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, on the normal, quasi-affine variety X, we analyse the maximal G-finite subalgebra OK of k(X). We also analyse the maximal G-finite subalgebra OK(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 π: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∩D≠∅ and the canonical morphism π:V→V//G has no exceptional divisors.