Is the following intersection complete? In $\mathbb{P}^n$ with coords $[z_0,z_1,...,z_n]$, denote by $g$ the action 
\begin{equation}
g:z_i \rightarrow z_{i+1},
\end{equation} 
with $z_{n+1}$ identified with $z_0$, so $g$ generates a group $G$ of order $n+1$, which acts on the homogeneous coordinates of $\mathbb{P}^n$. Suppose $f$ is an irreducible homogeneous polynomial of degree $k \geq 2$ such that $g^j.f$ are all different for $i=0,...,n$. We also assume that $f(1,1,...,1) \neq 0$ to avoid the trivial case! Is the intersection of these $n+1$ $g^j.f$ complete in $\mathbb{P}^n$?
If not, what could we say about the intersection? like dimension? degree?

I doubt whether anything so general can be said. Just to illustrate, let me give an example. Let $F_d=\sum z_i^d$ and consider $f=F_{d+1}-2z_0 F_d$ (I am working over complex numbers and the 2 above in particular assures $f(1,\ldots,1)\neq 0$). If $n$ is not too small, $f$ is irreducible, $g^if$ are distinct and the intersection of all these is just the intersection of $F_{d+1}, F_d$. I am sure more complicated examples can be constructed to make it impossible to say anything general about dimension and degree.
