student workshop
Thursday in HN 2.42: 10-11 lecture
Friday in HN 2.42: 10-11 lecture, 11-12 discussion space
20.02 – L1
Moduli problems, Yoneda lemma, fine moduli space as a representing object, universal family.
References: [1, §§0.1–0.3], [3, Chapter 1].
21.02
CANCELLED
24.02 – L2
Examples: families of endomorphisms.
Reference: [3, Chapter 2].
27.02 – L3
Guest lecturer: Anand Deopurkar. Local universal families, categorical quotients, orbit spaces.
Reference: [3, Chapters 2, 3].
28.02 – L4
Guest lecturer: Asilata Bapat. Introduction to quiver representations.
03.03 – L5
Algebraic groups and actions. Basic facts. Affine algebraic groups are linear. Quotients of linear algebraic groups are qproj varieties (without proof).
Reference: [3, Chapter 3 §1].
06.03 – L6
General theory of algebraic groups. Why we need reductivity.
Reference: [3, Chapter 3 §1].
07.03 – L7
Reductive groups and affine quotients.
Reference: [3, Chapter 3 §1–2].
10.03
NO LECTURE: CANBERRA DAY
13.03 – L8
Good and geometric quotients. Good quotients are categorical quotients.
Linearizations and GIT quotients.
14.03 – L9
Globalising affine quotients.
17.03 – L10
Linearisations and stability.
20.03 – L11
GIT quotients. Existence of a good quotient of the semistable locus and a geometric quotient of the stable locus. Hilbert–Mumford criterion for (semi)stability.
21.03 - L12
Groupoids, pseudofunctors, fibred categories.
24.03 – L13
STUDENT PRESENTATION
27.03 – L14
STUDENT PRESENTATION
28.03 – L15
STUDENT PRESENTATION
14.04 – L16
Examples of fibered categories: QCoh, reverse (fiberwise oppposite).
17.04 – L17
2-category of fibered categories, 2-Yoneda lemma. Categories fibered in groupoids (prestacks).
Grothendieck topologies.
22.04 – L18
Sheaves and stacks on sites.
Useful morphisms of schemes.
24.04 – L19
Fppf topology. Representable morphisms. Algebraic spaces and stacks.
28.04 – L20
Quotient stacks, their algebraicity. Étale equivalence relations.
01.05 – L21
Quotient sheaves. Algebraic spaces <=> étale equivalence relations.
Example: "spiral" algebraic space.
Presentations of algebraic spaces and stacks.
02.05 – L22
Sheaves on stacks.
05.05 – L23
08.05 – L24
09.05 – L25
12.05
Beyond GIT. Moduli problems as pseudofunctors. Coarse moduli spaces, Keel-Mori theorem.
15.05
Good moduli spaces. Universality of good moduli spaces.
16.05
Grothendieck abelian categories, base change and tensor product.
19.05
Moduli of objects in abelian categories.