Ag.algebraic geometry – dimension of the orbit with respect to invariant functions

$ def CC { mathbb {C}} $Let $ K = CC (x_1, ldots, x_n) $ and let $ G $ to be a countable group of automorphisms of K $; in the cases that are important to me, $ G cong mathbb {Z} $. Then the domain of $ G $-invariants, $ K ^ G $, is an extension of $ CC $ from a certain degree of transition $ d $. I would like to know in what circumstances I can say that the closure of Zariski $ G $-bit of a generic $ n $-tuple to the dimension $ n-d $.

I want to be a little careless about what I mean by generic, but I do not think that there is anything at the bottom of this question. For "generic" $ x in CC ^ n $, all rational cards $ g $ are defined at $ x $so that we can consider $ {g (x) } _ {g in G} subset CC ^ n $ and take his Zariski closure. Each function in $ K ^ G $ is constant on this Zariski closure, so the Zariski closure has a dimension $ leq n-d $.

Question Can I conclude that the dimension is generic $ n-d $?