GIT 20-22

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


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).


-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.


  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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