2021 Fall. Beyond GIT

8/31

Moduli problems, Yoneda lemma, fine moduli space as a representing object, universal family.

9/7

Examples: families of endomorphisms.

9/9

Local universal families, categorical quotients, orbit spaces. Algebraic groups and actions.

9/14

Basic facts. Affine algebraic groups are linear. Quotients of linear algebraic groups are qproj varieties (without proof).

9/16

Reductive groups. Quotients of affine varieties.

9/21

Orbits are open in their closure. Good and geometric quotients. Good quotients are categorical quotients.

9/23

Linearizations and GIT quotients.

9/28

Groupoids, pseudofunctors, fibered categories.

9/28

CANCELED.

10/05

Examples of fibered categories: QCoh, reverse (fiberwise oppposite). 2-category of fibered categories, 2-Yoneda lemma. Categories fibered in groupoids (prestacks).

10/07

STUDENT PRESENTATION (Emma, Andrew): following King's paper "Moduli of representations of finite-dimensional algebras".

10/12

STUDENT PRESENTATION (Elijah): Straightening and unstraightening.

10/14

FALL BREAK.

10/19

Topologies.

10/21

Algebraic spaces and stacks.

10/26

Presentations of algebraic spaces and algebraic stacks.

10/28

STUDENT PRESENTATION (Yidi): introduction to group schemes.

11/02

STUDENT PRESENTATION (Ming, Deependra): examples of stacks.

11/04

Sheaves on stacks.

11/09

Beyond GIT. Moduli problems as pseudofunctors. Coarse moduli spaces, Keel-Mori theorem. Good moduli spaces.

11/11

STUDENT PRESENTATION (Ben, Marc): properties of good moduli spaces.

11/16

Universality of good moduli spaces.

11/18

STUDENT PRESENTATION (Ming, Emma): criterion for existence of good moduli spaces.

11/23

Grothendieck abelian categories, base change and tensor product.

11/30

STUDENT PRESENTATION (Deependra): numerical invariants and stability on algebraic stacks.

Moduli of objects in abelian categories.

12/2

12/7

STUDENT PRESENTATION (Ben, Marc): derived group schemes.

12/9

STUDENT PRESENTATION (Andrew): moduli of stable bundles on elliptic curves.