OBSERVABILITY IN INVARIANT THEORY II: DIVISORS AND RATIONAL INVARIANTS.

Let G x X → X be an action of the connected algebraic group G on the irreducible, affine variety X. We discuss the relationship between [k[X]^G] and k(X)^G, where [k[X]^G] denotes the quotient field of k[X]^G. We are particularly interested in the following three questions. (1) When is the inclusion [k[X]^G] ⊆k(X)^G a finite extension of fields? (2) What is the role of G-invariant divisors? (3) What is the exact characterization of "observable in codimension one"? [ABSTRACT FROM AUTHOR]