# GIT 20-22

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

### 20

Thm: If $G$ is an affine algebraic Group then $Pic G$ is finite.

Intermediate results

1. $L$ a line bundle, remove the zero section to get $L^*$.  It can be given the structure of an algebraic group, and $L$ become $L^*$ linearizable.
2. For any $L \in Pic G$ there is a finite covering $\gamma \colon G' \to G$ such that $L$ pulls back to the structure sheaf.
3. Somehow $L^{deg \gamma}$ should be trivial, so conclude every element of $Pic G$ has finite order
4. $Pic G$ is a finitely generated abelian group.

Cor. If $L \in Pic(X)$ and $X$ is smooth with $G$ connected, then some power of $L$ is $G$ linearizable.

Thm: $G$ a connected affine algebraic group acting on a smooth quasiprojective variety $X$.  Then there exists a represenation $G \to GL(V)$ and a $G$ equivariant embedding $X \to \mathbb{P}(V)$.

### 21

-Clarification of meaning of stability in different context.

-Categorical quotient, good and geometric

Roughly a categorical quotient of a variety with a $G$ action is another scheme $Y$ with a universal $G$-equivariant map to it $\phi \colon X \to Y$.

Good means you require the first 2 of the following properties

1. $\phi$ is a surjective open submersion.
2. $\phi^\#O_Y(U) = O_X(\phi^{-1}(U))^G$
3. fibers are $G$ orbits.

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

### 22

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