# GIT lecture 14

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

### Excerpts

The setup is $G \subset GL(V)$ is reductive and it acts on an invariant subvariety $X \subset \mathbb{P}(V)$$X = \mbox{Proj} R$, want $X//G = \mbox{Proj} R^G$.

If $X$ is projective then $x \in X$ is semi-stable if $x \in X_f$ for some homogeneous $f \in R^G$ of positive degree.  It is stable if action on $\mathbb{P}(V)_f$ is closed; the notion of stability depends on the embedding into projective space.  $x \in X$ is unstable if it is not semi stable.

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

For $x \in X \subset \mathbb{P}(V)$, denote the affine cone over $X$ by $X_a \subset V$ and $x_a = kv - 0$ be all the representatives of $x$.

Prop. 14.5 $x$ is semistable iff $0 \not \in \overline{G\cdot v}$, and $x$ is stable iff $G\cdot v$ is closed and $x \in X$ is regular.

1/2 proof: Say $x \in X$ is stable.  Then by a result from lecture 11 we know there is $x \in X_f$ with $f$ homogeneous and invariant such that the action of $G$ is closed on $X_f$.  Now $f(v) \ne 0$ and $f$ is constant on orbits, so $f(G \cdot v) \ne 0$, so $G\cdot v \subset X_f$, and the action of $G$ is closed on $X_f$ so orbits are closed so $G\cdot v$ is closed.  Also, in the affine case showed $X^{stable}_f = X_f - \phi^{-1}(Z)$ where $\phi$ is the map to the affine quotient, and $Z$ is the image of the irregular points.  So $x$ wasn’t regular, it couldn’t be stable.

Conversely, the seperation lemma (which applies since $G\cdot v$ is closed) gives a homogeneous invariant $f$ such that $x \in X_f$.  It remains to show $x \in X_f - \phi^{-1}(Z)$.  But if $x \in \phi^{-1}(Z)$, then its in the fiber of an irregular point, so its orbit is in the closure of another orbit, so $G \cdot v$ wouldn’t be closed (hmm think).

To construct a quotient you want to locally construct a quotient $\mbox{Spec} R_f^G$ and then glue, but you can do this around unstable points. So just work with semi stable points: $X//G := X^{ss}//G$.

Another setup. (note projective stuff isn’t explicitly mentioned here) properly stable means stable with finite stabilizer. $T$ is an alg. torus of dimension $n$ that acts on $V$ of dim $N$.  Associated to $T$ are $T^\vee = \{\chi \colon T \to k^*\}$ and $P = \{ \lambda \colon k^* \to T\}$.  Composing gives a map $k^* \to k^*$ of the form $t \mapsto t^m$, for an integer $m$.  So there is a pairing $P \times T^\vee \to \mathbb{Z}$; it is non degenerate.

It can be shown the action of $T$ looks like $t.(x_1, ..., x_N) =$ $(\chi_1(t)x_1, ...., \chi_N(t)x_N)$ with $\cap_i \ker \chi_i$ finite (we assume the stabilizer of a generic point is finite). In any case, the support $supp(x) = \{\chi_i | x = (x_1, ..., x_N) \mbox{ and } x_i \ne 0\}$.

Prop 14.8 With the above setup, $x \in V$ with $supp(x) = \{\chi_1, ..., \chi_k\}$ then $x$ is semi-stable iff $0 \in C(\chi_1, ..., \chi_k)$ and properly stable iff

1. $\dim span\{\chi_1, ..., \chi_k\} = n$
2. $0 \in$ interior of $C(\chi_1, ..., \chi_k)$

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