# GIT lecture 17

This lecture (  GIT17-18) was given on 10/5/09

### Excerpts

Before it was disscussed how if you had a torus acting on a vector space then (semi)stability could be tested by checking if $0$ was in the cone of certain characters (forming the support; see prev. lectures).  This lecture discusses another criterion for (semi)stability.  Namely you look at the torus in $SL(n+1) = \{(t_0, ..., t_n) | \prod t_i = 1\}$.  And look at the lattice of one parameter subgroups $\Lambda = \{(\lambda_0, ..., \lambda_n| \sum_i \lambda_i = 0)\}$

In the case $\mathbb{P}(\mathbb{V}_{n,d})$ a point is represented by a homogeneous poly of degree d, say f.  Then you look at $a \in \mbox{supp} f$ and say its unstable if for some choice of coordinates there exists $\lambda$ such that $\langle \lambda, a \rangle > 0$ for all $a$.  Chance $\forall$ to $\exists$ and $>$ to $\le, <$ to get conditions for semi stability and stability.

The rest of the lecture looks in particular at degree 3 forms and singular points on it.