Variation of Instability in Invariant Theory

Abstract

When a reductive group acts on an algebraic variety, a linearized ample line bundle induces a stratification on the variety where the strata are ordered by the degrees of instability. In this thesis, we study variation of stratifications caused by different choices of linearized ample line bundles. This serves as a refinement of variation of GIT quotients, a subject well studied in the 90's by Doglachev, Hu and Thaddeus. For a representation of a reductive group, linearized line bundles correspond to the characters of the group. We provide sufficient conditions for two characters to induce the same stratification. We also formulate two types of walls that completely capture two kinds of wall crossing behaviours. We then compute an example coming from the moduli of ordered points on the projective line. Finally, we explore variation of stratifications that occur in the GIT quotient construction for projective toric varieties. We prove that the variation is intrinsic to the primitive collections and the relations among ray generators of the fans.