GIT lecture 18,19

These lecture (  GIT17-18GIT19 ) were given on 10/7/09 and 10/9/09

Excerpts

There was an example of points in \mathbb{P}^n

proposition: S \subset \mathbb{P}^n with |S| = d is stable (resp. semi-stable) with respect to the action of SL(n+1) iff for any linear projective subspace Z, we have

|S \cap Z|/d < (\dim Z + 1)/(n+1)

resp, |S\cap Z|/d \le (\dim Z + 1) /(n+1)

Linearization.

Say \sigma \colon G \times X \to X is an action.  For a line bundle L on X a linearization is an action \bar \sigma \colon G \times L \to L s.t the following square commutes

\begin{array}{ccc} G \times L & \xrightarrow{\bar \sigma} & L \\ \downarrow & \mbox{} & \downarrow \\ G \times X & \xrightarrow{\sigma} & X \end{array}

You get a left action on the total space and this results in a right action on functions.

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