I am currently working through chapter 5 Rick Miranda book on Riemann surfaces. On page 157 he makes the comment that if $f_0, \dots, f_n $ are meromorphic functions and $ \psi: X \rightarrow \mathbb{P}^n$ is the holomorphic function defined $\psi(x) = [ f_0 (x) : \dots : f_n (x) ]$ then the function $\psi$ has nondegenerate image iff the coordinate functions $\{f_i\}$ are linearly independent.
It seems like a stupid question but what exactly does he mean by non-degenerate image here and why it does it follow that the $f_i$ must be linearly independent. Is he trying to say that the meromorphic function $[f : g : f + g]$ is constant? I don't think he is talking about being 1-1 or an embedding here, he just uses the word 'nondegenerate'.
Thanks
Nondegenerate means not contained in a hyperplane. It's quick to see that this is equivalent to the coordinate functions being linearly independent: a linear dependence relation gives a hyperplane that the variety is contained in and vice-versa.