Algebra Seminar

To a linear action of a complex reductive group G on a projective variety X, geometric invariant theory associates two open subsets of X, whose elements are respectively the stable and semistable points for the action, and a projective variety X//G with a G-invariant surjective morphism from the set of semistable points to X//G, whose restriction to the set of stable points identifies an open subset of X//G with the orbit space X^s/G. Moduli spaces in algebraic geometry can often be constructed as such orbit spaces.

Also associated to the linear action, there is a canonical G-invariant stratification of X with the set of semistable points as its open stratum. In the construction of moduli spaces of bundles over curves this stratification is given by classifying bundles according to the type of their Harder-Narasimhan filtrations. The aim of this talk is to discuss some refinements of this stratification with X^s as the open stratum, and associated refinements of the Harder-Narasimhan filtration of a bundle over a curve.