# 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.