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
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