Local invariants and exceptional divisors of group actions

We consider linear algebraic groups and algebraic varieties defined over the field k. We always assume that k is algebraically closed. Starting with an action G x X -> X, on the normal, quasi-affine variety X, we analyse the maximal G-finite subalgebra O-K of k(X). We also analyse the maximal G-finite subalgebra O-K(p) of k[X](p), where p is a height-one G-invariant prime ideal of k[X]. We use our findings to assess the behaviour of the canonical map pi : U -> U//G Spec(O(U)(G)) for a G-invariant, open subset U of X. It turns out that for any G-invariant divisor D, there is a G-invariant, open subset V such that V boolean AND D not equal phi and the canonical morphism pi : V -> V//G has no exceptional divisors. (C) 2015 Elsevier Inc. All rights reserved.