Mumford's Geometric Invariant Theory was developed to construct quotients in algebraic geometry. The unstable points are discarded to obtain a better quotient. However, the unstable locus has interesting stories. In this talk, I will introduce a stratification on the unstable locus in the affine setting; provide examples and results on variations of stratifications from Cox's GIT quotient construction of projective toric varieties;
how the variation is captured by certain kinds of closed subsets in the space of characters of the group acting on the affine variety in general.
Variation of Stratifications in Invariant Theory