# 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 $$n-d$$?