# GIT lecture 14

This lecture (GIT14) was given on 9/28/09.

### Excerpts

The setup is is reductive and it acts on an invariant subvariety . , want .

If is projective then is semi-stable if for some homogeneous of positive degree. It is stable if action on is closed; the notion of stability depends on the embedding into projective space. is unstable if it is not semi stable.

If is just quasi projective the definitions of semi-stable and stable are strengthened: you require to be affine (for semistability) and the action on needs to be closed (for stable).

For , denote the affine cone over by and be all the representatives of .

Prop. 14.5 is semistable iff , and is stable iff is closed and is regular.

1/2 proof: Say is stable. Then by a result from lecture 11 we know there is with homogeneous and invariant such that the action of is closed on . Now and is constant on orbits, so , so , and the action of is closed on so orbits are closed so is closed. Also, in the affine case showed where is the map to the affine quotient, and is the image of the irregular points. So wasn’t regular, it couldn’t be stable.

Conversely, the seperation lemma (which applies since is closed) gives a homogeneous invariant such that . It remains to show . But if , then its in the fiber of an irregular point, so its orbit is in the closure of another orbit, so wouldn’t be closed (hmm think).

To construct a quotient you want to locally construct a quotient and then glue, but you can do this around unstable points. So just work with semi stable points: .

Another setup. (note projective stuff isn’t explicitly mentioned here) properly stable means stable with finite stabilizer. is an alg. torus of dimension that acts on of dim . Associated to are and . Composing gives a map of the form , for an integer . So there is a pairing ; it is non degenerate.

It can be shown the action of looks like with finite (we assume the stabilizer of a generic point is finite). In any case, the support .

Prop 14.8 With the above setup, with then is semi-stable iff and properly stable iff

- interior of

Note the reversed role of 0 being an element of something for checking semi stability.

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- October 1, 2009 / 9:40 pm

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