Orbits and invariants of visible group actions

We consider actions G?×?X?→?X of the affine, algebraic group G on the irreducible, affine, variety X. If kX] ^G ]?=?kX]] ^G we call the action visible. Here A] denotes the quotient field of the integral domain A. If the action is not visible we construct a G-invariant, birational morphism φ: Z?→?X such that G?×?Z?→?Z is a visible action. We use this to obtain visible open subsets U of X. We also discuss visibility in the presence of other desirable properties: What if G?×?X?→?X is stable? What if there is a semi-invariant f ∈ kX] such that G?×?X _f ?→?X _f is visible? What if X is locally factorial? What if G is reductive?