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.

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