Seminar in Algebraic Geometry and Number Theory

Abstract: Let G be a reductive group acting on a projective variety X.  In Mumford's Geometric Invariant Theory (GIT),  the formation of a quotient in this situation depends on the choice of a G-linearized ample line bundle on X. In "Variation of GIT" (VGIT), one studies various aspects of this dependence. 

I will report on joint work with K. Hulek and Z. Zhang, and explain how some central results in VGIT can be extended to a relative (and equivariant) setting. Our main motivation was to apply these results to the study of certain Hilbert scheme degenerations constructed as relative GIT quotients. In the second part of the talk, I will indicate how VGIT can be used to provide a conceptual understanding of the (semi)stable locus in this quotient construction.