# GIT 20-22

These lectures (GIT20-22 ) were given the week of 10/12/09 – 10/16/09.

### 20

Thm: If is an affine algebraic Group then is finite.

Intermediate results

- a line bundle, remove the zero section to get . It can be given the structure of an algebraic group, and become linearizable.
- For any there is a finite covering such that pulls back to the structure sheaf.
- Somehow should be trivial, so conclude every element of has finite order
- is a finitely generated abelian group.

Cor. If and is smooth with connected, then some power of is linearizable.

Thm: a connected affine algebraic group acting on a smooth quasiprojective variety . Then there exists a represenation and a equivariant embedding .

### 21

-Clarification of meaning of stability in different context.

-Categorical quotient, good and geometric

Roughly a categorical quotient of a variety with a action is another scheme with a universal -equivariant map to it .

Good means you require the first 2 of the following properties

- is a surjective open submersion.
- fibers are orbits.

Having 1,2,3 means its also a geometric quotient.

### 22

- certain affine quotients turn out to be good categorical quotient
- restricting to stable points gives geometric quotient
- Analogous results 1,2 for projective quotients based on a linearization.
- some examples

## About this entry

You’re currently reading “GIT 20-22,” an entry on Math Meandering

- Published:
- October 17, 2009 / 7:23 pm

- Category:
- GIT (course)

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